JP3783058B2 - Method and system for inverse estimation of wave direction spectrum from radar image - Google Patents

Description

  The present invention relates to the direction of waves from an image representing the intensity of backscattered waves from the sea surface observed by an imaging radar (synthetic aperture radar, actual aperture radar, etc.) mounted on a flying object such as an aircraft or an artificial satellite. The present invention relates to a method and system for inverse estimation of a spectrum.

A Synthetic Aperture Radar (SAR) mounted on an aircraft or an artificial satellite is configured to irradiate a ground surface or sea surface with a microwave and receive reflection and backscattering from the surface.
The irradiation direction of the microwave is usually transverse to the traveling direction of the aircraft or the satellite and obliquely downward. For this reason, it is possible to avoid left and right ambiguity of reflection and backscattering with respect to the traveling direction. Due to this left-right ambiguity, a sensor that irradiates microwaves directly underneath cannot specify whether reflection or backscattering is from left or right.
Also, imaging of the ground surface and the sea surface is not performed by moving the antenna like a general radar and sweeping, but using a beam that is wide in the vertical direction and narrow in the horizontal direction, with the antenna fixed, Sweeps the surface of the earth and the sea surface in a band as the satellite moves.

  The SAR is an active sensor that can observe the ground surface and the sea surface without requiring sunlight. Therefore, observation can be performed regardless of the season or day and night. In addition, since the microwave used in the SAR has a high transmittance in the atmosphere, regular observation can be performed without being affected by clouds. As a result, it is possible to observe the state of the sea surface under a typhoon or a low atmospheric pressure that could not be observed with a conventional visible / near infrared sensor.

Visible and near-infrared sensors have a lower spatial resolution on the ground surface as the observation altitude increases. Usually, this type of sensor has a spatial resolution of 1/4 when the observation altitude is doubled.
Since the spatial resolution of the SAR is determined by the length of the microwave pulse used for observation, the bandwidth of the pulse, the size of the antenna, etc., it is independent of the observation altitude and the wavelength of the microwave used. Therefore, it can be observed with high spatial resolution even from the observation altitude of the artificial satellite. In addition, SAR has extremely high spatial resolution compared to other microwave sensors by pulse compression or synthetic aperture processing.

  The SAR measures the intensity of reflected / backscattered waves from the sea surface by irradiating microwaves obliquely downward. Microwave reflection and scattering are strongly influenced by the roughness and slope of the sea surface. In particular, when the sea surface is observed with the SAR, it is possible to observe a state in which the magnitude of the incident angle of the microwave continuously changes due to the waves existing on the sea surface. By analyzing the change in the incident angle of the microwave, wave information (wave height, wave direction, period, direction spectrum) can be obtained.

Various studies have been conducted on the relationship between the two-dimensional wave number spectrum obtained by analyzing the spectrum of the SAR image and the direction spectrum of the waves, and a formulation for relating the two has been made. For example, Hasellman (1991) formulates in consideration of non-linear effects generated during synthetic aperture processing based on quasi-linear theory.
In order to obtain the wave direction spectrum from the two-dimensional wave number spectrum of the SAR image, these relational expressions must be inversely analyzed. Until now, there are the following methods for analyzing this inverse problem.

  Hasselmann, K.M. and S. Hasselmann, “On the Nonlinear Mapping of an Ocean Wave Spectra into a Synthetic Approach Radar Image Spectrum and It's Inversion”, J. Am. Geophys. Res. , Vol. 96, no. C6, 1991, pp. 10713-10799.   Engen, G.M. , H .; Johnson, H.C. E. Krogstad, S.M. F. Barstow, “Directional wave specular by inversion of ERS-1 synthetic approach radar ocean imagery”, IEEE Trans. On Geoscience and Remote sensing, Vol. 32, no. 2, 1994, pp. 340-352   Takashi Izumimiya, Hiroyuki Iba, “Existence and Characteristics of Inverse Analysis Based on Forward Linear Theory of Synthetic Aperture Radar”, Coastal Engineering Papers, Vol. 47, 2000, pp. 1326-1330

The inverse analysis method of Non-Patent Document 1 is an analysis method that depends on the initial condition given by the wave estimation, and therefore cannot be analyzed when there is no wave estimation given as the initial condition. The analysis method depends on the so-called initial value (analysis method having an initial value problem).
In the inverse analysis methods of Non-Patent Documents 2 and 3, artificial processing is performed to uniquely determine the solution, and since the inverse problem is not completely solved, the direction spectrum of the waves is not necessarily accurate. Cannot be estimated. Further, the accuracy greatly varies depending on the magnitude of noise present in the SAR image.

  Therefore, the present invention reversely estimates the direction spectrum of a wave from a radar image typified by a synthetic aperture radar based on statistics and informatics with high accuracy without depending on the initial value and noise in the image. It is an object to provide a method and system.

In order to solve the above problems, a wave direction spectrum inverse estimation method from a radar image according to the present invention comprises the following arrangement.
That is, in the method of inversely estimating the wave direction spectrum from the image obtained by irradiating the sea surface with radio waves and processing the intensity of the backscattering, the wave direction spectrum is a discrete constant value in the exponent part. This is expressed as an exponential function with a function, which is applied to the relational expression between the wave direction spectrum based on quasi-linear theory and the two-dimensional wave number spectrum of the radar image, and is expanded in the coordinates of the radar azimuth and frequency. The wave direction spectrum is obtained from the intensity image of the backscattered wave , and the wave direction spectrum is expressed by an exponential function having a discrete constant value function in the exponent part. The integral equation obtained as a result of application to the relational expression between the direction spectrum and the two-dimensional wave number spectrum of the radar image is logarithmically limited to the wave direction of the wave in the semicircular direction. The By discretizing, by approximating the integral equations in the nonlinear algebraic equations, to smooth the variation of the two-dimensional wave number spectrum of the image, it can contribute to the improvement of the estimation accuracy of wave directional spectrum.

  The a priori condition of the direction spectrum of the wave with a smooth distribution is incorporated into the prior distribution of the Bayesian model, and the weighting coefficient of the prior condition is determined by minimizing the ABIC (Akaike Bayesian Information Criterion). The wave direction spectrum may be obtained by successive approximation using the maximum likelihood method to correspond to an objective determination of excessive prior conditions and weighting factors for the prior conditions.

Further, the wave direction spectrum inverse estimation system from the radar image of the present invention is a system for inversely estimating the wave direction spectrum from the image obtained by irradiating the sea surface with radio waves and processing the intensity of the backscattering. The wave direction spectrum is represented by an exponential function having a discrete constant value function in the exponent part, and this is expressed by a relational expression between the wave direction spectrum based on the quasi-linear theory and the two-dimensional wave number spectrum of the radar image. Applied with radar azimuth and frequency coordinates, equipped with a calculation unit that calculates the wave direction spectrum from the intensity image of the backscattered wave by introducing a Bayesian model , and the wave direction spectrum is discretely constant in the exponent part It is expressed as an exponential function having a value function, and is obtained as a result of applying it to the relational expression between the wave direction spectrum based on the quasilinear theory and the two-dimensional wave number spectrum of the radar image. Relative integral equation that takes the logarithm of wave of wave direction is limited to a semicircular direction, by discretizing the integral, it characterized that you approximate the integral equation to the nonlinear algebraic equations.

  According to the present invention, a wave direction spectrum can be back-estimated from a radar image with a small amount of calculation and high accuracy. In particular, it is possible to improve the estimation accuracy by minimizing the influence of speckle noise generated in the image in actual observation.

Various studies have been conducted on the relationship between a two-dimensional wave number spectrum obtained by spectral analysis of a radar image such as SAR and the direction spectrum of a wave, and a formulation relating the two has been made.
In the present invention, the wave direction spectrum is inversely estimated from the two-dimensional wave number spectrum using the following formula formulated based on the quasilinear theory proposed by Hasellman (1991).

_{Here, k = (k a, k} r) are waves of wave vector, k (= | k |) is the wave number, _{k a} is the azimuth direction wavenumber, _{k r} is in the range direction wavenumber, R represents a satellite watching The distance from the point, V is the moving speed of the satellite, S _SAR (k) is the two-dimensional wave number spectrum of the SAR image, S _w (k) is the wave direction spectrum, and T _v (k) and T _s (k) are Each represents a modulation function and is formulated as follows.

Here, ω is the wave frequency, k _radar is the wave number of the microwave used in the SAR, θ is the incident angle of the microwave to the sea surface, M _titl (k), M _hydro (k), M _vb (k ) Represent a slope modulation function, a hydraulic modulation function, and a velocity bunching modulation function, respectively.
The slope modulation function M _titl (k), the hydraulic modulation function M _hydro (k), and the velocity bunching modulation function M _vb (k) are formulated as shown below.

する。これにより、方向スペクトルＳ（ｆ，θ）は、指数部に離散的一定値関数を有する指数関数を用いて以下のように定義する。

To do. Accordingly, the direction spectrum S (f, θ) is defined as follows using an exponential function having a discrete constant value function in the exponent part.

Here, I is the number of frequency divisions, J is the number of direction divisions, and δ _{i, j} (f, θ) represents a delta function. The delta function δ _{i, j} (f, θ) satisfies the following relationship.

  Further, the frequency f and the direction angle θ in Expression 7 are discretized by the following expressions, respectively.

If Equation 7 is substituted into Equation 1, Equation 1 becomes an integral equation including unknowns X = (x _1,1 ,..., X _{I, J} ) ^t .
However, since the integral exists in the exponent part in Equation 1, the wave direction spectrum cannot be easily estimated from Equation 1.
Therefore, in the present invention, the logarithm of both sides of Equation 1 is taken and the equation is modified as follows.

By deforming as described above, the fluctuation of the two-dimensional wave number spectrum of the radar image such as SAR obtained by observation becomes smooth, and the estimation accuracy of the direction spectrum to be estimated is improved.
In particular, in actual observation, a noise component called speckle noise, which fluctuates in time and space, is mixed in the image, and the fluctuation of the two-dimensional wave number spectrum of the image is increased. For this reason, the accuracy of the direction spectrum estimated due to this large fluctuation is significantly reduced. In the present invention, by taking the logarithm, the influence of this noise is minimized and the estimation accuracy is improved.

There is a 180 degree ambiguity in the direction of the waves identified from the image.
The second term of the right side of equation 10 to express the ambiguity of 180 ^{_{degrees, {| T S (k)}} | 2 S W (k) + | T S (-k) | 2 S W (-k) }.
In the present invention, in the directional spectrum of estimating, by limiting the waves of wave direction to 0-180 degrees in the second term of equation 10 _| omitted _{2 S W (-k) | T} S (-k) I was able to do that. That is, Equation 10 can be modified as follows.

By discretizing the integral of Equation 11, the integral equation can be approximated by a non-linear algebraic equation.
In the present invention, in order to calculate the discretized integral of Equation 11, the direction spectrum value of the wave to be calculated is calculated by interpolating as follows using the direction spectrum on the lattice points in the vicinity of the four waves. .

きるようにした。
さらに、画像の２次元波数スペクトルの誤差Ｅε_ｋ４を考慮し、最終的には、式１の積分方程式は、次式で表される未知数Ｘ＝（ｘ_１，１，・・・，ｘ_Ｉ，Ｊ）^ｔを含む非線形代数方程式で近似される。

I was able to do it.
Further, considering the error Eε _k 4 of the two-dimensional wave number spectrum of the image, the integral equation of Equation 1 is finally an unknown X = (x _1,1 ,..., X _I expressed by the following equation: _{, J} ) approximated by a nonlinear algebraic equation including ^t .

は、次式で与えられる。Here, F _k (X) is a vector function obtained by discretizing the integral equation for each wave number, Safic

Is given by:

The wave direction spectrum is approximated as a discrete constant value function, but the energy correlation in each section is not considered. From the linearity of the wave, it is considered that the energy distribution in each minute section is independent, but it is difficult to think that the direction spectrum has a discontinuous energy distribution. In general, the direction spectrum is regarded as a smooth continuous function.
Therefore, here, as an expression of an assumption that the direction spectrum S (f, θ) is smooth, an a priori condition is defined as follows using an 8-direction differential operator.

  In addition, for the upper limit (i = I) and lower limit (i = 1) of the frequency to be analyzed, the prior conditions are defined as follows using a differential operator in five directions.

  As described above, using Equation 15 and Equation 16, the prior conditions can be expressed as follows.

However, x _{i, 0} = x _{i, J} , x _{i, −1} = x _{i, J−1} . Expression 13 can be expressed as a matrix by introducing an action matrix D as follows.

Equation 18 represents that the smaller the value, the smoother the estimated value of the direction spectrum.
Therefore, it is desirable that the estimated value of the direction spectrum S (f, θ) has a large likelihood (Formula 14) within a range where Formula 18 is not so large.
When this is formulated, using an appropriate parameter u ² (super parameter, weight coefficient),

X = (x _1,1 ,..., X _{I, J} ) ^t may be obtained.
This is the Bayesian inference method, and the posterior distribution p _POST ()

In this relational expression, the following expression is assumed as a prior distribution of X = (x _1,1 ,..., X _{I, J} ) ^t .

Given u ² , the X that maximizes Equation 21 is determined independently of λ ² ,

を最小化することにより得られる。

Is obtained by minimizing.

That is, this corresponds to the inverse problem of inversely estimating the direction spectrum from the two-dimensional wave number spectrum of the image reduced to the minimum value search problem expressed by Equation 22.
Here, the first term of Expression 22 represents objective error energy, and the second term represents subjective error energy.
The objective error energy represents the degree of coincidence between the observed value and the estimated value, and the smaller this value, the better the observed value and the estimated value match. In contrast, the subjective error energy represents how well the estimated directional spectrum satisfies the a priori condition.

The super parameter u is a constant that gives a weight of subjective error energy. If the value becomes smaller, the estimated direction spectrum is estimated to match the two-dimensional wave number spectrum of the image. The same result as the direction spectrum estimated by the least square method without considering the probability distribution (a priori condition) is obtained. On the other hand, when the value of the super parameter u is increased, the estimated direction spectrum becomes subjective (a priori condition), and the value of the two-dimensional wave number spectrum of the image is not reflected in the estimation.
The hyperparameters u and λ ² are desirable from the viewpoint of both accuracy and smoothness of the solution by minimizing ABIC (Akaike's Bayesian information criterion, Akaike (1980)) expressed by the following equation: Is decided.

てＸ_０のまわりでＦ_ｋ（Ｘ）をＴａｙｌｏｒ展開すると、When estimating the directional spectrum, the minimization of Equation 17 and the integration and minimization of Equation 18 must be performed. However, in this case, it is impossible to do them analytically.
Therefore, the equation 22 is linearized using Taylor's differential correction method, and repeated calculation is performed.

And Taylor expansion of F _k (X) around X ₀

It becomes.
When Expression 13 is displayed in matrix using Expression 24, the following is obtained.

まず、適当なｕ^２及びＸの初期値Ｘ_０を与えて、式２８で示される最小２乗法によ

First, an appropriate initial value X ₀ of u ² and X is given, and the least square method expressed by Equation 28 is used.

Here, the first approximate solution X ⁽¹⁾ is calculated by ^first applying X ₀ and applying the least square method to Equation 28. Next, replace X ₀ in Equation 26 with X ⁽¹⁾ , apply the least squares method to Equation 28, and replace X ⁽²⁾ with

Using the obtained result, ABIC for a given u ² is calculated by the following equation.

Then, repeating the above calculation by changing the u ² variously.

Ｓ（ｆ，θ）が得られる。
なお、本発明では、適用する際の数値計算の利便性を考え、全ての計算ケースで初期値Ｘ_０を０として計算し、数値計算の安定性を確認している。また、式２８の繰り返し計算の収束基準としては、（ｋ＋１）ステップ目における未知数の変化の大きさ‖Ｘ^{（ｋ＋１）}−Ｘ^（ｋ）‖と（ｋ）ステップ目における未知数の大きさ‖Ｘ^（ｋ）‖との比が１０^−２以下に

また、超パラメータｕ^２の設定に際しては、一般には下式を用い、格子探索法による繰り返し計算を行っている。

S (f, θ) is obtained.
In the present invention, in consideration of the convenience of numerical calculation when applied, the initial value X0 is calculated as ₀ in all calculation cases, and the stability of the numerical calculation is confirmed. Further, as the convergence criterion of the iterative calculation of Expression 28, the magnitude of the unknown change at the (k + 1) -th step ‖X ^{(k + 1)} −X ^(k) ‖ and the magnitude of the unknown at the (k) step ‖X ^{( k)} ‖ a ratio of ^{10 -2}

In setting the super parameter u ² , generally, the following equation is used, and the iterative calculation is performed by the lattice search method.

In order to clarify the estimation accuracy and problems of the present invention, verification by numerical simulation was performed.
In this verification, the two-dimensional wave number spectrum of the SAR image was calculated from the model wave direction spectrum, and the wave direction spectrum was inversely estimated by the wave direction spectrum inverse estimation method according to the present invention. The accuracy and problems of the estimation method proposed in the present invention were verified by comparing the estimation result with the direction spectrum of the model wave given in the initial condition of the calculation.
The direction spectrum S _S (f, θ) of the model wave given as the initial condition of the calculation is multiplied by the Bretschneider-Mitsuiyu type frequency spectrum S _f (f) shown below and the light-easy type direction distribution function G _S (θ). The combined ones were used.

をそれぞれ表している。Here, f is the frequency, θ is the wave direction, H _1/3 is the significant wave height, T _1/3 is the significant period, θ ₀ is the main wave direction,

Respectively.

FIG. 1 shows an example of the direction spectrum of a model wave. A direction spectrum in the case of a main wave direction of 0 degree, a significant wave height of 2 [m], a significant period of 6 [s], and a concentration parameter 80 is shown.
FIG. 2 shows a two-dimensional wave number spectrum of the SAR image calculated by substituting the wave direction spectrum shown in FIG. FIG. 2 is obtained by two-dimensional spectral analysis of an image obtained by observing the sea surface with SAR.
FIG. 3 shows the result of inverse estimation of the wave direction spectrum from the two-dimensional wave number spectrum (FIG. 2) of the SAR image.
Comparing FIG. 1 and FIG. 3, it can be seen that the shape of the wave direction spectrum is very similar. Moreover, it can be seen that the significant wave heights and the significant periods of the two are almost the same, and the wave direction spectrum can be accurately estimated from the two-dimensional wave number spectrum of the SAR image.
4 to 7 show the inverse estimation results when the main wave direction is changed (90 degrees, 180 degrees, 225 degrees, 270 degrees). From these figures, it can be seen that the wave direction spectrum can be accurately back-estimated from the two-dimensional wave number spectrum of the SAR image regardless of the wave direction.

  According to the present invention, it is possible to suppress the influence of speckle noise generated in an image in actual observation, and to reversely estimate the wave direction spectrum from the radar image with a practical level of high accuracy. As a result, it is useful in fields such as oceanography, coastal engineering, and fisheries, as well as weather forecasts and marine accident prevention.

  Graph showing an example of the direction spectrum of model waves   Graph of 2D wave number spectrum of SAR image   Graph of the result of inverse estimation of wave direction spectrum from SAR image two-dimensional wave number spectrum   A graph of the true value and its inverse estimation result when the main wave direction is changed (90 degrees)   Graph of true value and its inverse estimation result when main wave direction is changed (180 degrees)   A graph of the true value when the main wave direction is changed (225 degrees) and its inverse estimation result   A graph of the true value and its inverse estimation result when the main wave direction is changed (270 degrees)

Claims (3)

A method of inversely estimating the direction spectrum of a wave from an image obtained by radiating radio waves onto the sea surface with a radar and processing the intensity of the backscattering,
The directional spectrum of waves is represented by an exponential function having a discrete constant value function in the exponent part,
This is applied to the relational expression between the wave direction spectrum based on the quasi-linear theory and the two-dimensional wave number spectrum of the radar image,
A wave direction spectrum inverse estimation method from a radar image, which is developed with radar azimuth and frequency coordinates, introduces a Bayesian model and obtains the wave direction spectrum from the intensity image of the backscattered wave ,
The wave direction spectrum is expressed by an exponential function having a discrete constant value function in the exponent part, and is obtained as a result of applying this to the relational expression between the wave direction spectrum based on the quasilinear theory and the two-dimensional wave number spectrum of the radar image. For the integral equation
By taking the logarithm, limiting the wave direction of the waves to the semicircular direction, and discretizing the integral,
Approximate integral equation to nonlinear algebraic equation. Wave direction spectrum inverse estimation method from radar image. A priori condition of the direction spectrum of waves with a smooth distribution is incorporated into the prior distribution of the Bayesian model,
The weighting factor of the a priori condition is determined by minimizing ABIC (Akaike Bayesian Information Standard),
The Bayesian maximum likelihood, wave directional spectrum inverse estimation method from the radar image according to claim 1, obtained by successive approximation the directional spectrum of waves. A system that reversely estimates the direction spectrum of waves from an image obtained by radiating radio waves onto the sea surface with radar and processing the intensity of the backscattering,
The directional spectrum of waves is represented by an exponential function having a discrete constant value function in the exponent part,
This is applied to the relational expression between the wave direction spectrum based on the quasi-linear theory and the two-dimensional wave number spectrum of the radar image,
And developed with radar azimuth and frequency coordinates, in wave directional spectrum inverse estimation system from radar images, characterized in that it comprises an arithmetic unit for determining the wave direction spectrum from backscatter intensity image by introducing a Bayesian model There,
The wave direction spectrum is expressed by an exponential function having a discrete constant value function in the exponent part, and is obtained as a result of applying this to the relational expression between the wave direction spectrum based on the quasilinear theory and the two-dimensional wave number spectrum of the radar image. For the integral equation
By taking the logarithm, limiting the wave direction of the waves to the semicircular direction, and discretizing the integral,
An inverse wave direction spectrum estimation system from radar images that approximates integral equations to nonlinear algebraic equations.

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