RU2459220C1 - Microseismic zoning method - Google Patents

Abstract

FIELD: physics.

SUBSTANCE: microseismic zoning method involves placing analysed and reference observation points on areas with different geological engineering conditions. The reference observation points are also placed under business facilities. The distance between observation points does not exceed 1/3-1/4 of the wavelength of seismic vibrations with the highest frequency, which form information-bearing amplitude fluctuations, and the distance between profiles is equal to 1/3-1/4 of the minimum spatial period of information-bearing amplitude fluctuations of the high-frequency range of seismic vibrations. High-frequency seismic vibrations which form information-bearing amplitude fluctuations of seismic vibrations are selected on space-time volumes of discrete measurements by constructing a Cayley tree. A collection of measurements is automatically selected from the entire range of measurements, having minimum measurement error. The measurements with minimum measurement error are used to establish the extreme value of the seismic vibration based on which damage by the extreme value of seismic vibrations is determined.

EFFECT: high reliability of assessing seismic hazard.

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Description

The invention relates to the field of instrumental seismic research in seismic micro-zoning of territories of civil and industrial construction.

The objective of the invention is to increase the reliability of seismic hazard assessment by taking into account the influence of lateral heterogeneity of the rock base and deeper horizons of the geological section, as well as determining potential damage to objects of economic activity located in the seismic activity zone.

A known method of seismic microzoning (patent RU No. 1251694 [1]), which is implemented as follows.

In the area subject to seismic micro-zoning, an engineering-geological mapping is performed and areas with different engineering-geological conditions are identified. The investigated and reference points of observation, equipped with three-contact seismological registration equipment, are placed uniformly throughout the studied area. Observe and determine the options for dynamic oscillation parameters for various frequencies of a given frequency range of studies from different focal zones, for sectors not exceeding 30 °. According to the measurement results, a range of quantitatively significant (for example, J> 0.5 points) high-frequency variations of dynamic parameters is established. In accordance with the data obtained for all selected areas of the approach, data is used in the highest frequency range f _n . Taking into account the identified directions to the potentially dangerous focal zones, support profiles are laid and perpendicular to them, along which the first group of additional observation points is placed. At the same time, the distance between them is chosen that does not exceed 1 / 3-1 / 4 of the wavelength of the high-frequency range of studies.

Seismic oscillations propagating from earthquakes (explosions and other sources) from potentially dangerous and other focal zones are recorded, but along the directions closest to the directions to potentially dangerous focal zones, taking into account that in the general case a potentially dangerous focal zone is some limited volume of the geological environment, located at a certain distance from the observation points. According to the results of observations of seismic waves along the reference profiles, the dynamic parameters of the oscillations are determined and graphs of their variations are constructed separately for different directions of the approach of the waves from this focal zone. Moreover, the variations of the dynamic parameters are determined relative to the observations at the reference point located in the geotechnical conditions, taken as the reference for the seismic microzoning for the given territory, as well as relative to the average values of the dynamic parameters, determined depending on the task for a certain set of data within the study area or for the whole territory of micro-zoning. Spatial filtering of the data is carried out along the profiles and the periods obtained are used to determine the periods of high-frequency quantitatively significant spatial variations of the dynamic oscillation parameters (for example, more than 0.5 points).

The distances between the observation points are chosen equal to the minimum spatial periods of quantitatively significant variations.

In accordance with the obtained value of the distance between the observation points at the seismic micro-zoning site along the directions parallel to the reference profile, observations are made.

According to the second embodiment, this method is implemented as follows. Under conditions of large lateral heterogeneity of the geological section, the periods of spatial variations in the dynamic parameters over the area can vary greatly. In accordance with this, after determining the high-frequency spatial variations on the reference profiles, the distance between the additional observation points is selected. Due to differences in the internal structure of the rock foundation, these periods in a part of the territory turn out to be different from the values in the rest of the territory.

Performing a three-component registration of seismic oscillations along an orthogonal profile network oriented to potentially dangerous focal zones does increase the reliability of the classification of a possible earthquake.

However, due to the fact that in the known method the determination of dynamic parameters is carried out by analyzing only the most high-frequency seismic vibrations, the achievement of the technical result, which consists in increasing the reliability of the forecast, is possible only with time-stable oscillatory processes and in the absence of interference caused by acoustic and hydrodynamic noise of natural and technogenic in nature. And if in land conditions, with some assumptions, the use of this method has a positive technical effect, then in marine conditions it is practically not applicable to predict the possibility of a tsunami because it is impossible to determine the nature of the bottom deformation at considerable distances (large outbreak size), and A significant tsunami wave occurs only with its vertical or inclined movements. False alarms lead to large material losses.

The objective of the proposed technical solution is to increase the reliability of determining the occurrence of an earthquake.

The problem is solved due to the fact that in the method of seismic microzoning, including the placement of the studied and reference points of observation in areas with different engineering and geological conditions, registration of seismic vibrations from earthquakes from potentially dangerous and other focal zones, determination of the dynamic parameters of seismic vibrations and their variations in each observation point under study relative to the reference ones in a given frequency range of studies, in which additional component registration of seismic vibrations along an orthogonal profile network oriented to potentially dangerous focal zones, the distance between observation points not exceeding 1/3-1 / 4 of the wavelength of the most high-frequency seismic oscillations forming informative amplitude variations, and the distance between profiles is 1/3 1/4 of the minimum spatial period of informative amplitude variations of the high-frequency range of seismic vibrations, in which, unlike the prototype, reference observation points they are also placed under the objects of economic activity, when determining the dynamic parameters of seismic vibrations and their variations in each observation point relative to the reference ones, the distribution of measurement coordinates is taken into account in a given frequency range of studies, while high-frequency seismic vibrations that form informative variations in the amplitudes of seismic vibrations are distinguished by spatial temporal volumes of discrete measurements by constructing a Cayley tree, and n selection of measurements from the entire array of measurements having a minimum measurement error, from measurements having a minimum measurement error, the magnitude of the seismic oscillation extremum is determined, which determines the damage from the extreme value of seismic oscillations.

The essence of the proposed method is illustrated by drawings (figure 1, 2).

Figure 1. Cayley tree of binary partitioning of the variability range of seismic characteristics.

Figure 2. The image of the cubic grid at n = 2, M = 4.

A specific example of the implementation of the method.

Seismic micro-zoning is carried out on an area of 1.5 km ² . Ground conditions are represented by three different sites: rock, pebble deposits and loose sediments. A network of seismic stations is installed on the seismic micro-zoning area and records from career explosions located at a distance of 100 km are recorded. According to the registration results, a range of high-frequency quantitatively significant variations is determined. A profile is broken up with a distance between registration points of about 200 m with the profile oriented in the direction to the location of the explosions. According to the results of registration of vibrations, the period of spatial variations (400-600 m) is determined by the profile. An orthogonal observation network with a distance between registration points of 150-200 m is split.

On one of the sections of the seismic micro-zoning site, due to the peculiarities of the internal structure of the rock base, the distance between the registration points thickens to 100 m.

Unlike the prototype, in the inventive method, reference observation points are also placed under the objects of economic activity (offshore terminals, onshore drilling complexes, etc.).

When determining the dynamic parameters of seismic vibrations and their variations in each observation point relative to the reference, in a given frequency range of studies, the distribution of the coordinates of the measurements is taken into account, while high-frequency seismic vibrations, which form informative variations in the amplitudes of seismic vibrations, are distinguished by spatial-temporal volumes of discrete measurements by constructing Kaylee tree (figure 1).

In this case, in contrast to the prototype, in which real numbers are used in the analysis of seismic vibrations, the proposed method uses p-adic numbers in the analysis of seismic vibrations (Vladimirov BC, Volovich IV, Zelenev E.I. p-Adic analysis and mathematical physics. M.: Nauka, 1994. - 352 pp. Kozyrev SV Methods and applications of ultrametric and p-adic analysis: from burst theory to biophysics.Modern problems of mathematics.M .: MIAN, 2008. - 170 p. Fridman AA World as space and time. - Izhevsk: SRC "Regular and chaotic din Amika ", 2001. - 96 pp. Suppes P., Luce D.H. Foundation of Measurement. - San Diego: Academic Press. - 1990. - 125 pp.).

The motivation for using numerical systems other than real numbers in seismic studies is related to the following main circumstances.

Firstly, the system of real numbers is infinite, while in reality seismic oscillations always lie in a limited range of values.

Secondly, as a result of any measurements of seismic characteristics, quantitative values are obtained in rational numbers. Indeed, in any physical dimension, only ultimate accuracy can be achieved in principle (Harmut X. Application of information theory methods in physics. - M .: Mir, 1989. - 344 p.), I.e. operate only with numbers that have a finite number of characters (decimal or, for example, binary). These are rational numbers. However, starting from the field of rational numbers you can get either a system of real numbers, or one of the systems of p-adic numbers. By Ostrovsky’s theorem, other numerical systems cannot be obtained from rational numbers. Historically, real numbers have been given priority, and p-adic peers with formal properties having equal rights with them have much less use. This can be explained by the fact that p-adic numbers were discovered 800 years later than the actual ones - at the end of the 19th century, they are not studied at universities and they have unusual properties with respect to real numbers.

Thirdly, the spatiotemporal structure of seismic characteristics is fundamentally heterogeneous. Their dynamics due to the heterogeneity of the structure of the earth's relief, and even more so of the hydrosphere, is turbulent. Vortex formations are traced on all scales of space and time and form a complex hierarchical structure in size and lifetime. The state of the hierarchy has a natural representation in the form of a tree-like graph structure (Benzi R., Biferale L., Trovatore E. Ultrametric Structure of Multiscale Energy Correlations in Turbulent Models // Physical Review Letters, V.79, No. 9, 1997. - pp. 1670 -1673). It is impossible to describe such structures with real numbers, because the most important feature of the material continuum is its homogeneity.

Fourth, the variability of seismic characteristics is clearly related to the considered spatio-temporal scale. In a first approximation, this dependence is described by a power function:

where δX is the average variability of the seismic characteristic in the time interval Δt and in the space interval with the volume ΔV, α, β are real parameters, large zero. Note that for relation (1), a correspondence with tree structures is found (Olemskaya A.I., Flat A.Ya. Use of the fractal concept in condensed matter physics. // UFN, T.163, No. 12, 1993. - p.1 -fifty).

Actual numbers used in measurements cannot fundamentally describe seismic characteristics at the required spatio-temporal scale. This is a "point" data. However, in practice they are implicitly used to describe the state of the environment for spatio-temporal regions. The consequence of this is that, in principle, there is no possibility of comparing the values of the characteristics measured in various fields. For example, it is incorrect to compare the values of temperatures measured even at one time in different cities, because areas of cities and intervals of variability in them are different.

Fifth, the use of real numbers leads to the problem of matching the values of measurements carried out by instruments with different resolutions. Indeed, when rounding or refining a numerical value, it is possible to change the immediately previous (previous) signs of the number in relation to the position to be changed.

Let us present one of the possible ways of transition to the representation of acoustic characteristics by p-adic numbers, the use of which removes the above disadvantages that arise when using real numbers. We describe this transition using three consecutive steps: introducing a hierarchical structure, determining the ultrametric distance on it, and switching to p-adic numbers.

Let a seismic characteristic X be given and an allowable range of variation of its values X∈ [X _min , X _max ] = I be determined. For example, we will carry out the simplest binary division of this range; it corresponds to the Cayley graph shown in Fig. 1. At each crushing level, each subband of this level is assigned a symbol 1 or 0. Each interval at the n-level of crushing is described by a sequence of zeros and ones that are arranged in order of passage from the tree root to the required interval, for example, for n = 3, x ₂ = {0,0,1}.

Crushing can be carried out not only binary, but for any m> 1, m is natural.

In addition, crushing can continue indefinitely. Then each point of the partition x has an infinite number of coordinates

where each coordinate takes a finite number of values

Denote the sequence space (3) by the symbol Z _m . We introduce the distance ρ between the elements of this space as follows. We fix the real number 0 <q <1. Let be

x = (α ₀ , α ₁ , α ₂ , ..., α _n , ...), y = (β ₀ , β ₁ , β ₂ , ..., β _n , ...) ∈Z _m .

Put

This function is a distance (metric) and not even Archimedean. This distance is called ultrametric. This distance satisfies the enhanced triangle inequality:

In order to find the distance ρ _m (x, y) between two sequences of digits x and y, you need to find the first position k such that the sequences have different digits at this position. The choice of the constant q does not play any role. Standard selection: q = 1 / m. In this way

Example. For a binary partition, m = 2. Let x = (0,1,0, ...) and, y = (0,1,1, ...). Here k = 2 and, therefore, ρ ₂ (x, y) = 1/4.

Note that up to this point, no number systems have yet appeared, only now we proceed to the introduction of numbers. We do it as follows. Note that the point x = (α ₀ , α ₁ , α ₂ , ..., α _k , ...) of the space Z _m can be identified with a "number"

This series converges in the metric space Z _m . In particular, finite sequences x = α ₀ α ₁ ... α _k can be identified with natural numbers x = α ₀ + α ₁ m + ... + α _k m ^k .

Therefore, the set of all finite sequences can be identified with the set of natural numbers. Moreover, the set of natural numbers is a dense subset of Z _m : any x∈Z _m can be approximated with arbitrary precision by natural numbers. If x is a natural number, then | x | _m = m ^-k , if and only if x is divisible by m ^k and not divisible by m ^{k + 1} .

The set Z _m is called the set of m-adic integers.

On the set of m-adic integers Z _m, one can introduce algebraic operations, namely addition, subtraction and multiplication. These operations are natural extensions of standard operations on the set of natural numbers N = {0,1,2,3, ...}. Note that the division in Z _{m is} not correctly defined (it is a numerical ring, but not a numerical field) and the apparatus of mathematical analysis is not developed for these numerical systems.

To obtain complete numerical non-Archimedean systems (p-adic numbers), it is necessary to expand the numerical sets Z _m . Consider an expression of the form:

where α _j = 0, 1, ..., m-1 and s = 0, ± 1, ± 2, .... Denote the set of all such expressions by Q _m . Put | x | _m = m ^-s if α _s ≠ 0. This is a natural continuation of the valuation given on Z _m . If s = 0, 1, 2, ..., then x = α _s m ^s + ... + α _j m ^j + .... Then | x | _m = m- ^s . On the other hand, if s = -1, -2, ..., then x = α _-k / m ^k + ... + α _j m ^j + ..., where k = -s. Then | x | _m = m ^-s = m ^k .

Here we note that the system of real numbers R consists of expressions of the form:

Moreover, the decimal base is usually used, i.e. m = 10. In the real case, there can only be a finite number of terms with positive powers at m, and in the m-adic case there can be an infinite number of terms with positive powers at m. For negative degrees, the opposite is true. We introduce addition, subtraction, and multiplication by Q _{m in} exactly the same way as by Z _m , continuing the standard operations defined on the set of finite sums:

Note that | xy | _m ≤ | x | _m | y | _m . If m = p is a prime, then (as for the standard module in the real case) | xy | _p = | x | _p | y | _p . In general, the division by Q _{m is} not defined. However, if m = p, then the division is defined correctly and, moreover, there is an apparatus for mathematical analysis and probability theory. Numerical systems of type Q _p are called p-adic numbers.

The construction carried out is quite possible to carry out according to available contact and remote measurements of seismic vibrations. In this case, it is possible to limit ourselves to weaker numerical systems Z _m . Note the characteristic drawback of the presented hierarchy of numerical systems: the absence of isomorphism between systems with different m, and their large variety - m - is any natural number greater than 1, i.e. infinite number (as well as prime).

It seems true that the laws of nature should not depend on the numerical systems used to describe it, i.e. must appear in any of them. However, it is also true that in some numerical systems these laws have a simpler form, which allows them to be detected faster.

Next, the selection of measurements is automatically distinguished from the entire array of measurements that have a minimum measurement error, for measurements that have a minimum measurement error, the magnitude of the seismic oscillation extremum is established, which determines the damage from the extreme value of seismic oscillations.

In the practice of processing seismic observations, there is the task of estimating the average value of the seismic characteristic in a certain spatio-temporal volume from discrete observations. Each discrete observation is fixed by four coordinates: time, latitude, longitude, and depth (height). In other words, the value of a seismic characteristic is generally a function defined in a four-dimensional coordinate space. In special cases, this dimension can take smaller values. The dimension is determined by the number of coordinates of the observation points, the values of which change in the array of observations. For example, for typical tasks of calculating the average value of seismic characteristics from discrete observations:

by time-discrete measurements at a geographical point with fixed coordinates at a fixed depth;

by simultaneous observations in a certain spatial region at a fixed depth;

according to observations in a certain area at different points in time and at different points at a fixed depth;

according to observations in a certain fixed region for a certain time interval in a certain interval of depths.

The numbering of tasks corresponds to the dimension of the space of the task of hydrometeorological characteristics.

Currently, the arithmetic average for any task is calculated by direct summation of all measured values falling into the considered spatio-temporal volume, and divided by the total number of measurements.

In fact, this method of calculating the arithmetic mean of the seismic characteristic does not allow us to obtain the minimum possible error in estimating the mean. The reason for this is that in the adopted method, the distribution of observation points over the spatio-temporal volume is ignored. Moreover, the accuracy of estimating the average value of the applied method decreases with increasing dimension of the space of observation coordinates.

Let us prove this and give an adequate method for calculating the arithmetic mean value of the seismic characteristic, minimizing the error.

From the mathematical point of view, the task of determining the average value of a seismic characteristic from the values of the characteristic at discrete spatiotemporal points is the problem of estimating the arithmetic mean of a continuous function in a spatio-temporal volume from its values at discrete points. The error in the estimation of the arithmetic mean, subject to the isotropy of the variability of the function, is determined by the structure of the "unevenness" of the location of the measurement points in the spatio-temporal volume. The smallest error is given by points uniformly distributed over the volume under consideration. Points are called uniformly distributed in an n-dimensional unit cube if in any hypercube the number of points is proportional to the volume of the hypercube (I. M. Sobol. Multidimensional quadrature formulas and Haar functions. - M., Nauka, 1969. - 288 p.).

Formally, this is defined as follows. We denote by K ^{n the} unit cube in n-dimensional space: K ⁿ consists of all points P with Cartesian coordinates P = (x ₁ , ..., x _n ) that satisfy the inequalities 0≤x _j ≤1 (j = 1, 2, ..., n). Consider a sequence of points P ₀ , P ₁ , ..., P _j , ... belonging to a cube K ^{n of} dimension n, and denote by S _N (G) the number of points P _i with numbers 0≤i≤N-1 belonging to G. The sequence points P ₀ , P ₁ , ..., P _i , ... - is called uniformly distributed in K ⁿ (abbreviated pp), if for any nth box π with edges parallel to the coordinate axes,

where V _π is the volume of the parallelepiped π. It can be proved that if G is an arbitrary domain located in K ⁿ and having volume V _G , then it follows from (12) that

.

.

Thus, for large N, the number of points p.p. sequences belonging to any domain G is proportional to the volume G.

By Weyl theorem (Sobol IM Multidimensional quadrature formulas and Haar functions -. Nauka, Moscow, 1969. -. 288) to {P _i} was pp, it is necessary and sufficient that any integrable by Riemann, the function f (P) satisfies the relation

Expression (2) is an estimate of the arithmetic mean. The error estimate (13) is determined by the expression

where C is a constant different for functions with different variability, D is the deviation of the distribution of points from p.p.

Deviation D is defined as follows. Consider in K ^{n a} grid consisting of N arbitrary points P ₀ , P ₁ , ..., P _N-1 . To each point P of K ⁿ we associate a parallelepiped π _P with the diagonal OP (O is the origin). The volume V _{P of} this parallelepiped is equal to the product x ₁ ... x _{n of the} coordinates of the point P. The deviation of the grid P ₀ , P ₁ , ..., P _N-1 is the number

where the upper bound is taken over all P∈K ⁿ .

In order for the sequence of points to be p.p., it is necessary and sufficient that, as N → ∝

The characteristic D (P ₀ , ..., P _N-1 ) is a very complex function of the structure of the points. The upper bound is D≤N, its lower bound is still not known (with the exception of the case n = 1, when inf D = 1/2). There is an assumption that the best possible estimate of D for an n-dimensional grid consisting of N points is

It should be noted that in most cases, researchers mistakenly believe that cubic grids (they are called “regular”, “uniform”, etc.) are always very “good” and the goal of collecting information is to obtain measurements on such a grid. The cubic grid for N = M ² points is specified by the coordinates

where i ₁ , i ₂ , ..., i _n - independently run through the values 1, 2, ..., M. Figure 2 shows a cubic grid with n = 2, M = 4.

будет максимальным, например, в точке P'=(1/2M,1,1,…,1), когда S_N(π_p')=0, NV_P'=N/2M=М^n-1/2. Следовательно,It is easy to verify that for such grids the value

will be maximum, for example, at the point P '= (1 / 2M, 1,1, ..., 1), when S _N (π _p' ) = 0, NV _{P '} = N / 2M = M ^n-1 /2. Hence,

- такой же порядок соответствует случайным сеткам, состоящим из N независимых случайных точек, равномерно распределенных в Kⁿ. Значит, при n≥3 сетки (18) асимптотически хуже случайных.It follows from formula (8) that for n = 1, cubic grids are optimal. However, with increasing n, the uniformity of the grids (18) worsens and the orders in formula (19) approach the worst, equal to N. Already at n = 2, order (19) is equal

- the same order corresponds to random grids consisting of N independent random points uniformly distributed in K ⁿ . Therefore, for n≥3, grids (18) are asymptotically worse than random ones.

, которое образует сетку с наибольшей степенью равномерного распределения. Решение этой задачи будем искать следующим образом. Определим подходящую опорную сеть точек, имеющую p.p. Сравнивая координаты точек наблюдений с координатами точек опорной p.p. сетки, найдем искомое подмножество

, которое будет давать минимальную ошибку с оценке среднего арифметического, в соответствии с (14).Let K ⁿ set a set of observation points for the hydrometeorological characteristics of P ₀ , P ₁ , ..., P _N-1 . This set of points is far from pp. It is necessary to define such a subset of points

which forms the grid with the greatest degree of uniform distribution. We will seek a solution to this problem as follows. We define a suitable reference point network with pp. Comparing the coordinates of the observation points with the coordinates of the points of the reference pp grid, we find the desired subset

, which will give a minimum error with the estimation of the arithmetic mean, in accordance with (14).

In computational mathematics, many variants of grids close to pp are constructed. For our task, it is advisable to choose a grid with a sequence of points Q ₀ , Q ₁ , ..., Q _i , ... that satisfies three requirements:

uniform distribution of the grid should be asymptotically optimal;

uniform distribution of points should be observed not only as N → ∝, but already for small N;

the algorithm for calculating the points Q _i should be quite simple.

These requirements are satisfied, for example, by the so-called LP _τ - sequences (I. M. Sobol. Multidimensional quadrature formulas and Haar functions. - M., Nauka, 1969. - 288 p.).

Without setting out a theoretical justification for p.p. properties of these sequences, we present a simple calculation algorithm.

In this algorithm, the coordinates (q _i1 , ..., q _in ) of the point Q _i from the LP _τ - sequence are calculated by the formula

- двоично-рациональные числа вида

, числители которых табулированы.where i = e _m ... e ₂ e ₁ is the representation of i in the binary system,

- binary rational numbers of the form

whose numerators are tabulated.

для 1≤s≤20 и 1≤j≤4, что позволяет легко вычислять точки Q_i размерности n≤4 в количестве N≤2²⁰.Table 1 presents

for 1≤s≤20 and 1≤j≤4, which makes it easy to calculate points Q _{i of} dimension n≤4 in the amount of N≤2 ²⁰ .

Table 1.

Value table

s n one 2 3 four one one one one one 2 one 3 one 3 3 one 5 7 one four one fifteen eleven 5 5 one 17 13 31 6 one 51 61 29th 7 one 85 67 81 8 one 255 79 147 9 one 257 465 433 10 one 771 721 149 eleven one 1285 823 719 12 one 3855 4091 3693 13 one 4369 4125 3841 fourteen one 13107 4141 11523 fifteen one 21845 28723 16641 16 one 65535 45311 49925 17 one 65537 53505 16671 eighteen one 196611 250113 83229 19 one 327685 276231 515921 twenty one 983055 326411 482707

The algorithm for selecting representative points can be represented as follows:

Given: a set of N observation points P ₀ , P ₁ , P _N-1 in an n-dimensional cube K ⁿ , n = 1, 2, 3, 4. Each observation point is represented by the normalized coordinates P = (x ₁ , ..., x _n ), 0≤x _j ≤1, j = 1, 2, ..., n.

, которые дают наименьшую ошибку в оценке среднего арифметического функции, измеренной в этих точках (т.е наиболее репрезентативный набор точек измерений для оценки среднего или, что тождественно, образуют сетку с наибольшей степенью равномерного распределения).Required: Define a subset of points

which give the smallest error in the estimation of the arithmetic mean function measured at these points (i.e., the most representative set of measurement points for estimating the mean or, identically, form a grid with the greatest degree of uniform distribution).

Initiation: T _k = T ₀ = ⌀ - a set of candidate points at the k step.

Solution flow:

Calculate N points Q _k (k = 1, ... N) LP _τ - sequences according to formula (20).

среди множества точек

, и добавить ее в набор точек-кандидатов на k, образуя

.Sequentially for each k from 1 to N for each point Q _k find the point closest to the Euclidean metric

among many points

, and add it to the set of candidate points on k, forming

.

Calculate the deviation D _k = D (T _k ) for the point P _i according to the formula (4).

Sequentially for each k from 1 to N find D _0k = max (D ₁ , ..., D _k ).

Find k ₀ for which there is a minimum value of D _0k .

является искомым множеством точек наблюдений, дающим минимальную ошибку в оценке среднего арифметического. Таким образом, при расчете среднего арифметического значения сейсмической характеристики по дискретным наблюдениям в некотором пространственно-временном объеме для минимизации погрешности необходимо учитывать распределение координат измерений. На основе специального математического аппарата предложен метод расчета среднего арифметического значения гидрометеорологической характеристики по дискретным наблюдениям в некотором пространственно-временном объеме, дающий минимальную теоретическую погрешность в оценке, причем метод «автоматически» выявляет тот поднабор измерений из всего массива данных, который дает минимальную погрешность в оценке среднего арифметического. Для корректности вычислений необходимо ввести обычным путем локальные декартовы координаты для рассматриваемой пространственно-временной области.Result: point set

is the desired set of observation points giving a minimal error in the estimation of the arithmetic mean. Thus, when calculating the arithmetic mean of the seismic characteristic from discrete observations in a certain spatio-temporal volume, in order to minimize the error, it is necessary to take into account the distribution of measurement coordinates. Based on a special mathematical apparatus, a method is proposed for calculating the arithmetic mean of the hydrometeorological characteristic from discrete observations in a certain spatio-temporal volume, which gives the minimum theoretical error in the estimation, and the method “automatically” identifies that subset of measurements from the entire data array that gives the minimum error in the estimate arithmetic mean. For the calculation to be correct, it is necessary to introduce in the usual way local Cartesian coordinates for the considered spatio-temporal region.

In the problems of accounting for the influence of seismic phenomena on objects of economic activity, as well as in the design of various coastal structures and equipment, in the problems of assessing the reliability of the operation of technical means, the choice of the distribution function of seismic characteristics for evaluating extreme values that determine the degree of danger of the influence of seismic conditions is of great importance.

We show that the extreme values of seismic characteristics are distributed not according to the exponential law, as is characteristic of normally distributed random variables, but according to a power law. This leads to the fact that the damage from extreme values is much higher than what is usually estimated using the normal distribution law. The rationale can be the classical theory of extrema (Leadbetter M., Rotsen X., Lindgren G. Extremums of random sequences and processes. - M .: Mir, 1989. - 392 p.), Which considers the distribution of the maximum

n independent and identically distributed random variables ξ, with the distribution function F (x) for large values of n.

The main result of this theory states that if for some sequences of normalizing constants a _n > 0, b _{n the} random variable a _n (М _n -b _n ) has a non-degenerate limit distribution function G (x), then this function G (x) should have one of three possible forms

In particular, it is proved that the “tails” of all distribution functions F (x) have only two types:

- exponential (e ^-x ) for type 1 (for example, for the normal distribution law);

- power law (x ^-α , α> 0) for types 2 and 3.

We show that the extreme values of acoustic characteristics are distributed according to a power law. The logic of the proof is as follows.

Direct statistical estimates of the series of observations cannot evaluate the behavior of the “tail” of the distribution due to the rarity of extreme events. We use an indirect technique, namely, that the behavior of the “tails” of the distributions splits the entire set of non-degenerate distributions of random variables into two equivalence classes - power and exponential. Moreover, linear statistics do not violate this partition. We choose such statistics L ( a _n , b _n ) for the time course of the acoustic characteristic ξ (n), which leads to a random variable η (n) distributed according to some law F * (y), for which the distribution type is known from the classical theory of extrema her tail. Then the same type of tail distribution will be for the extrema of the acoustic characteristic.

As linear statistics we use the statistics of Gerst, which has the form (Feder E. Fractals. - M .: Mir, 1991. - 260 p.)

,

,

Where

,

,

.

.

Let us estimate the distribution function F * (y) of the random variable η obtained using this transformation.

Processing time series of seismic characteristics leads to the following power law

F * (y) ~ τ ^H ,

where 1/2 <H <1.

Therefore, Gerst statistics determines a random function distributed according to a power law, which, as proved in the theory of extrema, has a power distribution of the tail. This means that the “tails” of seismic characteristics have a power-law distribution.

The damage from the extreme value of the seismic characteristic is obviously a certain power function of the magnitude of the extremum

Q ~ x ⁿ , n≤1.

If, as a quantitative assessment of the influence of extreme values of seismic characteristics during design, we use the mathematical expectation of damage

,

,

where f (x) is the distribution density function of the tail of the seismic characteristic, then we obviously get:

при любом n. Для наблюдаемых сейсмических процессов α~1, и вероятнее всего большое влияние "хвоста" распределения на величину ущерба от возникновения экстремального значения сейсмической характеристики.For the normal distribution law

for any n. For the observed seismic processes, α ~ 1, and most likely a large influence of the “tail” of the distribution on the damage caused by the occurrence of extreme values of the seismic characteristic.

The method can be implemented on broadband acoustic transducers having industrial applicability, for example, of the type ЭХД-17 or ЭХД-20 and computer technology that implements the algorithms described in the description of mathematical software.

Information sources

1. Patent RU No. 1251694.

Claims (1)

A method of seismic micro-zoning, including the placement of the studied and reference observation points in areas with different engineering and geological conditions, registration of seismic vibrations from earthquakes from potentially dangerous and other focal zones, determination of the dynamic parameters of seismic vibrations and their variations in each investigated observation point relative to the reference in a given research frequency range, in which three-component recording of seismic oscillations is additionally carried out along the orthogonal network of profiles oriented to potentially dangerous focal zones, and the distance between the observation points does not exceed 1 / 3-1 / 4 of the wavelength of the most high-frequency seismic oscillations, which form informative variations in amplitudes, and the distance between the profiles is 1 / 3-1 / 4 the minimum spatial period of informative amplitude variations of the high-frequency range of seismic vibrations, characterized in that the observation reference points are also placed under the objects of economic activity, p and the determination of the dynamic parameters of seismic vibrations and their variations in each observation point relative to the reference in a given frequency range of studies take into account the distribution of the measurement coordinates, while high-frequency seismic vibrations that form informative variations in the amplitudes of seismic vibrations are distinguished from the spatio-temporal volumes of discrete measurements by constructing a tree Kayleigh, at the same time, automatically select the selection of measurements from the entire array of measurements having a minimum According to measurements with a minimum error, the maximum error of the measurements is determined by the magnitude of the extremum of the seismic oscillation, which determines the damage from the extreme value of the seismic oscillations.

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