Accuracy Assessment of Forward Intersection by a Graphical Method
Abstract Generated abstract
This paper addresses the practical problem of assessing the accuracy of points determined by forward intersection, for which existing analytical procedures are described as cumbersome and therefore often unused. It derives coordinate and directional mean-square error expressions in terms of side accuracy vectors, shows their equivalence to the conventional error ellipse formulation, and proposes a graphical construction for obtaining directional errors, coordinate errors, and ellipse elements without first computing the ellipse analytically. Numerical examples compare the graphical results with analytical calculations and rigorous adjustment, showing close agreement. The method is also extended to multiple forward intersections, where graphically determined coordinate weights are used to compute weighted mean coordinates for triangulation points.
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Accuracy Assessment of Forward Intersection by a Graphical Method
V. I. AKULOV
(Presented by the scientific seminar of the departments of mine surveying and geodesy)
Forward intersection is widely used in practice. The accuracy of the intersection is usually not analyzed, which is explained by the complexity of the methods known in the literature for assessing accuracy. Thus, for example, with the analytical method the root-mean-square errors of the coordinates of point \(C\) (Fig. 1) are computed by the formulas:
\[ m_x=\pm \frac{m_3}{\sqrt{P_x}}; \tag{1} \]
\[ m_y=\pm \frac{m_3}{\sqrt{P_y}}, \tag{2} \]
where \(P_x, P_y\) are the weights of the abscissa and ordinate of point \(C\);
\(m_3\) is the root-mean-square error of measuring an angle, whose weight is equal to unity.
To determine the weights \(P_x, P_y\), a system of normal equations of the form is composed
\[ [paa]\,\delta x + [pab]\,\delta y = 0; \]
\[ [pab]\,\delta x + [pbb]\,\delta y = 0. \]
The coordinates of point \(C\) are determined directly from the measured angles \(\beta_1\) and \(\beta_2\); therefore the free terms of the normal equations and the corrections \(\delta x\) and \(\delta y\) are equal to zero.
The coefficients \(a\) and \(b\) are determined by the known formulas
\[ a_1=-\frac{\rho \sin \alpha_{AC}}{S_{AC}}, \qquad b_1=\frac{\rho \cos \alpha_{AC}}{S_{AC}}; \]
\[ a_2=-\frac{\rho \sin \alpha_{BC}}{S_{BC}}, \qquad b_2=\frac{\rho \cos \alpha_{BC}}{S_{BC}}, \]
where \(\alpha_{AC}, \alpha_{BC}\) are the direction angles of the sides \(AC\) and \(BC\);
\(S_{AC}, S_{BC}\) are the lengths of the sides \(AC\) and \(BC\);
\[ \rho = 206265''. \]
The weights \(P_x\) and \(P_y\) are equal to
\[ P_x=P_y\frac{[Paa]}{[Pbb]}; \tag{3} \]
\[ P_y=[Pbb.1]. \tag{4} \]
In the graphical method it is necessary to know the elements of the mean-square error ellipse, which are computed by the formulas
\[ \tg 2\varphi_1=\frac{-2[Pab]}{-([Paa]-[Pbb])}; \tag{5} \]
\[ A^2=m_\beta^2\frac{[Paa]+[Pbb]+w}{2D}; \tag{6} \]
\[ B^2=m_\beta^2\frac{[Paa]+[Pbb]-w}{2D}; \tag{7} \]
\[ W=-\frac{2[Pab]}{\sin 2\varphi_1} =\frac{-([Paa]-[Pbb])}{\cos 2\varphi_1}; \tag{8} \]
\[ D=[paa][pbb\cdot 1], \tag{9} \]
where \(A, B\) are the major and minor semiaxes of the mean-square error ellipse;
\(\varphi_1\) is the direction angle of the major axis of the mean-square error ellipse;
\(D\) is the determinant of the system of normal equations.
Mean-square errors of the coordinates of a point in the graphical-analytical method are determined from the curve of mean-square errors
\[ P^2=A^2\cos^2\Theta+B^2\sin^2\Theta, \tag{10} \]
where \(P\) is the radius vector;
\(\Theta\) is the angle between the major axis of the mean-square error ellipse and the radius vector \(P\).
The article proposes a graphical method for assessing the accuracy of a forward intersection.
To justify the graphical method, let us establish the mean-square errors of the coordinates of point \(C\) (Fig. 1) from a forward intersection.
Fig. 1. Graphical determination of coordinate errors in a forward intersection.
The coordinates of point \(C\) are equal to
\[ X_C=X_A+\frac{b\sin\beta_2}{\sin(\beta_1+\beta_2)} \cos(\alpha_{AB}-\beta_1); \tag{11} \]
\[ Y_C=Y_A+\frac{b\sin\beta_2}{\sin(\beta_1+\beta_2)} \sin(\alpha_{AB}-\beta_1), \tag{12} \]
where \(b, \alpha_{AB}\) are the length and direction angle of the fixed side \(AB\).
The mean square errors of the coordinates of point \(C\), without taking into account the errors of the fixed points, will be
\[ m_x^2 = \frac{m_{\beta_1}^2 b^2 \sin^2 \beta_2 \cos^2(\alpha_{AB}+\beta_2)} {\rho^2 \sin^4(\beta_1+\beta_2)} + \frac{m_{\beta_2}^2 b^2 \sin^2 \beta_1 \cos^2(\alpha_{AB}-\beta_1)} {\rho^2 \sin^4(\beta_1+\beta_2)}; \tag{13} \]
\[ m_y^2 = \frac{m_{\beta_1}^2 b^2 \sin^2 \beta_2 \sin^2(\alpha_{AB}+\beta_2)} {\rho^2 \sin^4(\beta_1+\beta_2)} + \frac{m_{\beta_2}^2 b^2 \sin^2 \beta_1 \sin^2(\alpha_{AB}-\beta_1)} {\rho^2 \sin^4(\beta_1+\beta_2)}, \tag{14} \]
where \(m_{\beta_1}, m_{\beta_2}\) are the mean square errors of the measurements of the angles \(\beta_1\) and \(\beta_2\).
The mean square linear error of the position of point \(C\) is equal to
\[ M = \pm \sqrt{m_x^2 + m_y^2}, \tag{15} \]
or
\[ M = \pm \frac{b\sqrt{m_{\beta_1}^2\sin^2\beta_2 + m_{\beta_2}^2\sin^2\beta_1}} {\rho \sin^2(\beta_1+\beta_2)}. \tag{15a} \]
With equal-accuracy measurement of the angles, the error \(M\) will be
\[ M = \pm \frac{m_\beta b\sqrt{\sin^2\beta_1+\sin^2\beta_2}} {\rho \sin^2(\beta_1+\beta_2)}. \tag{15b} \]
Let us denote
\[ r_{BC} = \pm \frac{m_{\beta_1} b \sin\beta_2} {\rho \sin^2(\beta_1+\beta_2)}; \tag{16} \]
\[ r_{AC} = \pm \frac{m_{\beta_2} b \sin\beta_1} {\rho \sin^2(\beta_1+\beta_2)}; \tag{17} \]
where \(r_{BC}, r_{AC}\) are the accuracy vectors of the sides \(BC\) and \(AC\), respectively.
Since
\[ AC=\frac{b\sin\beta_2}{\sin(\beta_1+\beta_2)}; \]
\[ BC=\frac{b\sin\beta_1}{\sin(\beta_1+\beta_2)}, \]
then
\[ r_{BC}=\pm \frac{m_{\beta_1}AC}{\rho\sin(\beta_1+\beta_2)}; \tag{16a} \]
\[ r_{AC}=\pm \frac{m_{\beta_2}BC}{\rho\sin(\beta_1+\beta_2)}. \tag{17a} \]
The mean square errors of the coordinates of point \(C\) through the accuracy vectors of the sides are equal to
\[ m_x^2 = r_{AC}^2 \cos^2\alpha_{AC} + r_{BC}^2 \cos^2\alpha_{BC}; \tag{18} \]
\[ m_y^2 = r_{AC}^2 \sin^2\alpha_{AC} + r_{BC}^2 \sin^2\alpha_{BC}. \tag{19} \]
Hence
\[ m_x^2 + m_y^2 = r_{AC}^2 + r_{BC}^2 \]
or
\[ M = \pm \sqrt{r_{AC}^2 + r_{BC}^2}. \tag{20} \]
We shall write formulas (18, 19) as one formula, namely
\[ m_{\alpha}^2 = r_{AC}^2 \cos^2(\alpha-\alpha_{AC}) + r_{BC}^2 \cos^2(\alpha-\alpha_{BC}), \tag{21} \]
where \(m_{\alpha}\) is the root-mean-square error of the position of the point in the direction with direction angle \(\alpha\).
Equation (21) is the equation of the curve of root-mean-square errors of a point from a forward intersection, corresponding to equation (10).
Let us transform equation (21).
Function (21) attains a maximum and a minimum when \(\alpha\) is equal to \(\varphi_1\) and \(\varphi_2\), determined by the formulas
\[ \tan 2\varphi_1 = \frac{r_{AC}^{2}\sin 2\alpha_{AC}+r_{BC}^{2}\sin 2\alpha_{BC}} {r_{AC}^{2}\cos 2\alpha_{AC}+r_{BC}^{2}\cos 2\alpha_{BC}}; \tag{22} \]
\[ \varphi_2=\varphi_1\pm 90^\circ . \tag{23} \]
Denote
\[ \alpha=\varphi_1+\Theta, \tag{24} \]
where \(\Theta\) is the angle between the given direction and the direction in which the error \(m_{\alpha}\) is maximal.
Then equation (21) will take the form
\[ m_{\alpha}^{2}=A^{2}\cos^{2}\Theta+B^{2}\sin^{2}\Theta; \tag{25} \]
\[ A^{2}=r_{AC}^{2}\cos^{2}(\varphi_1-\alpha_{AC})+ r_{BC}^{2}\cos^{2}(\varphi_1-\alpha_{BC}); \tag{26} \]
\[ B^{2}=r_{AC}^{2}\sin^{2}(\varphi_1-\alpha_{BC})+ r_{BC}^{2}\sin^{2}(\varphi_1-\alpha_{BC}). \tag{27} \]
Equation (25), as is known, expresses the polar of the root-mean-square ellipse of errors with semiaxes \(A\), \(B\) and with the direction angle of the major axis equal to \(\varphi_1\).
Formulas (26) and (27) may be written as follows:
\[ A^{2}=\frac{1}{2}(M^{2}+W); \tag{26a} \]
\[ B^{2}=\frac{1}{2}(M^{2}-w); \tag{27a} \]
\[ W= \frac{r_{AC}^{2}\sin 2\alpha_{AC}+r_{BC}^{2}\sin 2\alpha_{BC}} {\sin 2\varphi_1}. \tag{28} \]
The direction angle of the major axis of the root-mean-square ellipse of errors and the value \(w\) can also be determined graphically.
For this purpose a quadratic polygon is constructed with sides \(r_{AC}^{2}\) and \(r_{BC}^{2}\) and with direction angles \(2\alpha_{AC}\) and \(2\alpha_{BC}\), respectively. The direction angle of the closing side of the quadratic polygon is equal to \(2\varphi_1\), and the length of the closing side is equal to \(w\).
Formula (21) shall be written as
\[ m_{\alpha}^{2}=r_{AC_{\alpha}}^{2}+r_{BC_{\alpha}}^{2}; \tag{21b} \]
\[ r_{AC_{\alpha}}=r_{AC}\cos(\alpha-\alpha_{AC}); \tag{29} \]
\[ r_{BC_{\alpha}}=r_{BC}\cos(\alpha-\alpha_{BC}), \tag{30} \]
where \(r_{AC_{\alpha}}, r_{BC_{\alpha}}\) are the projections of the accuracy vectors of the sides \(AC\) and \(BC\) onto the given direction with direction angle \(\alpha\).
Equation (25), by analogy with equation (21b), may be represented in the form
\[ m_a^2=A_a^2+B_a^2; \tag{25a} \]
\[ A_a=A\cos(\alpha-\varphi_1); \tag{31} \]
\[ B_a=B\cos(\alpha-\varphi_2), \tag{32} \]
where \(A_a, B_a\) are the projections of the semiaxes of the root-mean-square error ellipse onto the specified direction with direction angle \(\alpha\).
It is necessary to note that equations (10), (21b), and (25a) express one and the same curve, namely the curve of the root-mean-square errors of a point determined by forward intersection.
Using formula (21b), the accuracy assessment of forward intersection can be carried out easily and quickly by a graphical method.
The graphical method for assessing the accuracy of forward intersection is performed in the following sequence.
From the given angles \(\beta_1\) and \(\beta_2\), the position of point \(C\) (Fig. 1) relative to the fixed points \(A\) and \(B\) is determined on the diagram. From the diagram, the lengths of the sides \(AC\) and \(BC\) are found graphically. From the sides \(AC\) and \(BC\), the perpendicular displacements of point \(C\) relative to the sides \(AC\) and \(BC\) are computed by the formulas
\[ U_{AC}=\pm \frac{m_{\beta_1}AC}{\rho}; \tag{33} \]
\[ U_{BC}=\pm \frac{m_{\beta_2}BC}{\rho}. \tag{34} \]
Thus, for example, for point \(C\) (Fig. 1) we have
\[ U_{AC}=\pm \frac{5\cdot 56\cdot 40000}{206265}=\pm 54\ \text{mm}; \]
\[ U_{BC}=\pm \frac{5\cdot 81\cdot 40000}{206265}=\pm 79\ \text{mm}, \]
where \(m_{\beta_1}=m_{\beta_2}=\pm 5''\),
\(40000\) is the denominator of the scale in which the diagram is drawn.
On the diagram, at a convenient scale, the perpendicular displacements of point \(C\) are laid off; for example, in Fig. 1 the perpendicular displacements \(U_{AC}=CD'\) and \(U_{BC}=E'C\) are laid off at a scale of \(1:2\).
From the points \(D'\) and \(E'\), the straight lines \(D'D\) and \(E'E\) are drawn parallel to the sides \(AC\) and \(BC\), respectively, until they intersect the extensions of the sides \(BC\) and \(AC\), respectively.
The segments \(CD\) and \(CE\), as follows from formulas (16a, 17a, 33, and 34) and from the construction, are the accuracy vectors of the sides \(BC\) and \(AC\) at the scale in which the perpendicular displacements \(U_{AC}\) and \(U_{BC}\) are laid off.
To determine the projection of the accuracy vectors of the sides onto a specified direction with direction angle \(\alpha\), for example, onto the specified direction \(D_7E_7\) (Fig. 1) with direction angle \(\alpha_7\), we project the accuracy vectors of the sides onto the specified direction and obtain the points \(D_7\) and \(E_7\). We lay off the segment \(CD_7\) on the line \(E_7E\) \((E_7 7_1=CD_7)\) and obtain the point \(7_1\). The hypotenuse of the right triangle \(CE_7 7_1\) is the mean error of the position of point \(C\) in the direction with direction angle \(\alpha_7\). We transfer the point \(7_1\), with radius \(C7_1\), onto the line \(D_7E_7\), and obtain two diametrically opposite points \(7\) and \(7'\). The segments \(C7\) and \(C7'\)
correspond, as follows from the construction, to the root-mean-square error of the position of point \(C\) in the specified direction with direction angle \(\alpha_7\), at the scale in which the perpendicular displacements of point \(C\) are plotted.
Points 7 and \(7'\) lie on the curve of root-mean-square errors of point \(C\).
Having constructed, for a series of specified directions, for example at intervals of \(15^\circ\) (Fig. 1), counting from the axis of abscissas, points 1, 2, etc., analogously to points 7 and \(7'\), and connecting them by a smooth curve, we obtain the curve of root-mean-square errors for point \(C\). For example, in Fig. 1 the curve of root-mean-square errors is constructed at a scale of \(1:2\).
From the curve of root-mean-square errors one can graphically determine the elements of the root-mean-square error ellipse for point \(C\), i.e., the major and minor semiaxes and the direction angle of the major axis. For example, for point \(C\) (Fig. 1) the elements of the root-mean-square error ellipse are equal to
\[ A = CK = \pm 90\ \text{mm}; \]
\[ B = CF = \pm 50\ \text{mm}; \]
\[ \varphi_1 = 20^\circ \]
Analytical computation by formulas (5, 6 and 7) or (22, 26a and 27a) gives
\[ A = \pm 91\ \text{mm}; \qquad B = \pm 51\ \text{mm} \qquad \text{and} \qquad \varphi_1 = 20^\circ 07'. \]
The root-mean-square errors of the coordinates of point \(C\), corresponding to the root-mean-square errors of the position of point \(C\) along the coordinate axes, are determined analogously to those for the direction with direction angle \(\alpha_7\). Thus, for example, for point \(C\) (Fig. 1) the coordinate errors will be
\[ m_x = C1 = \pm 87\ \text{mm}; \]
\[ m_y = C7 = \pm 58\ \text{mm}. \]
Analytical computation by formulas (1, 2) gives
\[ m_x = \pm 87\ \text{mm}; \]
\[ m_y = \pm 57\ \text{mm}. \]
From a comparison of the results of the accuracy assessment of forward intersection (Fig. 1), performed by graphical and analytical methods, it is evident that the proposed graphical method gives good agreement with analytical computations.
Thus, using the accuracy vector of the sides, one can easily and quickly, for a forward intersection, find graphically both the error of the point in any specified direction and construct the curve of root-mean-square errors without knowing the elements of the root-mean-square error ellipse.
If a triangulation point of class IV is determined by multiple forward intersection with measurement of angles (Fig. 2), then the coordinates of the point, with accuracy sufficient for practical purposes, may be computed by the formulas
\[ X_0 = \frac{[P_x X]}{[P_x]}; \tag{35} \]
\[ Y_0 = \frac{[P_y Y]}{[P_y]}, \tag{36} \]
where \(X, Y\) are the coordinates of the point from a single forward intersection;
$P_x$, $P_y$ are the weights of the abscissa and ordinate of a point from a single forward intersection.
The weights $P_x$ and $P_y$ are equal to
\[ P_x=\frac{k}{m_x^2}; \tag{37} \]
\[ P_y=\frac{k}{m_y^2}, \tag{38} \]
where $m_x$, $m_y$ are the root-mean-square errors of the coordinates of a point from a single intersection;
$k$ is a constant number.
The errors $m_x$ and $m_y$ for each independent single forward intersection are determined by the graphical method considered above.
With angles measured with equal accuracy, the weights $P_x$ and $P_y$ are more conveniently determined by the formulas
\[ P_x=\frac{k}{R_x^2}; \tag{39} \]
\[ P_y=\frac{k}{R_y^2}, \tag{40} \]
where $R_x$, $R_y$ are the conditional errors of the abscissa and ordinate of a point from a single forward intersection.
Fig. 2. Graphical determination of conditional coordinate errors in multiple forward intersection.
The conditional errors $R_x$ and $R_y$, for example for the first intersection $ADB$ (Fig. 2), are equal to
\[ R_{x_1}=\sqrt{R_{AD_x}^2+R_{BD_x}^2}; \tag{41} \]
\[ R_{y_1}=\sqrt{R_{AD_y}^2+R_{BD_y}^2}, \tag{42} \]
where \(R_{AD_x}, R_{AD_y}\) are the projections of the conditional accuracy vector of side \(AD\) onto the axes of abscissas and ordinates;
\(R_{BD_x}, R_{BD_y}\) are the projections of the conditional accuracy vector of side \(BD\) onto the axes of abscissas and ordinates.
The conditional accuracy vectors of the sides, for example for the first intersection (Fig. 2), are equal to
\[ R_{AD_1}=\frac{BD}{\sin(\beta_1+\beta_2)}; \tag{43} \]
\[ R_{BD_1}=\frac{AD}{\sin(\beta_1+\beta_2)}. \tag{44} \]
The conditional accuracy vectors of the sides and the conditional coordinate errors of a point from a single forward intersection are determined graphically.
Thus, for example, to determine the conditional accuracy vectors for the first intersection (Fig. 2), we draw through point \(D\), perpendicular to the sides \(DA\) and \(DB\), the straight lines \(DE'_1\) and \(DF'_1\). With a compass we transfer points \(A\) and \(B\) onto the straight lines \(DE'_1\) and \(DF'_1\). From points \(E'_1\) and \(F'_1\) we draw the straight lines \(E'_1E_1\) and \(F'_1F_1\), parallel respectively to the sides \(AD\) and \(BD\), until they intersect the extensions of the sides \(BD\) and \(AD\). The segments \(DE_1\) and \(DF_1\), at the scale of the diagram, are equal to the conditional accuracy vectors of the sides \(BD\) and \(AD\), i.e.
\[ R_{AD_1}=DF_1 \qquad \text{and} \qquad R_{BD_1}=DE_1. \]
Projecting the conditional accuracy vectors of the sides \(AD\) and \(BD\), i.e. the segments \(DF_1\) and \(DE_1\), onto the coordinate axes, we obtain the projections of the accuracy vectors of the sides onto the coordinate axes, for example,
\[ R_{AD_x}=DF_{1x}, \qquad R_{AD_y}=DF_{1y} \qquad \text{and} \qquad R_{BD_x}=DE_{1x}, \qquad R_{BD_y}=DE_{1y}. \]
We construct two right triangles \(a_1b_1c\) and \(a_2b_2c\) (Fig. 2), the first with legs \(ca_1=R_{AD_x}\) and \(cb_1=R_{BD_x}\), and the second with legs \(ca_2=R_{AD_y}\) and \(cb_2=R_{BD_y}\). The hypotenuses of the indicated triangles will correspond to the conditional coordinate errors of the point, namely
\[ R_{x_1}=a_1b_1; \]
\[ R_{y_1}=a_2b_2. \]
The conditional coordinate errors \(R_x\) and \(R_y\) are expressed in centimeters or in millimeters. Thus, for example, for the first intersection (Fig. 2)
\[ R_{x_1}=a_1b_1=67\ \text{mm}, \qquad R_{y_1}=a_2b_2=38\ \text{mm}. \]
The conditional coordinate errors of point \(D\) from the second independent forward intersection \(BDC\) (Fig. 2), found in the same way as for the first, are equal to \(R_{x_2}=40\ \text{mm}\) and \(R_{y_2}=55\ \text{mm}\).
The weights of the coordinates of point \(D\) (Fig. 2) from the separate independent intersections, computed by formulas (39, 40), are equal to
a) from the first intersection:
\[ P_{x_1}=\frac{\kappa}{R_x^2}=\frac{1000}{67^2}=0.22; \]
\[ P_{y_1}=\frac{\kappa}{R_y^2}=\frac{1000}{38^2}=0.69; \]
b) from the second intersection:
\[ P_{x_2}=\frac{\kappa}{R_{x_2}^{2}}=\frac{1000}{40^{2}}=0.62; \]
\[ P_{y_2}=\frac{\kappa}{R_{y_2}^{2}}=\frac{1000}{55^{2}}=0.33. \]
The coordinates of point \(D\) from the first and second independent intersections (Fig. 2) are
\[ X_{D_1}=1859.476\ \text{m}, \qquad Y_{D_1}=928.846\ \text{m}; \]
\[ X_{D_2}=1859.644\ \text{m}, \qquad Y_{D_2}=928.770\ \text{m}^{1}). \]
The weighted mean coordinates of point \(D\), computed by formulas (35) and (36), will be
\[ X_0=1859.600\ \text{m}; \]
\[ Y_0=928.821\ \text{m}. \]
From a rigorous adjustment, the coordinates of point \(D\) are
\[ X=1859.590\ \text{m}; \]
\[ Y=928.833\ \text{m}. \]
The coordinates of point \(D\), computed as the arithmetic mean from the first and second intersections, as recommended in the Handbook of Mine Surveying, will be
\[ X'=1859.560\ \text{m}; \]
\[ Y'=928.808\ \text{m}. \]
From a comparison of the coordinates of point \(D\), it is evident that the weighted mean coordinates computed by formulas (35, 36) are close to the coordinates from the rigorous adjustment.
Conclusions
-
The accuracy assessment of forward intersections must be carried out by the graphical method.
-
In a multiple forward intersection of a class IV triangulation point, the weighted mean values from the individual independent forward intersections may be taken as the final coordinates.
The weights of the point coordinates from the individual intersections are determined graphically.
- A point determined by forward intersection must be determined from those fixed points for which the linear error of the point, computed by formula (20), has the minimum value.
\(^{1}\) Handbook of Mine Surveying, Ugletekhizdat, p. 294, 1955.