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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

A new electromagnetic approach for the simulation of polarimetric SAR images is proposed. It starts from Maxwell's equations, employs the spectral domain full-wave technique, the moment method, and the stationary phase method to compute the far electromagnetic fields scattered by multilayer structures. A multilayer structure is located at each selected position of a regular rectangular grid of coordinates, which defines the scene area under imaging. The grid is determined taking into account the elementary scatter size and SAR operational parameters, such as spatial resolution, pixel spacing, look angle and platform altitude. A two-dimensional separable “sinc” function to represent the SAR spread point function is also considered. Multifrequency sets of single-look polarimetric SAR images are generated, in L-, C- and X-bands and the images are evaluated using several measurements commonly employed in SAR data analysis. The evaluation shows that the proposed simulation process is working properly, since the obtained results are in accordance with those presented in the literature. Therefore, this new approach becomes suitable for carrying out theoretical and practical studies using polarimetric SAR images.

Retrieval of targets' biophysical and geophysical parameters is one of the main goals of microwave remote sensing, having been the subject of numerous studies. Understanding the electromagnetic reflective properties of targets is a key to the correct interpretation of microwave imaging data. In this sense an image simulator might become a powerful tool for remote sensing researchers, since the use of simulated images may improve considerably the knowledge on several synthetic aperture radar (SAR) applications.

The applicability of synthesized images ranges from theoretical considerations to practical problems. For instance, from SAR simulated images it is possible to develop dedicated algorithms for filtering, segmenting or classifying images. Simulated images can also be used in remote sensing inversion techniques, in the identification of the scattering mechanisms intrinsic to a set of pixels, and in sensor calibration procedures, among others.

There are several ways to synthesize an image; for instance, in [

According to [

The paper is organized as follows. A general theory behind the electromagnetic model is outlined in Section 2. In this section the moment method is also developed and applied to a particular structure. The generation of polarimetric SAR images is detailed in Section 3, where SAR images for a simple multilayer structure are simulated. In Section 4, the simulated polarimetric images are evaluated based on intrinsic properties of amplitude and polarimetric SAR data; results of different classification approaches are also analyzed. Finally, the conclusions are drawn in Section 5.

The electromagnetic model is based on the determination of the electromagnetic fields scattered by a multilayer planar structure that is excited by plane waves. The structure under analysis is composed of _{g}_{g}_{n}_{n}_{n}_{N}

The electromagnetic fields in a multilayer structure are determined through the methodological approach described in [^{jωt}_{n}_{n}_{n}_{n}

Using the following constitutive relations for each of these media
_{n}_{n}_{nϑ}_{x}_{y}_{nϑ}_{x}_{y}_{x}_{y}_{n}

Interesting relations among the amplitudes of the transformed fields are derived by introducing the inverse Fourier transform of _{vς}_{x}_{y}_{v}_{g}

Once the Green's functions are derived, the next step is to set up integral equations constrained to the required boundary conditions. The integral equation is a statement of the boundary condition requiring that the total electric field tangential to the each of the perfectly conducting surfaces is zero [_{v}^{s}_{v}_{v}^{i}_{v}^{r}_{v}^{i}_{v}_{v}_{L}_{L}_{m}_{v}_{x}_{y}_{n}_{v}_{x}_{y}_{n}

The double integrations in

The far electromagnetic fields scattered by the multilayer structure are computed based on asymptotic expressions, which are derived from the stationary phase method [_{0}, _{xe}_{0} sin_{ye}_{0} sin_{0} is the wave number of the excitation wave and

The approach described above is now applied to a structure, consisting of four layers (_{2} (_{2} =_{1} + _{2}) and oriented along the _{0} = _{0}_{2}, _{1} = _{1}_{1}, and _{2} (_{2}−_{1}). Notice that only the amplitudes
_{0}_{z}_{x}_{y}_{0}_{z}_{x}_{y}_{pm}_{pm}_{1}, _{2}, _{3} and _{pm}

_{0}(.) stands for the zero-order Bessel function of the first kind and Δ

From _{pm}_{1} =_{2} = 263.82 mm, _{r}_{1} = _{r}_{2} = 2.33, tan_{1} = tan_{2} = 1.2×10^{-4}, _{rg}_{g}^{-1}. Notice that the layers 1 and 2 have the same electric characteristics, and the variations of the real and the imaginary parts of the Green's function become more numerous and more abrupt as the frequency increases. These graphics illustrate that neither the analytical nor the numerical treatment of this kind of function is an easy task.

The first double integral of _{1} and _{2} paths, is depicted in _{1} represents the parabolic path and _{2} the path along the real axis. The advantage of contour _{0} and 1.1_{0}_{rm}_{rm}_{r}_{1}, _{r}_{2}})^{1/2}.

According to [_{2} of a four-layer structure can be accurately characterized by rooftop subdomain basis functions with twenty expansion modes and taking into account the edge condition. After that, the integrations are computed using the 96-point Gauss-Legendre quadrature rule, by truncating the upper limit of the

To exemplify the process of polarimetric SAR image simulation, the four-layer structure presented in Section 2.3 is considered. The parameters that can be varied in this structure are: the thickness of each confined layer, the dielectric characteristics of each layer (except for free space), and the size, orientation and location of the dipoles. Meanwhile, to define an image region having similar electromagnetic characteristic in its pixels, only dipole orientation is varied whereas the other parameters are constant.

Image generation begins with the definition of a regular rectangular grid of coordinates over the scene to be imaged. The grid size is determined by the dipole size and by SAR operational parameters, such as spatial resolution, pixel spacing and the extension of the area to be imaged. The coordinates of the rectangular grid determine the possible positions that any structure can occupy. Mutual interaction among the dipoles can be avoided by the definition of a guard band around each one. In

Considering that the multilayer structure is excited by (vertically and horizontally) polarized plane waves, the far electric field scattered by the structure is computed based on the electromagnetic model and the stationary phase method. In the spherical coordinate system, this field is given by _{0}d_{m}_{m}_{m}_{m}_{m}_{m} sin_{m}

In order to form an image pixel the radar return is calculated by the vector summation of the individual fields weighted by a separable two-dimensional sinc(.) function _{0} is a proportionality constant and the required spatial resolutions (in the azimuth and range directions) for the SAR image are expressed by _{a}_{r}

Multifrequency sets of single-look polarimetric SAR images have been generated in the L-, C- and X-bands, corresponding to 1.25, 5.3 and 9.6 GHz respectively. The acquisition geometry is particularized for a monostatic sensor flying at an altitude of 6,000 m (airborne platform altitude) and 35° grazing angle imaging a 290 m × 290 m area terrain. For this imaging geometry, the look angle between near- and far-range changes less than 1°, that is a small variation around of the look angle at the center swath width. The 3.0 m spatial resolution and 2.8 m pixel spacing were set in the range and the azimuth directions. In the simulation process the elementary scatterer was represented by a 50 mm × 1 mm electric dipole printed on the interface _{2} (see the structure in

The simulated images are based on a phantom image (an idealized cartoon model), which contains five different regions. The phantom image is depicted in

Differentiation among the image regions is based on the local orientation of the dipoles and the electric characteristics of the layers. The dipole local orientation is relative to the azimuth-axis (_{0} and that the confined layers (layers 1 and 2 of the structure) have the same electric characteristics and thickness (_{i}

The first part of the SAR data analysis carried out is visual inspection (qualitative analysis). It shows that all images present a granular appearance typical of speckle noise. Mainly in the HV and VV channels, the boundary between some regions is unclear, due to the speckle noise effect. The mean backscatter plots, shown in

In the L-band, regions C and D of the HH channel have similar backscatter mean values, off by about 0.28 dB, which makes extremely difficult the separation between these regions. The largest difference between the mean backscatter levels among regions B, C and D of the HV channel is about 1.73 dB, for which the visual distinction between regions is already problematic. For the VV channel, regions A, D and E, as well as regions B and C, present the same separability issue. This fact can be confirmed by visual inspection of the HV and the VV channels, where distinction between regions becomes a hard task, except for region A of the HV channel.

In the C-band, the ability to discriminate the image regions is an issue for regions C and D in the HH channel, for the pairs of regions A-C and D-E in the HV channel, and for regions B and C in the VV channel. The largest difference between the mean backscatter levels for these regions reaches 1.94 dB. For the HH channel in X-band, all regions are visually distinguishable since the boundaries between any two contiguous regions can be clearly identified. This statement is not true however for regions B and C of the HV channel, as well as for regions A and E and for regions B and C of the VV channel, where the largest difference between the mean backscatter levels reaches 2.5 dB. Consequently, it can be stated that, for any variation in the backscatter mean levels that is less than 2.5 dB, the two corresponding regions will not be visually distinguishable.

The purpose of this section is to carry out a quantitative analysis in the simulated images aiming at the validation of the simulation methodology. The analysis will be performed through statistical tests and feature extraction from SAR amplitude and polarimetric data. An application employing the simulated data is also shown using two classification procedures.

Within the SAR image processing community, the multiplicative model is widely used to describe statistically the data [

In order to test the hypothesis that the generated SAR data has a square root of gamma distribution, a ^{2} goodness-of-fit test was performed for all channels using only one sample of each image region. This sample has size of 1200 pixels since it was formed by grouping the training and test sample sets. The test was applied to amplitude data and their resulting p-values are listed in ^{2} goodness-of-fit test for each channel in each band are shown in

Another important quantity, commonly used within the SAR literature, is the equivalent number of looks, which can be estimated from the moments of the square root of gamma distribution. The estimated

Under the linear detection and for single look data the ratio of the standard deviation and the expected value (the _{v}^{1/2} = 0.5227. This value can be obtained from the moments of the Rayleigh distribution, which is a particular case of the square root of gamma when _{0}+_{1}×_{0} and _{1} are the estimated intercept and slope, respectively. In the case under analysis it is expected that the intercept should be zero and the slope should be equal to 0.5227.

The adjusted linear model (_{0}+_{1}×_{0}) are always around zero and, in general, the values of slope (_{1}) are over estimated, but around the 0.5227 value.

A Student's t-test for the intercept being equal to zero and the slope being equal to 0.5227 was performed. The number of samples used in each linear fit and their respective p-values for statistical analysis of the two parameters of the regression are shown in

A major problem in analyzing polarimetric SAR data arises from the complexity of the scattering mechanisms that give rise to features in the different polarization parameters. A lot of work has been done for modeling polarimetric radar backscatter for various types of targets. In [

The

The histograms of

The mean values of

The polarimetric images analysis follows by deriving some polarimetric features from the standard Cloude-Pottier eigenvalue/eigenvector target decomposition [

The eigenvalue/eigenvector decomposition of the coherency matrix into elementary mechanisms (i.e. single, double and volume scattering) is employed in order to identify the global mean scattering mechanism. From the eigenvalue/eigenvector can be defined the ᾱ-angle, which ranges from 0° to 90° and is used to represent physical scattering mechanism. Furthermore, eigenvalues can be combined to form the anisotropy (

From the eigenvalue analysis it was observed that for all image regions their coherency matrix has only one nonzero eigenvalue (coherency matrix with rank 1). It leads to a zero entropy, which complies with a deterministic scattering process (or pure target), characterizing a single scattering matrix equivalent descriptor. It means that the region does not depolarize the incident wave, and in this case the anisotropy is zero also.

A pointwise estimation was employed to form an ᾱ image in all bands and their histograms are illustrated in

Digital classification is one of the most extensively used tools in remote sensing applications. Using this tool the discriminatory capability of polarimetric generated images is quantitatively evaluated. The classification procedure used is based on the Iterated Conditional Modes (ICM) algorithm [

The ICM method is a contextual procedure that, in order to classify every pixel, uses both the observed value in the corresponding co-ordinate and the classification of the surrounding sites. In order to incorporate the context within a statistical framework, a Markovian model is used for the classes. This model is known in the literature as Potts model [_{ξ}_{s}_{s}_{s}

In order to evaluate the discriminatory capability of the five image regions two classification approaches based on the ICM classifier were applied. The first one takes into account the bivariate distribution of the HH and VV intensities channels developed in [

The results of the classification are shown in

The classification results can be considered excellent in both approaches and for all bands, since all regions could be well distinguished from the others with little confusion among them. The results show the sensibility of the HH and VV channels and the attributes

A two-sided statistical z-test was performed to evaluate the equality between all pair of

This paper presented an electromagnetic way to simulated polarimetric SAR images, starting from Maxwell's equations. Images were simulated with five different regions, and their electromagnetic characteristics were used to distinguish them. The generated images were evaluated according to several measurements commonly employed in SAR data analysis. Firstly, the evaluation analysis consisted of statistical tests to amplitude data, showing that the data are adequately fitted by a square root of gamma distribution, which is the characteristic distribution of the multilook amplitude SAR data. Secondly, the equivalent number of looks was estimated, proving that the simulated data have only one look as simulated. It was checked by using a simple regression linear model whether the mean value (

The authors are grateful to FINEP-CAPTAER project for the partial support and Nilson Rabelo, M.Sc. for his fruitful suggestions. The authors would also like to thank the reviewers for their useful comments.

Geometry of the planar structure with

Geometry of a planar structure with four layers (lateral view).

The 3-D Green's function at 1.25 GHz: (a) real part and (b) imaginary part.

The 3-D Green's function at 9.6 GHz: (a) real part and (b) imaginary part.

Parabolic deformed integration path.

Rectangular grid with a few selected dipole positions and a zoomed area.

Phantom image and sample locations.

L-band amplitude simulated polarimetric SAR images: (a) HH, (b) HV and (c) VV channels.

C-band amplitude simulated polarimetric SAR images: (a) HH, (b) HV and (c) VV channels.

X-band amplitude simulated polarimetric SAR images: (a) HH, (b) HV and (c) VV channels.

Mean backscatter values for each image region, per channel: (a) L-, (b) C- and (c) X-bands.

L-band fits: (a) HH - region B, (b) HV - region A and (c) VV - region E.

C-band fits: (a) HH - region A, (b) HV - region E and (c) VV - region D.

X-band fits: (a) HH - region D, (b) HV - region C and (c) VV - region B.

Linear fit to L-band sample data: (a) HH, (b) HV and (c) VV channels.

Linear fit to C-band sample data: (a) HH, (b) HV and (c) VV channels.

Linear fit to X-band sample data: (a) HH, (b) HV and (c) VV channels.

L-band

C-band

X-band

Alpha image histogram: (a) L-, (b) C- and (c) X-bands.

Classified images using bivariate HH-VV distribution: (a) L-, (b) C- and (c) X-bands.

Classified images using the

Region characteristics.

_{r} |
_{r} |
_{r}_{g} |
_{g} |
|||
---|---|---|---|---|---|---|

Red | 2.33 | 1.2×10^{-4} |
5.0 | 2.0×10^{-1} |
10° | |

Magenta | 2.33 | 1.2×10^{-4} |
5.0 | 2.0×10^{-1} |
30° | |

Cyan | 2.33 | 1.2×10^{-4} |
5.0 | 2.0×10^{-1} |
TR | |

Blue | 4.00 | 1.2×10^{-1} |
8.0 | 2.0×10^{+1} |
TR | |

Green | 2.33 | 1.2×10^{-4} |
8.0 | 2.0×10^{+1} |
TR |

P-values for the ^{2} goodness-of-fit.

| |||||||||
---|---|---|---|---|---|---|---|---|---|

22.59 | 73.35 | 35.35 | 67.36 | 48.03 | 48.53 | 84.72 | |||

72.05 | 90.44 | 46.18 2 | 68.9 | 59.72 | 51.89 | 41.01 | |||

31.52 | 87.14 | 79.49 | 38.00 | 24.35 | 36.07 | 46.24 | 78.61 | ||

72.66 | 72.78 | 77.65 | 11.51 | 68.92 | 52.18 | 33.27 | |||

68.24 | 52.09 | 46.94 | 41.74 | 57.41 | 48.68 | 44.35 |

Estimated Equivalent Number of Looks (

L | C | X | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

| ||||||||||

Average ENL | Region | HH | HV | VV | HH | HV | VV | HH | HV | VV |

0.984 | 0.972 | 0.958 | 1.003 | 1.022 | 1.037 | 1.047 | 1.068 | 1.084 | ||

1.118 | 1.117 | 1.113 | 0.985 | 0.986 | 0.988 | 1.083 | 1.087 | 1.093 | ||

0.981 | 1.041 | 1.113 | 1.049 | 1.028 | 0.999 | 0.917 | 1.030 | 1.005 | ||

1.020 | 1.050 | 1.017 | 0.944 | 0.976 | 0.979 | 0.934 | 1.019 | 1.040 | ||

1.072 | 0.998 | 1.011 | 0.978 | 1.038 | 1.008 | 0.985 | 0.994 | 1.034 | ||

| ||||||||||

1.035 | 1.035 | 1.042 | 0.992 | 1.010 | 1.002 | 0.993 | 1.040 | 1.051 | ||

| ||||||||||

1.038 | 1.001 | 1.028 |

Statistic for the regression linear fit.

| ||||
---|---|---|---|---|

_{0} |
_{1} | |||

59 | 53.02 | 16.26 | ||

56 | 36.99 | 39.37 | ||

59 | 8.44 | 10.64 | ||

57 | 90.74 | 44.36 | ||

59 | 74.73 | 49.60 | ||

59 | 16.45 | 20.60 | ||

58 | 80.63 | 0.87 | ||

60 | 42.95 | 29.03 | ||

58 | 94.03 | 59.67 |

19.228 |
99.63 | -9.612 |
99.79 | 6.274 |
97.05 | |

19.211 |
99.67 | -9.290 |
99.99 | 6.294 |
97.27 | |

15.312 |
54.24 | -8.177 |
90.97 | -4.154 |
99.83 | |

-0.078 |
63.92 | 1.596 |
99.64 | -8.012 |
98.94 | |

31.236 |
30.56 | -17.912 |
33.58 | 10.733 |
99.08 |

Confusion matrix (# of pixels) using the bivariate HH-VV distribution (L-Band).

| ||||||
---|---|---|---|---|---|---|

0 | 0 | 2 | 0 | 402 | ||

0 | 0 | 1 | 0 | 401 | ||

0 | 0 | 61 | 14 | 473 | ||

0 | 0 | 2 | 6 | 338 | ||

0 | 0 | 0 | 6 | 386 | ||

| ||||||

400 | 400 | 400 | 400 | 400 |

^{−5}

Confusion matrix (# of pixels) using the bivariate HH-VV distribution (C-Band).

| ||||||
---|---|---|---|---|---|---|

0 | 1 | 0 | 0 | 401 | ||

0 | 1 | 0 | 0 | 396 | ||

0 | 5 | 0 | 0 | 397 | ||

0 | 0 | 6 | 0 | 406 | ||

0 | 0 | 0 | 0 | 400 | ||

| ||||||

400 | 400 | 400 | 400 | 400 |

^{−6}

Confusion matrix (# of pixels) using the bivariate HH-VV distribution (X-Band).

| ||||||
---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 400 | ||

0 | 0 | 0 | 0 | 396 | ||

0 | 4 | 1 | 0 | 403 | ||

0 | 0 | 2 | 0 | 401 | ||

0 | 0 | 0 | 0 | 400 | ||

| ||||||

400 | 400 | 400 | 400 | 400 |

^{−6}

Confusion matrix (# of pixels) using the Gaussian bivariate distribution for

| ||||||
---|---|---|---|---|---|---|

0 | 0 | 3 | 0 | 403 | ||

0 | 1 | 0 | 0 | 401 | ||

0 | 0 | 45 | 0 | 420 | ||

0 | 0 | 17 | 1 | 349 | ||

0 | 0 | 7 | 21 | 427 | ||

| ||||||

400 | 400 | 400 | 400 | 400 |

^{−5}

Confusion matrix (# of pixels) using the Gaussian bivariate distribution for

| ||||||
---|---|---|---|---|---|---|

1 | 0 | 0 | 1 | 402 | ||

0 | 0 | 0 | 0 | 398 | ||

0 | 0 | 0 | 2 | 402 | ||

0 | 0 | 0 | 0 | 400 | ||

0 | 1 | 0 | 0 | 398 | ||

| ||||||

400 | 400 | 400 | 400 | 400 |

^{−6}

Confusion matrix (# of pixels) using the Gaussian bivariate distribution for

| ||||||
---|---|---|---|---|---|---|

1 | 0 | 0 | 0 | 401 | ||

0 | 0 | 0 | 0 | 397 | ||

0 | 0 | 0 | 2 | 401 | ||

0 | 2 | 1 | 0 | 403 | ||

0 | 0 | 0 | 0 | 398 | ||

| ||||||

400 | 400 | 400 | 400 | 400 |

^{−6}