RU2517983C1 - Method of profiling bottom deposits - Google Patents

Abstract

FIELD: physics.

SUBSTANCE: method of profiling bottom deposits involves mounting radiating and receiving antennae of a profile recorder on a towed carrier, wherein the radiating and receiving antennae of the profile recorder are separately mounted on the carrier, and the receiving antenna used is a K-element receiving antenna directed along the longitudinal axis of the carrier. The carrier is towed over the bottom. A pulsed acoustic phase-shift keyed signal, which is modulated by an M-sequence, is emitted. The reflected signal is received and undergoes correlation processing with a copy of the emitted acoustic phase-shift keyed signal, wherein amplification and correlation processing of the received signal is carried out with a K-channel receiving circuit. After amplification and correlation processing of signals received by each element of the K-element receiving antenna, Q values of the complex amplitude of the received signal $$S_{qp}^{(к)}$$

are generated. A matrix is formed from Q row elements. Time delay is calculated for each emission time t_pn and arrival time t_q. Time shift and coherent summation operations are repeated for the entire array of data for each element of the receiving antenna. Coherent summation of K signals received by the K-element receiving antenna is performed for each time of arrival of received signals t_q and time of emission t_p. Graphic construction of the profile of bottom deposits is then performed based on the delay time of the reflected signal.

EFFECT: higher resolution of the method of profiling in a longitudinal direction while maintaining a sufficiently large depth of profiling and high resolution in the vertical direction.

2 dwg

Description

The invention relates to the field of geophysics and hydroacoustics and can be used to study the structure of bottom sediments in the shelf zone of the oceans, as well as to study the characteristics of sound propagation in the bottom layer of the shallow sea.

A known method of profiling bottom sediments, implemented in the work of Kasatkin B.A., Kosareva G.V., Larionova Yu.G. Study of the bottom of the Amur Bay with a high-resolution profilograph. Proceedings of the Russian Acoustic Society. M .: GEOS, 2001, v. 2, p. 18-22. The known solution is based on the principle of reflection of broadband acoustic impulses propagating in water and soil from all interfaces, such as the water-seabed interface, as well as the interface between the individual layers of marine sedimentary rocks.

The method includes emitting broadband pulsed acoustic signals in the frequency range 3-8 kHz, receiving reflected signals, correlating them in real time, and constructing a profile of bottom sediments from the delay time of reflected signals from the layer interfaces. The use of broadband pulsed acoustic signals makes it possible to realize a sufficiently high resolution of the profiling method in this method. The disadvantage of this method is the relatively small profiling depth.

A known method of profiling bottom sediments, in which to increase the depth of profiling while maintaining high resolution as a pulsed acoustic sounding signal, a phase-shifted signal modulated by an M-sequence is used (RF Patent No. 23560696, IPC G01V 1/38, 2007). This method is the closest to the claimed solution.

This method of profiling sediments includes installing a receiving-emitting antenna of the profilograph on a towed carrier and towing it above the bottom, emitting a phase-manipulated signal modulated by the M-sequence, receiving the reflected signal, correlating it with an acoustic copy of the emitted signal, and then graphically plotting the profile of the sediment according to the delay time of the reflected signal. In this method, to increase the profiling depth, a phase-manipulated signal with a sufficiently large base, with sufficient energy, is used, and the resolution of this method in the vertical direction remains high due to the high signal-to-noise ratio at the input of the receiving path, which is also provided by the selection of a phase-shifted signal with a sufficiently large base.

The disadvantage of this profiling method is the low resolution in the longitudinal direction, i.e. in the direction of towing the profiler antenna. This is because the profilograph antenna at the operating profiling frequencies has a weak directivity in the vertical plane, as a result of which the profiling point object is displayed on the profilogram as a characteristic parabola, which makes it difficult to correctly identify the profiling objects. To increase the resolution in the longitudinal direction, either increase the aperture of the antenna of the profilograph, which significantly increases the mass-dimensional characteristics of the profilograph, or use the technique of synthesizing the aperture. Such a technique as applied to the profiling problem is given, for example, in the work of A.I. Zakharov, V.I.Kaevitser, V.M. Razmanov, V.N. Raskatov, Application of aperture synthesis methods in low-frequency echo sounders-profilographs. Proceedings of the VIII International Conference Applied Technologies of Hydroacoustic and Hydrophysics. St. Petersburg, Nauka, 2006, p.143-147. However, the aperture synthesis algorithms described in this paper do not take into account the differences in the speed of sound propagation in water and bottom sediments and the associated refraction of sound rays at the water-seabed interface, and therefore are very approximate. As a result, they are quite effective only when the synthesized aperture is small and when the bottom sediments are an unconsolidated sediment in which the speed of sound is close to the speed of sound in water.

Another disadvantage of the described method lies in the fact that in the region of incidence angles smaller than the critical one, and the profiler operates precisely in this range of incidence angles, there is a backward return wave and associated birefringence. This phenomenon also distorts the image of real profiling objects on the profilogram and degrades the resolution in depth of scattering objects.

The objective of the invention is to increase the resolution of the method of profiling in the longitudinal direction by synthesizing the aperture taking into account the refraction of rays at the water-seabed interface and increasing the resolution in the vertical direction by compensating for the birefringence effect that always occurs at incidence angles less than critical.

, соответствующих Q элементам пространственного разрешения в интервале времен прихода отраженного акустического фазоманипулированного сигнала t_q∈(t₀, t_MAX), q=1, 2, Q, где $$Q=\raisebox{1ex}{$(t_{MAX}-t_0)$}\!\left/ \!\raisebox{-1ex}{$\Delta t$}\right.$$

, $$t_0=\raisebox{1ex}{$2h$}\!\left/ \!\raisebox{-1ex}{$c_1$}\right.$$

, $$t_{MAX}=t_0+\raisebox{1ex}{$2H$}\!\left/ \!\raisebox{-1ex}{$c_2$}\right.$$

(c₁, c₂ - предварительно определенные эффективная скорость звука в воде и в донных отложениях соответственно, h - высота антенны профилографа над дном, $$\Delta t=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\Delta f$}\right.$$

, Δf - полоса частот акустического фазоманипулированного сигнала, Н - предполагаемая глубина профилирования морского дна). Из Q элементов - строк формируется матрица из Q×(2N+1) - значений комплексной амплитуды принятого сигнала $$S_{qpn}^{(к)}$$

соответствующих Q элементам пространственного разрешения по временам прихода t_q и временам излучения t_pn=t_p±nT q∈(1, Q); n∈(0, N). Для каждого момента времени излучения t_pn и времени прихода t_q вычисляются временные задержки The problem is solved by the method of profiling bottom sediments, including installing a receiving-emitting antenna of the profilograph on a towed carrier and towing it above the bottom, emitting a pulsed acoustic phase-shifted signal modulated by the M-sequence, receiving a reflected signal, its correlation processing with a copy of the emitted acoustic phase-shifted signal and subsequent graphical construction profile of bottom sediments by the delay time of the reflected signal. To compensate for the spatial distortions associated with the weak directivity of the profilograph antenna, and therefore, to increase the resolution in the longitudinal direction and in the depth of the scattering objects, the radiating and receiving antennas of the profilograph are mounted separately from each other on the carrier, and oriented along the longitudinal axis of the carrier K is the elementary receiving antenna, and the amplification and correlation processing of the received signals is produced by the K - channel at capacitive tract. After amplification and correlation processing of the signals received by each element of the K - element receiving antenna, for each transmission of the pulse signal emitted at time t _p = t ₀ + pT, T is the repetition rate of the profiler radiation pulses, p = 1, 2, 3 ... , from the processed data array, Q values of the complex amplitude of the received signal are formed $$S_{qp}^{(\mathrm{to})}$$

corresponding to Q spatial resolution elements in the interval of arrival times of the reflected acoustic phase-manipulated signal t _q ∈ (t ₀ , t _MAX ), q = 1, 2, Q, where $$Q=\raisebox{1ex}{$(t_{MAX}-t_0)$}\!\left/ \!\raisebox{-1ex}{$\Delta t$}\right.$$

, $${\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="00000044.png"\; state="\{\{state\}\}">t</figure-callout>}}_0=\raisebox{1ex}{$2h$}\!\left/ \!\raisebox{-1ex}{$c_{\mathrm{one}}$}\right.$$

, $$t_{MAX}={\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="00000044.png"\; state="\{\{state\}\}">t</figure-callout>}}_0+\raisebox{1ex}{$2H$}\!\left/ \!\raisebox{-1ex}{$c_2$}\right.$$

(c ₁ , c ₂ are the predefined effective speed of sound in water and in bottom sediments, respectively, h is the height of the profiler antenna above the bottom, $$\Delta t=\raisebox{1ex}{$\mathrm{one}$}\!\left/ \!\raisebox{-1ex}{$\Delta f$}\right.$$

, Δf is the frequency band of the acoustic phase-manipulated signal, N is the estimated depth of the seabed profiling). From Q elements - rows, a matrix is formed from Q × (2N + 1) - values of the complex amplitude of the received signal $$S_{qpn}^{(\mathrm{to})}$$

corresponding to Q spatial resolution elements with respect to arrival times t _q and radiation times t _pn = t _p ± nT q∈ (1, Q); n∈ (0, N). For each moment of radiation time t _pn and arrival time t _q time delays are calculated

$$\Delta t_{qn}=\Delta t_{qn}^{(\mathrm{one})}+\Delta t_{qn}^{(2)}$$

, $$\Delta t_{qn}^{(\mathrm{one})}=t_0[\frac{\mathrm{one}-\cos {\theta }_{\mathrm{eleven}}}{\cos {\theta }_{\mathrm{eleven}}}+\frac{\mathrm{one}-\cos {\theta }_{21}}{\cos {\theta }_{21}}-\frac{t_q-t_0}{t_0}]$$

,

, $$t_0=\frac{2h}{c_1}$$

, $$\cos {\theta }_{11}=\sqrt{1-{\sin }^2{\theta }_{11}}$$

, $$\cos {\theta }_{21}=\sqrt{1-c_{21}^2{\sin }^2{\theta }_{11}}$$

, $$t_q=\frac{2h}{c_{\mathrm{one}}}+\frac{2z_q}{c_2}$$

, $${\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="00000044.png"\; state="\{\{state\}\}">t</figure-callout>}}_0=\frac{2h}{c_{\mathrm{one}}}$$

, $$\cos {\theta }_{\mathrm{eleven}}=\sqrt{\mathrm{one}-{\sin }^2{\theta }_{\mathrm{eleven}}}$$

, $$\cos {\theta }_{21}=\sqrt{\mathrm{one}-c_{21}^2{\sin }^2{\theta }_{\mathrm{eleven}}}$$

,

, $${\chi }_0=\frac{{\overline{r}}_n}{\sqrt{1+{\overline{r}}_{{}_n}^2}}$$

, $${\overline{r}}_n=\frac{nUT}{c_1{t}_0}$$

, $${\overline{z}}_q=c_{21}\frac{t_q-t_0}{t_0}$$

$$\sin {\theta }_{\mathrm{eleven}}={\chi }_0[\mathrm{one}-{\overline{z}}_q\frac{{(\mathrm{one}-{\mathrm{<figure-callout\; id="0"\; label="\chi "\; filenames="00000044.png"\; state="\{\{state\}\}">\chi </figure-callout>}}_0^2)}^{3/2}(c_{12}^2-{\mathrm{<figure-callout\; id="0"\; label="\chi "\; filenames="00000044.png"\; state="\{\{state\}\}">\chi </figure-callout>}}_0^2)}{{(c_{12}^2-{\mathrm{<figure-callout\; id="0"\; label="\chi "\; filenames="00000044.png"\; state="\{\{state\}\}">\chi </figure-callout>}}_0^2)}^{3/2}+c_{12}^2{\overline{z}}_q{(\mathrm{one}-{\mathrm{<figure-callout\; id="0"\; label="\chi "\; filenames="00000044.png"\; state="\{\{state\}\}">\chi </figure-callout>}}_0^2)}^{3/2}}]$$

, $${\mathrm{<figure-callout\; id="0"\; label="\chi "\; filenames="00000044.png"\; state="\{\{state\}\}">\chi </figure-callout>}}_0=\frac{{\overline{r}}_n}{\sqrt{\mathrm{one}+{\overline{r}}_{{}_n}^2}}$$

, $${\overline{r}}_n=\frac{nUT}{c_{\mathrm{one}}{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="00000044.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}$$

, $${\overline{z}}_q=c_{21}\frac{t_q-t_0}{{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="00000044.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}$$

$$\Delta t_{qn}^{(2)}=t_0[\frac{\cos {\theta }_{\mathrm{eleven}}-\cos {\theta }_{12}}{\cos {\theta }_{\mathrm{eleven}}\cos {\theta }_{12}}+c_{12}^2\frac{\cos {\theta }_{21}-\cos {\theta }_{22}}{\cos {\theta }_{\mathrm{eleven}}\cos {\theta }_{12}}\frac{\sin \left({\theta }_{12}-{\theta }_{\mathrm{eleven}}\right)}{\cos {\theta }_{\mathrm{eleven}}{\cos }_{22}+\cos {\theta }_{12}\cos {\theta }_{21}}]$$

; $$c_{21}=\raisebox{1ex}{$c_2$}\!\left/ \!\raisebox{-1ex}{$c_1$}\right.$$

, $$c_{21}=\raisebox{1ex}{$c_1$}\!\left/ \!\raisebox{-1ex}{$c_2$}\right.$$

, $$\cos {\theta }_{22}=\sqrt{\mathrm{one}-c_{21}^2{\sin }^2{\theta }_{12}}$$

; $$c_{21}=\raisebox{1ex}{$c_2$}\!\left/ \!\raisebox{-1ex}{$c_{\mathrm{one}}$}\right.$$

, $$c_{21}=\raisebox{1ex}{$c_{\mathrm{one}}$}\!\left/ \!\raisebox{-1ex}{$c_2$}\right.$$

,

, $$\sin {\theta }_{12}=[\mathrm{one}-\frac{{\overline{z}}_q^2}{c_{12}}\frac{f_{\mathrm{one}}(c_{12},{\overline{r}}_n)}{\mathrm{one}+{\overline{z}}_q^2{f}_2\left(c_{12},{\overline{r}}_n\right)}]\sqrt{\frac{c_{12}^2-{\overline{z}}_q^2}{\mathrm{one}-{\overline{z}}_q^2}}$$

,

$$f_{\mathrm{one}}(c_{12},{\overline{r}}_n)=\frac{c_{12}^2(\mathrm{one}-c_{12}^2)}{2c_{12}(c_{12}-{\overline{r}}_n\sqrt{\mathrm{one}-c_{12}^2})},$$

, $$f_2(c_{12},{\overline{r}}_n)=f_{\mathrm{one}}\left(c_{12},{\overline{r}}_n\right)[\frac{3-7c_{12}^2}{2c_{12}(\mathrm{one}-c_{12}^2)}-2\frac{\sqrt{\mathrm{one}-c_{12}^2}+{\overline{r}}_n{c}_{12}^2}{c_{12}\sqrt{\mathrm{one}-c_{12}^2-{\overline{r}}_n\left(\mathrm{one}-c_{12}^2\right)}}]$$

,

(U is the predefined velocity of the carrier of the antenna of the profilograph, Z _q is the depth of the reflective layer). For each moment of time of arrival of the received signals t _q and the radiation time t _pn, 2N + 1 signals are shifted in phase, shifted along the time scale by

$$〈S_{qp}^{(k)}〉=\frac{\mathrm{one}}{2N+\mathrm{one}}{\displaystyle \sum _{n=0}^N{S}_{qpn}^{(k)}(t_p+\Delta t_{qn})}\left(\mathrm{one}\right)$$

the time shift and common-mode summing operations are repeated for the entire data array p> N + 1, q≤Q for each element of the receiving antenna. For each moment of arrival time of the received signals t _q and the radiation time t _{p, the} K signals received by the K - element receiving antenna are shifted in phase by the value of kT

$$〈S_{qp}〉=\frac{\mathrm{one}}{K}{\displaystyle \sum _{k=\mathrm{one}}^K〈S_{qp}^{(k)}〉(t_{qp}+kT)}\left(2\right)$$

moreover, the repetition period of the radiation pulses T, the length of an individual element K - element receiving antenna L and the carrier speed of the profiler antenna U are related by the relation L = TU.

The invention is illustrated by drawings, where Fig. 1 shows a radiation treatment of double refraction of rays at the water-seabed interface z ₀ = h, r _n = nUT, Fig. 2 shows a diagram of the formation of a synthesized aperture.

As a result of such processing of the entire data array, spatial distortions associated with the weak directivity of the profilograph antenna, as well as with birefringence, are compensated, which is equivalent to an increase in the directivity of the profilograph antenna in the vertical plane and an improvement in its resolution in the longitudinal direction and in the depth of the scattering objects.

The method of profiling bottom sediments is implemented as follows.

When profiling the sedimentary layer of the seabed, the effective values of the speed of sound in water and the sedimentary layer of the seabed are preliminarily measured or calculated, which are included as parameters in the signal processing algorithms. The radiating and receiving antennas of the profilograph are mounted on the carrier separately from each other. As a receiving antenna, a K-element receiving antenna oriented along the longitudinal axis of the carrier is used. The profiling process includes installing the receiving-emitting antenna of the profilograph on a towed carrier, the movement of the carrier of the antenna of the profilograph above the bottom with a speed U, which is in a certain ratio with the length of an individual element of the antenna L and the repetition period of the radiation pulses T, the radiation of a pulsed acoustic phase-shifted signal modulated by M- sequence, receiving the reflected signal K - element receiving antenna, oriented along the profiler antenna carrier, gain and co relational processing received signals K - channel reception path.

To increase the resolution of the method of profiling in the longitudinal direction, the entire array of data obtained at the output of the K-channel receiving path is processed by synthesizing the aperture taking into account the refraction of rays at the water-seabed interface and the effects of birefringence according to formula (1) for each element K - element receiving antenna and common-mode addition of signals received by individual elements of the K - element receiving antenna, according to formula (2), followed by graphical construction of the d Fat nnyh the time delay of the reflected signal.

With such a structure of the receiving antenna of the profilograph, the receiving path of the profilograph and the proposed signal processing algorithms, the resolving power of the profilograph in the longitudinal direction will be equal to the length of one antenna element L, and the effective synthesized aperture is (2N + 1) KL.

Thus, the proposed method differs from the known novelty of the solution of the problem and allows you to significantly increase the resolution of the profiling method both in the longitudinal direction and in the vertical direction.

Claims (1)

A method for profiling bottom sediments, including installing a receiving-emitting antenna of a profilograph on a towed carrier and towing it above the bottom, emitting a pulsed acoustic phase-shifted signal modulated by the M-sequence, receiving a reflected signal, correlating it with a copy of the emitted acoustic phase-shifted signal and subsequent graphical construction of the profile of bottom sediments the delay time of the reflected signal, characterized in that the emitting and receiving antennas ofilografa mounted on the carrier separately from each other, and as a receiving antenna is used oriented along the longitudinal axis of the carrier K - element receiving antenna, and the gain and correlation processing of received signals produce K - channel reception path; after amplification and correlation processing of the signals received by each element of the K - element receiving antenna, for each transmission of the pulse signal emitted at time t ₀ = t ₀ + pT, T is the repetition rate of the profiler radiation pulses, p = 1, 2, 3 ... , from the array of processed data form Q values of the complex amplitude of the received signal $$S_{qp}^{(\mathrm{to})}$$

, in the interval of arrival times of the reflected acoustic phase-manipulated signal t _q ∈ (t ₀ , t _MAX ), q = 1, 2, Q, where $$Q=\raisebox{1ex}{$(t_{MAX}-t_0)$}\!\left/ \!\raisebox{-1ex}{$\Delta t$}\right.$$

, $$t_0=\raisebox{1ex}{$2h$}\!\left/ \!\raisebox{-1ex}{$c_{\mathrm{one}}$}\right.$$

, $$t_{MAX}=t_0+\raisebox{1ex}{$2H$}\!\left/ \!\raisebox{-1ex}{$c_2$}\right.$$

, (c ₁ , c ₂ - predefined effective speed of sound in water and in bottom sediments, respectively, h is the height of the profiler antenna above the bottom, $$\Delta t=\raisebox{1ex}{$\mathrm{one}$}\!\left/ \!\raisebox{-1ex}{$\Delta f$}\right.$$

, Δf is the frequency band of the acoustic phase-manipulated signal, N is the estimated depth of the seabed profiling), from Q elements - rows form a matrix of Q × (2N + 1) - values of the complex amplitude of the received signal $$S_{qpn}^{(\mathrm{to})}$$

corresponding to arrival times t _q and emission times t _pn = t _p ± nT q∈ (1, Q); n∈ (0, N), for each moment of radiation time t _pn and arrival time t _q time delays are calculated
$$\Delta t_{qn}=\Delta t_{qn}^{(\mathrm{one})}+\Delta t_{qn}^{(2)}$$

$$\Delta t_{qn}^{(\mathrm{one})}=t_0[\frac{\mathrm{one}-\cos {\theta }_{\mathrm{eleven}}}{\cos {\theta }_{\mathrm{eleven}}}+\frac{\mathrm{one}-\cos {\theta }_{21}}{\cos {\theta }_{21}}\frac{t_q-t_0}{t_0}]$$

,
$$t_q=\frac{2h}{c_{\mathrm{one}}}+\frac{2z_q}{c_2}$$

, $$t_0=\frac{2h}{c_{\mathrm{one}}}$$

, $$\cos {\theta }_{\mathrm{eleven}}=\sqrt{\mathrm{one}-{\sin }^2{\theta }_{\mathrm{eleven}}}$$

, $$\cos {\theta }_{21}=\sqrt{\mathrm{one}-c_{21}^2{\sin }^2{\theta }_{\mathrm{eleven}}}$$

,
$$\sin {\theta }_{\mathrm{eleven}}={\chi }_0[\mathrm{one}-{\overline{z}}_q\frac{{(\mathrm{one}-{\chi }_0^2)}^{3/2}(c_{12}^2-{\chi }_0^2)}{{(c_{12}^2-{\chi }_0^2)}^{3/2}+c_{12}^2{\overline{z}}_q{(\mathrm{one}-{\chi }_0^2)}^{3/2}}]$$

, $${\chi }_0=\frac{{\overline{r}}_n}{\sqrt{\mathrm{one}+{\overline{r}}_{{}_n}^2}}$$

, $${\overline{r}}_n=\frac{nUT}{c_{\mathrm{one}}{t}_0}$$

, $${\overline{z}}_q=c_{21}\frac{t_q-t_0}{t_0}$$

$$\Delta t_{qn}^{(2)}=t_0[\frac{\cos {\theta }_{\mathrm{eleven}}-\cos {\theta }_{12}}{\cos {\theta }_{\mathrm{eleven}}\cos {\theta }_{12}}+c_{12}^2\frac{\cos {\theta }_{21}-\cos {\theta }_{22}}{\cos {\theta }_{\mathrm{eleven}}\cos {\theta }_{12}}\frac{\sin \left({\theta }_{12}-{\theta }_{\mathrm{eleven}}\right)}{\cos {\theta }_{\mathrm{eleven}}{\cos }_{22}+\cos {\theta }_{12}\cos {\theta }_{21}}]$$

,
$$\cos {\theta }_{22}=\sqrt{\mathrm{one}-c_{21}^2{\sin }^2{\theta }_{12}}$$

; $$c_{21}=\raisebox{1ex}{$c_2$}\!\left/ \!\raisebox{-1ex}{$c_{\mathrm{one}}$}\right.$$

, $$c_{21}=\raisebox{1ex}{$c_{\mathrm{one}}$}\!\left/ \!\raisebox{-1ex}{$c_2$}\right.$$

,
$$\sin {\theta }_{12}=[\mathrm{one}-\frac{{\overline{z}}_q^2}{c_{12}}\frac{f_{\mathrm{one}}(c_{12},{\overline{r}}_n)}{\mathrm{one}+{\overline{z}}_q^2{f}_2\left(c_{12},{\overline{r}}_n\right)}]\sqrt{\frac{c_{12}^2-{\overline{z}}_q^2}{\mathrm{one}-{\overline{z}}_q^2}}$$

,
$$f_{\mathrm{one}}(c_{12},{\overline{r}}_n)=\frac{c_{12}^2(\mathrm{one}-c_{12}^2)}{2c_{12}{(c_{12}-{\overline{r}}_n\sqrt{\mathrm{one}-c_{12}^2})}^2},$$

$$f_2(c_{12},{\overline{r}}_n)=f_{\mathrm{one}}\left(c_{12},{\overline{r}}_n\right)[\frac{3-7c_{12}^2}{2c_{12}(\mathrm{one}-c_{12}^2)}-2\frac{\sqrt{\mathrm{one}-c_{12}^2}+{\overline{r}}_n{c}_{12}}{c_{12}\sqrt{\mathrm{one}-c_{12}^2-{\overline{r}}_n\left(\mathrm{one}-c_{12}^2\right)}}]$$

,
(U is the predefined speed of the carrier of the profilograph antenna, z _q is the depth of the reflecting layer), 2N + 1 signals shifted in time scale by Δt _qn by the formula
$$〈S_{qp}^{(k)}〉=\frac{\mathrm{one}}{2N+\mathrm{one}}{\displaystyle \sum _{n=0}^N{S}_{qpn}^{(k)}(t_p+\Delta t_{qn})}$$

repeat the time-shift and common-mode summing operations for the entire data array p> N + 1, q≤Q for each element of the receiving antenna, for each moment of arrival time of the received signals t _q and radiation time t _p in-phase sum K signals received by the K - element receiving antenna, shifted in timeline by the value of kT, according to the formula
$$〈S_{qp}〉=\frac{\mathrm{one}}{K}{\displaystyle \sum _{k=\mathrm{one}}^K〈S_{qp}^{(k)}〉(t_{qp}+kT)}$$

moreover, the repetition period of the radiation pulses T, the length of an individual element K - element receiving antenna L and the carrier speed of the profiler antenna U are related by the relation L = TU.

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