A GEOMETRIC METHOD FOR STUDYING FAULTS AND ITS APPLICATION TO THE SEARCH FOR THE DISPLACED PART OF A DEPOSIT

Prof. I. A. Molchanov.

INTRODUCTION

The correct interpretation of disjunctive disturbances, i.e., the determination of the actual or true direction and amplitude of the relative displacement of adjacent blocks situated on one and the other side of the disturbance, is of enormous importance in solving various geological problems, especially in large-scale surveys and detailed investigations of ore-bearing regions, mine and shaft fields both in operation and being prepared for exploitation. It is likewise clear that only a correct interpretation of, if not all, then at least the most important of the disturbances encountered in a district can serve as a reliable basis for constructing tectonic schemes and for more or less broad geotectonic generalizations and conclusions.

But this question acquires especially great importance in the conduct of prospecting and prospecting-preparatory work, when it becomes necessary to locate the part of a bed or vein displaced by a disturbance. An erroneous solution of the question in this case means a loss of funds and time in futile searches for the displaced part of the deposit in a falsely indicated direction, and sometimes may even endanger the very life of the mine.

This question arises especially often in the exploration of vein deposits, which, by the very manner of their formation, are usually confined to tectonically disturbed regions. But it is also common in the setting of exploration of bedded deposits, in particular coal deposits, especially such basins as the Kuzbass, which is characterized by an especially complex tectonics. From the practice of exploration of coal deposits in the Kuzbass it is known that, even with detailed exploration of mine fields, only some of the largest disturbances can be established by drilling. During the cutting and preparation of mine workings, new disturbances are constantly encountered. The seam along which the working is being driven proves to be lost. The face has to be stopped.

until the encountered fault has been deciphered in one way or another, until it has been indicated in which direction the displaced part of the seam is located.

Sometimes this can be established quite simply, from the bends and turnings of the seam at the fault, or from other purely geological indications. But often these indications prove insufficient. It becomes necessary to resort to underground or surface drilling, which entails an expenditure of money and time, causes delays in the deployment of preparatory work, and, as a consequence, cannot fail to affect the regularity and productivity of the mine’s operation.

The method I propose for the geometrical solution of the question posed can provide very substantial assistance in all the cases mentioned above. Hence the interest in it on the part of exploration geologists and mine geologists is understandable, especially recently, in connection with the order of People’s Commissar L. M. Kaganovich on improving the geological service at mines.

The basic ideas developed in the present article were presented by me at the regional conference of mine surveyors of Western Siberia as early as February 1935 (7). When I delivered my report at that conference, I was still far from certain either of the originality of the method I proposed for interpreting disjunctives, since I had not had the opportunity to become acquainted with all the most recent mine-surveying and mining-geometrical literature, or of its practical significance for stratified deposits, especially under the conditions of the complex and distinctive tectonics of Kuzbass.

Over the past four years, however, it has become clear that the method I propose is indeed new, having no precedent either in geological or in mine-surveying literature. Moreover, the work carried out under my supervision by postgraduate student (now docent) A. A. Belitskii (2) on the study of disturbances in the field of the Koksovaya mine of the Prokopievsk district showed that this method can be successfully applied also to the deciphering of complex tectonic structures of the coal deposits of Kuzbass, and that it gives the geologist a new and quite reliable means for determining the direction in which the displaced part of a seam is located even in cases where the data are still insufficient for a complete solution of the disjunctive.

Now our method has already received general recognition in the Prokopievsk district. With its help more than 60 different faults have already been deciphered, 20 of them entirely new, and not once has the prescribed direction of exploratory working proved erroneous. Such unexpectedly brilliant results allow me to speak of this method with greater confidence and to recommend that it be tested on a broader scale.

BASIC CONCEPTS AND STATEMENT OF THE PROBLEM

Disjunctive dislocations or, more briefly, disjunctives are those tectonic disturbances that are associated with a rupture in the continuity of the rocks undergoing dislocation. These ruptures are expressed as cracks, sometimes as entire zones of fractures, along which the displacement of one block of the earth’s crust relative to another takes place, caused by its actual movement in space.

The displacement of one of two adjacent blocks relative to the other is usually accompanied by destruction and abrasion of the material compressed between them, and also by grinding and polishing of the surfaces of the crack as a result of their friction. Such polished surfaces are called slickensides, and the striations preserved on them may serve as indicators of the direction of the relative motion of the blocks.

It should be noted, however, that the true or absolute motions and displacements of the disrupted blocks in space are, for the most part, entirely unknown to us, since there are no reliable criteria for determining which of them moved and which remained at rest. To be sure, in particular and comparatively rare cases, the actively acting and, consequently, moving block may be established with some degree of probability on the basis of general geological considerations, with the application of the laws of mechanics and the strength of materials. But the mechanics of tectonic phenomena has not yet been developed to such an extent that we could solve this question without error in every individual case. General geotectonic considerations usually lead us to the conclusion that both adjacent blocks, separated by a disjunctive, participate simultaneously in the movement, and moreover often in the same direction, only one of them moving somewhat faster and being displaced farther in that direction than the other.

The fracture along which this displacement occurs, in accordance with the terminology established in mining-geometric literature, we shall call the fault plane, although this term, as I. N. Ushakov rightly points out (23—143), may not be altogether successful.

Generally speaking, the fault plane is a curved surface. But, just as in the case of beds and veins, which likewise do not represent ideal planes, its comparatively small ...

the segment that usually falls within our field of vision can, without great error, be regarded as a plane, and one may speak of its strike and dip. The masses displaced by it, depending on their position relative to the dislocator, are called the footwall and hanging wall, or the walls of the disjuncture.

By the term displacement one usually understands the rupture and noncoincidence in position of some primary structural element, situated in one wall of the disjuncture, with respect to its continuation in the other. Such a primary structural element may be a bed, a vein, a dike, a bedding plane, and, in general, a boundary between two rocks (a contact), as well as another, previously formed fracture, provided that it is possible to identify its parts situated on either side of the dislocator.

It is clear that in a homogeneous, structureless medium the movement of the walls of a disjuncture cannot produce a visible displacement of structural elements, owing to their absence, since every point of one wall may be identified with any point of the other. When structural elements are present, however, the latter will be ruptured and displaced in one wall with respect to their continuation in the other; and this visible displacement is, properly speaking, the only reliable criterion for determining the magnitude and direction of the actual relative movement of the walls of the disjuncture.

It is true that, as already mentioned, the striation on the plane of the dislocator may give some idea of the direction of the relative movement of the walls of the disjuncture. However, this sign cannot be considered entirely reliable. In any case it is clear that movements observed in nature of several meters, and sometimes even many tens of meters, could not have formed in a single act. One must suppose that they formed as a result of repeated, but comparatively small, displacements that occurred along the given fracture generally in one, though not necessarily an entirely identical, direction. The striation on the plane of the dislocator will be preserved chiefly as that which corresponds to the last of these elementary displacements, and therefore its direction may deviate more or less strongly from the direction of the general resultant vector of all these elementary displacements. As for the total amplitude of displacement, i.e. the length of this resultant vector, it can be determined only from the displacement of parts of the primary structural elements situated on either side of the dislocator.

Thus, the relative motion of one of two adjoining masses, disrupted by a disjuncture, in a definite direction—as a process—results in its displacement in space by a definite distance. This actual relative displacement of one of the masses causes visible dis-

…of the structural elements situated in it, relative to their continuation—in the other one¹.

Thus, these two similar-sounding terms denote entirely different concepts, and therefore they must be clearly distinguished. At the same time, the concepts themselves should not be confused, as is sometimes done in presentations of the question in the mining-geometrical literature.

As is evident from the foregoing, for a complete characterization of a disjunctive dislocation it is necessary and sufficient to know the direction and magnitude of the actual displacement of one of its blocks relative to the other. The magnitude and direction of the displacement, however, are determined by the visible offset of the structural elements situated on either side of the fault. Consequently, in order to resolve the question fully, it is necessary to find the functional dependence between the actual relative displacement of the blocks of a disjunctive dislocation and the visible offset of the primary structural elements situated in them—beds, veins, and so forth.

It is clear that such a dependence between offset and displacement undoubtedly exists, and that it is reciprocal. Consequently, it will make it possible to solve both the direct and the inverse problems of faulting; that is, from a given offset of structures situated in its blocks, to determine their actual relative displacement, and, conversely, from a given displacement of the blocks of a disjunctive dislocation, to determine the position of the primary structural element displaced by it.

Let us now see how mine surveyors and geologists approach the solution of this question.

Brief review of the literature.

In the literature on faults we find two sharply expressed trends, characterized by entirely different approaches to the treatment of the question under consideration. One trend is reflected in the geological literature, the other in the mine-surveying literature.

Geologists and geotectonists pose tectonic problems on a large scale and approach their solution from a principled point of view. They are interested, first of all, in the questions: first, the mechanism of formation of disjunctive dislocations—in particular, whether they are formed as a result of compressive or tensile stresses arising in the earth’s crust; second, the determination of the phases and directions of the orogenic pressure that caused the tectonics of a given region, their age, and so forth. All this is, of course, of more theoretical interest and provides a broad field for various kinds of hypotheses and speculations. Con—

concrete forms of disjunctives are described and analyzed by them comparatively rarely. If this question does sometimes arise, it is posed only in the sense of the direct problem of the disjunctive, and is solved chiefly by purely geological methods, without the use of geometric constructions.

The mine-surveying literature presents an entirely different picture.

Finding themselves in conditions of responsible work in mines, mine surveyors, in the absence of mine geologists, had long been compelled to study particular disturbances, and moreover in the most utilitarian aspect. Unlike geologists, they chiefly had to deal with the inverse problem of the disjunctive—with the question of where and how to look for the displaced part of a bed or vein. It is clear that, for solving such a question, posed in this sharp form, purely speculative constructions alone are far from sufficient. Something more real is required here, something that can be obtained only by measurements and calculations. This, in fact, determined both the general direction of the mine surveyors’ thought and the research methodology they chose, employing methods of geometry and mathematical analysis. In the course of time they developed their own terminology, distinct from the geological one, their own original classification, and their own, purely geometric, methodology for studying disturbances; thus, in the end, this branch of mine surveying separated out into an independent science under the name of mining geometry¹.

Especially vigorous development of the question of disturbances begins at the start of the present century. To this period belong both the development of mathematical methods of investigation and the formation of the modern classification of displacements. The beginning was laid in 1903 by Gausse (27). Having analyzed all possible combinations of the interrelations between the bed and the displacer, he derived his classification of displacements, numbering more than 100 different forms. Leontovskii (5), in 1905, revised this classification and reduced the number of forms to 71. Bauman (1), in 1907, subjected it to further reduction, showing that the number of all possible combinations, if they are considered in space without relation to the plane of the horizon, does not exceed 43. This last classification is considered generally accepted in mining geometry, although in practice hardly anyone uses it. Later, in 1924, Dalinkevich (4) proposed still another classification, built on a somewhat different principle, in which only 26 types are counted, but this classification likewise has not yet become widespread.

All these classifications, apart from their dismaying cumbersomeness, complex and artificial nomenclature, ignoring the indi-

terms that have become established in mining suffer from one further important defect, which completely nullifies their practical significance. All of them consider not the fault as a whole, not the disjunctive dislocation as such, but only the displacement of a bed or vein caused by it, entirely disregarding the movement of the wings of the disjunctive dislocation by which this displacement was caused. Meanwhile, it is clear that one and the same fault, with a definite magnitude and direction of movement of its wings, will produce different visible displacements of the beds or veins it intersects, if only the conditions of their occurrence are not identical, and, consequently, will fall into different categories of these classifications. In other words, each fault will receive as many different names as the number of different beds or veins it intersects. On the other hand, determining the form of displacement for one bed in no way predetermines the form of the displacements of another bed faulted by the same dislocator, if only the conditions of its occurrence differ even slightly. Since this is so, the determination of the form of displacement loses all practical significance.

The situation is not much better with the methodology of investigation. In studying a fault, the direction of displacement of its wings in mining geometry is always taken to be the conventional direction perpendicular to the trace of the bed on the plane of the dislocator. This conventional direction is used even in the case when the actual (true) direction can be determined. In general, in mining geometry very little importance is attached to determining the actual displacement of the wings of a disjunctive dislocation. Gutt, for example, states directly that “in mining, knowledge of the true direction of displacement plays a secondary role; it is much more important to know the geometrical position of the rocks that have undergone displacement” (3—145). Ushakov adds to this that this does not make sense either, since “displacements in different directions may create the same visible position of the wings of a deposit in certain sections” (23—184)¹. In general, he for some reason considers this factor to be purely geological, having no direct significance for the geometrical solution of the problem, and for its determination indicates exclusively geological methods (23—185).

Little attention is paid in mining geometry to geometrical methods for determining the actual (true) displacement of the wings of a disjunctive dislocation. This question remains in the background and is almost wholly undeveloped.

The underestimation of the significance of the actual displacement of the wings of a disjunctive dislocation in the search for the displaced wing of a bed or vein is most clearly reflected in hopeless attempts to derive, from an analysis of the disposition of structures, rules for searching for the displaced part of a bed or vein.

The first attempts to create such rules belong to remote antiquity and, for their time, were quite intelligible. One such rule—the so-called “old miners’ rule”—was recorded by Agricola as early as 1657.¹ It seems to us far more surprising that analogous “rules” were drawn up repeatedly later as well—by Carnall (25), Leontovskii (5), Bauman (1), Aubel (24), and others, and, in very recent times, by Dalinkevich (4) and Gornokh (29). Meanwhile, Gefer-Heimhalt had long ago proved the futility of rules of this kind, since they are valid only for vertical displacements and take no account at all of the horizontal component (28—114).

In the later manuals on mining geometry, Gutt (3) and Ushakov (23), although they do not give any “rules” of their own, devote much attention to a systematic analysis of existing ones, as if not rejecting either their significance or the possibility of creating more perfect ones. Yet for anyone not devoid of spatial imagination it ought to be clear that deriving a universal rule for prospecting from the mutual position of the seam and the displacer is as hopeless a matter as inventing a perpetual-motion machine.

However, such rules as Dalinkevich’s rule or the analogous “second rule” of Gornokh are indeed universal. The latter reads literally as follows: “If the deposit lies deeper, the working is driven into the lying side; if higher, then into the hanging side” (29—55). Gutt (3—170), citing this rule in German and in Russian translation, considers it necessary to clarify the author’s wording still further.² Here one does not know what to marvel at more: the profundity of the author’s thought (and of his interpreter), or the geometric character of the rule itself. If mining geometers, after all their complex mathematical investigations, arrive at such profoundly thoughtful conclusions, this can testify neither in favor of the science they represent nor of the methods by which it operates.

In general, we are left with the impression that mining geometry, in the matter of investigating faults, despite the enormous amount of work it has done, is on a false path. The study of displacement phenomena apart from the phenomena of movement that generate them has led mining geometry into a dead end. Such a one-sided method of studying faults is unscientific and antidialectical. It has not given, and could not have given, anything more valuable than the rule of Gornokh cited above, since “not a single phenomenon in nature can be understood if it is taken in isolation—

in isolation, apart from the surrounding phenomena, for any phenomenon, in any domain of nature, can be turned into nonsense if it is considered without connection with the surrounding conditions, in isolation from them.”¹

We obviously could not follow such a clearly defective method.

The geological literature on faults is considerably poorer. Indeed, in not one of the major manuals of general and field geology do we find a methodology for investigating faults, for determining the direction and amplitude of displacement of their blocks, let alone an analysis of the displacements caused by them. Likewise, in works on structural geology and geotectonics we do not find a treatment of this question, although some authors (6, 12, 13) devote much attention to the mechanics of tectonic phenomena. M. M. Tetyaev, however, makes a timid attempt to determine the vertical amplitude of a fault from the visible horizontal displacement of the blocks of the primary structure (11—112—116), but he confines himself to considering only vertical displacements (in fact even only normal faults), completely disregarding the influence of the horizontal component. Meanwhile Spurr (37, 38), who earlier than other geologists began to concern himself with questions of mine tectonics, long ago demonstrated that vertical displacements are only a particular, and therefore rather rare, case of tectonic faults. The overwhelming majority of faults are the result of an inclined displacement of the blocks, in which the horizontal component (the strike-slip element) often plays a very substantial, sometimes even dominant, role. Similarly, Ransome (33), Evans (26), and others cite numerous facts of large horizontal displacements (strike-slips), whose amplitude sometimes reaches many hundreds of meters, even as much as 3 km. Therefore Tetyaev’s attempt may be regarded rather as a statement of the problem than as its solution.

However, American geologists, especially those engaged chiefly in the study of deposits, long ago came to the conclusion that a more thorough investigation of faults was necessary, with the application of more exact geometrical techniques, and that a corresponding classification had to be worked out. In 1907, in the pages of the journal Economic Geology (29), an extensive discussion on this subject arose, prompted by an article by Ransome (33). A whole series of prominent American geologists—Jaggar, Spurr, Fairchild, B. Willis, J. Ried, Cushing, Tolman, Chamberlin, Evans, and others (26)—took part in this discussion, publishing articles under the common title “How Should Faults Be Named and Classified.”

Of these, the articles by Cushing and Tolman (26) are the most substantial from our point of view. The first points out

the need to distinguish the geometric elements of a disjunctive structure, while the latter gives a fairly detailed analysis of them.

Somewhat later, in 1909, H. Ried pointed out the possibility of a graphical solution of several basic questions concerning disjunctive structures, including the determination of the direction and amplitude of the actual relative displacement of the blocks (34).

In 1911 Tolman (39) developed this question in greater detail and gave a fuller and more elegant solution, applying the method of projections with numerical marks, which we also use with a few minor modifications. Unfortunately, the method he proposed did not receive due reflection in the geological literature. Nor was it reflected in the work of the special commission that was convened as a result of this discussion by the Geological Society of America in 1912 under the chairmanship of H. Ried, in order to work out a rational classification of faults.

In the official report of this commission (35) we find a clearly developed terminology for the geometric elements of faults, though without indicating their functional dependence on one another, and a fairly complete classification of disjunctive structures built on a geometric principle. This classification is based on the feature of the direction of the actual displacement of the rock masses situated on one side and the other of a tectonic fissure, relative to one another. At the same time, the authors emphasize that they stopped precisely at the “geometric” and not at the “genetic” principle of classification, so that a fault could be described insofar as it can be directly observed, without the necessity of necessarily establishing the forces that caused its formation. All the more so since the indisputable establishment of these forces is by no means always possible, and since the so-called “compression faults” can sometimes be formed without the participation of compression, and “tension” faults without “tension” (35—164).

This view seems to us entirely correct, and we therefore lay it at the basis of all our further reasoning.

Among Russian geologists, only one professor, Academician M. A. Usov, dealt thoroughly with questions of local tectonics, or, more precisely, the tectonics of deposits. In a series of articles (14—19 and 21) he gave a detailed description of the most important faults in almost all the deposits being worked in the Kuzbass, using the rich material of underground workings, to which geologists had until then paid little attention. This extensive material enabled him to work out his own methods of investigation and even to propose his own classification of faults (20 and 22).

Unlike the American geologists, he bases this classification on the genetic principle, limiting himself, however, to the consideration only of tangential disjunctive structures, which enjoy predominant development in the Kuzbass. In this

the classification provides, among other things, for the division of types of disjunctives not only according to the character of the acting forces, but also according to the direction of the absolute displacement of the walls. Thus, for example, forms are distinguished that are produced by the movement of the hanging wall upward along the inclined plane of the displacer—a reverse fault and a “reverse overthrust”—from wholly similar forms produced by movement of the footwall downward, beneath the hanging wall—a “underthrust.” It is clear that the diagnosis of the latter form is extremely difficult, and the author himself, when giving examples of such dislocations, is not fully confident in the correctness of their determination.

While giving due credit to the important advantages of classifications built on the genetic principle, we nevertheless think that the introduction of this principle into the classification of disjunctives does not correspond to the present level of our knowledge of phenomena of this kind. Quite apart from the instability of the foundations for determining the absolute movements of rock masses in space, even the question of the manifestation of one or another stress participating in the formation of a disjunctive is for the most part a disputed question, as is rightly noted in the work of the commission of American geologists.

In a brief characterization of M. A. Usov’s classification, let us note two more points.

First, it does not take into account the influence on the character of displacement of the horizontal component. True, one cannot say that the author entirely ignores the possibility of horizontal displacements. On the basis of his own observations and those of his collaborators, he gives several concrete examples of exclusively horizontal displacements—strike-slips (22–41). In exactly the same way he also indicates the possibility of “complex,” i.e. oblique, displacements, of the type of reverse-strike-slip faults and overthrust-strike-slip faults, etc.; however, in the classification he proposes these forms do not find their reflection. Proceeding from Willis’s simplified experiments on the crushing of blocks and his theory of the ellipsoid of stresses, M. A. Usov finds no genetic prerequisites for the occurrence of horizontal displacements. Therefore these displacements could not fit into the classification proposed by M. A. Usov (22–43).

Meanwhile, upon careful investigation of the striations on the plane of the displacer, it turns out that the overwhelming majority of the faults of the Kuzbass¹ represent oblique displacements, in which the horizontal component is not only almost always present, but sometimes even exceeds the vertical.

Second, this classification, like, incidentally, all the others, does not take into account the influence of the angle formed by the strike line of the bed with the strike line of the displacer; yet, as recent studies show, in concordant disjunctives, especially in the case when the displacer dips more gently than the bed,

this angle acquires special significance. With a certain change in it, one and the same displacement of the wings of a disjunctive fault may produce two entirely different forms of displacement.

This latter circumstance, together with the established predominance of inclined displacements, makes it possible easily and simply to decipher a number of enigmatic disturbances. Thus, for example, in order to explain how one and the same disjunctive fault in two adjacent beds can produce two different types of displacement (22—17, Fig. 14), slight changes in the attitude of the bed or of the displacer are sufficient to obtain both a “normal” and a “reverse” thrust on one and the same displacer.

Fundamental principles adopted by us in solving the question of disjunctive faults

An undoubtedly great and indisputable merit of Academician M. A. Usov is the development of the foundations of a genetic classification of disjunctive faults. However, despite the long and thorough treatment of the vast and varied material on the tectonics of the Kuzbass, M. A. Usov could not provide an exhaustive classification that would encompass all forms, even only of tangential disjunctive faults. In particular, strike-slip and complex thrust–strike-slip disturbances were not reflected in his classification, although they apparently have a fairly wide distribution in the Kuzbass and are caused, in general, by the same tangential stresses. This was clear to M. A. Usov himself when he pointed out that “unfortunately, a reliable criterion has not yet been worked out for determining the vectoriality of movements in strike-slip displacements,” and therefore there is no possibility of establishing which wing of the disturbance actually moved (22—42).

If such difficulties are encountered in a region so comparatively well studied tectonically as the Kuzbass, then how great and insurmountable will they be in the investigation of disturbances in other, less studied regions and districts?

All this makes it evident that, despite all the advantages and methodological significance of the genetic principle in the classification of disjunctive faults, its practical application confronts us, at the present time, with insurmountable difficulties. This compels us, for the time being, to take the path of their geometric classification.¹

We believe that at present only a mass investigation of disjunctive faults, with careful determination of the actual relative displacement of their wings, will make it possible to approach the determination of the absolute (and, of course, not quite complete) direction of movement of more or less large blocks of the earth’s

…cracks in space, and at the same time the stresses by which they are caused. But for this, a further long and painstaking effort is required to accumulate primary factual material. This material will be the more valuable the more accurately the actual relative displacements of the walls of each disjunctive structure are determined.

Proceeding from these considerations, we share the view of American geologists as to the necessity of geometric methods for studying faults, but we take a further step forward by establishing a functional dependence between the geometric elements of displacements and the shifts caused by them. This enables us to derive a general formula for a disjunctive structure, embracing all possible cases of straight translational disjunctive faults, thereby reducing the entire diversity of phenomena to unity. This formula makes it possible to solve mathematically both the direct and the inverse problems of a disjunctive structure, and its study makes it possible to establish what data are necessary and sufficient for their complete or partial solution. On the other hand, analysis of this formula and of all the variables entering into it makes it possible to give an exact classification of faults, to derive and mathematically substantiate each of the types distinguished in it.

In deriving this formula we make the following assumptions.

1. The absolute displacements of the walls of a disjunctive structure are for the most part unknown to us. Therefore, we can speak only of their relative displacements, as the sole factor determining the visible shift of the structural elements displaced by them.

2. Although the displacer generally represents a curved surface, nevertheless, considering it within limited bounds of space, we shall take it to be a plane, just as we shall the elements of the primary structure displaced by it—beds, veins, boundaries between formations, etc., which in what follows, for brevity, we shall simply call a bed.

3. For the same reasons—the limited nature of the space within which the given disjunctive structure is considered—we take the blocks displaced by it to be rigid systems not subject to bending forces.

4. Finally, we confine ourselves to considering only straight translational faults, since we believe that rotational faults—helical, cylindrical, etc.—are in general rare and have as yet been very poorly studied. As for hinged faults, although they are common, the angle of rotation caused by them is for the most part so small that it can have a significant effect only on very gently dipping structures.

It is clear that these assumptions introduce a certain error into all our reasoning; however, they are entirely natural and unavoidable in the study of any more or less complex phenomenon of nature.

In doing so, we assume that each fault is characterized by three factors: 1) the displacement of the blocks in space relative to one another; 2) the intersection of the bed with the fault plane or, in other words, the mutual position of these two planar elements in space; 3) the offset of the bed, i.e., the visible result of the combined effect of the first two.

Let us first consider each of these separately.

Analysis of displacements

Since we have agreed to consider only relative displacements of the blocks, it is essentially quite immaterial to us which of them actually moved and which remained at rest. At the same time, for convenience of analysis, we shall always take the movable block to be the hanging wall of the disjunctive fault.

It is clear that any point of the hanging wall lying in the fault plane may move in any direction, provided it does not leave this plane. If we consider only translational displacements, in which each point of the moving block is displaced parallel to, and by the same distance as, any other point of it, then the total amplitude of displacement of any point chosen by us in the fault plane will be equal and parallel to the amplitude of the total relative displacement of the hanging wall as a whole.

Thus, the total relative displacement of the hanging wall in space may be represented as a vector \(R\), lying in the fault plane and connecting the initial position of an arbitrarily chosen indicator point \(O\) with its final position \(M\) (Fig. 1).

Let us now resolve this vector into two components in the directions of strike and dip of the fault plane (Fig. 1). Then the first of them, \(OL\), will obviously represent the strike-slip component, which we shall denote by the letter \(\lambda\), while the second, \(ON\), will represent the inclined height of displacement \(h\), i.e., the component of reverse or normal displacement, depending on whether it is directed upward or downward along the inclined fault plane. The total displacement will naturally be called a reverse-slip or normal-slip strike-slip displacement.

If we agree that the fault plane is in front of the observer and dips toward him (Fig. 2), and take movement of the indicator point upward and to the right as positive, and downward and to the left as negative, then all possible displacements can easily be accounted for and quite precisely characterized: either by the projections of the displacement vector \(R\) on the lines of strike and dip of the fault plane, \(h\) and \(\lambda\), with the corresponding signs placed before them; or, directly, by the magnitude of the vector \(R\) and the angle \(\gamma\) made by this vector with the positive direction of the strike line of the fault plane (Fig. 1).

Fig. 1. Resolution of the displacement vector \(R\) into components in the plane of the displacer and in space. Fig. 2. Diagram of possible displacements of a point in the plane of the displacer. Fig. 3. Diagram of the intersection of the bed \(AB\) and \(CD\) with the displacer \(ST\) in plan.

In general, the whole variety of conceivable displacements can be represented by a simple table (Table 1 and Fig. 2).

Classification of displacements
Table 1.

Quadrant or line	Angle \(\gamma\)	Elements: shift \(\lambda\)	Elements: throw \(h\)	Name of type of displacement
OA	0	\(+R\)	0	\(+\) shift
AOB	0–90	\(+R\) to 0	0 to \(+R\)	\(+\) reverse-oblique slip
OB	90	0	\(+R\)	reverse fault
BOC	90–180	0 to \(-R\)	\(+R\) to 0	\(-\) reverse-oblique slip
OC	180	\(-R\)	0	\(-\) shift
COD	180–270	\(-R\) to 0	0 to \(-R\)	\(-\) normal-oblique slip
OD	270	0	\(-R\)	normal fault
DOA	270–360	0 to \(+R\)	\(-R\) to 0	\(+\) normal-oblique slip

From this table it is clear that the pure forms—reverse fault, normal fault, and strike-slip—are special cases of dislocations; moreover, reverse fault and normal fault are identical forms, differing from one another only in sign. Therefore, a normal fault may be regarded as a negative reverse fault, or conversely.

From this one may draw the conclusion that the general case of dis—

of displacement is a reverse strike-slip fault, characterized by the elements \(\pm h\) and \(\pm \lambda\), which are related to the total amplitude of displacement by the following simple relations:

\[ h = R \operatorname{Sin}\gamma \ldots (a);\quad \lambda = R \operatorname{Cos}\gamma \ldots (b); \]

\[ \text{and}\quad R=\sqrt{h^{2}+\lambda^{2}}\ldots (c)\ldots\ldots (1) \]

The vector \(R\) is a quantity that is always positive, while the signs of \(h\) and \(\lambda\) are determined by the sign of the corresponding function of the angle \(\gamma\), as the latter varies from \(0\) to \(360^\circ\).

Further resolving the element of the upthrow \(h\), which represents the inclined height of displacement, into vertical and horizontal components in a plane perpendicular to the strike line of the displacer (Fig. 1), we obtain the vertical height of displacement \(OP=H\) and the second, horizontal, component \(OK=b\), which we shall call the width of displacement.

Denoting the angle of dip of the displacer by \(\beta\), we obtain for \(H\) and \(b\), respectively:

\[ H=h\operatorname{Sin}\beta=R\operatorname{Sin}\beta\operatorname{Sin}\gamma \]

\[ \text{and}\quad b=h\operatorname{Cos}\beta=R\operatorname{Cos}\beta\operatorname{Sin}\gamma \ldots\ldots (2) \]

Analysis of Intersections.

The second factor determining the character of a disjunctive disturbance is the mutual position of the planes of the bed and the displacer in space. As is easy to imagine, these mutual relations between the bed and the displacer may also be infinitely varied, and this variety is, so to speak, of a double order. If one approaches the question from a purely mathematical point of view, then this double variety may be represented, on the one hand, by the variability of the dihedral angle formed by the considered planes of the bed and the displacer, which in mine-surveying and mining-geometrical literature has received the name angle of intersection (or crossing), and by the angle of inclination of the line of intersection to the horizon. This procedure, strictly speaking, is the one used in mining geometry in the analysis of disturbances; moreover, it is also the basis of Prof. Bauman’s classification of displacements.

On the other hand, the mutual position of structures in space may be characterized with the same degree of accuracy by the dip angles of the bed and the displacer and by the angle of intersection of their strike lines. This angle is also called by some authors the angle of intersection. However, in order to avoid misunderstandings, we shall call it the angle of encounter.

The first method, from the point of view of pure mathematics, is of course more correct; however, the second has indisputable

advantages. First, in this case we operate with concepts customary for miners and geologists—strike and dip—for denoting the position of planes in space; moreover, all these geometric elements are determined easily, simply, and directly, whereas there one has to determine the dihedral angle and the inclination of the line of intersection, which is at times a rather complex and painstaking task. Second, this method considerably simplifies the analysis of all possible combinations of the mutual positions of the planes under consideration, reducing the whole diversity of the phenomenon to unity. Finally, it makes it possible to derive a general functional dependence between the geometric elements of displacement and displacement in a simple and visual form, without resorting to complicated methods of higher mathematics.

When the strike lines of the bed \(AB\), \(CD\), meet the strike line of the fault \(ST\), four angles are obtained, forming two pairs of adjacent angles, pairwise equal to one another (Fig. 3). From these two pairs we must choose some one pair of equal angles, or one of them, as the independent variable (argument) for further analysis. As such, we shall agree always to take that angle relative to the vertex of which the arrowheads indicating the dip of the bed and of the fault have opposite signs. In the example given (Fig. 3), such an angle will be \(TCD\), or the equal angle \(ABS\). This angle we shall call the angle of meeting, denoting it by the Greek letter \(\omega\) (omega).

From the diagram (Fig. 3) it follows that the angle of meeting may vary continuously from \(0\) to \(180^\circ\), after which the forms of intersection are repeated. The dip angles of the bed and of the fault can evidently vary only within the limits from \(0\) to \(90^\circ\), determining the inclination of the line of intersection to the horizon. When the dip angle of the bed (or of the fault) is equal to 0, the line of intersection will be horizontal. When both the bed and the fault simultaneously have vertical dip, the line of intersection assumes a vertical position. In this limiting and exceptional case the angle \(\omega\) loses its orienting significance, but at the same time the distinction between the wings of the disjunctive disturbance is also lost—on the one hand, and the visible displacement of the bed can be due only to the presence of a strike-slip component, since it is clear that the vertical component of displacement cannot cause any displacement of the bed—on the other. In all other cases the line of intersection will be inclined, and its projection onto the horizontal plane will pass inside the angle \(\omega\) chosen by us.

These simple considerations show that the dip angles of the bed and of the fault do not play an essential role in determining the character of the disjunctive disturbance. The dominant significance in this respect belongs to the angle of meeting \(\omega\). Depending on the magnitude of this angle, all disturbances may be divided into the following 5 groups.

Table 2.

Classification of faults by angle of intersection.

Class	Value of angle $\omega$	Name of the fault
I	0	Longitudinal discordant
II	0—90	Oblique (diagonal) discordant
III	90	Transverse
IV	90—180	Oblique (diagonal) concordant
V	180	Longitudinal concordant

As is evident from this table, the general case in the present aspect will be oblique, or diagonal, faults¹, whereas transverse and longitudinal faults are special cases, when the variable parameter $\omega$ reaches its limiting values.

Since, for a given position of the bed and the dislocator, any displacement may occur, the angle of intersection should be regarded as the generic feature of a fault, and the direction of displacement as the specific one. In this case all conceivable translational faults fit into a simple and clear classification scheme, being defined with complete precision generically—by the character of the crossing—and specifically—by the character of the relative displacement of the blocks—the names. Such a natural² classification has indisputable advantages in comparison with the cumbersome, artificial, and purely formal classifications appearing in the mine-surveying and mining-geometrical literature. But, as we shall see below, it still requires certain additions.

Analysis of the elements of displacement.

By the term displacement, as was already indicated above, we understand the visible noncoincidence of one part of a bed, located—

…in one of the wings of the fault, relative to its continuation—in the other. Unlike displacement, whose magnitude (amplitude) and orientation in the plane of the dislocation are strictly determined by the direction and magnitude of the relative movement of the wings of the disjunctive, the apparent displacement of the half-planes of the bed may be represented in any horizontal, vertical, or inclined section and measured in an arbitrarily chosen direction. Of these sections the most characteristic are:

1. A horizontal section (plan of the working horizon, bed map, etc.).

2. A vertical section across the strike of the bed (a section along the line of boreholes or of a crosscut).

3. A vertical section across the strike of the fault plane (less interesting), and

4. A section in the plane of the fault plane, the one usually used by mine surveyors.

Of the directions in which the magnitude of displacement may be measured, the following are of greatest interest: for vertical sections—1) vertical, 2) horizontal, and 3) normal to the traces of the bed in the given section; for a horizontal section—1) along the strike of the fault plane and 2) across the strike of the bed; for a section in the plane of the fault plane—1) along the strike of the fault plane, 2) along the dip of the fault plane, and 3) normal to the traces of the bed (Baumann’s direction of “conditional displacement”).

It is not difficult to see that all these quantities are mutually connected through the dip angles of the bed and of the fault plane and the angle of intersection between them by a simple functional dependence, so that, for a given position of the bed and fault plane, knowing one of them it is easy to determine all the rest. In practice, however, only the following of these quantities are of importance.

1. Normal displacement—$N$, i.e., the distance between the half-planes of the bed (or their continuations), measured in a direction perpendicular to the plane of the bed. This will be the shortest distance between the displaced parts of the bed in duplications.

2. Vertical displacement—

\[ Z = \frac{N}{\operatorname{Cos}\alpha} \cdots \cdots \cdots \cdots \cdots \cdots \cdots (3a), \]

where $\alpha$ is the dip angle of the bed. This is the vertical distance between the half-planes of the bed (or their continuations), obviously the same for any vertical section.

1. Horizontal displacement (measured across the strike of the bed)—

\[ l = \frac{N}{\operatorname{Sin}\alpha} \cdots \cdots \cdots \cdots \cdots \cdots (3в). \]

It is the shortest horizontal distance between the displaced parts of the bed in cases of duplication.

1. Displacement along the strike of the dislocator—

\[ L=\frac{l}{\operatorname{Sin}\omega} =\frac{N}{\operatorname{Sin}\alpha \cdot \operatorname{Sin}\omega} \ldots\ldots\ldots (3e), \]

where \(\omega\) is the angle of intersection of the bed with the dislocator. This will be the shortest horizontal distance between the displaced parts of the bed in extension.

In what follows, in deriving the formula for the disjunctive displacement, we shall use only one of these elements, namely, the horizontal displacement—\(l\), since it is easily taken directly from the plan and is more convenient to use in graphical constructions, especially by the method of projections with numerical marks.

Derivation of the general formula for a disjunctive displacement.

Having become acquainted in general outline with the character of the quantities determining the form of a disjunctive displacement, we come to the conclusion that in every disjunctive displacement we must distinguish three principal factors: 1) displacement, which is a linear quantity, or more precisely—a vector definitely oriented in space; 2) intersection, or the mutual arrangement of the planes of the bed and the dislocator in space, determined by angular quantities; and 3) offset, likewise representing a linear and therefore vector quantity. The latter is evidently a function of the first and is connected with it through the angular parameters determining the character of the intersection. It is the form of this function that we must establish.

As is evident from the preceding analysis of possible displacements and intersections, the entire conceivable doubly infinite variety of disturbances reduces to unity, the general type of disjunctive displacement being an oblique (diagonal) reverse-strike-slip fault with displacement elements \(\pm h\) and \(\pm \lambda\), which, together with the elements of attitude of the dislocator plane, determine the position of the complete displacement vector in space. Therefore, in order to derive the dependence between the actual displacement of the blocks in space and the visible offset of the bed caused by this displacement, we may confine ourselves to an analysis of this general type of disturbance. However, let us first consider separately the influence of the horizontal and vertical components of the general inclined displacement, replacing, for convenience, the inclined height of displacement \(h\) by the vertical one—

\[ H=h\sin\beta \ldots\ldots\ldots (2a). \]

Let us first suppose that the disturbance is an oblique strike-slip fault (Fig. 4) and that the position of the half-planes of the bed on either side of the dislocator is known. In that case the ampli-

from which the total relative displacement is \(R=\lambda=BC\), and the visible horizontal displacement is \(l=BE\); from triangle \(BCE\) it is easily established that, for any angles of intersection,

\[ l=\pm \lambda \operatorname{Sin}\omega \ldots \ldots \ldots \ldots \ldots (4). \]

Let us note that \(\operatorname{Sin}\omega=\operatorname{Sin}(180-\omega)\) is always a positive quantity; the sign of \(\lambda\) is determined by the direction of displacement, while the sign of \(l\) depends on the convention as to which direction of this quantity we shall regard as positive. We shall clarify this question a few lines below (see p. 161).

Fig. 4.

Let us now suppose that the fault under consideration is a normal fault or a reverse fault, i.e., such a fault in which the hanging wall moved upward or downward along the inclined plane of the fault, without undergoing lateral displacement, so that the displacement vector coincides with the line of dip of the fault. The displacement caused by such movement is more conveniently analyzed on a block diagram (Fig. 5).

Fig. 5. Block diagram of an oblique reverse fault.

Let ABCD and EFGH represent the half-planes of a bed disrupted by the fault plane MNST; moreover, the hanging wall SCN has moved parallel to itself in such a way that point D of the bed, which previously coincided with point E, moved into its present position along the line ED, coinciding with the line of dip of the fault. Dropping from point D the perpendicular DD′ to the horizontal plane P belonging to the footwall of the fault, and joining points D and E, we obtain the right triangle DD′E, whose plane is obviously vertical, since it passes through the plumb line DD′ and is perpendicular to the strike line of the fault ST, because at the same time it passes through its dip line ED. In this triangle the leg

Fig. 6. Diagram of an oblique reverse fault. Letter designations are the same as in Figure 5.

DD′ is the vertical height of the reverse fault—H, the hypotenuse DE is its inclined height—h, the second leg ED′ is its width—b, and the angle at point E is the dip angle of the fault β. Hence we obtain the dependence already known to us:

\[ \mathrm{b}=\mathrm{h}\operatorname{Cos}\beta=\mathrm{H}\operatorname{ctg}\beta \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (2) \]

If we now cut the uplifted wall by the plane P, after first projecting onto it the traces of the bed CD and of the fault MN on the plane Q, then we obtain the picture presented in Fig. 6, where the letter designations are preserved. It is not difficult to see that the fault in plan looks entirely analogous to a strike-slip fault, but is caused by phenomena of a different order.

The quantity l, i.e. the apparent horizontal displacement, here is composed of two elements: the quantity k, caused by the bed’s displacement down the dip as a result of its being cut by a plane at a deeper horizon, and the quantity x, caused by the oblique angle of intersection of the bed and the fault:

\[ \mathrm{l}=\mathrm{k}+x \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (5) \]

The quantity k, as is evident from triangle ACC′ (Fig. 5), is determined by the equation:

\[ \mathrm{k}=\mathrm{H}\operatorname{Ctg}\alpha \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (6) \]

Let us note here that in the upthrown limb the throw always occurs in the direction of dip of the bed, while in the downthrown limb it is the reverse. Therefore, if we agree to measure the apparent horizontal displacement \(l\) always from its footwall limb toward the hanging wall and, when this direction coincides with the direction of dip of the bed, to regard the quantity \(l\) as positive, and in the opposite case as negative, then the signs of \(l\) and \(H\) in the general case must be the same.

The quantity \(x\) is easily derived from the right triangle ELD (Fig. 6), in which the hypotenuse ED is equal to the width of the thrust fault—\(b\), and the angle at point E is equal to the angle of intersection \(\omega\) (or to its supplement to \(180^\circ\)), as the angle formed by mutually perpendicular sides. Consequently, taking (2) into account, we may write:

\[ x = b \operatorname{Cos}\omega = H \operatorname{Ctg}\beta \operatorname{Cos}\omega \ldots\ldots\ldots (7) \]

Substituting the derived values of \(k\) and \(x\) into equation (5), we obtain:

\[ l = H(\operatorname{Ctg}\alpha + \operatorname{Ctg}\beta \operatorname{Cos}\omega) \ldots\ldots\ldots (8) \]

This equation expresses the dependence between the height of the true relative displacement, with vertical movement of the limbs—\(H\), and the horizontal displacement of the bed caused by it—\(l\).

Here, however, some clarification is required. Depending on the angle of intersection and on the relative displacement of the limbs, the quantity \(x\) may enter equation (5) with different signs. But if we adhere to the rule we have adopted, namely: to take as the angle of intersection \(\omega\) that one of the angles in which the arrows indicating the dip of the bed and of the fault strive to turn the half-planes of the bed and of the fault toward one another, then it turns out that the sign before the second term of the right-hand side of equality (8) remains positive in all cases of dislocations, as is readily verified from consideration of the diagrams (Figs. 7 and 8).

From the diagrams (Fig. 7) it is seen that in discordant thrust faults, i.e. when \(\omega < 90^\circ\), the quantity \(x\) is added to the throw \(k\), which increases the apparent horizontal displacement of the bed—\(l\). Therefore the validity of expression (8) is obvious. In concordant thrust faults the quantity \(x\) is subtracted from \(k\), but at the same time the sign of the function of the angle \(\omega\) also changes, since this angle becomes greater than \(90^\circ\). Therefore the sign before the second term of expression (8) in general does not change.

When the direction of relative displacement of the limbs changes, i.e. in the case of normal faults, the apparent horizontal displacement of the structure occurs in the opposite direction. The sign of \(l\) changes to the opposite one, but at the same time the sign of \(H\) also changes, so that expression (8) does not change its form in this case either (Fig. 8).

We shall now proceed to consider the general form of dislocation,

Normal reverse faults

Discordant reverse faults

Fig. 7-a, 7-b. Diagrams of reverse faults under various combinations of bed and dislocator. \(ST\)—trace of the dislocator on the lower horizon; \(MN\)—the adopted form of the pipe of the hanging wall in a reverse fault. \(FE\)—initial position of the bed in the footwall; \(ED\)—line of displacement (slip) of point \(D\) when it is moved from \(E\) to \(D\). \(DC\)—trace of the bed on the plane of the uplifted wing (hanging wall); \(BA\)—trace of the bed in the hanging wall on the plane of the original horizon after displacement. A circle with a dot denotes the uplifted wing.

In the diagrams:
- Discordant faults
- Concordant faults
- Footwall
- Hanging wall
- \(\theta = k + x;\ \omega < 90^\circ\)
- \(\theta = k + x;\ \omega > 90^\circ\)
- \(\theta = k - x;\ \omega < 90^\circ\)
- \(\theta = k - x;\ \omega > 90^\circ\)

Fig. 8a. Fig. 8b.

Figs. 8a, 8b. Diagrams of faults for various combinations of the bed and the fault plane. \(CD\)—initial position of the bed in the footwall. \(DE\)—line of displacement (slip). \(EF\)—position of the displaced part of the bed in the hanging wall after the faulting. \(AB\)—trace of bed \(CD\) on the horizon of the downthrown block. A circle with a dot indicates the relatively uplifted block. A circle with a cross indicates the downthrown block.

when the relative displacement of the hanging wall occurs in an arbitrary direction.

As we have already seen, any arbitrary translational displacement may be regarded as composite and resolved into two elementary, mutually perpendicular displacements along the strike and dip lines of the displacer. Each of these elementary displacements will produce the corresponding horizontal displacement of the structure under consideration, so that the total effect will be their algebraic sum.

\[ l = l_1 + l_2, \]

where \(l_1\) is the apparent horizontal displacement caused by the element of reverse faulting (or normal faulting), and \(l_2\) by the element of strike-slip. Substituting here the values of \(l\) that we have derived for vertical and horizontal displacements respectively, we obtain the general formula of the disjunctive in the following form:

\[ l = \mathrm{H}\,(\mathrm{Ctg}\alpha + \mathrm{Ctg}\beta\,\mathrm{Cos}\omega) \pm \lambda\,\mathrm{Sin}\omega \qquad \ldots \qquad (9); \]

The actual sign of the last term of this formula depends exclusively on the sign of \(\lambda\), i.e., on the direction of the strike-slip, since \(\mathrm{Sin}\omega\), for all possible variations of the angle of intersection from \(0\) to \(180^\circ\), remains a positive quantity. Therefore, for positive values of \(\mathrm{H}\) and \(\lambda\), the last term will increase the apparent horizontal displacement of the bed when the inclination of the bed coincides with the positive direction of strike-slip. And since the inclination of the bed in the uplifted wall always occurs in the direction of its dip, an increase of \(l\) will occur only in the case when the dip of the bed coincides with the positive direction of strike-slip. If, however, the bed dips toward negative values, then the last term must be subtracted from the first, since the displacements caused by the unambiguous \(\mathrm{H}\) and \(\lambda\) will be directly opposite to one another.

Hence follows the rule of signs: when the bed dips toward positive values of strike-slip, the sign before the last term of formula (9) is taken as \((+)\); when the bed dips toward negative values, it should be taken as \((-)\).

Analysis of the Formula of the Disjunctive

The formula (9) derived by us has universal significance, since it is applicable to any translational disjunctive, irrespective of the conditions of intersection of the bed and the displacer and of the directions of displacement. Relating the linear quantities \(\mathrm{H}\), and \(l\) by means of angular parameters that depend on the relative position of the structures under consideration, it makes it possible to analyze the influence of these latter. Therefore, before proceeding to the solution of the direct and inverse problem of the disjunctive, let us try to analyze the influence of changes in these parameters, taking as a basis the influence of changes in the angle of intersection \(\omega\).

The angle of intersection, according to the conditions we have adopted, may vary within the limits from \(0\) to \(180^\circ\). For \(\omega = 0\), i.e., in the case of longitudinal discordant disturbances, the formula for the disjunctive takes the form:

\[ l = H(\operatorname{Ctg}\alpha + \operatorname{Ctg}\beta) \tag{10} \]

This shows that the influence of the strike-slip component, as was to be expected, in this case produces no visible displacement of the bed. The quantity \(l\) is a function only of the height of displacement and of the angles \(\alpha\) and \(\beta\). Since the angles \(\alpha\) and \(\beta\), as the angles of dip of the bed and of the displacer, vary within the limits from \(0^\circ\) to \(90^\circ\), their functions are always positive; the sign of \(l\) will be identical with the sign of \(H\), and the absolute magnitude of \(l\) will increase with an increase in the absolute value of the height \(H\) and with a decrease in the angles \(\alpha\) and \(\beta\) (Figs. 9 and 10).

As the angle of intersection changes from \(0\) to \(90^\circ\), \(\cos \omega\) gradually decreases from \(1\) to \(0\). Therefore the expression in parentheses, serving as the coefficient of \(H\), will gradually decrease, while remaining positive throughout.

Fig. 9.

Fig. 10.

At the same time, \(\sin \omega\), serving as the coefficient of \(\lambda\), gradually increases, as a result of which the influence of the strike-slip element correspondingly increases.

It is clear that this influence may either strengthen or weaken the amplitude of the visible horizontal displacement \(l\). Therefore, in some cases it may turn out that the displacement of the bed caused by vertical movement will be counterbalanced by an equal displacement, directed in the opposite direction, caused by the strike-slip element. As a result, no visible displacement of the bed will occur, since the displacement vector will go along the line of intersection of the bed with the displacer.

This, obviously, will take place when:

\[ H(\operatorname{Ctg}\alpha + \operatorname{Ctg}\beta \cos \omega) \pm \lambda \sin \omega = 0 \tag{11} \]

whence

\[ \frac{H}{\lambda} = \mp \frac{\sin \omega}{\operatorname{Ctg}\alpha + \operatorname{Ctg}\beta \cos \omega} \tag{11 a} \]

where the upper sign is for beds dipping toward positive values of \(\lambda\), and the lower sign for those dipping toward negative values.

When the angle of intersection reaches \(90^\circ\), we obtain a transverse fault, and the formula of the disjunctive takes the form:

\[ l = H \operatorname{Ctg}\alpha \pm \lambda \tag{12} \]

In this case the influence of the slip element reaches its maximum value. The influence of the vertical displacement \(H\) decreases with increasing dip angle of the bed, while the dip angle of the displacer plays no role.

The absence of visible displacement will occur under the condition:

\[ \frac{H}{\lambda} = \operatorname{Tg}\alpha \tag{13} \]

when the bed dips toward negative values, and under:

\[ \frac{H}{\lambda} = -\operatorname{Tg}\alpha \tag{13a} \]

in the opposite case.

With a further increase of the angle of intersection from \(90^\circ\) to \(180^\circ\), we enter the domain of oblique concordant faults. \(\operatorname{Cos}\omega\) assumes negative values, varying from \(0\) to \(-1\), while \(\operatorname{Sin}\omega\) remains positive. The expression:

\[ (\operatorname{Ctg}\alpha + \operatorname{Ctg}\beta \operatorname{Cos}\omega) \tag{14} \]

which serves as the coefficient of \(H\), may in this case take both positive and negative values, and at the boundary between them become \(0\). This property, inherent only in concordant disjunctives, deserves special consideration.

When the bed dips more gently than the displacer, then \(\operatorname{Ctg}\alpha\) will be greater than \(\operatorname{Ctg}\beta\), since these functions decrease as the angles increase. All the more so,

\[ |\operatorname{Ctg}\alpha| > |\operatorname{Ctg}\beta \operatorname{Cos}\omega| \]

because the cosine of an angle is always less than \(1\). Therefore, when the bed dips more gently than the displacer, an oblique concordant disjunctive obeys the same laws as those derived for oblique discordant faults, with only the difference that the visible horizontal displacement caused by the vertical component will, all other conditions being equal, always be smaller than in the latter. The limit of this, obviously, will be the equality

\[ \operatorname{Ctg}\alpha = -\operatorname{Ctg}\beta \operatorname{Cos}\omega \tag{15} \]

In this case the vertical component will not produce a visible displacement of the bed, and the latter, if it exists, must be due exclusively to the slip element.

When

\[ |\operatorname{Ctg}\alpha| < |\operatorname{Ctg}\beta \operatorname{Cos}\omega| \tag{15a} \]

the sign of expression (14) will change to the opposite, and the displacement of the bed caused by the vertical movement will occur—

goes in the opposite direction, causing, in the case of an upthrow, not a compression of the bed but its extension (Fig. 11).

Such disturbances, which differ substantially in the character of the displacements they produce, are more conveniently separated into special varieties, assigning them the name thrusts, when the hanging wall moves upward along the inclined plane of the displacer, and reverse slips, when the hanging wall moves downward. The latter form, however, is extremely rare, and, in the mechanism of its formation, can probably arise only as a result of bending of the surface of the displacer in a thrust. Such cases are indicated, among others, by M. A. Usov for certain disturbances of the Kuzbass, but they have not yet been proved with complete certainty, since an entirely analogous form may result from oblique displacement under the influence of a horizontal component.

Fig. 11.

The condition for the absence of visible displacement for thrust-slips, in the presence of both vertical and horizontal components of the complex displacement, will be:

\[ H(\operatorname{Ctg}\alpha+\operatorname{Ctg}\beta\operatorname{Cos}\omega)=+\lambda\operatorname{Sin}\omega \tag{16} \]

for beds dipping in the direction of negative values of the slip, and

\[ H(\operatorname{Ctg}\alpha+\operatorname{Ctg}\beta\operatorname{Cos}\omega)=-\lambda\operatorname{Sin}\omega \tag{16a} \]

for beds dipping in the direction of its positive values. It is clear that in disturbances of this type the slip element plays a relatively large role, since the coefficient of \(H\) is comparatively small and even becomes 0 for finite values of \(\alpha\) and \(\beta\).

When \(\omega\) reaches its extreme limit—\(180^\circ\), \(\operatorname{Sin}\omega\) becomes 0, and \(\operatorname{Cos}\omega=-1\), and the formula for the disjunctive displacement takes the form:

\[ l=H(\operatorname{Ctg}\alpha-\operatorname{Ctg}\beta) \tag{17} \]

This will be the formula for a longitudinal concordant disjunctive displacement, differing from formula (10) only by the sign before the second term of the expression in parentheses. However, this small difference has essential significance.

When \(\operatorname{Ctg}\alpha>\operatorname{Ctg}\beta\), i.e., when the displacer dips more steeply than the bed it disturbs, the sign of \(l\) coincides with the sign of \(H\). Likewise,

As in unconformable faults, in a reverse fault we obtain overlap, and in a normal fault—a gap (Figs. 12 and 13), and both are directly reflected in the formula in the form of the second term on the right-hand side of equality (17).

As the dip angle of the dislocator approaches the dip angle of the bed, the magnitude of the apparent displacement decreases, and when these angles are equal it becomes 0. As a result we obtain so-called bedding-plane displacements, which do not cause any apparent displacement of the bed.

When the dip angle of the dislocator becomes smaller than the dip angle of the bed, the sign of the expression in parentheses changes to the opposite, and the quantities \(H\) and \(l\) become of opposite sign. On the bed this is reflected in the fact that, for positive movement of the hanging wall, a gap is obtained, and for negative movement—overlap (Figs. 14 and 15). In short, here, as in the preceding genus of oblique conformable faults, we obtain two special varieties, which we propose to call overthrust and over-normal fault.

Fig. 12.

Fig. 13.

Fig. 14.

Fig. 15.

Morphological Classification of Disjunctives.

As the foregoing analysis of the angular parameters of the general formula for translational disjunctives shows, the angle of intersection \(\omega\) has a decisive influence on the form of the fault. This justifi—

determines its choice as a generic classificatory feature. At the same time this analysis shows that the dip angles of the bed and of the fault plane play an almost subordinate role in discordant faults, and only in concordant ones is their influence fairly substantial. This latter circumstance compels us to supplement the classificatory scheme given above with newly revealed forms as additional types of concordant faults. Thus, our classificatory scheme ultimately assumes the following form (see Table 3).

As is evident from this table, we operate with a very limited number of terms to designate the various faults; moreover, the overwhelming majority of these terms are considered generally accepted in the geological literature and are used in the sense that we, too, assign to them. The only new point may be considered the introduction of the concepts “positive” and “negative” for shifts, which is absolutely necessary in order to distinguish the direction of horizontal displacements, and the terms thrust and overthrust for certain types of concordant faults.

The term thrust appears in the geological literature, and by it are usually meant vertical faults of the reverse-fault type, only with gentle angles of the fault plane. However, no clear distinction between these two similar concepts is given. We give the concept thrust a narrower, but entirely concrete, content, understanding by it only concordant faults in which the hanging wall moves upward along the inclined plane of the fault, and in which the fault plane dips much more gently than the structures displaced by it. It is clear that such a division is to some extent conventional, as indeed is also the purely geological definition of this term (more gentle, as compared with a steeper one—a reverse fault), since it depends not only on the dip angle of the fault plane, but also on the dip angle of the bed displaced by it. Nevertheless, the introduction of special names for faults that produce fundamentally different forms of displacement is, in our view, highly desirable.

The same considerations also prompted the introduction of a new term for faults directly opposite to a thrust. After much hesitation I settled on the term overthrow (peresbros), borrowing it from mine geometry, since there is no suitable term in the geological literature. In doing so I proceeded from the following considerations. In the character of displacement of the hanging wall, this fault is analogous to a normal fault. On the other hand, here, instead of a gap, an overlap is obtained. Thus the fault represents, as it were, a normal fault with overlap; hence the word formation of the term—overthrow.

Thus, our classification embraces all conceivable variety of intersections of a bed with a fault plane and all possible cases of relative displacement of the walls of the fault, distinguishing

Table 3.

Classification of disjunctive disturbances

No.	Displacement	Skew	Inconformable: longitudinal, \(\omega = 0\)	Inconformable: oblique (diagonal), \(0 < \omega < 90\)	Transverse, \(\omega = 90\)	Concordant steep, \(\operatorname{ctg}\alpha > \operatorname{ctg}\beta \cos\omega\): oblique (diagonal), \(90 < \omega < 180\)	Concordant steep, \(\operatorname{ctg}\alpha > \operatorname{ctg}\beta \cos\omega\): longitudinal, \(\omega = 180\)	Concordant gentle, \(\operatorname{ctg}\alpha < \operatorname{ctg}\beta \cos\omega\): oblique (diagonal), \(90 < \omega < 180\)	Concordant gentle, \(\operatorname{ctg}\alpha < \operatorname{ctg}\beta \cos\omega\): longitudinal, \(\omega = 180\)
1.	\(H = 0;\ \lambda > 0;\ \gamma = 0\)	—	—	—	—	Right lateral	Right oblique	Right reverse-oblique	Right reverse
2.	\(H > 0;\ \lambda > 0;\ 0 < \gamma < 90^\circ\)	—	—	—	—	Positive upthrow	Positive upthrow-oblique	Positive overthrust-oblique	Positive overthrust
3.	\(H > 0;\ \lambda = 0;\ \gamma = 90^\circ\)	—	—	—	—	Upthrow	Upthrow	Thrust	Thrust
4.	\(H > 0;\ \lambda < 0;\ 90 < \gamma < 180^\circ\)	—	—	—	—	Negative upthrow	Negative upthrow-oblique	Negative thrust-oblique	Negative thrust
5.	\(H = 0;\ \lambda < 0;\ \gamma = 180^\circ\)	—	—	—	—	Left lateral	Left oblique	Left thrust-oblique	Left thrust
6.	\(H < 0;\ \lambda < 0;\ 180 < \gamma < 270^\circ\)	—	—	—	—	Negative downthrow	Negative downthrow-oblique	Negative normal-oblique	Negative normal fault
7.	\(H < 0;\ \lambda = 0;\ \gamma = 270^\circ\)	—	—	—	—	Downthrow	Downthrow	Normal fault	Normal fault
8.	\(H < 0;\ \lambda > 0;\ 270 < \gamma < 360^\circ\)	—	—	—	—	Positive downthrow	Positive downthrow-oblique	Positive normal-oblique	Positive normal fault

simultaneously and the character of the displacements of the primary structure produced by them. The latter circumstance, it is true, may be regarded both as an advantage of our classification and as its shortcoming. It is clear that a reflection of the character of the displacement—whether it is compression or extension—is highly desirable. On the other hand, this leads to the fact that one and the same disturbance, in one case, according to one of the structural elements displaced by it, may be characterized, for example, as an upthrow, and according to another—as a thrust. But this is quite inevitable in any classification in which the attribute of displacement figures. However, in contrast to mine-surveying and mining-geometric classifications, which operate exclusively with the kind of displacements, our classification is based, first of all, on the actual relative movement of the blocks; and the numerical expressions of the elements of displacement make it possible to foresee the character of the displacement of any structural element—a bed, vein, etc.—if the conditions of its occurrence are known.

The proposed classification is sufficiently universal, simple, and convenient to use. Difficulties may arise only in certain limiting, and therefore exceptionally rare, cases, namely: 1) when

the dislocator dips perfectly vertically and, consequently, its hanging and footwall blocks are indistinguishable; and 2) when the dislocator has a perfectly horizontal position. In the first case these will be the so-called neutral dislocations, in which either block may be taken as the hanging wall, which introduces a certain arbitrariness into determining the form of the disjunctive disturbance. But after the choice of the hanging wall has been made on one basis or another, determining the kind and type of disjunctive disturbance no longer presents any difficulty.

As for the second case, for ordinary disjunctive disturbances in the strict sense of this concept it represents, in fact, no more than a mathematical abstraction. In nature, disturbances of this sort appear only on large scales and are called nappes. But this already pertains to phenomena of another order, which we shall exclude from our consideration.

Direct problem.

The formula derived by us:

\[ l = H(\operatorname{Ctg}\alpha + \operatorname{Ctg}\beta \operatorname{Cos}\omega) \pm \lambda \operatorname{Sin}\omega \tag{9} \]

makes it possible easily and quickly to solve both the direct and the inverse problems of a disjunctive disturbance. By the direct problem, as was already indicated above, we mean the determination of the direction and amplitude of displacement of the blocks of a disjunctive disturbance from the visible displacement of the primary structural elements disturbed by it.

Let us first analyze this problem and consider what data are necessary and sufficient for its solution. Of the components entering into formula (9), the magnitude of the visible horizontal displacement \(l\) is given by the conditions. For this, evidently, the position of the bed on both sides of the dislocator must be known. The dip angles of the bed and of the dislocator are obviously also known, as is the angle of intersection formed by the lines of their strike, which can easily be measured on the plan. Thus, \(H\) and \(\lambda\) remain unknown.

It is clear that one equation with two unknowns is insoluble. Consequently, the data that we have just considered are insufficient. In addition to it, it is necessary to have a second independent equation in order for the problem to be solvable. Here two cases are possible.

1st case.

Two beds are given (or any two primary structural elements) disturbed by one and the same dislocator but possessing different conditions of occurrence, and the positions of their blocks on both sides of the dislocator are known (Fig. 16).

In such a case, for each of them we can write an independent equation. In our example these will be

\[ \begin{aligned} l_1 &= \mathrm{H}\,(\mathrm{Ctg}\,\alpha_1+\mathrm{Ctg}\,\beta\,\mathrm{Cos}\,\omega_1)-\lambda\,\mathrm{Sin}\,\omega_1\\ l_2 &= \mathrm{H}\,(\mathrm{Ctg}\,\alpha_2+\mathrm{Ctg}\,\beta\,\mathrm{Cos}\,\omega_2)-\lambda\,\mathrm{Sin}\,\omega_2 \end{aligned} \tag{18} \]

Here the sign before the last term in both equations is taken as \((-)\), since both seams dip to the left, toward negative values of \(\lambda\). The quantities \(l_1\) and \(l_2\) must be taken with the sign \((+)\), since both seams in the hanging wall have been displaced in the direction of their dip.

Fig. 16.

Solving these two equations obviously presents no difficulties. Indeed, suppose that in our example:

\[ \alpha_1=50^\circ;\quad \alpha_2=65^\circ;\quad \beta=60^\circ;\quad \omega_1=130^\circ;\quad \omega_2=120^\circ;\quad l_1=40;\quad \text{and } l_2=10\ \text{m}. \]

Then the coefficients of the unknowns for the first equation will be:

\[ \begin{array}{lll} \mathrm{Ctg}\,50^\circ=0{,}839 & \mathrm{Ctg}\,60^\circ=0{,}577 & \mathrm{Sin}\,130^\circ=0{,}766 \\ \underline{\phantom{\mathrm{Ctg}\,50^\circ=}\ -0{,}372} & \mathrm{Cos}\,130^\circ=-0{,}643 & \\ +0{,}467 & \underline{\phantom{\mathrm{Cos}\,130^\circ=}\ -0{,}372} & \end{array} \]

And for the second, respectively—

\[ \begin{array}{lll} \mathrm{Ctg}\,65^\circ=0{,}466 & \mathrm{Ctg}\,60^\circ=0{,}577 & \mathrm{Sin}\,120^\circ=0{,}866 \\ \underline{\phantom{\mathrm{Ctg}\,65^\circ=}\ -0{,}289} & \mathrm{Cos}\,120^\circ=-0{,}500 & \\ +0{,}177 & \underline{\phantom{\mathrm{Cos}\,120^\circ=}\ -0{,}289} & \end{array} \]

Substituting the numerical values of the coefficients into our equations, we obtain:

\[ \begin{aligned} 0{,}467\,\mathrm{H}-0{,}766\,\lambda &= 40\\ 0{,}177\,\mathrm{H}-0{,}866\,\lambda &= 10 \end{aligned} \tag{18a} \]

Hence:

\[ H=\frac{40 \cdot 0.866-10 \cdot 0.766}{0.467 \cdot 0.866-0.177 \cdot 0.766}=100 \]

and \(\lambda=+9\);

Having determined \(H\) and \(\lambda\), it is easy to compute all the remaining elements of the displacement, since we already know that the inclined height is

\[ h=\frac{H}{\sin \beta}=\frac{100}{\sin 60^\circ}=\frac{100}{0.866}=115\ \text{m}, \]

the total amplitude:

\[ R=\sqrt{h^2+\lambda^2}=\sqrt{13225+81}=115.3 \]

and the angle

\[ \gamma=\operatorname{Arctg}\frac{h}{\lambda}=\operatorname{Arctg}\ 12.8=85^\circ 30' \]

Thus, the fault under consideration will be a positive thrust-strike fault with \(h=+115\ \text{m}\) and \(\lambda=+9\ \text{m}\), or \(R=115.3\) and \(\gamma=85^\circ 30'\).

The same problem can also be solved graphically, so that the two methods, mutually supplementing one another, may serve as checks on each other. The graphical solution is more visual, but less exact, especially at steep angles of dip of the bed and of the displacer.

Graphical solution

For the graphical solution one uses the method of projections with numerical elevations. For illustration, let us consider the same example (Fig. 17).

Let \(ST\) be the trace of the displacer on the plane of the plan, \(AB\) and \(CD\) the traces of the displaced wings of one bed with dip angle \(\alpha_1\), and \(EF\) and \(GH\) the displaced wings of a second bed with dip angle \(\alpha_2\), and let \(\beta\) be the dip angle of the displacer.

First of all, let us find the lines of intersection of the beds with the displacer. To do this, imagine a second horizontal plane at an arbitrary distance \(r\) below the plane of the plan. The traces of the beds and of the displacer on this plane will evidently be parallel to the corresponding traces on the plane of the plan, but will be offset from them down the dip by a distance \(a=r\operatorname{Ctg}\alpha\), where \(\alpha\) is the dip angle of the corresponding planes of the bed or of the displacer.

For the graphical determination of these distances, draw off to the side two parallel lines \(HH\) and \(OO\) (Fig. 17-II) at an arbitrarily chosen distance \(r\) from one another. At an arbitrary point \(M_1\) on the line \(OO\), set off the dip angle of the first bed, \(\alpha_1\), and extend the line thus obtained to its intersection with the second parallel \(HH\), and from the point \(N_1\) obtained, drop the perpendicular \(NO_1\) to the first. Then the segment \(M_1O_1\), equal to \(r\operatorname{Ctg}\alpha_1\), will obviously represent the required quantity \(a_1\), by which the trace of the first bed will be offset on the lower horizon. We proceed in exactly the same way to determine \(a_2\) and \(a_3\)—the corresponding distances for the second bed and the displacer.

Now we lay off from the line \(ST\) the found segment \(a_3\) in the direction of dip of the displacer, and through its end draw the line \(S_1T_1\), parallel to \(ST\). This line will be the trace of the displacer on the second horizon. In an analogous way we construct the traces \(A_1B_1\) and \(E_1F_1\), respectively, of the first and second beds of the footwall, and extend them to their intersections with the trace of the displacer \(S_1T_1\). Next we connect the obtained points \(BB_1\) and \(FF_1\) by straight lines and extend them to their mutual intersection at the point \(L\). Then the lines \(BL\) and \(FL\) will be the lines of intersection of the first and second beds

Fig. 17.

with the plane of the displacer in its footwall, and the point \(L\) the point of meeting of the lines of their mutual intersection with the plane of the displacer.

By a completely identical construction we find the lines \(CW\) and \(GW\), representing the lines of intersection of the same beds in the hanging wall of the displacer, and \(W\)—the point of meeting of the lines of their intersection with the plane of the displacer.

It is not difficult to see that the points \(L\) and \(W\) lie in the plane of the displacer and represent corresponding points which, before the disturbance, were in immediate proximity and practically coincided with one another. Continuing to take the hanging wall of the disjunctive as the movable wall, we conclude that the point \(W\), which before the disturbance coincided with the point \(L\), during the disturbance

has moved into position W. Connecting point L with W, we shall obviously obtain the vector of the total relative displacement of the hanging wall \(R\), or, in the plane of the map, its horizontal projection.

To determine the true magnitude of the vector \(R\), one may make use of the rotation of the plane of the fault about the line ST until it coincides with the plane of the map. But, in order not to clutter the drawing, one may use another device.

Let us resolve the vector \(R\), obtained by us in projection, into two components in the plane of the fault—one along the strike of the fault (the strike-slip component), and the other perpendicular to the strike of the fault (the dip-slip component). The strike-slip component KW, being a horizontal line, is projected onto the plane of the map without distortion. Therefore its magnitude is determined directly on the map. The dip-slip component (the inclined height of the displacement) coincides with the line of dip of the fault, and its projection onto the horizontal plane \(KL = b\) represents the width of the displacement.

Knowing the magnitude \(b\) and the dip angle of the fault, it is not difficult also to determine the magnitude \(h\), as the hypotenuse of a right triangle constructed on the given leg \(b\) and the adjacent acute angle \(\beta\), as has been done in Fig. III (Fig. 17). At the same time the vertical height of the displacement \(H\) is also determined, as the second leg of this triangle.

Finally, having determined \(h\) and \(\lambda\), it is not difficult to find also the full amplitude of the displacement \(R\), as the hypotenuse of a right triangle constructed on the legs \(h\) and \(\lambda\), simultaneously determining the angle \(\gamma\) (Fig. 17—IV).

Thus the problem posed is solved completely, both analytically and graphically; moreover, we obtain, if not an absolutely exact value—owing to the assumptions indicated above—then in any case a value close to the truth for the amplitude of the actual relative displacement of the wings of the disjunctive and the direction of this displacement.

2nd case.

Let us now assume that only one bed is given, whose position on both sides of the fault is known, and the striation on the plane of the fault.

In such a case, for the displaced bed, as before, we may write the equation:

\[ l = H(\operatorname{Ctg}\alpha + \operatorname{Ctg}\beta \operatorname{Cos}\omega) \pm \lambda \operatorname{Sin}\omega \tag{9} \]

in which the sign before the last term is determined, according to the rule given above, from the scheme of the arrangement of the bed and the fault. From measurements of the striation on the plane of the fault we additionally compose the following equations:

\[ \frac{h}{\lambda} = \operatorname{Tg}\gamma \tag{19} \]

\[ \mathrm{H}=h\operatorname{Sin}\beta \tag{2} \]

Solving these equations with respect to \(\mathrm{H}\), \(h\), and \(\lambda\) no longer presents any great difficulty. However, in this case it is more convenient to use the following device. We know that:

\[ \lambda=R\operatorname{Cos}\gamma;\qquad h=R\operatorname{Sin}\gamma \tag{1} \]

and

\[ \mathrm{H}=h\operatorname{Sin}\beta=R\operatorname{Sin}\beta\operatorname{Sin}\gamma \tag{2} \]

Substituting the quantities \(\mathrm{H}\) and \(\lambda\), determined by these expressions, into our equation (9), we obtain:

\[ l=R(\operatorname{Ctg}\alpha\operatorname{Sin}\beta\operatorname{Sin}\gamma+\operatorname{Cos}\beta\operatorname{Cos}\omega\operatorname{Sin}\gamma\pm\operatorname{Cos}\gamma\operatorname{Sin}\omega) \tag{20} \]

Only one unknown, \(R\), enters into this equation, since all the angular parameters and \(l\) are given by the conditions. Hence we determine the full amplitude of displacement \(R\) directly:

\[ R=\frac{l}{\operatorname{Ctg}\alpha\operatorname{Sin}\beta\operatorname{Sin}\gamma+\operatorname{Cos}\beta\operatorname{Cos}\omega\operatorname{Sin}\gamma\pm\operatorname{Cos}\gamma\operatorname{Sin}\omega} \tag{21} \]

It is clear that the sign before the last term of the denominator is determined by the same rule as in equation (9). It turns out that this equation gives not only the direct dependence of \(R\) on \(l\), but also makes it possible to establish the direction of displacement quite definitely, even in the case when the vectorial direction of the striation is unclear.

Indeed, in practice it often happens that the striation on the plane of the displacer is manifested quite distinctly, but the vectorial direction of motion remains in doubt. Or, in other words, the angle made by the striation with the displacer is determined alternatively: either \(\gamma_1\), or \(\gamma_1 \pm 180^\circ\). Substituting either of these alternative values of \(\gamma\) into our equation (21), we obtain a certain value of \(R\). Since, according to the condition adopted by us, the angle \(\gamma\) is measured from the positive direction of the strike line of the displacer counterclockwise to the displacement vector, which is always taken as a positive quantity, it is clear that the quantity \(R\) cannot have negative values. Consequently, if the value of \(R\) obtained by us is positive, then the angle \(\gamma\) has been taken correctly. If, however, we have obtained a negative value for the vector \(R\), this means that the angle \(\gamma\) must be changed to \(\gamma \pm 180^\circ\). But, since:

\[ \operatorname{Sin}\gamma=-\operatorname{Sin}(\gamma\pm180^\circ) \]

\[ \operatorname{Cos}\gamma=-\operatorname{Cos}(\gamma\pm180^\circ) \]

then such a substitution will not cause a change in the numerical value of \(R\), but only a change of its sign to the opposite one, since \(\operatorname{Sin}\gamma\) and \(\operatorname{Cos}\gamma\) enter each term of the denominator of formula (21) once.

This, in turn, shows that, when striation is present on the plane of the fault, even when the direction of movement along it cannot be established, the disjunction can be resolved completely from the apparent displacement of only one primary structural element. Hence the importance of measuring the direction of the striation is clear.

On the other hand, it is also clear that in this case the problem has only an approximate solution. As was already indicated above, the striation on the plane of the fault may fail to coincide with the direction of the total displacement of the wings. Consequently,

Fig. 18. I. Diagram of the attitude of the bed and the fault. II. Striation on the plane of the fault.

the values of \(R\), \(H\), \(\lambda\), etc., obtained by this method may deviate more or less appreciably from their true values.

Let us consider a practical example of solving the problem.

Let there be given a bed \(ABCD\), disturbed by a fault \(ST\) (Fig. 18), and the position of the stria on the plane of the fault, the stria making with the strike line of the fault an angle \(\gamma\) equal to \(120^\circ\) or \(300^\circ\). The dip angle of the bed is \(\alpha = 50^\circ\), the dip angle of the fault is \(\beta = 70^\circ\), and \(l = -50\) m. From the diagram we determine the angle of intersection \(\omega = 130^\circ\) (Fig. 18), and write the formula:

\[ R = \frac{l}{\operatorname{Ctg}\alpha \operatorname{Sin}\beta \operatorname{Sin}\gamma + \operatorname{Cos}\beta \operatorname{Sin}\gamma \operatorname{Cos}\omega - \operatorname{Sin}\omega \operatorname{Cos}\gamma} \ldots \tag{22} \]

In the present case the sign before the last term is taken as \((-)\), since the bed dips to the left, toward the negative values of \(\lambda\). Then we write out the functions of the angles, taking \(\gamma = 120^\circ\).

\[ \begin{array}{lll} \operatorname{Ctg}\ 50^\circ = 0.839 & \operatorname{Cos}\ 70 = 0.342 & \operatorname{Sin}\ 130 = 0.766 \\ \operatorname{Sin}\ 70^\circ = 0.940 & \operatorname{Cos}\ 130 = -0.643 & \operatorname{Cos}\ 120 = -0.500 \\ \operatorname{Sin}\ 120^\circ = 0.866 & \operatorname{Sin}\ 120 = 0.866 & \end{array} \]

\[ \begin{array}{ccc} +0.680 & -0.190 & -0.383 \end{array} \]

and, substituting their values in the formula, we obtain:

\[ R = \frac{-50}{+0.680 - 0.190 + 0.383} = \frac{-50}{+0.873} = -43.6\ \text{m} \]

For \(R\) a negative value was obtained. This shows that, for the adopted value \(\gamma = 120^\circ\), such a displacement cannot be obtained. Therefore another of its possible values must be taken, i.e. \(\gamma = 300\). Obviously, the numerical value of \(R\) will not change in this case; only the sign will change.

Thus we obtain:

\[ R = 43.6;\quad \gamma = 300^\circ, \]

whence we also determine the remaining elements of the displacement:

\[ \begin{aligned} \lambda &= R \operatorname{Cos}\gamma = +21.8\ \text{m}.\\ h &= R \operatorname{Sin}\gamma = -37.8\ \text{m}.\\ H &= h \operatorname{Sin}\beta = -35.5\ \text{m}. \end{aligned} \]

A simple, elegant, and quite definite solution of the problem is obtained: the disjunctive dislocation is a concordant positive normal-strike-slip fault with an amplitude of 43.6 meters, with its hanging wall having dropped by \(35.5\ \text{m}\) and shifted in the positive direction of the strike of the dislocator by \(21.8\ \text{m}\).

Graphical solution.

Let us solve the same problem graphically (Fig. 19), using the method of projections with numerical marks.

Let \(AB\) and \(CD\) be the traces of the displaced parts of the bed, and \(TS\) the trace of the dislocator on the plane of the plan. Let us imagine a second horizontal plane at an arbitrary distance \(r\) below the plane of the plan, and construct on this plane the traces of the displaced parts of the bed and of the dislocator. To do this, to the side we draw two parallel lines \(OO\) and \(NN\) at the chosen distance \(r\) from one another, take on one of them an arbitrary point \(N'\), and construct at it the angles \(\alpha\) and \(\beta\), corresponding to the dip angles of the bed and of the dislocator (Fig. 19-II), and the perpendicular \(N'O\). Then the segments of the second parallel \(OM_1\) and \(OM_2\), enclosed between the perpendicular and the corresponding inclined line, will be equal to:
\(OM_1 = r \operatorname{Ctg}\alpha\)—the distance between the projections onto the plan of the horizontal sections of the bed, and \(OM_2 = r \operatorname{Ctg}\beta\)—the distance between the projections of the sections of the dislocator.

We lay off the obtained quantities \(OM_1\) and \(OM_2\) in the direction of dip from the corresponding traces of the upper horizon and, drawing the lines \(C_1D_1\) and \(S_1T_1\), parallel to the latter, obtain the projections of the traces of the bed and of the dislocator on the plane of the lower horizon. By joining the points \(C\) and \(C_1\), we obtain the line of intersection of the hanging wall of the bed with the dislocator, and by drawing through the point \(B\) a line parallel to it, we obtain the line of intersection of the footwall with the dislocator.

It is clear that the displacement vector \(R\) must be located somewhere within the limits between these two parallel lines \(CC_1\) and \(BB_1\), the tail of this vector lying on the line \(BB_1\), which belongs to the footwall, and the head on the line \(CC_1\), which belongs to the hanging wall, since we have agreed to take as the movable wall theוואַ

to regard it always as a hanging one. This, naturally, restricts the range of possible values of \(\angle\gamma\).

In order to find the magnitude of the displacement vector \(R\), it is necessary to construct the direction of the striation on the fault plane. To do this we shall use the method of rotating planes. We shall bring the fault plane into coincidence with the plane of the plan by rotating it about the strike line \(ST\). From descriptive geometry it is known that the projection of any point belonging to the rotating plane will, during rotation, move along a line perpendicular to the axis of rotation. Therefore the point \(C'\), for example, must move along the line \(C_2C_0\), perpendicular to the axis of rotation \(ST\), and, when the fault plane is brought into coincidence with the plan, will occupy the position \(C_0\) at a distance from the axis equal to \(C_2C_0 = N'M'\). The line

Fig. 19.

of intersection \(CC'\) will thereby occupy the position \(CC_0\), the line \(BB_1\)—the position parallel to \(BB_0\), and the line \(T_1S_1\), respectively, \(T_0S_0\).

Now let us take on the line of intersection \(BB_0\), belonging to the footwall of the bed, any point, for example the point \(B_0\); through it draw the strike line of the fault \(B_0C_0\) (it has already been drawn in the construction) and construct at it the given angle \(\gamma\), which determines the direction of the striation. From the drawing it is seen that if we take the angle \(\gamma = 120^\circ\), then the vector drawn in this direction will not meet the line of intersection of the hanging wall, no matter how far we continue it. Consequently, the only possible direction of displacement will be displacement directly in

opposite side, toward \(B_0E\), which corresponds to the angle

\[ \gamma = 120^\circ + 180^\circ = 300^\circ. \]

Drawing the line \(B_0E\) to its intersection with the line of crosscut of the hanging wall of the bed, we obtain \(B_0E = R\)—the vector of the total amplitude of relative displacement. Dropping from point \(E\) the perpendicular \(EF\) to the strike line of the displacer \(S_0T_0\), we obtain \(EF = h\)—the inclined height of the displacement, while the segment \(B_0F\) gives us the horizontal component \(\lambda\). From the drawing it is clear that \(h\) is directed downward along the dip of the displacer and, consequently, indicates a “normal fault,” while \(\lambda\) is directed to the right, toward the positive direction of the strike line of the displacer, and therefore is a positive quantity.

Knowing the value \(h\) and the dip angle of the displacer \(\beta\), it is not difficult to construct also the vertical height \(H\), as the leg opposite the angle \(\beta\) in a right triangle constructed on the hypotenuse \(h\) and the acute angle \(\beta\).

Solution of the inverse problem of a disjunctive dislocation.

If the given disjunctive dislocation has in one way or another been resolved, i.e. if the direction and amplitude of the relative displacement of its wings are known, then determining the direction and magnitude of the apparent displacement of any structural element disturbed by it presents no difficulty and, again, may be done both analytically and graphically.

Let it be given that the disturbance is a positive reverse–strike-slip fault with amplitude \(R = 100\) m and \(\gamma = 36^\circ 50'\). The strike of the displacer is \(330^\circ\), dipping to NE \(\angle 70^\circ\). The bed, with strike \(300^\circ\), also dips to NE at an angle of \(55^\circ\), and is known only on the hanging-wall side of the displacer. It is required to find its displaced part.

In general form the problem is solved very simply by applying the formula already known to us:

\[ l = R \left(\operatorname{Ctg}\alpha \operatorname{Sin}\beta \operatorname{Sin}\gamma + \operatorname{Cos}\beta \operatorname{Cos}\omega \operatorname{Sin}\gamma \operatorname{Sin}\omega \pm \operatorname{Cos}\gamma\right) . . \quad (20) \]

Since \(R\) and all the angles whose functions enter into the formula are given, the value \(l\) is determined directly; moreover, the sign of \(l\) will quite definitely indicate in which direction one should search for the displaced part of the bed.

First of all we draw up a diagram of the position of the bed and the displacer, in order to determine the angle of intersection \(\omega\) and the sign before the last term of our formula (Fig. 20). From the drawing it is clear that the angle \(\omega = 150^\circ\), and that the sign before the last term must be taken as \((+)\), since the bed dips toward the positive values of \(\lambda\). Consequently, our formula, as applied to the given disjunctive dislocation, will take the following form:

\[ l = R \left(\operatorname{Ctg}\alpha \operatorname{Sin}\beta \operatorname{Sin}\gamma + \operatorname{Cos}\beta \operatorname{Cos}\omega \operatorname{Sin}\gamma + \operatorname{Cos}\gamma \operatorname{Sin}\omega\right) . . \quad (23) \]

Writing out the values of the angle functions, we obtain:

\[ \begin{array}{rclcrclcrcl} \operatorname{Ctg} 55^\circ &=& 0.700 &\quad& \operatorname{Cos} 70^\circ &=& 0.342 &\quad& \operatorname{Sin} 150^\circ &=& 0.500 \\ \operatorname{Sin} 70^\circ &=& 0.940 && \operatorname{Cos} 150^\circ &=& -0.866 && \operatorname{Cos} 36^\circ 50' &=& 0.800 \\ \operatorname{Sin} 36^\circ 50' &=& 0.600 && \operatorname{Sin} 36^\circ 50' &=& 0.600 &&& \\ \cline{1-3} \cline{5-7} && +0.395 && && -0.178 && && +0.400 \end{array} \]

Whence:

\[ l = 100\,(0.395 - 0.178 + 0.400) = +61.7\ \text{m}. \]

We have agreed to measure the magnitude \(l\) from the footwall toward the hanging wall across the strike of the bed and to regard it as positive when this direction coincides with the direction of dip of the bed. It follows that the head of the vector \(l\) must be located at point \(A\) (on the hanging wall of the bed); the vector must have the direction NE—in the direction of dip of the bed, since \(l\) is positive—while its tail must go to the SW, perpendicular to the strike line of the bed. Laying off in this direction the found magnitude \(l = 61.7\ \text{m}\), we obtain point \(K\), lying on the displaced wall of the bed \(CD\), whose strike direction will obviously be parallel to \(AB\).

Graphical solution.

The graphical solution of the same problem is shown in the drawing (Fig. 21). Here \(AB\) is the given wall of the bed, \(ST\) is the displacer, \(S_1T_1\) is the trace of the displacer, and \(A_1B_1\) is the trace of the bed on the lower horizon. \(AA_1\) is the line of intersection of the wall of the bed with the displacer¹. By rotating the plane of the displacer about its strike line \(ST\), we bring—

Fig. 20.

Fig. 21.

we bring it into coincidence with the plane of the plan. Then the line \(S_1T_1\) will move parallel to itself and will take the position \(S_0T_0\) at a distance from \(ST\) equal to \(A_2A_0 = r\sin\beta\). The point \(A_1\) will move along the perpendicular to the axis of rotation to the point \(A_0\). Taking it as the indicator point, we construct about it the vector \(R\) at the given angle \(\gamma\). Since the line \(AA_0\) belongs to the hanging wall of the bed, the head of the displacement vector \(R\) must lie on it. Consequently, the tail of the vector must go downward and to the left, if the plane of the fault is oriented so that it dips toward the observer.

Laying off in this direction the given value \(R = 100\) m, we obtain the tail of the vector at the point \(F_0\). Through the point \(F_0\) we draw the line \(F_0D\) parallel to \(AA_0\), and obtain the shifted trace of the sought part of the bed, and at the point \(D\) its intersection with the fault. The line \(CD\), parallel to \(AB\), will then be the trace of the sought part of the bed on the plane of the plan.

Thus, the inverse problem of the fault is solved very easily and simply, both graphically and analytically; moreover, we at the same time obtain both the direction in which the displaced part of the bed or vein must be sought, and the distance at which this displaced part lies from the given one. But for this we must know not only the conditions of occurrence of the bed and of the fault, but also the direction and amplitude of the displacement. Or, in other words, the inverse problem can be solved only after the direct problem has been solved, and its solution will be the more exact, the more exact the solution of the direct problem.

From the formula for the fault (20) there plainly follows the dependence of the magnitude \(l\) on the angles \(\alpha\), \(\beta\), and \(\omega\), determined by the conditions of occurrence of the bed and the fault. Consequently, if the elements of occurrence of the bed are changed while the other conditions remain constant, the magnitude \(l\) changes, and may even take opposite values. On the other hand, the magnitude (and direction) of the displacement \(l\) depends on the magnitude \(R\) and \(\gamma\), i.e., on the amplitude and direction of the displacement. Hence it is clear how vain and even ridiculous are the attempts of mine surveyors (and of mining geometers as well) to derive a universal rule for searching for the displaced part of a bed from the mutual relations of the bed and the fault, while neglecting the direction of the actual relative displacement of the wings of the fault.

We may be asked: what is to be done when there are insufficient data for solving the direct problem? To such a question one can answer only with another question: what is to be done when only one equation with two unknowns is given?

It is known from mathematics that such equations have an infinite set of solutions. In order to find a particular solution, a second equation must be set up, and for this it is necessary to obtain additional data. Such additional data, obviously, are also necessary in solving the problem of the fault. Otherwise it will not be a solution of the problem, but fortune-telling from coffee grounds.

Incidentally, in certain favorable cases our equation can render substantial assistance in searching for the displaced part of a seam even when the fault has not been completely deciphered.

A Partial Solution of the Problem

In the driving of the first level of the Koksovaya mine in the Prokopyevsk district, the preparatory workings very often encountered small faults, the greater part of which were met by the workings for the first time. This, naturally, caused much trouble for the mine administration, since the monotony of the coal-bearing strata and the absence of clear bends in the seams and other geological indicators—especially in the case of small faults—usually do not make it possible to establish the direction in which the displaced part of the seam should be sought. Likewise, the application of mine-surveying methods of search by no means always gave satisfactory results.

A. A. Belitsky, who at that time worked there as a mine geologist, had the idea of making use of our formula for determining the direction of exploratory-search workings toward the displaced part of the seam. In doing so he proceeded from the following considerations.

As is known, the overwhelming majority of the faults of the Kuzbass in general, and of the Prokopyevsk district in particular, are of the reverse-thrust character, i.e., they are characterized by displacement of the hanging walls, in general, upward along the inclined plane of the dislocator. Consequently, having established the striation on the plane of the dislocator—and this, with some skill, can almost always be done—we can assign to the angle $\gamma$ a quite definite value within the limits from 0 to $180^\circ$, since in reverse strike-slip faults and thrust strike-slip faults this angle does not go beyond the indicated limits.

Substituting the value of the angle thus obtained into our formula

\[ l = R(\operatorname{Ctg}\alpha \operatorname{Sin}\beta \operatorname{Sin}\gamma + \operatorname{Cos}\beta \operatorname{Cos}\omega \operatorname{Sin}\gamma \pm \operatorname{Sin}\omega \operatorname{Cos}\gamma) \tag{20}, \]

written, of course, with observance of the sign rule indicated above, and without carrying out the final calculations, we determine only the sign of the expression standing in parentheses. Since the magnitude $R$, under the conditions we have adopted, is always a positive quantity, it is obvious that the sign of the expression in parentheses, serving as the coefficient of $R$, must be identical with the sign of $l$. The sign of $l$ determines the direction in which the displaced part of the seam is located, as we have already seen above.

In general form this regularity may be formulated as follows.

If the magnitude $l$ is positive, then the hanging wall of the seam lies, with respect to the footwall, on the side indicated by the dip arrow of the latter; if the magnitude $l$ is negative, then it lies on the opposite side.

This regularity follows directly from the sign rule for \(l\) that we have adopted and, in essence, adds nothing new.

This method of Belitsky is very simple and convenient. It makes it possible very quickly to decide in which direction workings should be driven toward the displaced part of a seam. But it is not universal. Under the conditions of the Prokopyevsk district it gives very good results. In 20 cases out of 20, workings laid out by this method found the displaced part of the seam without error—incidentally, in 4 or 5 cases contrary to the constructions of the mine surveyors. It is quite possible that it will give equally good results in other districts of Kuzbass as well, since the general tectonics of this basin, though complex, is on the whole fairly uniform. But one should not forget that Belitsky’s method is based on the assumption of displacement of the hanging wall, although in an arbitrarily inclined direction (according to the orientation of the striae), yet in general upward along the inclined plane of the dislocator. Therefore it should give, and does give, correct results only for reverse strike-slip faults and thrust strike-slip faults. At the very first normal strike-slip fault encountered it will inevitably give an error, since the angle \(\gamma\) will change by \(180^\circ\) and all functions of this angle will change their sign to the opposite one. Disturbances of the normal strike-slip type, though rare, apparently do occur in Kuzbass; moreover, Belitsky himself proves the presence of a similar type of disturbance in the field of the Koksovaya mine.

Summary

The study of mine tectonics with precise recording of all elements of disturbances is a very important method for understanding the tectonics of a region, revealing the forms and direction of movements of the lithosphere. Therefore one cannot but regret that geologists still devote little attention to this interesting and important question.

These studies are of still greater importance in the practice of prospecting and working deposits, both vein and bedded, especially in such strongly dislocated regions as Kuzbass, where disturbances are encountered literally at every step. They provide guiding indications for the most rational placement of exploratory and preparatory workings and for the swift solution of the problem of searching for the displaced parts of a deposit.

These questions acquire especially important significance for mine geologists in connection with the implementation of the order of People’s Commissar L. M. Kaganovich on improving the geological service at mines. The work of mine geologists must be accompanied by the most effective, precise, and advanced methods for studying disturbances, with a clear spatial representation of all geometric elements and forms of dislocations, for which it is necessary

a rational classification of faults and a sufficiently detailed nomenclature of their geometric elements.

Unfortunately, in the Russian geological literature this question has scarcely been developed. The literature on mine surveying is somewhat richer in this respect, but the classifications and methods adopted in it for studying displacements do not withstand any serious criticism, since they consider only the forms of displacement of a bed or vein without analyzing the actual relative movement of the wings of a disjunction. Therefore, from the geologist’s point of view they are completely unacceptable, and in general their practical significance is extremely doubtful.

In the present work we make the first attempt to resolve the question of disjunctions in its full scope, and arrive at the following conclusions.

1. In every fault we must distinguish three basic elements: 1) the movement of the disrupted blocks in space relative to one another; 2) the intersection of the displacer with the bed or vein, as the primary structural element of the disrupted blocks; and 3) the displacement of individual parts of the bed or vein caused by this movement.

2. A geometric analysis of these elements shows that the first and the last of them are vector quantities, interconnected through the angular parameters of the intersection by a definite functional dependence.

3. This functional dependence is given by us in the form of an elementarily simple mathematical formula, which makes it possible easily and simply to solve both the direct and the inverse problem of the disjunction.

4. In parallel with this, we also give a graphical solution of the same problems in the simplest and most visual form, using the method of projections with numerical elevations.

5. Both methods—the graphical and the analytical—are elementarily simple and entirely independent of one another. The graphical solution is more visual, but not always convenient. Moreover, it gives less accurate results in comparison with the analytical one. It is therefore recommended for use in a preliminary, approximate solution of the problem, and also for checking gross computational errors in its analytical solution.

6. Analysis of the formula we have derived makes it possible to outline a simple and convenient scheme for the classification of disjunctions, which, by means of a combination of a small number of commonly used terms (without distorting their meaning), makes it possible to reflect the whole diversity of disjunctive phenomena, taking into account both the direction of movement of the blocks and the form of the displacement obtained thereby.

References

1. Bauman, V. I.—On the question of normal strike-slip faults and other displacements of veins and seams.—Proceedings of the Mining Institute. I, 1907.

2. Velitskii, A. A.—Methodology for searching for the displaced wing of a seam under the conditions of the Prokop'evsk district of Kuzbass. Bulletin of the Tomsk Industrial Institute, vol. 60, 1939.

3. Gutt, A. E.—A Course in Mining Geometry.—Kharkov, 1932.

4. Dalinkevich, M. A.—Mining Geometry.—Leningrad, 1924.

5. Leontovskii, P.—Mine-surveying problems. Elements of the occurrence of seams (Mining Geometry).—Bulletin of the Ekaterinoslav Higher Mining School. II, 1905.

6. Liese, Ch. K.—Structural Geology.—ONTI translation, 1935.

7. Molchanov, I. A.—Geometric analysis of translational disjunctives.—Mine-Surveying Collection. II, Tomsk, 1935.

8. Rumyantsev, V. S.—Tectonic disturbances observed on the NW margin of Kuzbass and their explanation. (An attempt to apply the theory of strength of materials to tectonics).—Mining Journal, 1928, Nos. 10 and 11.

9. The same.—On the tectonics of the northern extremity of Kuzbass.—Bulletin of TGRU. 49–2. Leningrad, 1930.

10. Sobolevskii, P. K.—On displacements. Proceedings of the First All-Siberian Mine-Surveying Congress. Tomsk, 1925.

11. Tetyaev, M. M.—Fundamentals of Geotectonics. ONTI, 1934.

12. Willis, B. and R.—Structural Geology.—Translation. Baku, 1932.

13. Willis, B. (Willis, B.).—Mechanics of Appalachian Structures.—ONTI, 1934.

14. Usov, M. A.—Tectonics of the Sudzhenka coal deposit.—Bulletin of the Siberian Geological Committee. 1–2. Tomsk, 1929.

15. The same.—Tectonics of the Anzherka deposit.—Ibid. 1–4, 1920.

16. The same.—Elements of the tectonics of the Leninsk district of Kuzbass. Tomsk, 1924.

17. The same.—Composition and tectonics of the southern district of Kuzbass. Novosibirsk, 1924.

18. The same.—Elements of the tectonics of Kuzbass.—Library of the Mining Journal No. 2, 1924.

19. The same.—Composition and tectonics of the Kemerovo deposit of Kuzbass.—Bulletin of the Siberian Branch of the Geological Committee. V–4. Tomsk, 1926.

20. The same.—The influence of tectonics on the industrial coal reserves in Kuzbass.—Proceedings of the All-Union Mining Scientific-Research Congress. VII, 1929.

21. The same.—Composition and tectonics of the Osinovka deposit of Kuzbass.—Bulletin of the West-Siberian Branch of the Geological Committee. X, 5. 1930, Tomsk.

22. The same.—Forms of disjunctive dislocations in the mines of Kuzbass.—Collection on the Geology of Siberia. Tomsk, 1933.

23. Ushakov, I. N.—Mining Geometry.—ONTI, 1937.

24. Aubel, F.—Vorlesungen aus der Markscheidekunde.—Leoben. 1912.

25. Carnall.—Die Sprünge in Steinkohlengebirge. Berlin.—1835.

26. Economic Geology. II, 1907.

27. Hausse.—Die Verwerfungen, in besondere ihre Konstruktion, Berechnung und Ausrichtung.—Zeitschr. f. d. Berg.-Hütt. und Salinenwesen, 1903.

28. Höfer von Heimhalt.—Die Verwerfungen.—Braunschweig, 1927.

29. Hornoch, A.—Das Verwerferproblem im Licht des Markscheiders. 1927.

30. Geikie, G.—Structural and Field Geology, 1905.

1. Lance, F. H.—Field Geology. London, 1923.
2. Nevin, Ch.—Principles of Structural Geology—N.-Y., 1931.
3. Ransome, F. H.—The Directions of Movement and the Nomenclature of Faults.—Economic Geology, 1906, No. 8.
4. Ried, H.—Geometry of Faults.—Bull. Geol. Soc. of Amer. XX, 1909.
5. Ried, H., Davis, Lawson and Ransome.—Report of the Committee on the Nomenclature of Faults.—Bull. Geol. Soc. of Amer. XXIV, 1913.
6. Schmidt, I.—Teorie der Verschiebungen alterer Gänge. Frankfurt, 1810.
7. Spurr, J. E.—The Measurement of Faults.—Journ. of Geol., 1897.
8. Onge.—Geology Applied to Mining.—N.-Y., 1904.
9. Tolman, C. F.—Graphical Solution of Fault Problems.—Mining and Scient Press, San Francisco, 1911.

1. For the construction of combined projections of two horizontal sections, see above. ↩↩↩↩↩↩↩↩↩

2. Such a classification could even be called genetic, since it takes into account all the principal factors in the formation of a fault, if some geologists did not invest this concept with the specific content of the “absolute” movements of the blocks of the disjunctive and of the forces, or more precisely stresses, by which these faults are conditioned. ↩↩