Rough Surface and Volume Scattering of Soil ... - IEEE Xplore

Rough Surface and Volume Scattering of Soil Surfaces, Ocean Surfaces, Snow, and Vegetation Based on Numerical Maxwell Model

of 3D Simulations

Abstract - In this paper, we give an overview and an update on the recent progress of our research group in Numerical Model of Maxwell equations 3D (NMM3D) on random rough surfaces and discrete random media and their applications in active and passive microwave remote sensing. The random rough surface models were applied to soil surfaces and ocean surfaces. The discrete random media models were applied to snow and vegetation. For rough surface scattering, we use the surface integral equations of PMCHWT (Poggio–Miller–Chang–Harrington–Wu–Tsai) that are solved by the MoM (Method of Moments) using the RWG (Rao-Wilton-Glisson) basis functions. The SMCG (Sparse Matrix Canonical Grid) method is used to accelerate the matrix column multiplications. In modelling the rough surfaces, we use the exponential correlation functions for soil surfaces and the DV (Durden-Vesceky) ocean spectrum for ocean surfaces. In scattering by terrestrial snow and snow on sea ice, we use the volume integral equations formulated with the dyadic half space Green’s function. The microstructure of snow is modelled by the bicontinuous media. In scattering by vegetation, we use the discrete

scatterers of cylinder. The NMM3D formulation is based on the Foldy-Lax multiple scattering equations in conjunction with the BOR (body of revolution) for a single scatterer. For rough surface scattering, simulations results are compared with AIEM (Advanced Integral Equation Model), SSA (small slope approximation), SPM (small perturbation method), and TSM (two scale model). For volume scattering by snow, results are compared with the Bic-DMRT (Bicontinuous Dense media radiative transfer). For scattering by vegetation, results are compared with DBA (distorted Born approximation) and RTE (Radiative Transfer Equation). Comparisons are also made with experiments.

INTRODUCTION In this paper, we give an overview, an update and present representative results of physical modeling of microwave scattering and emission using Numerical Maxwell Model of 3D (NMM3D) Simulations. Simulation results are obtained for soil surfaces, ocean surfaces, terrestrial snow and snow over sea ice, and vegetation. In the past, random rough surface

Leung Tsang 1, Tien-Hao Liao2, Shurun Tan 1, Huanting Huang1, Tai Qiao1,

and Kung-Hau Ding3

1Radiation Laboratory, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109, USA

2Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA

3Air Force Research Laboratory, Wright-Patterson AFB, Dayton, OH 45433, USA

scattering and random media volume scattering have been mostly studied with analytic methods [1-3]. In scattering by soil surfaces and ocean surfaces, classical analytic methods include physical optics and small perturbation method (SPM) [1-3]. More recent analytic methods include the Advanced Integral Equation Model (AIEM) [4], small slope approximation (SSA) [5,6], and Two Scale Model (TSM) [7,8]. In scattering by snow, models include the HUT [9], MEMLS [10], DMRT-ML [11], and QCA-MIE-DMRT [12-13]. In scattering by vegetation, the methods of distorted Born approximation (DBA) [14] and radiative transfer equation (RTE) [1-2,15] have been used. With the major advances of computers and computational electromagnetics, we have begun the work of full wave simulations of surface and volume scattering [16], initially for 2D problem and more recently 3D simulations of numerical solutions of Maxwell equations [16,17]. A review article of the work of our group on the subject was written in 2013 [17]. In this review paper, we give an update of the recent progress of our work since the 2013 paper. There are several application areas covered in this review paper, including rough soil surfaces, rough ocean surface, terrestrial snow, snow on sea ice, and scattering by vegetation. There are many papers and methods in each of these areas. Citing all these references, results and methods, will require a huge number of references with many research groups. Thus, the references are mostly from our group, as the purpose of this paper is to give an update of our results on NMM3D. Works of other groups are cited when there are comparisons of results.

Techniques of computational electromagnetics and commercial softwares have been tailored to deterministic problems such as radar cross sections of targets, scattering by periodic structures, discontinuities in waveguides, signal

integrity and EMC in PCBs etc. Problems in microwave remote sensing consist of random rough surfaces and random media that have distinct differences from deterministic problems. Firstly, the media are random because the positions of the scatterers are random in random media. In rough surfaces, the surface profiles are described by stochastic processes. The solutions of Maxwell equations have stochastic fluctuations so that averages of scattering cross sections are to be taken. Histograms, probability distribution of fields and scattering parameters and cross sections are also obtained from the simulations. Secondly, on taking averages, coherent effects arising from Maxwell equations that are crucial in deterministic, can be “washed out” and the wave interactions become incoherent. Monte Carlo simulations of many realizations need to be carried out and convergence of realizations need to be shown. Thirdly, the radar and radiometer pixels can be large, particularly from satellites. The pixels can be more than hundreds of thousands of square wavelengths. However, because far field wave interactions can become incoherent on averaging, it suffices to show, in simulations, the convergence of results with respect to sample size. The sample size needs not to be as large as hundreds of thousands of square wavelengths. On the other hand, the sample size cannot be too small as the statistical sample need to have many peaks and valleys in random rough surface scattering and many scatterers in volume scattering. Fourthly, deterministic problems of radar cross sections, waveguides etc., have well-defined geometry and positions of scatterers. The geometries in nature, such as random soil surfaces, ocean surfaces, ice grains in snow, branches and leaves in vegetation, do not have well-defined geometrical description. For example, considering thousands of branches of vegetation canopy in a statistical sample, every branch will have its own distinct geometry and position. We use stochastic characterizations such as Gaussian random processes with

exponential correlation functions, ocean spectra for rough surfaces, bicontinuous media, Metropolis shuffling methods, Lindenmayer tree growth model etc. However, stochastic characterizations are still approximate representations of nature. Fifthly, the emphasis of results of simulations are on the physical features of the microwave signatures, such as frequency dependence, polarization dependence, angular dependence, coherence effects and the relation of such dependences to the geometric characterizations of the media. Detailed geometries are usually not addressed if their effects do not show up in the physical features of microwave signatures. Sixthly, radar backscattering is a small number and cross polarization in backscattering is even a smaller number. In NMM3D simulations, a high percentage of the scattered power occur in the forward direction or specular direction. Backscattering is a small number in the simulation results compared with forward or specular scattering. The computational electromagnetic techniques need to be accurate enough to calculate such small numbers. For example, in random rough surface scattering, fine scale features exist in soil surfaces and ocean surfaces that have radii of curvature smaller or much smaller than a wavelength. Such fine scale features cause significant radar backscattering and cross polarization in backscattering.

In scattering by soil surfaces [18-21] and ocean surfaces [22], we note that both soil surfaces and ocean surfaces have fine scale features that are smaller than a wavelength and have larger rms slopes. In active remote sensing, the fine scale features, as noted earlier, are important. In passive remote sensing, the fine scale features increase the surface area so that the emissivity and brightness temperature increase. To take into account the fine scale features, we use surface integral equations which have unknowns as the tangential electric fields and the tangential magnetic fields on the rough surface. The surface integral equations are

solved by the method of Moments (MoM) with the RWG (Rao-Wilton-Glisson) basis functions. In formulating the surface integral equations, we use the PMCHWT (Poggio–Miller–Chang–Harrington–Wu–Tsai) formulations, which employ two Green’s functions: the Green’s function with air permittivity and the Green’s function with the dielectric permittivity. In PMCHWT, the assumptions of perfect electric perfect conductor or impedance boundary conditions are not made [23,24]. The MoM matrix equations are solved by iteration with pre-conditioning. The Sparse Matrix Canonical Grid (SMCG) [16-17] is used to accelerate matrix column multiplication. Although the physical problem is 3D waves scattering, yet the rough surface boundary has only 2 degrees of freedom. The SMCG method uses a Taylor expansion of the Green’s function about the flat surface so that a 2D Fast Fourier Transform (FFT) can be applied to the 3D electromagnetic scattering problem. The SMCG with the FFT feature has been implemented in the parallel computations [25].

In the scattering of waves by soil surfaces, we consider the soil to be a lossy dielectric medium. The random rough surface is characterized by Gaussian random process with exponential correlation functions [16,17,26]. The simulation results are compared with backscattering measurements at L, C, and X bands and passive microwave measurements at L band. The NMM3D simulations are in good agreement with measurements. In particular, it is shown that the results of the polarization ratio HH/VV are in better agreement with data than other models [20,21].

In scattering by ocean surfaces [22], the synthetic surfaces are generated by using the DV (Durden-Vesceky) ocean spectrum. A finite surface size is used and the surfaces are discretized. This means that the ocean spectrum is truncated between and where is inversely proportional to the surface size while

is inversely proportional to the discretizing

sampling. The scattering is then computed by solving the surface integral equations. Unlike the two scale composite surface model (TSM), there is no need to insert a dividing line between

and . Backscattering simulation results are compared to Aquarius satellite radar measurements for wind speeds at 5, 8 and 10 m/sec. The results show that NMM3D backscattering results are in good agreement with Aquarius data for VV, HH, HH/VV ratio and HV cross polarization.

In scattering by terrestrial snow and snow over sea ice, we use the bicontinuous media to simulate the microstructures of snow [27]. We first use NMM3D in the Bicontinuous-DMRT model [28-30]. In DMRT (dense media radiative transfer theory), we use NMM3D for a block of snow of several cubic wavelengths to calculate the effective permittivity, the extinction coefficient and the scattering phase matrix. Such computations include the near field coherent wave interactions of densely packed ice grains. Then these computed quantities are used in the radiative transfer equations which are incoherent. Thus, DMRT is a partially coherent model, consisting of a coherent part of solving Maxwell equations and an incoherent part of RT. The partial coherent model is based on the assumption that coherent far field interactions are “washed out” on averaging in random media. We recently [31, 32, 33, 34] also use NMM3D for the entire problem of a layer of snow over a dielectric half space. In such full wave simulations, the bistatic scattering parameters SVV, SHH, SVH, and SHV with amplitude and phase are calculated. The computations are performed using the discrete dipole approximation (DDA) applied to the volume integral equation with half space dyadic Green’s function. It is also shown that in spite of non-translational invariance in the vertical direction because of the reflective boundary, 3D FFT can still be applied. Full wave solutions exhibit coherent effects particularly for thin snow layer. Such effects in the results are beyond DMRT.

Comparisons are made with active and passive microwave measurements.

In vegetation canopy, previous approaches are based on distorted Born approximation (DBA) [14] and the radiative transfer equation (RTE) [1,2,15]. Both the distorted Born approximation and the RTE have the same assumption that the scatterers are statistically homogeneous in 3D. A consequence of that assumption is that there exists a per unit distance effective propagation constant for the layer of vegetation. The imaginary part of two times of the effective propagation constant is the per unit distance attenuation rate which is also called the extinction coefficient. We have begun a new approach using NMM3D [35]. We study the scattering of a vegetation layer consisting of many thin cylindrical scatterers. The full wave approach for solving Maxwell equations is based on the Foldy-Lax multiple scattering equations (FL) [1] combined with MoM for bodies of revolution (BOR) [36]. The transmission through a vegetation canopy of cylindrical scatterers is calculated using Monte Carlo simulation where the cylinders, as many as 500, are generated in each realization and the scattering is solved by FL-BOR with exact solutions of the matrix equation. A merit of FL-BOR is the much smaller number of surface unknowns than using RWG basis functions in the usual MoM codes. The results of transmission at C-band are compared with those of DBA and RTE. Two cases are studied: short cylinders and long cylinders. Long cylinders mean that the cylinder lengths are the same or comparable to the thickness of the vegetation layer. The NMM3D results for the case of long cylinders show that there are significant differences of transmission which is 6 dB than that from DBA/RTE.

In this overview, only representative results are given. Interested readers can find details in dissertations and journal papers in the listed references. The results of NMM3D for soil surfaces were published recently [21]. Thus, the

section III on soil surfaces are kept brief. The results of terrestrial snow, snow on sea ice, ocean surfaces and vegetation have not been published. For these latter 4 areas, more detailed descriptions and results are being prepared for journal publications. The outline of the paper is as follows. In Section II, we describe the equations governing the rough surface scattering problem. In Section III, we show the results of soil surfaces at C and X bands and comparison with experimental data. In Section IV, we show the results of ocean surfaces and comparisons with Aquarius data. In Section V, we describe the bicontinuous media for representing snow and Bicontinuous-DMRT results. In Section VI, we describe the full wave approach of a layer of snow over a dielectric half space. Results are compared with DMRT. In Section VII, we review the meaning of “tau”, the distorted Born approximation, RTE, and analytic multiple scattering theory. Then we describe the FL-BOR approach of vegetation scattering. Results of transmission and “tau” are compared with DBA and RTE.

ROUGH SURFACE SCATTERING FORMULATION

The rough surface scattering formulation in this section has been applied to both soil rough surfaces and ocean rough surfaces. Consider the 3-D scattering problem of a tapered wave, , incident on a 2-D dielectric rough surface with profile . The incident wave in direction is tapered in the wave vector domain to avoid edge diffraction. In the spectral domain, the incident wave is sharply centered at the incidence angle. Given the spectrum, the incident fields in the spatial domain are

ˆ,x y z

inc

i k x k y k zx y x y z

E r

dk dk e E k k e k (1)

The spectrum of the incident wave is in [16].

The Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) surface integral formulations are in [16].

1 2 1 2

inctan tantan

E r L L J r K K M r

(2)

1 2

1 22 21 2

1 1

inc

tan

tan

tan

H r

K K J r

L L M r

(3)

where and are the tangential magnetic field

and the tangential electric field, respectively. The operators and , , in (2) and (3) are defined by

'' , '

' '

m

m

mS

m

L X r

i X rds g r ri X r

(4)

' ' , 'm mS

K X r ds X r g r r (5)

in which is the scalar Green’s function of medium , with medium being the wave incident region and medium the region below rough surface. To solve the

PMCHWT equations, RWG basis functions and Galerkin matching are used to discretize the surface integral equations. In the initial work of 3-D simulations, we used pulse basis functions [25] which worked well for Gaussian correlation functions. However, when we subsequently solve problems of exponential correlation functions, we changed to RWG basis functions to better represent the tangential electric and magnetic fields. The use of RWG basis functions gives more accurate solutions that account for the fine scale features in exponential correlation functions [18,19]. It is to be noted that, for 3-D finite element method, volumetric basis functions are used [37] and for the method of extended boundary condition, complete domain basis functions of sines and cosines are used [38].

To accelerate the MoM solutions of rough surface scattering, we applied the following two methods: SMCG and pre-condition.

SMCG Method

In the Sparse Matrix Canonical Grid (SMCG) method [16-19, 22], the wave interactions are divided into near field and non-near field. The method decomposes the integral equation matrix into a sparse matrix, which represents near-field interactions, and the remainder of the matrix, which represents the non-near-field interaction part. Based on the decomposition of near-field and non-near-field interactions, the impedance matrix can be written as

near non nearZ Z Z (6)

The near field interactions are computed directly. The non-near-field part of the matrix

is rewritten in a Taylor series by expanding the Green’s function about a flat surface.

The Taylor series expansions are performed for the Green’s functions for

(7)

where . The

coefficients above, and , are translationally invariant in the horizontal directions. In equation (R1), M is the number of terms in the Taylor expansion. We have used up to 5th order in Taylor expansions.

Once the Taylor expansion is made on the flat surface, the matrix becomes translational invariant, making it Toeplitz. The memory requirement of SMCG is of instead of the usual requirement of in MoM and the computational complexity of SMCG is , where is the number of unknowns. Because the surface has 2 degrees of freedom, only 2D FFT is needed because of expansion about a flat surface. The 3D FFT is not needed for the 3D vector electromagnetic rough surface scattering.

Because of Taylor expansions, the SMCG method has been limited to small to moderate rms heights compared with wavelength. For the case of larger heights, two methods can be used (i) increasing the number of Taylor expansion, (ii) inserting an intermediate distance

. For this latter case, the impedance matrix is decomposed into three parts

. For , direct multiplication is iused, For

, SMCG is used. For between and , a rank based method of

multilevel UV [39,40] or Chebyshev interpolation have been used.

Preconditioning

The physically-based precondition is implemented to speed up the convergence of the iterative approach to the matrix equations of PMCHWT. The near field interactions play an

2' 21,2

20

, '4 '

iik r r Md

i m Rm R

zeg r r ar r

, ', 'dz f x y f x y1,2

m Ra 1,2m Rb

important role for the accuracy. This also makes it a good candidate for the preconditioner. Within specifying squared-region around each grid point, we generate a sparse matrix which includes all near field interactions. This sparse matrix is then applied to precondition the original impedance matrix. The preconditioner has several features. Firstly, generation of the matrix is memory efficient as in instead of . There are only small amounts of interactions considered for each grid point. Secondly, the preconditioner is independent of polarization so that one time generation can be applied to both vertical and horizontal polarizations. Thirdly, the procedure to calculate impedance matrix elements of the preconditioner is the same as the procedure in the strong interaction region in the original impedance matrix. We also make parallel implementation of preconditioning in NMM3D. In a previous paper, we show that preconditioning reduces more than 20 times of iterations and preconditioning is not sensitive to the roughness [21].

SCATTERING BY ROUGH SOIL SURFACES

Backscatters from Soil Surfaces

In this section, we summarize the results. We show NMM3D results for soil surfaces. Simulation results are compared with POLARSCAT data-1 measurements. L-band comparisons were made in previous study and in our previous review paper [17-19]. Comparisons are made with analytic models including SPM, SSA, and AIEM for C-band and X-band. The rms heights from measurement sites range from 0.32cm to 1.12cm and soil moistures range from 9% to 27%. The surface sizes used for NMM3D are 16 by 16 square wavelength ( with 16 points discretization per wavelength. Monte Carlo simulations were performed using 256 CPU

cores on the NSF Extreme Science and Engineering Discovery Environment (XSEDE). This includes TACC Stampede (2.7GHz Xeon E5-2680) and NICS Darter (2.6GHz Xeon E5-2600). Fifty realizations were simulated for each case. Cross polarization results are found in [21].

Figure 1 shows scatter plots of VV and HH for C-band and X-band. Comparisons are made for incidence angles from 20 to 50 degree. Root mean square error (RMSE), correlation (CORR), and mean bias error (BIAS) are calculated to show the differences among models. Results of NMM3D are in good agreement with measurement data.

We next make comparisons of polarization ratio, HH/VV, as the polarization ratio is strongly dependent on soil moisture. In Figure 2, we show the comparisons with POLARSCAT DATA1. Results of NMM3D show good agreement with measurement data. In the comparisons, results of NMM3D outperform results from other methods. The comparisons include rms error, correlations, and bias. We note that SPM has significant smaller HH/VV ratio than the data [21].

3.2 Improvement of Computational Accuracy and Updating of Results

The results on soil rough surfaces were published recently in Liao’s paper. [21] Computations were performed using 16 by 16 squared wavelength ( ) area at a discretization of 16 points per wavelength. The accuracy of results of some cases can be a concern. Because of the possible lack of accuracy, analytic methods may perform better in some cases. In computational electromagnetics, computational accuracy and computational efficiencies are subjects of continual investigations of our group. NMM3D results will be updated when larger surfaces and more number of points per wavelength are used.

SCATTERING BY OCEAN SURFACES

Ocean roughness spectra have been extensively used in electromagnetic modeling of ocean surface scattering. Common ocean spectra include the Pierson-Moskowitz, Durden-Vesecky (DV), Donelan-Banner-Jahne and Elfouhaily spectra [2]. These wind-dependent ocean wave height models are written in the form of the product of an isotropic part and an azimuthal anisotropic (wave direction) part.

1, , , , ,W k u S k u k uk

(8)

(9)

where is the omnidirectional spectrum with k being the spectral components and being the surface wind speed. is the azimuthal angle with respect to the wind direction. The function is the directional spreading function or angular dependence function which denotes the nonuniform distribution of wave roughness in azimuth. We have used Durden-Vesecky spectrum for the NMM3D simulations.

The microwave frequency is 1.26GHz, which was used for the Aquarius L-band scatterometer. The wavenumber of the microwave is 26 . From the Bragg scattering theory, the spectral component of the ocean spectrum that contributes to the backward scattering is at

02 sinBk k (10)

At the incidence angle ( ) of 40 degrees, the Bragg scattering wavenumber is at 33 .

In computer generated rough surface used in the surface integral equation approach, we need to specify the surface area specified by the surface lengths and in x and y directions

respectively. We choose surface lengths . In the discretization of the surfaces,

sampling intervals of are used. In the rough surface generations, these two numerical parameters will generate surfaces with spectral components in the range of

, where

ukx (11)

2

lkL (12)

In Figure 3, we show a log-log scale of the isotropic part of the Durden-Vesecky spectrum for a wind speed 5 m/s. The incident wavenumber , the Bragg wavenumber,

for surface lengths and samplings used in the NMM3D simulations are also shown.

In Figure 3, the lower limit corresponds to a surface size 64 64 and the upper limit corresponds to the discretization

1 / 3 2x y . The microwave wavenumber and Bragg wavenumber fall between the lower limit and the upper limit spectral components.

In Table I, we tabulate and values that correspond to the surface sizes and the discretization intervals used in the numerical simulations. With the increase of surface length, large scale ocean waves are incrementally included in the simulations.

For L-band NMM3D simulations, we account for roughness due to both short gravity waves (5 m and less) and shorter gravity-capillary waves (down to 2 cm or so). Since the wavelength of gravity waves increases with wind speed, a larger simulation surface is necessary larger wind speed. To account for the influence of fine capillary waves on backscatter, a small discretization of the surface is necessary.

, , 1 , cos 2k u k u

u

By varying surface length and discretization as shown in Table I, we can examine the effects of long gravity waves and short capillary waves on backscattering. In Table II, we showed the VV and HH for wind =8m/s at three incidence angles 29o, 39o, 46o. Results of two scale model (TSM) and Aquarius collected data are also listed. In the simulations, an isotropic DV spectrum is used for NMM3D. On the other hand, the TSM model results are based on anisotropic DV spectrum. The results in the table for TSM and Aquarius data are the results averaged over azimuthal angles. It is seen that the NMM3D results are in good agreement with Aquarius data with the maximum difference at 0.9dB for both VV and HH.

In

Table III, we showed the cross polarization VH results for wind speed 10m/s for upwind crosswind and downwind. The results show that TSM gives a lower VH compared with Aquarius data while the results of NMM3D are comparable with that of Aquarius. Table IV lists the polarization ratio for wind speed 10m/s for 3 incident angles. Each data point is obtained by averaging over all wind directions. The results show that with the increase of incident angle, the VV/HH ratio is also increasing. NMM3D gives better agreement with Aquarius data for VV/HH ratio than TSM.

BICONTINUOUS DMRT MODEL FOR SNOW COVER IN TERRESTRIAL SNOW

AND SNOW OVER SEA ICE Bicontinuous Media Model

The bicontinuous model has been used in the study of simulations of random porous structures [41-43]. It is based on a continuous representation of interfaces between inhomogeneities within the medium. The interfaces are constructed from a large number of stochastic, continuous, sinusoidal waves with random phases and in random directions. The random structure is defined by setting a level on the random process according to the required volume fraction of inhomogeneities. The morphology of bicontinuous medium is described by a random function which is a summation of large number of stochastic standing waves, [27, 41-44]:

1

1 cosN

n nn

S r rN

(13)

in which the vector and the phase are random variables and assumed to be independent. For a large , e.g., , becomes a Gaussian random process, with

and . In computer generation of bicontinuous medium, the phase

is chosen to be uniformly random between and , the unit vector is uniformly

distributed on a unit spherical surface, and the distribution of wavenumber is described by a probability density function , e.g., the Gamma distribution function [44]:

1 111

1

bb bbp e

b (14)

where is the Gamma function, is mean wavenumber, and b is the shape parameter. Next, a leveled-wave scheme of two phases was

introduced. Given the value , for every position , the Heaviside step function is defined as

1 for 0 otherwise

S rS r (15)

in which is the cutting level specifying the interfaces between two types of materials. The value of can be determined from the given fractional volume according to

1 1 erf2vf S r (16)

where denotes statistical average and is the error function. Thus, the entire

space is divided randomly into regions of and . To simulate discrete random

medium, we assign the region as one medium and the region as the second medium. For snow, one medium is ice and the other medium is air.

Because is a Gaussian random process, statistical properties of bicontinuous medium e.g. correlation functions, moments etc., are functions of the Gaussian random process and can be derived. The correlation function of bicontinuous model is given by [27,29].

2

1

mV m S

m

C r f C C r (17)

where and is the order Hermite polynomial. In Equation (25), is the correlation function of

0

sin S

rC r d p

r (18)

The analytical expression of for Gamma distribution is provided in [27]. For a general distribution, can be obtained by

numerical integration. The mean wavenumber is inversely proportional to the mean grain

size, and the parameter is related to the clustering effect. As decreases there will be more effectively large ice grain aggregates in the medium. In Figure 4 we illustrate two example cross-sections of computer-generated bicontinuous structures with different parameterizations.

Comparison of Correlation Function between Sticky Spheres Model, Multi-Size Spheres Model and Bicontinuous Models

In the past, we have used the sticky sphere model [12,13] and the multi-size model of spheres in the QCA-DMRT [12] and Foldy-Lax-DMRT models [45] to study the microwave interactions with dense media. The sticky sphere model is convenient in characterization through a single grain size and a stickiness parameter which represents the tendency of particles to form clusters. The multi-size spheres models on the other hand use size distributions of ice grains, which can, in principle, be measured in ground measurements. A question is to relate the bicontinuous media characterization to the sphere models which facilitates the determination of effective grain sizes from the bicontinuous model. The multi-size spheres model is characterized by pair distribution functions , where the indices and denote the sphere species with different sizes. The sticky sphere model is a special case of multi-size sphere model where only a single size is used but with stickiness. Recently, we have derived the correlation functions from the pair distribution functions of the sphere model [29, 30]. Thus we can compare the correlation function of bicontinuous media and the correlation functions of the sticky sphere and the multi-size spheres. Given a grain size distribution or a single grain size with stickiness, the Percus-Yevick pair distribution functions can be

calculated and the correlation functions are calculated from the pair distribution functions [29, 30].

In Figure 5, the correlation functions of the multi-size spheres, sticky spheres, and bicontinuous medium are compared with exponential correlation function. These functions are plotted in log-scale, so the exponential function is a straight line. The parameters are: 1) for multi-size spheres with inverse Gamma distribution: mean grain diameter and the shape factor

; 2) for sticky spheres: grain diameter , and stickiness ; 3) for the

bicontinuous media: the mean wavenumber , and shape factor . We

fix the fraction volume to be 30%. We can see that all functions have about the same correlation length, as they coincide at distances smaller than 1.0 mm. However, the exponential function decays linearly in log-scale. The other three correlation functions have tails at distances larger than 2.0 mm, which indicates large grain aggregation in the medium. In particular, the parameters of the single-size sticky case and the multi-size case are selected such that they match asymptotically the tail of bicontinuous correlation function. The tail of the correlation function is important because scattering increases significantly with grain size at microwave frequencies for terrestrial snow and snow over sea ice. For the case of sticky particle, the tail corresponds to the measure of stickiness [12,13]. For the case of multiple sizes, the tail corresponds to the larger grain sizes though they may be fewer in number. For the case of bicontinuous medium, the tail corresponds to aggregation. As demonstrated in [29, 30], the sticky sphere model, the multi-size sphere model with inverse Gamma size distributions, and the bicontinuous media with Gamma wavenumber distributions, when parameterized to have small to moderate correlation lengths but with a higher tail in correlation functions, predict non-secular scattering features of more forward scattering

pattern, weaker frequency dependence and grain-size dependence, which agree with controlled experiments [49]. On the other hand, non-sticky particle with small grain size, or random media model such as the strong fluctuation theory characterized by exponential correlation functions with small correlation lengths show close to Rayleigh scattering behaviors in terms of scattering pattern and frequency and size dependences. The bicontinuous media model, with most realistic microstructure representations of snow, shows most variability in its rich scattering features, covers the wide range of data variation, and is one of the best candidates in the development of operational SWE retrieval algorithms [51].

DMRT

In DMRT (dense media radiative transfer theory), NMM3D method is first applied to simulate a block of snow of several cubic wavelengths to obtain the effective permittivity, the extinction coefficient and the scattering phase matrix [27]. The existence of effective properties of effective permittivity, extinction coefficient, and phase matrix is based on the assumption that the probability distribution of the positions of the ice grains in snow are statistical homogeneous in 3D. This calculation based on statistical homogenization has been performed for multi-size spheres and for bicontinuous media. NMM3D simulations consider the near field interactions among ice grains. Then these computed quantities are applied to DMRT.

The DMRT equations are

, ,

, , , ,e

dI zcos I z S z

dz

(19)

, ,

, , , ,e

dI zcos

dzI z W z

(20)

in which is the extinction coefficient, is the specific intensity and

2 /2

0 0

2 /2

0 0

, ; ', ', , '

', ',

, ; ', ''

', ',

PS z d d sin

I z

Pd d sin

I z

(21)

2 /2

0 0

2 /2

0 0

, ; ', ', , '

', ',

, ; ', ''

', ',

PW z d d sin

I z

Pd d sin

I z

(22)

where is the phase matrix. The DMRT equations are the same as the RT equations except that the phase matrix, and extinction coefficients are calculated by NMM3D. Once the DMRT equations are solved [1,13,A4], we have the solution of specific intensity . The bistatic scattering coefficients are then computed.

Results from Bicontinuous DMRT Model

In this section, we show the results from the Bicontinuous media / DMRT model following the approach developed in [52] to solve the DMRT equations iteratively with cyclical corrections to include the backscattering enhancement effects. In the example, we set bicontinuous media parameters:

and ground relative permittivity . The bicontinuous media is illustrated in Figure 4 (a) and is characterized by an exponential correlation length of 0.165mm. At Ku band of 16.7GHz, it has an effective dielectric constant of 1.24, and a scattering coefficient of 0.2134/m. The scattering coefficient is equivalent to that of the sticky spheres with grain diameter of 0.438 mm and stickiness of 0.1. We consider the scattering results at 16.7GHz with incidence angle. Surface roughness is not considered here. The backscatter as a function of snow depth is shown in Figure 6. The backscatter, both co- and cross-pol, increases with the snow depth. The co- and cross-pol backscatter ratio decreases from ~10dB at 5cm snow depth to ~5dB at 1.5m snow depth. The backscatters for both co- and cross-pol increase by ~3dB when the snow depth is doubled. The cross-pol increases faster than co-pol at shallow depths. Also included in Figure 6 are the QCA/ DMRT results. The sticky sphere parameters are as reported earlier producing the same scattering coefficient as the bicontinuous media. The bicontinuous / DMRT shows lower co-pol backscatter as it is characterized by more forward scattering pattern than the sticky sphere model [29]. On the other hand, the bicontinuous/ DMRT shows higher cross-pol backscatter as it is characterized by non-zero cross-pol phase matrix components [27, 28] due to irregularities in the microstructure and multiple near- to intermediate field interactions. The overall effects are that the bicontinuous/ DMRT predicts smaller co-pol to cross-pol backscatter ratio than the QCA/ DMRT by ~3dB. Validations of the bicontinuous media / DMRT model have been performed in [28, 52, 53] using in situ ground observations of snowpack and microwave measurements.

Recently, the bicontinuous DMRT model has been used for the basis of a SWE radar retrieval algorithm [51].

FULL WAVE NMM3D NUMERICAL SIMULATIONS OF SNOW COVER OVER

A DIELECTRIC HALF SPACE

Recently, we have started studying microwave radiative transfer direct through Maxwell equations numerically for the entire snowpack over a half space of dielectric medium [31,32,52]. Note that radiative transfer equations are not used in such an approach. This approach has been applied to the snow cover sea ice for thin snow layers up to 20cm in snow thickness, at Ku band. Snow on sea ice is typically thin of 10~20 cm [42]. It is well known that snow on sea ice is a critical climate problem. Snow cover plays a key role in Arctic sea ice processes, and it has thinned significantly in the past two decades of a changing climate. The full wave approach gives the complete coherent solutions that can be important for thin snow layers. Full wave simulations also directly calculate the scattering parameters of , , and for the complete problem. On the other hand, the Bicontinuous-DMRT approach in the previous section gives the scattering coefficients/cross sections.

We consider the layered medium problem as shown in Figure 7(a). The half-space below the snow represents sea ice in the polar region or the soil below the terrestrial snow. For the snow on sea ice application, we assume that the sea ice thickness is larger than the penetration depth so that the sea ice can be modeled as a dielectric half space. The half-space dielectric causes reflection and such reflections are included in the numerical solution of Maxwell equations by using the half space dyadic Green’s function in the formulation of volume integral equation. Periodic boundary conditions are also used. Consider the 3D problem of a plane wave impinging upon a snow layer with infinite lateral extent emulated by periodic boundary conditions in both and directions. Ewald

summation method is used to compute the 3D dyadic Green’s function with 2D periodicity [47,48]. The 3D volume integral equation is

2

0

1

,

inc

d r

E r E r

rk dx dy dz

G r r E r

(23)

where is the electric field at point , is the incident plane wave with wave vector ,

is the wavenumber, is the relative permittivity at , and is the half-space dyadic Green’s function representing the field response at due to a current source at . Note that the integration limits over the extent of the snow layer are infinite in the horizontal x and y directions and finite in the vertical z direction from the snow/ground interface at to the snow/air interface at . Applying the Bloch wave conditions in the x and y directions for periodic medium gives

exppq i pqE r E r ik (24)

where is the projection of on the horizontal plane, and

1 2pq pa qa (25)

where and are integers, and and are lattice vectors (periods) in the horizontal plane. We choose and . With the periodicity in permittivity, we have

r pq rr r (26)

Then we can convert the integral domain to one period in x and y directions.

0

21

, ;

r

inc

P i

rE r E r k dr

G r r k E r

(27)

where denotes one lattice cell. The lattice cell centered at the origin can be defined as

, , and , where and are the periods along the and directions, respectively. is the periodic dyadic Green’s function related to the half-space dyadic Green’s function as follows,

,, ; , expP i pq i pq

p qG r r k G r r ik

(28)

The half-space dyadic Green’s function , where and represents the free-

space and the reflection response, respectively. We decompose into the free space response

and the reflection contribution ,

0, ; , ; , ;R

P i P i P iG r r k G r r k G r r k (29)

Singularity is only within when , and is thus limited to within

one unit period. We notice that and have different translational symmetry properties.

0 0, ; ;P i P iG r r k G r r k (30)

, ; , ;R R

P i P iG r r k G z z k (31)

The volume integral equation is re-arranged to become

0

0

2 0

2

, ; 1

, ; 1

inc

P i r

RP i r

E r E r

k dr G r r k r E r

k dr G r r k r E r

(32)

We discretize the snow volume into small cubes with cube edge length and volume , where the superscript denotes the -th cub. As in the discrete dipole approximation (DDA), we apply the pulse basis function to represent and define the dipole moment at the i-th cube as

1i r i ip V r E r (33)

Note that the matrix-vector multiplication of 0 , ;P i j i j

j iG r r k p is in the form of

convolutions in three dimensions with the translational invariance relation (30) and a canonical and uniform sampling grid of and

. The multiplication of the free space part is accelerated by 3D FFT which is the usually done [16,50]. We also note that we have a response part in the half space dyadic Green’s function. In the response part, the matrix-vector multiplication of , ;R

P i j i jj

G r r k p has the

form of convolution in the two horizontal dimensions and the form of correlation in the vertical dimensions as indicated in (31). This shows that the matrix-column multiplication product of the response part of the dyadic Green’s function with electric field vector can also be accelerated with 3D FFT, not 2DFFT.

We illustrate the numerical results from 3D full wave simulation. The full wave simulation results are compared with those of bicontinuous-DMRT model. The bicontinuous medium is with parameters of ,

, and . The random media is illustrated in Figure 4 (b) and it has a corresponding exponential correlation length of 0.36mm by fitting the short-range correlation function with an exponential function. On the other hand, the exponential function of the bicontinuous media decays much slowly than the exponential function, yielding an effective larger grain size and non-Rayleigh scattering behaviors. The snow layer has thickness

. The two horizontal directions are truncated at , applying periodic boundary conditions. The microwave frequency is set at Ku band of 17.2 GHz, at which the computational domain is of 8 x8 x5.8 . A plane wave is impinging upon the snow layer at 40 degree. The bottom half space has dielectric constant of 5+0.5i. The ice has dielectric constant of 3.2+i0.001. The numerical sizes of the problems are 150 by 150 cubes in the horizontal plane times 100 cubes in the vertical direction, giving 2.25 million cubes, and 7.75 million scalar electric field unknowns in the volume integral equation. The largest problem we have solved so far includes ~100 million cubes [31, 53]. More discussions are provided in [31, 34].

In Figure 8, we show the results of radar scattering problem. The solid curves are the results of the fully coherent approach of NMM3D with periodic boundary conditions; the dashed curves are the results of DMRT-BiC. The NMM3D results oscillates around the DMRT results demonstrating that the coherent layering effects cause multiple reflections and coherent wave interferences.

In Figure 9, we show the results of passive microwave remote sensing. The markers are the results of the NMM3D; the solid curves are the results of layered media emission theory with the snow layer being approximated by an effective permittivity , only considering coherent wave reflections at interfaces but completely ignoring volumetric

scattering effects. The dashed curves are the results of DMRT-BiC. The NMM3D results oscillates around the DMRT results with much weaker variation than the layered media emission results. The comparison shows that the damped oscillations and decreased brightness temperatures in the NMM3D results are due to scattering. The coherent layer effects are still exhibited in the weak oscillations, and the coherent layer effects are shown to be stronger for horizontal polarizations than vertical polarizations especially at larger observation angles.

It is also interesting to compare the angular oscillation patterns as demonstrated in Figure 8 for the backscatters and in Figure 9 for the brightness temperatures. We note from Figure 9 that for the angles between 10 to 40 degree, when the angular oscillation patterns of the effective media theory are relatively slow and when the effective optical thickness are relatively small, the full wave simulation results follow the pattern of the effective media results closely. These are accompanied by reversed patterns as demonstrated in Figure 8 for the backscatters, agree with the reciprocity relation between active and passive observations. The reciprocity relations are further revealed in the relative values between the two polarizations: the brightness temperature is in general larger in vertical polarization than in horizontal polarization due to the Brewster angle effects, while the vertical polarized backscatter is in general smaller than the horizontal polarized counterpart. The reversed angular response is less obvious as the angle gets larger, partially due to the increased optical thicknesses and the increased oscillation rates in the effective media results that are not resolved by the coarse sampling in the full wave simulations. These oscillations are not artifacts of finite horizontal simulation domains as demonstrated by convergence test with respect to the lateral periods [31, 56].

NMM3D OF VEGETATION Two physical models have been used to study microwave remote sensing of vegetation over the past several decades. They are the radiative transfer equation (RTE) and the distorted Born approximation (DBA). We first review how the transmissions are calculated in RTE and DBE and the respective assumptions. Consider an incident wave of intensity I0 impinging upon a layer of thickness d consisting of discrete scatterers (Figure 10). Note that in a vegetation canopy, the volume fraction of vegetation is no more than 1% with the remainder 99% being air. We first consider the case when the cylinders are short, meaning the cylinders’ lengths are much less than the layer thickness (Figure 10). The radiative transfer equation is

(34)

where is the per unit distance attenuation rate, also known as the extinction coefficient. Note that the attenuation rate is defined as per unit distance (Figure 10). The second term has the phase matrix . If we ignore the phase matrix for this discussion, then the RTE equation becomes

(35)

The solution of the equation for a layer of thickness d is

(36)

The factor accounts for the longer path length at an angle . The well known quantity "tau", from RTE, is

(37)

The transmissivity or transmission t is the fraction of power transmitted. Then using RTE

(38)

Giving

(39)

The per unit distance attenuation rate, , is the most fundamental quantity in RTE. However, it is to be noted that in Maxwell equations which we shall solve numerically for the entire layer of scatterers in this section, the computed solution is the transmission t. Solving Maxwell equations does not calculate . Next, we review distorted Born approximation which also has per unit distance attenuation rate embedded in the per unit distance effective propagation constant. The distorted Born approximation [14] has a mean wave with electric field E.

(40)

where is the incident electric field amplitude and s is the distance travelled by the mean wave. The per unit distance effective propagation constant K is complex.

(41)

where Re and Im represent the real part and imaginary part, respectively.

Since , and , thus

(42)

The DBA has the same form as RTE for attenuation rate by equating

(43)

Next, we show how the per unit distance attenuation rates are calculated in DBA and RTE. In DBA, using Foldy's approximation, the per unit distance effective propagation constant is

(44)

where is the forward scattering amplitude and n is the number of scatterers per unit volume. Then the per unit distance attenuation rate for DBA is

(45)

From optical scattering theorem [1],

(46)

where and are the absorption and scattering cross section of the scatterer, respectively.

Hence

(47)

The expression above is also used by RTE to calculate . Thus, DBA has the same formula as RTE for the per unit distance attenuation rate.

Next, we review the assumptions of DBA. DBA is a first order solution in analytic multiple scattering theory and the paper by Frisch [54] is the fundamental paper. We also note that Gyorffy [55] used similar treatment in his 1970 paper of electronic waves scattering by impurities in condensed matter physics. In Frisch paper [54], the lowest order approximation of the mass operator is (equation (6.40) in [54])

(48)

where is the probability density function of scatterers’ positions, is the scattering matrix and N is the number of scatterers.

Next, it is assumed in Frisch’s paper that

(49)

which is the condition of 3D statistically homogeneous. Then

(50)

And

(51)

In the far field, the scattering matrix is equal to . The above derivation gives the origins of the per unit distance concept. The underlying assumption is 3D statistically homogeneous as represented by . The per unit

distance attenuation rate in DBA and RTE is based on the assumption of 3D statistically homogeneous.

The case of short cylinders is shown in Figure 10. It is clear visually that the medium is 3D statistically homogeneous. Next we consider the case of long cylinders where the cylinder lengths are comparable to the layer thickness (Figure 11). Such cases are common in vegetation canopy such as grass, wheat, corn, the lower story of forests, etc. For the case in figure 11, the positions of the discrete scatterers are not 3D statistically homogenous. It may be treated as statistically homogeneous in the two horizontal directions, but not statistically homogenous in the vertical direction. Thus, the assumption of 3D statistically homogeneous in analytic multiple scattering theory of Frisch [54] and RTE is not obeyed by the case of long cylinders. In the following we shall use NMM3D simulations with the method of FL-BOR to calculate transmission for the two cases of figures 10 and 11. Note that the Foldy-Lax approach and ICA were combined previously for scattering by vertical cylinders with cylindrical wave expansions [56], and the model was used for rice canopy [57]. The advantages of using FL-BOR in the following are that the cylinders can have orientations and the scatterers can also be circular discs representing leaves.

The formulation of FL-BOR is described as follows. Consider an incident plane wave impinging upon N scatterers. The Foldy-Lax equations are derived from Maxwell equations and are exact relations without approximations [1]. The Foldy-Lax equations state that the final exciting field of scatterer m is the sum of incident wave and the scattered waves

from all other scatterers n except m itself [1].

1

Nm m nex inc scat

nn m

E E E (52)

The scattered field from scatterer n is calculated from the scattering T matrix of the scatterer n as [1, 58]

0n n

nscat exE G T E (53)

Note that equation (53) is of a single scatterer and the T matrix is that of an isolated scatterer. The Foldy-Lax multiple scattering relations are also illustrated in Figure Figure 12.

The T matrix completely characterizes the scattering of a single scatterer. In the past, the T matrix was usually put in the form of the spherical wave expansions. In general, other types of expansions can be used. For scattering by cylinders in vegetation, the T matrix part is obtained by another type of expansion which is the Body of Revolution MoM solution.

First, we express the scattered field from scatterer n in equation (52) using the Huygen’s principle [1]. We have

'

0 0

'1 0n

Nm mex inc

n dSn m

G i J rE E dS

G M r (54)

where N is the total number of scatterers modelling the vegetation particles, is the dyadic Green’s function, and and are the surface electric and magnetic currents, respectively.

Similarly, for magnetic field, we have

'0 0

'1 0n

Nm mex inc

n dSn m

G i M rH H dS

G J r (55)

Equations (54) and (55) are the Foldy-Lax multiple scattering equations based on Huygen’s principle. The next step is to relate

the surface currents to the exciting fields, using the BOR formulation [35,36].

1 1

E

exH

ex

EJ VZ Z W

HM V (56)

where 1

Z is the inverse of the impedance matrix for BOR. Note that the above equation is that of a single isolated scatterer as equation (53). Combining equation (54)-(56), the surface currents J and M are solved. From the surface currents, the scattered fields from all the cylinders can be calculated using the Huygen’s principle [1].

To illustrate the results using FL-BOR approach, we calculate the transmission through a vegetation canopy using Monte Carlo simulation. For each realization of vegetation layer, a large number of cylinders are generated which are either uniformly randomly distributed or located in randomly positioned clusters. The observation area, labeled as “NMM3D receiver” of , is put at 1cm below the center of the bottom boundary of the vegetation canopy as indicated in Figure 11. The transmission in each realization is defined as

(57)

where is the transmitted Poynting vector at the observation point.

and are the total fields which are the sum of the incident fields and the scattered fields from all the cylinders computed using FL-BOR as explained above.

The quantity is the power received by the NMM3D receiver put below the vegetation layer. To find the transmission through the whole vegetation layer, the NMM3D receiver should be moved around the vegetation area. An alternative way is keeping the NMM3D receiver at the same place and shuffling the

cylinders in the Monte Carlo simulations. Thus the transmission for the vegetation layer is obtained by averaging over all realizations,

(58)

We have used 100 realizations.

In RTE and DBA, the per unit distance attenuation rate is calculated using equation (47) where is the number of scatterers per m3, and are the absorption and scattering cross sections of a single cylinder, respectively. In Maxwell equations as well as the NMM3D simulations described above, we calculate transmission, not .

NMM3D full wave simulations are applied to the case where the short and thin cylinders whose lengths are much shorter than the canopy height are uniformly distributed in 3D in the vegetation layer with 720 cylinders in total. All the cylinders are vertically oriented. The average spacing between the cylinders is about 10cm (~1.8 ) The simulation area is . The radius and length of the cylinders are 1mm and 3cm, respectively. The total vegetation canopy height is 60cm. The incident wave is V-polarized at . The computed transmission through vegetation layer using NMM3D is 0.9097 while it is 0.8984 from the DBA/RTE as calculated using equation (47). The results from NMM3D simulations and DBA/RTE agree with each other for the case of short cylinders.

NMM3D simulations are next applied to the case as shown in Figure 11 where the lengths of the cylinders are equal to the thickness of the vegetation layer. The incident angle, polarization and the cylinders’ orientations are the same as those for the case of short cylinders. The density of the cylinders is 2122 per m2, corresponding to vegetation water content (VWC) 1kg/m2 which is a typical value for the grass field. The average spacing between cylinders is less than /2. Two distributions are simulated: one is uniformly distributed and the

other is distributed in clusters with five cylinders per cluster, as shown in Figure (a) and (b), respectively. 100 and 500 cylinders are used in simulation. The results for after averaging over all the realizations are presented in Table V. We notice that the case of 500 cylinders occupies a larger area than the case of 100 cylinders. Comparing the NMM3D results for the 100 and 500 cylinders cases, a sign of convergence in terms of simulation area (or total number of cylinders) is observed. The results from DBA/RTE are also listed for comparison. It can be seen that from NMM3D simulation for the 500 cylinders case is about 4.09 times (6.12 dB) larger than that from DBA/RTE. Thus, the DBA/RTE overestimates the attenuation though vegetation canopy for the long cylinder case.

Since in NMM3D, we calculate transmission , where is the incident angle. We

define “tau” using equation (39) that

(59)

for NMM3D. In DBA/RTE, we use equation (37) for ( ).

It is found that NMM3D for the clustered case has a value of tau that is 5 times smaller than DBA/RTE for the case of long cylinders. The comparisons of vegetation modelling using RTE/DBA and NMM3D are also presented in Table VI.

The recent work on NMM3D simulations has been motivated by the NASA SMAP (Soil Moisture Active Passive) mission in which distorted Born approximation and RTE are the basis of the baseline algorithm for both active and passive microwave remote sensing. It has also been motivated by using the Sentinel C-band active data in combination with SMAP passive data [59-61]. For radar scattering, the scattering from vegetation fields consists of volume scattering, double bounce scattering and surface scattering. The surface scattering

and the double bounce scattering are directly related to the soil permittivity and are important for the radar remote sensing of the soil moisture. The surface scattering is calculated by the bare surface scattering, as explained in section II, multiplied by the attenuation through the vegetation layer. Thus, the difference in the attenuation between DBA/RTE and NMM3D will result in larger surface scattering and larger double bounce scattering in NMM3D. For passive remote sensing, a stronger transmission means a stronger microwave emission radiometric signals from the soils below the vegetation. Thus, the much higher transmission will impact both active and passive microwave remote sensing of soil moisture. The paper report on the initial results of using NMM3D for vegetation. The past approaches have been limited to DBA and RTE. The initial results show significantly larger transmission than that of DBA/RTE for extended cylinders. The NMM3D simulation results in this paper are computed using a single PC with Intel Core i7-4790 CPU and 32 GB RAM. We are presently writing codes for parallel computing to facilitate studies of a larger number of cylinders over a larger surface area. With the implementation of parallel computing, many cases will be studied and look up tables (LUT) will be prepared.

CONCLUSIONS

In this paper, we present an update on the NMM3D simulations for both rough surface scattering and volume scattering with applications in microwave remote sensing. Simulations were conducted for soil surfaces, ocean surfaces, snow cover of terrestrial snow and snow over sea ice and vegetation. Simulation results are compared with analytical models of rough surface and volume scattering and with DMRT. The differences in results are discussed. Comparisons have been made with measurements. It is to be noted that microwave

radiative transfer are governed by Maxwell equations. RTE are only approximations of microwave radiative transfer. We also discussed effective properties such as attenuation rates per unit distances, effective phase matrices and the assumptions of statistically homogenous in 3 dimensions. With the continual advancement of computers and methods of computational electromagnetics, NMM3D results with larger sample size and improved accuracies should be forthcoming.

ACKNOWLEDGEMENT

This work used the Extreme Science and Engineering Discovery Environment (XSEDE) under the proposal, Large Scale Simulations of Multiple Scattering of Waves in Random Media and Rough Surface (TG-EAR100002), which is supported by National Science Foundation grant number ACI-1053575. The research in this paper was supported by NASA SMAP, Terrestrial Hydrology, Remote Sensing Theory, AIST, IIP and NSF Polar Science.

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Table I

UPPER AND LOWER LIMIT SPECTRUM COMPONENTS FOR VARYING SAMPLING POINTS DENSITIES AND SURFACE SIZES

Surface Length ( )

16 1/16 1.65 211 32 1/16 0.82 211 16 1/32 1.65 422 64 1/16 0.41 211

Table II

VV AND HH FOR WIND 8M/S AT INCIDENCE ANGLE 29O, 39O, 46O OF NMM3D TSM AND AQUARIUS DATA

Method VV(dB) HH(dB) TSM -10.05 -11.91

Aquarius -10.30 -12.30 NMM3D -10.70 -13.10

Method VV(dB) HH(dB) TSM -15.18 -20.05 Aquarius -14.9 -18.7 NMM3D -15.06 -19.0

Method VV(dB) HH(dB) TSM -17.24 -24.47

Aquarius -17.5 -23.3 NMM3D -17.34 -22.44

Table III

VH OF TSM, AQUARIUS, NMM3D FOR WIND SPEED 10 M/S FOR UPWIND, CROSSWIND AND DOWNWIND

Wind direction TSM Aquarius NMM3D

Upwind -35.11 -33.0 -33.76

Crosswind -36.15 -35.20 -35.27

Downwind -35.11 -33.80 -33.9

Table IV

COMPARISON OF NMM3D AND TSM OF VV/HH RATIO WITH AQUARIUS AT 29°, 39°, AND 46° AT WIND 10 M/S

Incidence angle TSM Aquarius NMM3D

29° 1.60 1.80 2.59

39° 4.67 3.70 4.04

46° 7.07 5.84 6.34

Table VTRANSMISSION THROUGH VEGETATION CANOPY

Cylinder No. L (m) (Area=L2) RTE/DBA NMM3D Uniform NMM3D Clustered 100 0.217 0.1722 0.6874 0.7527 500 0.485 0.1722 0.6124 0.7044

Table VIICOMPARISON FOR VEGETATION MODELLING AT MICROWAVE FREQUENCIES

RTE/DBA NMM3D Attenuation rate (attenuation

per unit distance) No attenuation rate

Transmission t t calculated by solving Maxwell equations

Extended cylinders Calculate and , and put in No attenuation rate

Heterogeneous/gaps/clusters Same as homogenous Not the same as homogeneous

Results of Can be very difference from NMM3D

Can be very different from RTE/DBA

(a) (b)

(c) (d)

Figure 1 Copolarized backscatter comparison of soil surface, VV and HH, with POLARSCAT DATA1 for incidence between and (a) VV at C-band (4.75GHz) (b)

HH at C-band (4.75GHz) (c) VV at X-band (9.5GHz), and (d) HH at X-band (9.5GHz)

Figure 3 Isotropic part of the Durden-Vesecky surface elevation spectrum (solid black line) for a wind speed 5 m/s. Magenta and blue lines denote lower and upper limit, respectively. Lower

limit corresponds to . Upper limit corresponds to 1 / 32x y . Red and Green lines denote microwave wave number and Bragg wavenumber , respectively.

(a) (b)

Figure 2 Polarization ratio comparison of soil surface, HH/VV, with POLARSCAT DATA1 for incidence between and (a) HH/VV at

C-band (4.75GHz) and (b) HH/VV at X-band (9.5GHz)

Figure 4 Cross section images of simulated bicontinuous structures of inhomogeneities with Gamma distributions of wavenumber. Each side of slices is equal to . The parameters are

(a, left) , , and volume fraction ; (b, right) , , and .

Figure 5 Correlation functions of multi-size spheres, sticky spheres, and bicontinuous models

Figure 6 Backscatter at Ku band as a function of snow depth

0.05 0.5 1 1.5 2 2.5

Snow depth (m)

-30

-25

-20

-15

-10

-5

backscattering coefficient | | Incident Angle = 40ø

DMRT-Bic(VV)DMRT-Bic(VH)DMRT-QCA(VV)DMRT-QCA(VH)

x

z

y

(a) active problem A (b) passive problem B

Figure 7 Illustration of the full wave simulation of a plane wave impinging upon a snowpack. Periodic boundary conditions are applied in the two horizontal directions. Half-space periodic Green’s function is used to take account for the influence from the bottom half-space. Field solution and bistatic scattering matrix with both magnitude and phase are obtained. (a) The

active scattering problem is solved directly. (b) the passive emission problem is derived from reciprocity using the field solution of the active scattering problem.

Figure 8 Backscatter as a function of incidence angle compared with DMRT results.

10 20 30 40 50 60

Incident Angle (°)

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

back

scat

terin

g (

dB)

Snow depth = 10cm, co-pol

Snow DDA PBC-VVSnow DDA PBC-HHSnow DMRT Bic-VVSnow DMRT Bic-HH

Figure 9 Brightness temperature as a function of observation angle compared to the results of DMRT and layered media emission

0 10 20 30 40 50 60 70

observation angle (°)

140

160

180

200

220

240

260

280

Brig

htne

ss te

mpe

ratu

re (

K)

Brightness temperature Tb, snow depth = 10cm

Snow DDA PBC-VSnow DDA PBC-HEffective media-VEffective media-HSnow DMRT Bic-VSnow DMRT Bic-H

Figure 10 Short cylinders statistically homogeneous in 3D. DBA/RTE use the attenuation per

unit distance .

Figure 11 Long cylinders with length comparable to layer thickness. DBA/RTE use the

attenuation per unit distance .

Figure 12 Illustration of Foldy-Lax multiple scattering relations

(a) (b)

Figure 13 Positions of the cylinders on the x-y plane in one realization for (a) uniformly distributed case, (b) clustered case. The scale of both axes is meter.