RU2510045C2 - Side-scanning phase sonar - Google Patents

Abstract

FIELD: physics.

SUBSTANCE: side-scanning phase sonar includes a radiator, a receiver, a sensor for measuring angle of roll, a control unit and a recorder, connected to a computer; the side-scanning phase sonar further includes a sensor for measuring angle of pitch, a program unit, a reflected signal spectrum analyser, a monitor; the computer is further connected to outputs of a ship echo sounder, a ship satellite navigation system receiver, the sensor for measuring angle of pitch, the program unit and the reflected signal spectrum analyser.

EFFECT: high accuracy of survey.

3 cl, 2 dwg

Description

The invention relates to the field of hydrography, in particular to technical means for determining the depths of a water area with a side-scan phase sonar, and can be used to take pictures of the relief of the bottom of the water area.

A known method of determining the depths of the water phase side-scan sonar (Shipbuilding abroad. Edition CRI Rumb, 1987, 76-80. [1]), including the radiation of the hydroacoustic signal to the bottom and receiving the reflected signals at two points located at a distance d along vertical measurement delay time t _of reflectance of x signals and the phase shift φ in these points, to determine the direction (α) receiving reflected signals of the formula

$$\alpha =\arcsin \frac{C}{fd}\cdot \frac{\varphi }{2\pi }\left(\mathrm{one}\right)$$

where f is the frequency of the sonar signal;

C is the speed of sound propagation in water.

The device measures the angle β of the side rolling of the antenna carrier and calculates the depths (Z) of the water area according to the formula

$$Z=\frac{C{t}_s}{2}\cdot \sin \gamma \left(2\right)$$

.Where $$\gamma =\alpha +\beta$$

.

However, this method has insufficient accuracy.

This is because when it is used, there is a significant error (m _γ ) in determining the direction (γ) of arrival of the reflected signals from the measured values of their phase shift φ at the receiving points.

The value of m _γ , based on formula (1), can be calculated by the formula

$$m_{\gamma }=\left(\mathrm{one}/{\left(\mathrm{one}-\varphi /2\pi \right)}^2\right)\cdot m_{\varphi }\left(3\right)$$

where m _φ is the measurement error φ.

Since a reliable determination of the direction of γ (without resolving the ambiguity) is possible only in the case of a small separation of d antennas not exceeding two wavelengths (λ), the possible range of the phase shift φ at d = λ will be 0-2π.

The error m _γ , as can be seen from formula (3), varies nonlinearly and for φ = 0 m _γ = -m _φ , and for φ = 2π it reaches a maximum and can be 2m _φ .

This circumstance causes a significant error (m _z ) for determining the depth of the water area z when using the known method.

The value of m _z , based on formula (2), can be calculated by the formula

$$m_z=\left(C{t}_s/2\right)\cdot \cos \phi \cdot m_{\phi },\left(\mathrm{four}\right)$$

Where $$\gamma =\alpha +\beta .$$

For example, when C = 1500 m / s t _z = 0.7 s γ = 30 ° cos φ = 87 m _γ = 2m _φ = 2 · 0.0017 (m _φ for modern phase side-scan sonars does not exceed 1 °). Then m _z will be 13 m or about 4% of the value.

The permissible error in determining the depth of the water area in accordance with regulatory documents for surveying the bottom topography is 1% of the depth Z.

There is also a method for determining the depths of a water area with a side-scan phase sonar (Stubbs A.K.: Telesounding a merhod of wide swathe a depth measurement International Hydrographic Review 1974, 51 Ns 1, pp23-59 [2]), including simultaneous radiation towards the bottom of the main and auxiliary hydroacoustic signals differing in frequency by a small amount, and receiving the reflected main and auxiliary signals at two points located vertically at a given distance d, measuring the time t _{c the} arrival delay of the common-mode main and auxiliary signals using two double-frequency channel receiver-measuring units, measuring the pitch angle of the antenna carrier by the device, determining from the received data the directions of arrival of in-phase main signals and the desired depths of the water by calculation.

Also known is a side-view phase sonar, which contains the first and second two-frequency antennas located vertically at a given distance (d), the transmitting unit, the first and second two-frequency multi-channel receiving and measuring units, a control unit, a calculator and a recorder, while the outputs of the first and second two-frequency antennas are connected respectively to the inputs of the first and second two-frequency multi-channel receiving and measuring units, the output of the transmitting unit is connected to the radiating antenna, the outputs of the first and second of multi-channel dual-frequency priemoizmeritelnyh units connected to the output of the calculator, the latest output is connected to the DVR.

The disadvantage of this method and device (phase sonar side view) is that they are difficult to use.

This is because in order to increase the accuracy of determining the depths of the water area in these technical solutions, reflected hydroacoustic signals are received at two points located vertically at a large distance (d = n λ) of several wavelengths compared to the analogue, and the signal delay time is measured , that is, signals in which the phase shift is absent or a multiple of an integer 2π.

The implementation of this action provides high accuracy in determining the depths of the water area, since with a phase shift of φ = 0, as can be seen from formula (3), the error m _γ will be minimal and equal to m _φ .

However, in this case, there is a need to solve a complex problem - the elimination of ambiguity in determining the directions of arrival of common-mode signals. To solve this problem when using the first known known method, it is necessary, in addition to the main ones, to perform the following additional complex actions:

to receive at the third point, located vertically on the auxiliary base at a given distance from the common point, the reflected sonar signals;

measure in the third and main points the delay time of the arrival of common-mode signals.

To carry out these actions, it is necessary to equip the side-view phase sonar with a receiving antenna and a receiving-measuring unit, that is, significantly complicate the structural diagram and increase the weight and size characteristics of the side-view phase sonar.

When using the known method for solving the same problem, it is necessary to perform, in addition to the main ones, the following additional steps:

- emit an auxiliary hydroacoustic signal at a given frequency that differs from the fundamental frequency;

- receive reflected auxiliary signals (common mode) in the same cases where the main common mode signal is received;

- measure the arrival delay time of auxiliary common-mode signals.

To carry out these actions, it is necessary to use an auxiliary phase sonar side view. That is, to implement this method, it is necessary to use almost two phase side-scan sonars operating at different frequencies, which causes considerable difficulty in using known methods and devices.

There is also a method of determining the depths of the water with a side-scan phase sonar and a side-view phase sonar for its implementation (patent RU No. 1829019A1, 07.23.1993 [3]), in which the technical result consists in simplifying the process of determining the depth of the water area with a side-view phase sonar.

In this case, the technical result is achieved by the fact that in the method for determining the depths of the water with a side-scan phase sonar, which includes emitting the hydroacoustic signal to the bottom and receiving the reflected signals at two points located vertically at a given distance, measuring the delay time of the arrival of common-mode signals, the pitch angle carrier antennas and determining from the data arrival directions θ _n in-phase signal and the desired depth of the waters by calculation, measurement of the time delay _of arrival t the reflected sonar signal vertically, the time delays (t _p ) of the arrival of the common-mode signals are separated if they are reflected from the flat bottom surface in each calculated (θ _p ) direction according to the formula

$$t_{p_n}=\frac{t_p}{\sin \left({\theta }_{p_n}+\beta \right)}+\frac{t_p\cdot \cos \left({\theta }_{p_n}+\beta \right)\cdot \sin \xi }{\sin \left({\theta }_{p_n}+\beta \right)\sin \left({\theta }_{p_n}+\beta +\xi \right)}\left(5\right)$$

where n = 1,2, 3 ... N;

N = d / λ is the number of calculated directions θ _p ;

ξ is the general angle of inclination of the bottom in the sensing strip.

Then, the convergence of the calculated t _n and measured t _p values is determined by the criterion (δ ² ) calculated by the formula

$${\delta }^2={\displaystyle \sum _{m=\mathrm{one}}^{M-\mathrm{one}}{\left[\left(t_{p_n+\mathrm{one}}-t_{p_n}\right)-\left(t_{c_m+\mathrm{one}}-t_{c_m}\right)\right]}^2}\left(6\right)$$

where M is the number of measured values of t _s for each number M of design directions θ _p .

In this case, the initial value of n is successively changed from 1 to K = N-M, and M calculated directions θ _p , characterized by a minimum value of δ, are taken for the sought-after directions of arrival of common-mode signals.

The side-view phase sonar for implementing the known method comprises a first and a second antenna located vertically at a given distance, a transmitting unit, a first and second receiving-measuring units, a control unit, a calculator and a recorder, while the outputs of the first and second antennas are connected respectively to the inputs of the first and the second receiving-measuring devices, the output of the transmitting unit is connected to the receiving-radiating antenna, the output of the first and second receiving-measuring units are connected to the input of the calculator, the output of the last one is connected to the registrar, and the control unit is connected to the first and second receiving and measuring units, the transmitting unit, the calculator and the registrar, and is equipped with a unit for determining the time delays of arrival of common-mode signals reflected from the flat bottom of the bottom in each calculated direction, the input of which is connected to the control unit, and the output is connected to the input of the calculator.

However, despite the achievement of simplifying the known method, it is burdened by significant calculations and laborious computational operations.

In addition, the selection of reflected signals from a flat bottom surface and their mathematical processing are associated with a number of assumptions, which significantly reduces the reliability of obtaining the final measurement results.

For example, studies of the distortion of the spectra of linear frequency-modulated (LFM) signals in marine soil depending on the depth of penetration into sea soil have shown that the first reflection is recorded in the bottom layer at the higher frequencies of the spectrum. As the depth increases (below the bottom surface), the spectrum expands toward lower frequencies, the low-frequency part of the spectrum increases and begins to prevail, and with a further increase in depth, the remaining low-frequency part gradually decreases. The maximum value of the intensity is achieved in the low-frequency part of the spectrum. For thick bottom sediments, features in the nature of the change are sometimes observed. These features consist in a relative increase in a certain depth range of the high-frequency part of the spectrum and are manifested in the appearance of a low-frequency spectrum between high-frequency spectra. Moreover, the predominance of the low-frequency part in this range is less significant for this type of soil. The spectrum forms become equalized and only then does the high-frequency part decrease and the low-frequency part decay. This feature is associated with a layered structure of precipitation and weak absorption in individual layers. This circumstance limits the application of the known method [3], since its implementation requires the presence of a flat bottom surface, free from bottom sediments.

A significant disadvantage of the device for implementing the known method is the fact that when calculating the time delays of arrival of the reflected waves, only the change in the pitching angles is taken into account, while the measurement results will also be burdened by the influence of pitching.

The objective of the proposed technical solution is to expand the functionality of the side-scan phase sonar to determine the depths of the water area, while increasing the reliability of shooting the bottom of the water area.

The problem is solved due to the fact that in the phase sonar side view, which contains the emitter, receiver, sensor for measuring the pitching angles, control unit and recorder connected to the computer, in contrast to the prototype, an additional sensor for measuring pitching angles, a program unit, reflected spectrum analyzer, monitor, calculator is additionally connected to the outputs of a marine echo sounder, ship receiver-indicator of satellite navigation systems, pitching angle measurement sensor, the program unit and the analyzer of the spectrum of the reflected signal and the monitor, the monitor is made in the form of a multifunction display with the ability to generate 3D images with a constant three-dimensional representation of the data, the program unit contains a normalization device, a histogram determination device, a polynomial coefficient calculation unit, an approximation accuracy estimation device, device for setting the degree of the approximating polynomial, decision making device, division unit, integrating device, coefficient allocation unit cents for the first two terms of the decomposition, the unit for computing the quotient.

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

Figure 1. The structural diagram of the side-scan phase sonar includes a calculator 1, a buffer device 2, an external memory 3, a control device 4, an adjustable pre-amplifier 5, a power amplifier 6, an emitter 7, a receiver 8, a bandpass filter 9, a pre-amplifier 10, an ADC 11, a recorder 12 Sensor 13 for measuring pitch angles, Sensor 14 for pitching angles, program unit 15, spectrum analyzer 16 of the reflected signal, monitor 17.

The transmitter 1 through the buffer device 2 is also connected to the outputs of the ship’s echo sounder 18, the ship’s receiver 19 of satellite navigation systems, the pitching angle sensor 14, the program unit 15 and the reflected spectrum analyzer 16.

Figure 2. The block diagram of the software block. The program unit 15 contains a normalization device 20, a histogram determination device 21, a polynomial coefficient calculation unit 22, an approximation accuracy estimation device 23, an approximating polynomial degree setting device 24, a decision making device 25, a division unit 26, an integrating device 27, a coefficient allocation unit 28 for the first two members of the decomposition, block 29 calculating the quotient, microprocessor 30.

Microprocessor 30 is designed based on a DSP processor and includes a memory unit, a timer, an interrupt controller, and communication ports.

Calculator 1 is made on the basis of a computing platform in the form of a system on a chip of the System on Chip (SoC) type and consists of a general-purpose processor, whose functions are to solve navigation equations and interface maintenance, and two processors with vector-matrix coprocessors, which are designed for complete real-time software processing of recorded signals.

Solving navigation problems involves making corrections in the analysis of recorded signals depending on speed, course, coordinates, angles of the onboard and keel qualities, depth under the keel of the vessel when performing bathymetric surveys.

Processors with a vector-matrix coprocessor (for example, type NM6403, NM6404) differ from general-purpose processors in that they have an additional vector-matrix coprocessor with a matrix size of at least 64 × 64, where multiplication with accumulation is implemented in hardware. These processors with a vector-matrix coprocessor are ideally suited for digital filtering of signals - multiplying their samples by weight coefficients and accumulating measurement results, as well as for solving the problem of calculating the correlation of input signals, the cross-correlation function of the signals and their support for the delay.

In addition, on the same chip (as part of a computing platform), a signal preprocessing unit (BPOS) is additionally implemented, which performs the functions of hardware support for software signal processing. BPOS implements digital notch filters of narrow-band interference with a finite impulse response (FIR filters). It also implements buffering and coherent accumulation of digitized samples of signals at time intervals defined by software, pre-sorting schemes for digital signal samples and a quadrator for constructing a frequency panorama using spectral methods based on the fast Fourier transform algorithm. In addition, analog-to-digital converters of the ADC of the input signals, operational memory and program memory blocks, an interface block, and bus of the intra-system information exchange are also placed in a computing platform made in the form of SoC. The satellite receiver 19 can use the signals of the navigation systems and their support systems GPS, GLONASS, GALILEO, SBAS and GBAS. The receiver 19 of satellite signals is functionally connected with sensors 13 and 14 for measuring the angles of pitching and pitching, which makes it possible to implement an inertial measuring system that can determine the angular and linear velocities and accelerations at the installation site of the emitter 7 and receiver 8, as well as the ship's course and necessary corrections.

The emitter 7 emits a pump wave with a frequency f _n . Since the pump frequency is quite high, the pump wave is reflected from the water-soil interface and propagates toward receiver 8. Due to the nonlinearity of the propagation medium, the pump wave will interact with low-frequency signals with a frequency F reflected from abnormal areas, including underwater objects. The result of the interaction will be waves with Raman frequencies f _n ± F, or a change in the phase of the pump wave.

In the calculator 1, commands are generated for sending emitting signals and the parameters of the device as a whole are set. Through the buffer device 2, information from the calculator 1 enters the external memory 3 and the control device 4. The control device 4 controls the operation of the external memory and the adjustable pre-amplifier 5. The power amplifier 6 amplifies the signal and feeds it to the emitter 7.

The distance between the emitter 7 and the receiver 8 is 1/1 _σ = 1. The signals from the receiver 8 are fed to a band-pass filter 9 and a pre-amplifier 10, which form a pre-processing unit. The signals are digitized by ADC 11. Next, the digital signal is fed to the program unit 15, namely, the normalization device 20 and the histogram determination device 21. The device 24 sets the degree of the approximating polynomial and controls the operation of the polynomial coefficient calculator 22. Next, the approximation accuracy is estimated on the approximation accuracy estimation device 23 and the data is transmitted to the decision-making device 25, which is controlled by the control unit 4. Moreover, if the accuracy does not satisfy a given threshold, the degree of polynomial increases. The increase occurs until the approximation accuracy is satisfactory. In block 26, the expression is divided for the probability density of the emitted signal, which is read from the external memory 3 and the obtained probability density in block 22.

The integrating device 27 presents the result to the coefficient extraction block for the first two terms of the expansion 28, and the quotient is calculated in block 29. The result obtained through the buffer element is displayed on the calculator 1.

The signals from the ADC 11 are recorded on the hard drive of the calculator 1. Each report is encoded in 14 bit format. The input of the algorithm receives data, based on which the normalized histograms of the probability density for each signal are determined. Then, using the polynomial approximation, an analytical expression of the probability density for each histogram is determined. Depending on the study (detection of defects in pipelines or search for leakages of the transported product), either the nonlinearity function or the values of moment functions that characterize the change in the shape of the distribution law (in the case of detection of foreign inclusions in the medium) are calculated.

The proposed method is based on the determination of nonlinear properties of the working medium by solving the inverse problem of converting the statistical characteristics of nonlinear waves.

As you know, the direct problem of transforming the distribution law when passing through a nonlinear system has the form:

W (p ₂ ; x, t) = W (p ₁ (p ₂ ); x, t) / | Ψ (p ₁ (p ₂ )) |, where p ₂ = Ψ (p ₁ ) - non-linear deterministic without inertial the transformation given by the deterministic function Ψ (p ₁ ); p1 = Φ (p2) is the branch of the function inverse to p ₂ = Ψ (p ₁ ).

Then the solution for the inverse problem, which consists in finding the expression for the function Ψ (p ₁ ), takes the form of the Stieltjes integral

$$\Psi \left(p_{\mathrm{one}};x,t\right)={\displaystyle \int W\left(p_{\mathrm{one}}\left(t\right)\right)/W}\left(p_2\left(x,t\right)\right)*d\left(p_{\mathrm{one}}\left(t\right)\right).$$

This formula describes the approach to determining the function that connects the reaction pressure of the medium p ₂ with the perturbation pressure p ₁ . It underlies the method for determining the nonlinear properties of the medium, which are described by the nonlinear function Ψ (p ₁ ).

The experimentally obtained changes in the probability density of acoustic pressure depending on the radiation intensity showed that changes in the shape of the distribution law are manifested in symmetry breaking. Due to the increase in radiation power, the absolute value of the probability density decreases.

In this case, the distribution law of low-frequency components at distances between the emitter and receiver 1, 2, and 3 1/1 _σ at the pump frequencies of 140 and 150 kHz practically does not change and its change is caused only by a change in the structure of the medium.

The experimentally obtained waveforms of the signals and the corresponding histograms obtained at different distances from the emitter showed that the main changes associated with the nonlinear properties of the propagation medium occur in the near zone. Therefore, the most significant change in the shape of the distribution law also occurs in the near zone of the emitter to a distance of 1/1 _σ = 1. A further change in the form of the distribution law is associated with the redistribution of energy in the wave and the generation of higher-frequency components. With increasing distance from the emitter, the process normalizes.

When a difference frequency wave propagates, the law of distribution of instantaneous values of acoustic pressure practically does not change, and its change is caused only by a change in the structure of the medium. The technical essence of this method is considered in the source: Kutsenko N.N. Abstract of dissertation for the degree of candidate of technical science. “Study of the laws of acoustic pressure distribution of parametric antenna pump transducers in nonlinear media” - Southern Federal University in Taganrog, 2009. (Website: library sfedu. Ru / referat / D212-208-23 / 01-04-06 / 2009121_D212 -208-01-04-06_KutsekoNN.pdf).

The design of the emitter 7 provides a discrete scan of the internal space, which is carried out by step-by-step review due to irradiation with a narrow directivity characteristic of the emitter of a limited area of space and receiving echo signals within the entire sector in which the survey is carried out. The review cycle is equal to the time interval between two successive emissions: T _review = 2x _max / s, where x _max is the maximum radiation range. Before each signal emission, the directivity characteristic of the emitter 7 is rotated by an angle equal to its width (search step). The total review time of a given sector is determined by the review cycle and the ratio of the sector value to the width of the directivity characteristic.

When detecting an underwater object located on the ground, the calculator 1 generates a command to form a high directivity, which provides a more reliable determination of the location of the identified underwater object.

The transmitter 1 through the buffer device 2 is also connected to the outputs of the ship’s echo sounder 18, ship receiver 19 of satellite navigation systems, pitching angle sensor 14, program unit 15 and the reflected spectrum analyzer 16.

The determination of time delays of arrival of common-mode signals reflected from a flat bottom surface is carried out by means of a marine echo sounder 18, which is a multi-beam echo sounder. Multi-beam echo sounders have incomparably higher performance and provide high detail in comparison with single-beam systems. The marine echo sounder 18 is a multi-beam echo sounder with a complex linear-frequency modulated signal and is designed to measure depths from 20 to 6000 m. The power of the received signals is scanned in range and angle. The nature of the change in power in the beam with a range depends on the shape of the bottom topography. Of the 32 receiving channels, 256 beams are formed, which makes it possible to obtain a quasi-continuous profile of the relief. The receiving antenna of the multi-beam echo sounder 3 of the frequency range 30 kHz consists of 32 elements.

The use of information from a multipath echo sounder allows you to abandon the unit for determining time delays, since such a task is solved by a ship echo sounder 18.

In addition, the current trend, for example, the construction of submarine pipelines at great depths, necessitates the use of high-frequency multipath echo sounders to obtain a detailed picture of the bottom topography. It is important for the pipeline designer that all landforms or artificial objects be identified during bathymetric instrumental surveys. Moreover, when using only one marine echo sounder to perform a bathymetric survey, there is a possibility of missing a dangerous bottom shape (hollow, bulge). During the bathymetric survey in the waters of the seas of the Arctic Ocean, omissions of local hazardous forms were noted when processing the measurement results. Further analysis of bathymetric survey data showed that the probability of missing a local dangerous bottom shape, depending on the speed of the vessel and the depth of the water area, can reach 0.45.

At shallow depths commensurate with towing height, it is possible to use pseudo-sonar images, in all other cases, and especially at great depths, only sonar imagery can provide the designer of subsea pipelines with useful information, which includes:

- identification of various objects on the seabed that may exist along the selected route of the pipeline, including deflections of the seabed, underwater obstacles, bedrock, sand waves, muddy streams and landslide structures;

- underwater cable communications along the selected pipeline route.

Monitor 16 is designed as a multi-function display

with the ability to create 3D images with a constant three-dimensional representation of data and is built on the basis of an Intel Core2Duo processor with RGB input ranges, digital video interfaces DVI-D, 3NTSC / PAL, etc.

Acoustic signals reflected from objects on the seabed are fed to a recorder 12 and are continuously recorded on a thermal printer tape or on magnetic media. These signals are reflections from various objects on the seabed, such as gravel, bedrock, underwater obstacles. The intensity of the reflected signal depends on the object from which this signal is reflected. For example, a signal representing the reflection from the cliff will be represented on the recording darker than the reflected signal from the sand. Studying the intensity of the reflected signals and images on the records allows you to conduct a geological and geophysical interpretation and evaluate the size of various objects at the bottom.

In addition, the linear dependence of the attenuation on the frequency of the degree of asymmetry of the spectrum of the reflected signal can determine the coefficient of the frequency dependence of the attenuation within the signal band.

Software, optionally also enables measurement of time delay of arrival in-phase signal and determining from the data lines θ _n joining inphase signals and the desired water depth by calculation, measurement of time t _of a delay arrival of the reflected sonar signal along the vertical, as is done in prototype, i.e. separate the time delays (t _p ) of the arrival of common-mode signals if they are reflected from a flat bottom surface for each calculated (θ _p ) direction using formula (5) and then determine the convergence of the calculated t _n and measured t _p values according to the criterion (δ) calculated by the formula (6).

The use of linearly-frequency-modulated signals makes it possible to obtain the frequency dependence of the coefficient of backward volume scattering during one pulse, i.e. to analyze the multi-frequency echo signal from the same scatterers simultaneously during carrier movement.

Information sources

1. Shipbuilding abroad. Edition of Central Research Institute Rumb, 1987, pp. 76-80.

2. Stubbs A.K: Telesounding a merhod of wide swathe a depth measurement International Hydrographic Review 1974, 51 Ns 1, p. P. 23-59.

3. Patent RU No. 1829019 A1, 07.23.1993.

Claims (3)

1. Phase side-scan sonar, comprising a transmitter, a receiver, a pitching angle measurement sensor, a control unit and a recorder connected to a computer, characterized in that a pitching angle measurement sensor, a program unit, a reflected signal spectrum analyzer, a monitor, a calculator are additionally introduced additionally connected to the outputs of the ship’s echo sounder, ship receiver of satellite navigation systems, pitching angle sensor, program unit and reflected spectrum analyzer . 2. The phase side-scan sonar according to claim 1, comprising a monitor, characterized in that the monitor is in the form of a multifunction display with the ability to form a 3D image with a constant three-dimensional representation of the data. 3. The phase side-scan sonar according to claim 2, comprising a program unit, characterized in that the program unit comprises a normalization device, a histogram determination device, a polynomial coefficient calculation unit, an approximation accuracy estimation device, a degree approximating polynomial setting device, a decision making device, a unit division, an integrating device, a coefficient extraction unit for the first two terms of the decomposition, a quotient calculation unit.

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