LOCAL-STATISTICAL CHARACTERISTICS OF THE DESTRUCTION OF A FOREST STAND BY A SHOCK WAVE

V. G. FAST

(Presented in May 1966 to the mathematical section of the scientific-technical conference of the AVTF, dedicated to the 70th anniversary of TPI)

As is known, trees are sensitive mainly to the action of the braking loads of a shock wave [1], and the forest damage caused by it is similar to the damage inflicted on trees by a hurricane or a strong wind. We have shown that if \(a\) is the aerodynamic pressure acting in the elementary forest plot under consideration, and \(\sigma_0^2\) is the variance of the component of the noise vector, then \(a=\Lambda\sigma_0\), where \(\Lambda\) is a dimensionless parameter of the distribution of the directions of tree fall, and \(\sigma_0=\mathrm{const}\) for homogeneous forest stands and is subject to experimental determination [2]. It turns out that, if \(a\) is large, then \(\Lambda \approx \dfrac{1}{s}\), where \(s\) is the standard deviation of the directions of tree fall from the mean, expressed in radians. In practice one may take

\[ a=\frac{\sigma_0}{s}, \quad \text{if } s^0=\frac{s\cdot 180^\circ}{\pi}\leqslant 16^\circ . \]

From experimental work on the dynamometry of trees, carried out by K. A. Lyubarskii, it follows that \(\sigma_0\) does not depend on age, species, or direction of fall, and that only a dependence on tree diameter is possible [3].

The latter dependence may be investigated in the following way. Let \(p(\varphi,a_1,a_2,\ldots,a_k)\) be the density function of the distribution of the directions \(\varphi\) of tree fall, depending on certain parameters \(a_1,a_2,\ldots,a_k\), and suppose that it preserves its form in passing from homogeneous forest stands with one mean tree diameter to stands with another mean diameter. A detailed analysis of the formation of the distribution curve shows that such an assumption is quite natural for forest stands of one region. Suppose, further, that the hypothesis \(H_\delta^{\sigma}\) consists in the assertion that the distribution density \(p(\varphi,\ldots)\) for trees of diameter \(\delta\) coincides with the distribution density of the directions \(\varphi\) of fall of trees of all possible diameters \(d\in[a,b]\). If \(H_\delta^{\sigma}\) is valid for all \(\delta\in[a,b]\), then for \(d\in[a,b]\) \(\sigma_0\) does not depend on tree diameters. The hypothesis \(H_\delta^{\sigma}\) may be tested by any of the nonparametric criteria based on the simplest functions of order statistics [4], if one uses empirical histograms of the directions of tree fall, on which estimates of tree diameters are available.

Let \(\alpha_{1/4}\) and \(\alpha_{3/4}\) be the quartiles of the empirical histograms. Denote by \(N_{\alpha_\rho}^{\alpha_q}(\delta)\) the number of trees of diameter \(\delta\) felled in the dir-

directions \(\alpha_p<\alpha<\alpha_q\) \((0\le p\le 1,\quad 0\le q\le 1,\quad \alpha_1=\alpha_0+2\pi)\). The hypothesis \(H_0^\delta\) can be rephrased as follows: if

\[ z=N_{\alpha_{1/4}}^{\alpha_{3/4}}(\delta)-\left[N_{\alpha_0}^{\alpha_{1/4}}(\delta)+N_{\alpha_{3/4}}^{\alpha_1}(\delta)\right], \]

then \(P(z>0)=P(z<0)=\dfrac{1}{2}\), where \(P(A)\) denotes the probability of the event \(A\).

As the competing hypothesis to \(H_0^\delta\), the two-sided alternative

\[ P(z>0)\ne \frac{1}{2} \]

is considered.

The described method for testing the hypothesis \(H_0^\delta\) was applied by us to the destruction caused by the shock wave of the Tunguska meteorite. A sample consisting of 51 histograms was analyzed. The diameter \(d\) of trees at a height of 1.3 m was estimated in a three-grade system: thin (9–15 cm), medium (15–30 cm), and thick (over 30 cm). Application of the two-sided sign test with significance level \(2\gamma=0.05\) led to the conclusion that, for thin and medium trees, \(H_0^\delta\) cannot be rejected, whereas for thick trees it is immaterial whether one accepts \(H_0^\delta\) or \(P(z>0)>\dfrac{1}{2}\). This somewhat indeterminate answer compels one to turn to stronger tests. As such, we used the run-count test [4], which takes into account not only the sign but also the magnitude of \(z\). It turns out that not only the significance level \(\gamma=0.05\), but also \(\gamma=0.10\), does not allow rejection of the hypothesis \(H_0^\delta\) for all \(\delta>8\) cm.

Thus, it may be considered that, in the case of taiga destruction by the shock wave of the Tunguska meteorite, obtaining a definite distribution of the directions of tree fall at different points required one and the same aerodynamic pressure, irrespective of the mean diameter of the trees growing there, and that the value \(x_0\) does not depend on the diameters of the trees.

REFERENCES

1. The Effects of Nuclear Weapons, translated from English, Voenizdat, 1963.
2. V. G. Fast. Advances in Meteoritics, Novosibirsk, pp. 16–18, 1966.
3. A. P. Boyarkina, D. V. Demin, I. T. Zotkin, V. G. Fast. Meteoritics, XXIV, pp. 112–128, 1964.
4. L. N. Bolshev, N. V. Smirnov. Tables of Mathematical Statistics, Nauka, 1965.