On the Nonlinear Mapping of an Ocean Wave Spectrum into a New Polarimetric SAR Image Spectrum

Yanmin Zhang; Yunhua Wang; Qiaohui Xu

doi:10.1175/JPO-D-20-0045.1

Jump to Content Jump to Main Navigation

Journal of Physical Oceanography

Abstract
1. Introduction
2. Method
3. The tilt MTFs
4. The retrieval algorithm
5. The retrieved results and discussions
6. Conclusions
The Detailed Derivation of Eq. (9)

ProCite

RefWorks

Reference Manager

BibTeX

Zotero

EndNote

View raw image

Fig. 1.

The values of (a) $P_{k hhvv}^{S_{1}}$ , (b) $P_{k hhvv}^{S_{2}}$ , and (c) $P_{k hhvv}^{S_{3}}$ . Here, their values have been normalized with F_k.

View raw image

Fig. 2.

A comparison between the empirical and theoretical tilt MTFs.

View raw image

Fig. 3.

The empirical tilt MTFs for different wind speeds for (left) hh and (right) vv polarization.

View raw image

Fig. 4.

A comparison of the empirical $| T_{k hhvv}^{t_e} (u, θ_{i}, ϕ_{i}) |$ and $| T_{k hhvv}^{t} |$ for different wind speeds and wind directions.

View raw image

Fig. 5.

(a) Vertically (vv) polarized image of the SAR ID 1 data in Table 1; (b) attitude information of the SAR image.

View raw image

Fig. 6.

A comparison between (a) the first-guess spectrum and (b) the retrieved spectrum from the SAR ID 1 data.

View raw image

Fig. 7.

The corresponding 512 × 512 pixel size images selected from SAR ID 1 image: (a) hh-polarized NRCS, (b) vv-polarized NRCS, and (c) the new polarimetric image.

View raw image

Fig. 8.

The retrieved spectra from the SAR ID 1 image: (a) the retrieved spectrum by using the new nonlinear mapping and the empirical MTF [i.e., Eq. (27)], (b) the retrieved spectrum by using the new nonlinear mapping and the theoretical MTF [i.e., Eq. (24)], (c) the retrieved spectrum by using the vv-polarized image and the MPI algorithm (Hasselmann and Hasselmann 1991), (d) the 2D wave spectrum obtained by the ECMWF ECWAM model.

View raw image

Fig. 9.

As in Fig. 5, but for the SAR ID 2 image.

View raw image

Fig. 10.

As in Fig. 7, but for the SAR ID 2 image.

View raw image

Fig. 11.

As in Fig. 8, but for the SAR ID 2 image.

View raw image

Fig. 12.

As in Fig. 5, but for the SAR ID 3 image.

View raw image

Fig. 13.

As in Fig. 7, but for the SAR ID 3 image.

View raw image

Fig. 14.

As in Fig. 8, but for the SAR ID 3 image.

	All Time	Past Year	Past 30 Days
Abstract Views	230	0	0
Full Text Views	427	198	21
PDF Downloads	438	166	23

Numerical Simulations of the 2013 Alberta Flood: Dynamics, Thermodynamics, and the Role of Orography

Authors:

Shawn M. Milrad

Kelly Lombardo

Eyad H. Atallah

, and

John R. Gyakum

Assessment of Vertical Mesh Refinement in Concurrently Nested Large-Eddy Simulations Using the Weather Research and Forecasting Model

Authors:

Jeffrey D. Mirocha

and

Katherine A. Lundquist

Diagnosing the Horizontal Propagation of Rossby Wave Packets along the Midlatitude Waveguide

Authors:

Gabriel Wolf

and

Volkmar Wirth

The Structure, Evolution, and Dynamics of a Nocturnal Convective System Simulated Using the WRF-ARW Model

Authors:

Benjamin T. Blake

David B. Parsons

Kevin R. Haghi

, and

Stephen G. Castleberry

Toward 1-km Ensemble Forecasts over Large Domains

Authors:

Craig S. Schwartz

Glen S. Romine

Kathryn R. Fossell

Ryan A. Sobash

, and

Morris L. Weisman

Editorial Type:: Article

Article Type:: Research Article

On the Nonlinear Mapping of an Ocean Wave Spectrum into a New Polarimetric SAR Image Spectrum

Yanmin Zhang

Yanmin Zhang College of Information Science and Engineering, Ocean University of China, Qingdao, China

Search for other papers by Yanmin Zhang in
Current site
Google Scholar
PubMed

Yunhua Wang

Yunhua Wang College of Information Science and Engineering, Ocean University of China, Qingdao, China
Laboratory for Regional Oceanography and Numerical Modeling, Qingdao National Laboratory for Marine Science and Technology, Qingdao, China

Search for other papers by Yunhua Wang in
Current site
Google Scholar
PubMed

, and

Qiaohui Xu

Qiaohui Xu College of Information Science and Engineering, Ocean University of China, Qingdao, China

Search for other papers by Qiaohui Xu in
Current site
Google Scholar
PubMed

Online Publication:: 17 Oct 2020

Print Publication:: 01 Nov 2020

DOI:: https://doi.org/10.1175/JPO-D-20-0045.1

Page(s):: 3109–3122

Received:: 29 Feb 2020

Accepted:: 22 Jul 2020

Published Online:: 17 Oct 2020

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

Download PDF

Free access

Abstract

A new nonlinear transformation relation is derived to describe the mapping of a two-dimensional ocean wave spectrum into a new polarimetric synthetic aperture radar (SAR) image spectrum. It is a further expansion and improvement of Hasselmann’s work. First, the nonlinear mapping relation proposed is derived on the basis of a new polarimetric SAR image instead of the conventional single-polarization SAR image. Second, the nonlinear mapping relation no longer includes the complex hydrodynamic modulation transfer function (MTF). Third, the traditional tilt MTF, which is not accurate enough for the retrieval of sea wave spectrum, is replaced by an empirical tilt MTF derived on the basis of the C-band geophysical model function [i.e., C-band synthetic aperture radar (CSAR) normalized radar cross section (NRCS) model]. A sea wave spectrum retrieval algorithm is then proposed that is based on the new nonlinear mapping and the empirical tilt MTF. The retrieved spectra from C-band polarized RADARSAT-2 SAR images are compared with the results obtained by the ECMWF Ocean Wave Model (ECWAM) and buoy measurements.

Corresponding author: Yunhua Wang, yunhuawang@ouc.edu.cn

Abstract

Corresponding author: Yunhua Wang, yunhuawang@ouc.edu.cn

Keywords: Microwave observations; Remote sensing; Satellite observations

1. Introduction

Because of its high spatial resolution, the synthetic aperture radar (SAR) has been widely used for sea wave monitoring since the launch of Seasat satellite in 1978. Up to now, several methods have been proposed to retrieve sea wave spectrum, wave slope and significant wave height by SAR images. The work of Gonzalez and Beal (1979) showed that sea wave structures on Seasat image can be used to infer sea wave parameters. Based on the SAR images acquired by Seasat, the results of Beal et al. (Beal 1980; Beal et al. 1983) showed that two-dimensional sea wave spectrum can be retrieved by L-band SAR images. In addition to Seasat SAR images, scholars also carried out many investigations on sea wave spectrum retrieval algorithm based on airborne and shuttle borne SAR image (Monaldo and Lyzenga 1988; Beal et al. 1986; Monaldo and Beal 1998; Brüning et al. 1988; Monaldo et al. 1993; Melsheimer et al. 1998). The successful launch of ERS-1 and ERS-2 satellites further stimulated the study of wave spectrum inversion based on SAR images (Tilley and Beal 1994; Holt et al. 1998; Kuo et al. 1999; Krogstad and Barstow 1999). The sea wave information retrieved using ERS-1/-2 SAR images had been successfully assimilated into the wave prediction models (Breivik et al. 1998; Heimbach et al. 1998). The launches of RADARSAT-1, Envisat Advanced Synthetic Aperture Radar (ASAR), and RADARSAT-2 satellites once again promoted the development of SAR wave spectrum retrieval technology (Li et al. 2002; Ardhuin et al. 2004; He et al. 2004, 2006; Zhang et al. 2010). SAR imaging mechanism of sea waves is the basis of wave retrieval algorithm. Alpers (1985) and Hasselmann et al. (1985) pointed out that there were three modulation mechanisms in SAR sea wave imaging theory, that is, the tilt modulation, the hydrodynamic modulation and the velocity bunching modulation. Using these three modulation mechanisms, a famous SAR sea wave retrieval algorithm was proposed by Hasselmann and Hasselmann (1991) on the basis of the nonlinear mapping of an ocean wave spectrum into a copolarized SAR image spectrum. Later, the modified nonlinear inversion method, the cross-spectrum method and so on had been proposed successively to retrieve ocean wave spectra from SAR images (Engen and Johnsen 1995; Schulz-Stellenfleth et al. 2005, 2007; Schuler et al. 2004).

In the retrieval algorithms above, the complex hydrodynamic modulation transfer function (MTF) is always involved. Unfortunately, the hydrodynamic MTF has not been well understood and the relaxation rate in the hydrodynamic MTF is still poorly known experimentally. Up to now, the values of the relaxation rate estimated by various investigators differ by almost one order of magnitude (Caponi et al. 1988). To overcome this problem, a method, which is originally used in topographic measurements, had been applied to retrieve ocean wave slopes and wave spectra (Schuler et al. 2004). In this method, the azimuth component of ocean wave slope was estimated based on the polarimetric orientation angle; on the other hand, the wave slope along SAR range direction was estimated using the alpha parameter from the Cloude–Pottier polarimetric scattering decomposition theorem. Using the POLSAR image intensity, He et al. proposed a new method to retrieve ocean wave slopes without the need of the complex hydrodynamic MTF (He et al. 2004, 2006; Zhang et al. 2010). In this method, the ocean wave slope along SAR range direction can be retrieved by an algorithm which is only related with the copolarization tilt MTFs. It should be pointed out that the method in He et al. (2006), Zhang et al. (2010) was obtained based on the linear modulation theory. However, for generally case, the velocity bunching modulation of the ocean wave is so strong that the linear modulation theory is no longer valid.

In this paper, based on the SAR polarimetric parameter defined by He et al. (2006), a further extension of Hasselmann’s work is addressed. In section 2, a new nonlinear transformation relation is derived to describe the mapping of a two-dimensional ocean wave spectrum into the spectrum of the polarimetric image. In section 3, empirical tilt MTFs for copolarization echoes are proposed. In section 4, the retrieval algorithm based on the new nonlinear transformation relation and the empirical tilt MTFs is presented. Section 5 presents the retrieved results and the discussion. The conclusions of this paper are given in the final section.

2. Method

He et al. (2006) and Zhang et al. (2010) proposed a polarimetric SAR wave retrieval algorithm without estimating the hydrodynamic MTF. If we completely neglect the effect of the velocity bunching modulation, the polarimetric image can be defined as

I_{hhvv}^{R} = \frac{σ_{hh} (r, t)}{{\bar{σ}}_{hh}} - \frac{σ_{vv} (r, t)}{{\bar{σ}}_{vv}} = \sum_{k} (T_{k hhvv}^{t}) ζ_{k} \exp (i k \cdot r - i ω t) + c . c .

(1)

where

T_{k hhvv}^{t} = T_{k hh}^{t} - T_{k vv}^{t}

T_{k hh}^{t}

and

T_{k vv}^{t}

denote the tilt MTFs for horizontal and vertical polarization cases, respectively. Essentially,

I_{hhvv}^{R}

is a real aperture radar (RAR) polarimetric image. For water waves, each scattering element in SAR image has an additional displacement ξ along SAR azimuth direction due to the orbital velocity. The relationship between the additional displacement ξ and the orbital velocity is (Hasselmann and Hasselmann 1991)

ξ = β \sum_{k} T_{k}^{υ} ζ_{k} \exp (i k \cdot r - i ω t) + c . c .,

(2)

where β is the ratio of the SAR slant range to the flight speed of the SAR platform, ω = (g|k|)^1/2 is the angular frequency of the water wave, ζ_k denotes the Fourier transform coefficients of the water surface Z(r, t), k denotes the wavenumber of water wave, and

T_{k}^{υ}

is the range velocity transfer function:

T_{k}^{υ} = - ω (\sin θ \frac{k_{l}}{| k |} + i \cos θ) .

(3)

Here, k_l is the projection of wavenumber in the range direction of SAR image. Considering the effect of ξ, the real SAR image can be obtained by mapping each facet in RAR image

I_{hhvv}^{R}

at position r′ into its corresponding position

r = r^{'} + ξ (r^{'}) \hat{y}

(Hasselmann and Hasselmann 1991); thus

I_{hhvv}^{s} (r) = \int I_{hhvv}^{R} (r^{'}) δ [r - r^{'} - ξ (r^{'}) \hat{y}] d r^{'},

(4)

where

\hat{y}

denotes the unit vector along SAR azimuth direction.

Integrating out the δ function, Eq. (4) yields

I_{hhvv}^{s} (r) = {[I_{hhvv}^{R} (r^{'}) | \frac{d r^{'}}{d r} |]}_{r^{'} = r - ξ (r^{'}) \hat{y}} .

(5)

Using Eq. (5), the Fourier transform coefficient and the spectrum of the SAR image

I_{hhvv}^{s} (r)

can be expressed as

\begin{matrix} I_{k hhvv}^{s} = \frac{1}{A} {\int [I_{hhvv}^{R} (r^{'}) | \frac{d r^{'}}{d r} |]}_{r^{'} = r - ξ (r^{'})} \exp (- i k \cdot r) d r and \\ = \frac{1}{A} \int I_{hhvv}^{R} (r^{'}) \exp {- i k \cdot [r^{'} + ξ (r^{'}) \hat{y}]} d r^{'} \end{matrix}

(6)

\begin{matrix} P_{k hhvv}^{s} = 〈 I_{k hhvv}^{s} I_{k hhvv}^{s *} 〉 \\ = \frac{1}{A^{2}} \int \exp [- i k \cdot (r^{″} - r^{'})] 〈 \exp {- i k \cdot \hat{y} [ξ (r^{″}) - ξ (r^{'})]} \times I_{hhvv}^{R} (r^{″}) I_{hhvv}^{R *} (r^{'}) 〉 d r^{'} d r^{″}, \end{matrix}

(7)

respectively. In Eqs. (6) and (7), A denotes the area of the integral region. Because ξ and

I_{hhvv}^{R}

satisfy the Gaussian processes, using the following identical equation

〈 \exp {- i k \cdot \hat{y} [ξ (r^{″}) - ξ (r^{'})]} I_{hhvv}^{R} (r^{″}) I_{hhvv}^{R *} (r^{'}) 〉 = {- \frac{\partial 〈 \exp [- i k \cdot \hat{y} ξ (r^{″}) + i k \cdot \hat{y} ξ (r^{'}) + i α_{1} I_{hhvv}^{R} (r^{″}) + i α_{2} I_{hhvv}^{R *} (r^{'})] 〉}{\partial α_{1} \partial α_{2}} |}_{α_{1} = α_{2} = 0}

(8)

and after a tedious but straightforward mathematical derivation (see the appendix for the detailed derivation), it is obtained that

P_{k hhvv}^{s} = - \frac{1}{A^{2}} \exp (- k_{y}^{2} 〈 ξ^{2} 〉) \int \exp [- i k \cdot (r^{″} - r^{'})] \exp [k_{y}^{2} ρ_{ξ ξ} (r^{″} - r^{'})] {- k_{y}^{2} [ρ_{ξ I} (0) - ρ_{ξ I} (r^{'} - r^{″})] [ρ_{ξ I} (0) - ρ_{ξ I} (r^{″} - r^{'})] - ρ_{I I} (r^{″} - r^{'})} d r^{'} d r^{″},

(9)

with

〈 ξ^{2} 〉 = β^{2} \int {| T_{k}^{υ} |}^{2} F (k) d k and

(10)

ρ_{ξ I} (0) = \frac{1}{2} \int {{(T_{k hhvv}^{t})}^{*} β T_{k}^{υ} F (k) + (T_{- k hhvv}^{t}) {(β T_{- k}^{υ})}^{*} F (- k)} d k .

(11)

Here, k_y is the spatial wavenumber along SAR azimuthal direction, and ρ_γχ() denotes the correlation function between the Gaussian processes γ and χ. For stationary processes, the correlation function depends only on r = r″ − r′. Then, through the variable substitution, Eq. (9) is simplified as

P_{k hhvv}^{s} = \exp (- k_{y}^{2} 〈 ξ^{2} 〉) \int \exp (- i k \cdot r) \exp [k_{y}^{2} ρ_{ξ ξ} (r)] [ρ_{I I} (r) + k_{y}^{2} ρ_{ξ I}^{2} (0) - k_{y}^{2} ρ_{ξ I} (- r) ρ_{ξ I} (0) - k_{y}^{2} ρ_{ξ I} (0) ρ_{ξ I} (r) + k_{y}^{2} ρ_{ξ I} (- r) ρ_{ξ I} (r)] d r .

(12)

Expanding the exponential

\exp [k_{y}^{2} ρ_{ξ ξ} (r)]

by Taylor series, it is obtained that

\exp [k_{y}^{2} ρ_{ξ ξ} (r)] = 1 + k_{y}^{2} ρ_{ξ ξ} (r) + \frac{k_{y}^{4} ρ_{ξ ξ}^{2} (r)}{2!} + \dots .

(13)

Substituting Eq. (13) into Eq. (12), we can obtain that

P_{k hhvv}^{s} = \exp (- k_{y}^{2} 〈 ξ^{2} 〉) \int \exp (- i k \cdot r) [1 + k_{y}^{2} ρ_{ξ ξ} (r) + \frac{k_{y}^{4} ρ_{ξ ξ}^{2} (r)}{2!} + \dots] [ρ_{I I} (r) + k_{y}^{2} ρ_{ξ I}^{2} (0) - k_{y}^{2} ρ_{ξ I} (- r) ρ_{ξ I} (0) - k_{y}^{2} ρ_{ξ I} (0) ρ_{ξ I} (r) + k_{y}^{2} ρ_{ξ I} (- r) ρ_{ξ I} (r)] d r .

(14)

If we neglect the constant term and the higher-order terms above second order, an approximate expression of Eq. (14) is obtained as

P_{k hhvv}^{s} \approx \exp (- k_{y}^{2} 〈 ξ^{2} 〉) \int \exp (- i k \cdot r) [ρ_{I I} (r) - k_{y}^{2} ρ_{ξ I} (- r) ρ_{ξ I} (0) - k_{y}^{2} ρ_{ξ I} (0) ρ_{ξ I} (r) + k_{y}^{4} ρ_{ξ I}^{2} (0) ρ_{ξ ξ} (r)] d r .

(15)

Integrating Eq. (15) over r, a new nonlinear transformation relation between the two-dimensional ocean wave spectrum and the image spectrum of

I_{hhvv}^{s} (r)

is derived as

P_{k hhvv}^{s} = P_{k hhvv}^{s 1} + P_{k hhvv}^{s 2} + P_{k hhvv}^{s 3},

(16)

P_{k hhvv}^{s 1} = \exp (- k_{y}^{2} 〈 ξ^{2} 〉) ({| T_{k hhvv}^{t} |}^{2} \frac{F_{k}}{2} + {| T_{- k hhvv}^{t} |}^{2} \frac{F_{- k}}{2}),

(17)

P_{k hhvv}^{s 2} = 2 \exp (- k_{y}^{2} 〈 ξ^{2} 〉) k_{y}^{2} ρ_{ξ I} (0) β \cos θ \times [i ω T_{k hhvv}^{t} \frac{F_{k}}{2} + {(i ω T_{- k hhvv}^{t})}^{*} \frac{F_{- k}}{2}], and

(18)

P_{k hhvv}^{s 3} = \exp (- k_{y}^{2} 〈 ξ^{2} 〉) k_{y}^{4} β^{2} ρ_{ξ I}^{2} (0) ({| T_{k}^{υ} |}^{2} \frac{F_{k}}{2} + {| T_{- k}^{υ} |}^{2} \frac{F_{- k}}{2}) .

(19)

For the case of dξ(r′)/dr′ ≪ 1, Eq. (5) yields that

I_{hhvv}^{s} (r) = {I_{hhvv}^{R} (r^{'}) [1 - \frac{d ξ (r^{'})}{d r^{'}}]} \approx I_{hhvv}^{R} (r^{'}) .

(20)

Then, the linear transformation relation between the two-dimensional ocean wave spectrum and the image spectrum of

I_{hhvv}^{s} (r)

is derived as

P_{k hhvv}^{s} = {{| T_{k hhvv}^{t} |}^{2} \frac{F_{k}}{2} + {| T_{- k hhvv}^{t} |}^{2} \frac{F_{- k}}{2}} .

(21)

Equation (21) is completely consistent with the conclusion in He et al. (2006) and Zhang et al. (2010).

To show the effects of the three terms on the right-hand side of Eq. (16), the normalized $P_{k hhvv}^{s 1}$ , $P_{k hhvv}^{s 2}$ , and $P_{k hhvv}^{s 3}$ are presented in Fig. 1. According to the usual parameters of the satellite SAR and sea waves, the values of β, ⟨ξ²⟩ and ρ_ξI(0) in the three terms are set to be 94.83, 1692.5, and 2.79, respectively. The results in Fig. 1b and Fig. 1c demonstrate that the effects of $P_{k hhvv}^{s 2}$ and $P_{k hhvv}^{s 3}$ can be neglected when sea waves travel along SAR range direction (i.e., k_x direction). However, with the increase of k_y, the velocity bunching effect has a more significant impact on SAR images, which leads to the larger values of $P_{k hhvv}^{s 2}$ and $P_{k hhvv}^{s 3}$ , in this case, the influences of $P_{k hhvv}^{s 2}$ and $P_{k hhvv}^{s 3}$ in Eq. (16) cannot be ignored.

3. The tilt MTFs

At present, there is no objection to the expression of the velocity bunching modulation function. Therefore, in Eq. (16), the MTF

T_{k hhvv}^{t}

becomes the key factor that affects the accuracy of the retrieved sea spectrum by the mapping relationship. In the previous works, based on the traditional Bragg resonance theory, the expressions of the theoretical tilt MTFs for hh and vv polarizations were derived as (Wright 1968; Lyzenga 1986)

T_{k hh}^{t} = 8 i k_{l} {(\sin 2 θ)}^{- 1} and

(22)

T_{k vv}^{t} = 4 i k_{l} \cot θ {(1 + \sin^{2} θ)}^{- 1},

(23)

respectively. Then, the difference between the copolarization tilt MTFs is obtained as

T_{k hhvv}^{t} = T_{k hh}^{t} - T_{k vv}^{t} = i k_{l} \frac{8 \tan θ}{1 + \sin^{2} θ} .

(24)

However, it is well known that the traditional Bragg resonance theory cannot accurately describe the backscattering field from sea surface. Therefore, the tilt modulation functions derived based on the Bragg theory also cannot accurately reflect the change of scattering field with sea surface slope. Compared with the Bragg resonance theory, more accurate scattering coefficient [normalized radar cross section (NRCS)] can be evaluated by the empirical geophysical mode function. Mouche and Chapron (2015) proposed an empirical C-band synthetic aperture radar (CSAR) model to evaluated the scattering coefficient

σ_{hh}^{e} (u, θ, ϕ)

and

σ_{vv}^{e} (u, θ, ϕ)

for hh and vv polarizations. Based on the CSAR model, the values of the copolarization tilt MTFs can be evaluated by

T_{k hh}^{t_e} (u, θ_{i}, ϕ_{i}) = {i k_{l} \frac{1}{σ_{hh}^{e} (u, θ_{i}, ϕ_{i})} \frac{\partial σ_{hh}^{e}}{\partial θ} |}_{θ = θ_{i}, ϕ = ϕ_{i}} and

(25)

{T_{k vv}^{t_e} (u, θ_{i}, ϕ_{i}) = i k_{l} \frac{1}{σ_{vv}^{e} (u, θ_{i}, ϕ_{i})} \frac{\partial σ_{vv}^{e}}{\partial θ} |}_{θ = θ_{i}, ϕ = ϕ_{i}}

(26)

and then the difference of the copolarization tilt MTFs is given by

T_{k hhvv}^{t_e} (u, θ_{i}, ϕ_{i}) = T_{k hh}^{t_e} (u, θ_{i}, ϕ_{i}) - T_{k vv}^{t_e} (u, θ_{i}, ϕ_{i}) .

(27)

First, it should be pointed out that the values of MTFs in Figs. 2–4 have been normalized by spatial wavenumber ik_l.

In Fig. 2, the empirical tilt MTFs evaluated by Eq. (25) and Eq. (26) are compared with the results of the theoretical models [i.e., Eqs. (22) and (23)]. Here, the wind speed is set to be 9 m s⁻¹. Just as predicted, there are significant differences between the empirical and the theoretical tilt MTFs. From Eq. (22) and Eq. (23) we can also find that the theoretical copolarization tilt modulation functions are not sensitive to wind speed and wind direction. However, the curves in Fig. 3 illustrate that the empirical tilt MTFs are sensitive to wind speed. The values of the empirical tilt MTFs decrease with wind speed. On the other hand, wind direction also has some influence on the empirical tilt MTFs. In general, the value of the MTF reaches its maximum/minimum when the radar looks along the upwind/downwind direction, respectively, except for the vv-polarization MTF at lower wind speed.

In Fig. 4, the values of

| T_{k hhvv}^{t_e} (u, θ_{i}, ϕ_{i}) |

for different wind speed and wind direction are compared with

| T_{k hhvv}^{t} |

. From Fig. 4 we can find that the value of

| T_{k hhvv}^{t} |

evaluated based on the Bragg resonance theory is significant larger than the values of

| T_{k hhvv}^{t_e} (u, θ_{i}, ϕ_{i}) |

. And the values of

| T_{k hhvv}^{t_e} (u, θ_{i}, ϕ_{i}) |

for different wind speeds and wind directions increase with incident angle. If the wind speed is lower, such as u = 5 m s⁻¹, the value of

| T_{k hhvv}^{t_e} (u, θ_{i}, ϕ_{i}) |

at larger incident angles do not change regularly with wind direction. However, if wind speed is higher, the differences between the values of

| T_{k hhvv}^{t_e} (u, θ_{i}, ϕ_{i}) |

for various wind directions become not significant. Then, the average value

\bar{| T_{k hhvv}^{t_e} (u, θ_{i}, ϕ_{i}) |}

[= \frac{| T_{k hhvv}^{t_e} (u, θ_{i}, 0) + T_{k hhvv}^{t_e} (u, θ_{i}, π / 2) + T_{k hhvv}^{t_e} (u, θ_{i}, π) |}{3}]

can be used to replace

| T_{k hhvv}^{t_e} (u, θ_{i}, ϕ_{i}) |

. From Fig. 4d we can find that the average value

\bar{| T_{k hhvv}^{t_e} (u, θ_{i}, ϕ_{i}) |}

is almost independent of wind speed. To facilitate the application, a polynomial fitting function of

\bar{| T_{k hhvv}^{t_e} (u, θ_{i}, ϕ_{i}) |}

is given by

F (θ_{i}) = A θ_{i}^{3} + B θ_{i}^{2} + C θ_{i} + D,

(28)

where the parameters A = 6.9094 × 10⁻⁶, B = 0.0022, C = −0.0501, D = 1.9070 and the incident angle θ_i is in degrees. From Fig. 4d, it is found that the results calculated by the fitting function agree well with the values of

\bar{| T_{k hhvv}^{t_e} (u, θ_{i}, ϕ_{i}) |}

. Multiplying F(θ_i) by ik_l, a new empirical function for the difference of the copolarization tilt MTFs is obtained as

T_{k hhvv}^{t_f} = i k_{l} (A θ_{i}^{3} + B θ_{i}^{2} + C θ_{i} + D) .

(29)

4. The retrieval algorithm

The cost function used in this paper has the same expression as that proposed by Hasselmann and Hasselmann (1991); that is,

J = \int {(P_{k hhvv}^{S} - {\hat{P}}_{k hhvv}^{S})}^{2} d k + μ \int {[\frac{F (k) - \hat{F} (k)}{B + \hat{F} (k)}]}^{2} d k,

(30)

where

P_{k hhvv}^{S}

and

{\hat{P}}_{k hhvv}^{S}

are the observed and fitted image spectra of

I_{hhvv}^{S}

, respectively. In this work, we use the wave spectrum retrieved by the linear mapping relationship [i.e., Eq. (21)] as the first-guess spectrum

\hat{F} (k)

. Here, F(k) denotes the optimal fit wave spectrum, which would minimize the cost function J. The values of the other two parameters μ and B are the same as those suggested by Hasselmann and Hasselmann (1991). The overall procedure of the retrieval algorithm can be summarized as follows:

Transfer the slant range hh-/vv-polarized SAR images into ground range images, and select a subimage of 512 × 512 pixel size from the region with clear wave textures; then a new polarimetric image is constructed by using Eq. (1).
Retrieve $\hat{F} (k)$ by using Eq. (21) as the first-guess spectrum $\hat{F} {(k)}_{i = 0}$ , and remove the directional ambiguity with model wave direction; here, the subscript i denotes the number of iterations.
Evaluate the initial values of parameters ⟨ξ²⟩_i=0 and ρ_ξI(0)_i=0 based on $\hat{F} {(k)}_{i = 0}$ .
Define wave spectrum F(k)_i=1 and evaluate ${\hat{P}}_{k hhvv}^{S}$ based on the new nonlinear mapping relation [i.e., Eq. (16)].
If F(k)_i=1 minimizes the cost function J (in the actual process, when the value of J is less than 10⁻⁵, we can consider that the cost function reaches the minimum value), F(k)_i=1 is considered to be the retrieved wave spectrum of the first iteration; then, go to step 3 to calculate ⟨ξ²⟩_i=1, ρ_ξI(0)_i=1 and repeat steps 4 and 5.
After n iterations, if $\int | F {(k)}_{i = n} - F {(k)}_{i = n - 1} | d k \leq 0.01$ , the iteration is stopped and F(k)_i=n is considered to be the final retrieved wave spectrum.

5. The retrieved results and discussions

In this section, 10 fully polarimetric SAR images acquired by C-band RADARSAT-2 are used to retrieve ocean wave spectrum based on the new nonlinear mapping between the ocean wave spectrum and the new polarimetric SAR image spectrum. The wave parameters retrieved by the algorithm in section 4 are compared with the data of the ECMWF Ocean Wave Model (ECWAM) and the buoys measurements. The detailed information of the SAR images and the buoys is given in Table 1.

Table 1.

The RADARSAT-2 fully polarimetric SAR images and NDBC buoys; here, ID is the identifier.

Figure 5 presents the location of the SAR ID 1 image. The incident angle at the center of this image is about 38.18°. In Fig. 6, the first-guess spectrum evaluated by Eq. (21), in which the impact of the velocity bunching effect has not been considered, is compared with the retrieved spectrum by the algorithm in section 4. Just as expected, in comparison with the first-guess spectrum, the retrieved spectrum changes significantly, especially the spectral density corresponding to the larger k_y. This is consistent with the conclusion drawn from Fig. 1.

To verify the validity of the algorithm, in the following, the retrieved spectra are also compared with the results obtained by the MPI method (Hasselmann and Hasselmann 1991). The retrieved sea wave parameters, such as significant wave height (SWH), wavelength, and wave direction, from C-band polarized RADARSAT-2 SAR images are also compared with the data obtained by the ECWAM model and the buoy measurements.

Figures 7a and 7b present the corresponding 512 × 512 pixel size hh- and vv-polarized scattering coefficient (NRCS) images selected from the SAR ID 1 image. The new polarimetric image $I_{hhvv}^{S}$ , which has been filtered by a 3 × 3 Gaussian filter to suppress the speckle noise, is shown in Fig. 7c. From Fig. 7, we find that the textures of the swell are clearly visible on the copolarized NRCS images and the new polarimetric image. As shown in Table 1, the wind speed measured by the buoy at the location of the image is only 3.7 m s⁻¹, so there is no wind-wave texture on the SAR images.

In Figs. 8a–c, the sea wave spectra retrieved by different algorithms with the same first-guess spectrum are presented. For comparison, the 2D wave spectrum obtained by the ECWAM model is also presented in Fig. 8d. In comparing the results in Figs. 8a–c with the spectrum obtained by the ECWAM model (https://www.ecmwf.int/en/forecasts/datasets/browse-reanalysis-datasets), one can find that the dominant wavelengths and propagation directions retrieved by different algorithms are all consistent with the results obtained by the ECWAM model. However, there are significant differences between the SWH obtained by different retrieval algorithms.

Figure 9 presents the location of the SAR ID 2 image. The incident angle at the center of this image is about 23.22°. The corresponding 512 × 512 pixel size copolarized NRCS images and the new polarimetric image $I_{hhvv}^{S}$ are shown in Figs. 10a–c. At the location of the SAR ID 2 image, the wind direction and wind speed measured by the buoy (46005) are 60° and 9 m s⁻¹, which is strong enough to generate significant wind-waves. From Figs. 10a–c, we can find the textures of the swell. However, the textures of the wind wave on the copolarized NRCS images and the polarimetric image become very ambiguous and indistinguishable because of the strong nonlinear velocity bunching effect along SAR azimuth direction.

The retrieved spectra from the SAR ID 2 data based on the different algorithms are shown in Fig. 11. We can find that the propagation directions of the dominant wave obtained by different algorithms are consistent with the results obtained by the ECWAM model. However, the wavelength of the dominant wave and the SWH retrieved by the SAR ID 2 data have larger errors than those retrieved by the SAR ID 1 data. The reason why the error of the retrieved wavelength and SWH in Figs. 11a and 11b increases is that the textures on $I_{hhvv}^{s}$ image are less sensitive to the slope of the sea surface with the decrease of the incident angle. Moreover, from the spectra in Figs. 11a–c, we can find that only the spectrum of the swell has been retrieved, which means that the wind-driven spectrum cannot be effectively retrieved based on the SAR images of the SAR ID 2. The result in Fig. 11d shows that the wind-driven spectrum is also not predicted by the ECWAM model.

Figure 12 presents the location of the SAR ID 3 image. The incident angle at the center of the SAR ID 3 image is about 33.26°. The corresponding 512 × 512 pixel size copolarized NRCS images and the new polarimetric image are shown in Figs. 13a–c. The retrieved spectra from the SAR ID 3 data based on the different algorithms are shown in Fig. 14. The ECWAM model spectra in Fig. 14d show that there are two wave modes at the location of the SAR ID 3 image. The first wave mode travels along 118.68°; the wavelength and the SWH are 243.98 and 2.01 m. The second wave mode travels along 5.02°, and the dominant wavelength and the SWH are 521.87 and 0.12 m, respectively. The retrieved spectra in Fig. 14a and Fig. 14b show that the spectrum of the first wave mode can be well retrieved from the new polarimetric image. However, the second wave mode presented in Fig. 14d is not effectively retrieved, perhaps because the slope of the second wave mode is too small to be imaged on the new polarimetric image. However, as shown in Fig. 14c, the spectra of the two wave modes are both retrieved from the vv-polarized NRCS image by the MPI method because the vv-polarized NRCS image is more sensitive to wave slope than the new polarimetric image.

To further analyze the accuracy of the wave parameters obtained by different algorithms, the comparison of SAR retrievals with the ECWAM data and buoy measurements for the SWH, as well as the wavelength and propagation direction of the dominant wave are given in Table 2. Here, based on the new polarimetric image, the significant wave height retrieved by using the empirical and the traditional tilt MTFs are denoted by SWH₁ and SWH₂, respectively. SWH₃ denotes the significant wave height retrieved based on the MPI algorithm. Parameters λ₁ and ϕ₁ denote the wavelength and the propagation direction of the dominant wave retrieved from the new polarimetric image based on the algorithm in section 4. Parameters λ₂ and ϕ₂ denote the wavelength and the propagation direction of the dominant wave retrieved from the vv-polarized NRCS image based on the MPI algorithm (Hasselmann and Hasselmann 1991). SWH, λ, and ϕ are the significant wave height, the wavelength, and the propagation direction, respectively, of the dominant wave from the ECWAM model or the buoy measurements. Bias and std dev represent the deviation and standard deviation between the inversion results and ECMWF data (buoy data), respectively. As can be seen from the results in Table 2, SWH₁ has a smaller bias and standard deviation than SWH₂ and SWH₃ relative to the significant wave heights obtained by the ECWAM model or the buoy measurements. For the wavelength and the propagation direction of the dominant wave, the results extracted from the new polarimetric image and the vv-polarized NRCS image have little difference in accuracy.

Table 2.

The wave parameters retrieved by different algorithms are compared with the corresponding wave parameters provided by ECWAM and NDBC buoys.

6. Conclusions

On the basis of the polarimetric SAR image proposed by He et al. (2006), we derive a new nonlinear transformation relation to describe the mapping of the sea wave spectrum into the polarimetric SAR image spectrum without the need for the complex hydrodynamic modulation transfer function. Using the empirical CSAR model (Mouche and Chapron 2015), new tilt MTFs are also proposed to replace the traditional tilt MTFs obtained by the Bragg scattering theory. We then present a new retrieval algorithm of sea wave spectrum on the basis of the new nonlinear mapping relation and the empirical tilt MTF. To verify the validity of the algorithm, the retrieved spectra and the corresponding sea wave parameters from 10 C-band polarized RADARSAT-2 SAR images are compared with the data obtained by the ECWAM model and the buoy measurements. From the results in this work, the following conclusions can be drawn:

In the nonlinear mapping relation, when sea wave travels along SAR range direction, the influence of the terms which are related to the velocity bunching effect can be neglected, in this case, the nonlinear mapping relation is simplified to the linear mapping relation proposed by He et al. (2004, 2006). However, with the increase of SAR azimuth wavenumber of sea waves, the influences of the terms related to the velocity bunching effect cannot be ignored.
The comparisons illustrate that there are significant differences between the new and the traditional tilt MTFs. The difference of the copolarization tilt MTFs obtained by the Bragg theory (i.e., $| T_{k hhvv}^{t} |$ ) is obviously larger than the empirical model, i.e., $| T_{k hhvv}^{t_e} (u, θ_{i}, ϕ_{i}) |$ . If we use $| T_{k hhvv}^{t_e} (u, θ_{i}, ϕ_{i}) |$ instead of $| T_{k hhvv}^{t} |$ in the inversion algorithm, the relative error between the retrieved SWHs and the buoy measurements or the results obtained by the ECWAM model become obviously smaller. Thus, to improve the accuracy of the retrieval algorithm, the traditional tilt MTF should be replaced by the empirical model.
Using the retrieval algorithm proposed in this work, the spectra of sea swells with larger amplitude can be retrieved effectively from the new polarimetric SAR images. However, the swell with very small amplitude are not effectively retrieved because the slope of this swell is too small to be imaged on the new polarimetric SAR image. However, the spectrum corresponding to the small amplitude swell can be retrieved from the vv-polarized NRCS image by the MPI method because the vv-polarized NRCS image is more sensitive to wave slope than the new polarimetric image. Moreover, the texture of the wind wave is also very ambiguous in the new polarimetric SAR image, and the corresponding wind wave spectrum is not effectively retrieved, too.

For the algorithm based on the new polarimetric SAR image without estimating the complex hydrodynamic modulation transfer function, as compared with the copolarized NRCS, the sensitivity of the polarimetric SAR image data to sea wave slope is significantly reduced, thus reducing the contrast of the wave texture in the polarimetric SAR image, which is a disadvantageous factor in the wave retrieval algorithm. Therefore, future research is needed to determine the extent to which the new retrieval algorithm is capable of measuring a more general wind-driven wave spectrum or a complex intersecting bimodal wave system.

Acknowledgments

Funding support has been provided by the National Key Research and Development Program of China (2017YFB0502700), the Natural Science Foundation of China (Grants 41576170 and 41976167), and the National Natural Science Foundation of China-Shandong Joint Fund for Marine Science Research Centers (Grant U1606404).

APPENDIX

The Detailed Derivation of Eq. (9)

Assuming that k_y is the azimuth component of the wavenumber vector k, then Eq. (8) can be rewritten as

〈 \exp {- i k \cdot \hat{y} [ξ (r^{″}) - ξ (r^{'})]} I_{hhvv}^{R} (r^{″}) I_{hhvv}^{R *} (r^{'}) 〉 = {- \frac{\partial 〈 \exp [- i k_{y} ξ (r^{″}) + i k_{y} ξ (r^{'}) + i α_{1} I_{hhvv}^{R} (r^{″}) + i α_{2} I_{hhvv}^{R *} (r^{'})] 〉}{\partial α_{1} \partial α_{2}} |}_{α_{1} = α_{2} = 0} .

(A1)

Because ξ and

I_{hhvv}^{R}

are both Gaussian processes, the characteristic function, that is, the ensemble average on the right-hand side of Eq. (A1), can be found by utilizing (Parzen 1962)

〈 e^{i A_{1} X_{1}} e^{i A_{2} X_{2}} e^{i A_{3} X_{3}} e^{i A_{4} X_{4}} 〉 = \exp (- \frac{1}{2} \sum_{j = 1}^{4} \sum_{k = 1}^{4} A_{j} A_{k} 〈 X_{j} X_{k}^{*} 〉) .

(A2)

Then, it is obtained that

〈 \exp [- i k_{y} ξ (r^{″}) + i k_{y} ξ (r^{'}) + i α_{1} I_{hhvv}^{R} (r^{″}) + i α_{2} I_{hhvv}^{R *} (r^{'})] 〉 = \exp [- k_{y}^{2} 〈 ζ^{2} (r^{'}) 〉 + k_{y}^{2} 〈 ζ (r^{'}) ζ (r^{″}) 〉 + k_{y} α_{1} 〈 ζ (r^{'}) I_{hhvv} (r^{'}) 〉 + k_{y} α_{2} 〈 ζ (r^{'}) I_{hhvv} (r^{″}) 〉 - k_{y} α_{1} 〈 ζ (r^{″}) I_{hhvv} (r^{'}) 〉 - k_{y} α_{2} 〈 ζ (r^{″}) I_{hhvv} (r^{'}) 〉 - \frac{1}{2} α_{1}^{2} 〈 I_{hhvv}^{2} (r^{'}) 〉 - α_{1} α_{2} 〈 I_{hhvv} (r^{'}) I_{hhvv} (r^{″}) 〉 - \frac{1}{2} α_{2}^{2} 〈 I_{hhvv}^{2} (r^{″}) 〉] .

(A3)

Substituting Eq. (A3) into Eq. (A1), we can obtain that

〈 \exp {- i k \cdot \hat{y} [ξ (r^{″}) - ξ (r^{'})]} I_{hhvv}^{R} (r^{″}) I_{hhvv}^{R *} (r^{'}) 〉 = - \exp [- k_{y}^{2} 〈 ζ^{2} (r^{'}) 〉 + k_{y}^{2} 〈 ζ (r^{'}) ζ (r^{″}) 〉] {k_{y}^{2} [〈 ζ (r^{'}) I_{hhvv} (r^{'}) 〉 - 〈 ζ (r^{″}) I_{hhvv} (r^{'}) 〉] [〈 ζ (r^{'}) I_{hhvv} (r^{″}) 〉 - 〈 ζ (r^{″}) I_{hhvv} (r^{'}) 〉] - 〈 I_{hhvv} (r^{'}) I_{hhvv} (r^{″}) 〉} .

(A4)

Substituting Eq. (A4) into Eq. (7), Eq. (9) is obtained.

REFERENCES

Alpers, W., 1985: Theory of radar imaging of internal waves. Nature, 314, 245–247, https://doi.org/10.1038/314245a0.
- Crossref
- Search Google Scholar
- Export Citation
Ardhuin, F., F. Collard, and B. Chapron, 2004: Wave spectra from ENVISAT’s synthetic aperture radar in coastal area. Proc. 2004 ISOPE Conf., Toulon, France, International Society of Offshore and Polar Engineers, ISOPE-I-04-319, https://www.onepetro.org/conference-paper/ISOPE-I-04-319.
Beal, R., 1980: Spaceborne imaging radar: Monitoring of ocean waves. Science, 208, 1373–1375, https://doi.org/10.1126/science.208.4450.1373.
- Crossref
- Search Google Scholar
- Export Citation
Beal, R., D. G. Tilley, and F. M. Monaldo, 1983: Large and small scale spatial evolution of digitally processed ocean wave spectra from SEASAT synthetic aperture radar. J. Geophys. Res., 88, 1761–1778, https://doi.org/10.1029/JC088iC03p01761.
- Crossref
- Search Google Scholar
- Export Citation
Beal, R., and Coauthors, 1986: A comparison of SIR-B directional ocean wave spectra with aircraft scanning radar spectra. Science, 232, 1531–1535, https://doi.org/10.1126/science.232.4757.1531.
- Crossref
- Search Google Scholar
- Export Citation
Breivik, L., M. Reistad, H. Schyberg, J. Sunde, and H. E. Krogstad, 1998: Assimilation of ERS SAR wave spectra in an operational wave model. J. Geophys. Res., 103, 7887–7900, https://doi.org/10.1029/97JC02728.
- Crossref
- Search Google Scholar
- Export Citation
Brüning, C. W., W. Alpers, L. F. Zambresky, and D. G. Tilley, 1988: Validation of a synthetic aperture radar ocean wave imaging theory by the Shuttle Imaging Radar-B experiment over the North Sea. J. Geophys. Res., 93, 15 403–15 425, https://doi.org/10.1029/JC093iC12p15403.
- Crossref
- Search Google Scholar
- Export Citation
Caponi, E. A., D. R. Crawford, H. C. Yuen, and P. G. Saffman, 1988: Modulation of radar backscatter from the ocean by a variable surface current. J. Geophys. Res., 93, 12 249–12 263, https://doi.org/10.1029/JC093iC10p12249.
- Crossref
- Search Google Scholar
- Export Citation
Engen, G., and H. Johnsen, 1995: SAR-ocean wave inversion using image cross spectra. IEEE Trans. Geosci. Remote Sens., 33, 1047–1056, https://doi.org/10.1109/36.406690.
- Crossref
- Search Google Scholar
- Export Citation
Gonzalez, F. I., and R. Beal, 1979: Seasat synthetic aperture radar: Ocean wave detection capabilities. Science, 204, 1418–1421, https://doi.org/10.1126/science.204.4400.1418.
- Crossref
- Search Google Scholar
- Export Citation
Hasselmann, K., and S. Hasselmann, 1991: On the nonlinear mapping of an ocean wave spectrum into a synthetic aperture radar image spectrum and its inversion. J. Geophys. Res., 96, 10 713–10 729, https://doi.org/10.1029/91JC00302.
- Crossref
- Search Google Scholar
- Export Citation
Hasselmann, K., R. K. Raney, W. J. Plant, W. Alpers, R. A. Shuchman, D. R. Lyzenga, C. L. Rufenach, and M. J. Tucker, 1985: Theory of synthetic aperture radar ocean imaging: A MARSEN view. J. Geophys. Res., 90, 4659–4686, https://doi.org/10.1029/JC090iC03p04659.
- Crossref
- Search Google Scholar
- Export Citation
He, Y., W. Perrie, T. Xie, and Q. Zou, 2004: Ocean wave spectra from a linear polarimetric SAR. IEEE Trans. Geosci. Remote Sens., 42, 2623–2631, https://doi.org/10.1109/TGRS.2004.836813.
- Crossref
- Search Google Scholar
- Export Citation
He, Y., H. Shen, and W. Perrie, 2006: Remote sensing of ocean waves by polarimetric SAR. J. Atmos. Oceanic Technol., 23, 1768–1773, https://doi.org/10.1175/JTECH1948.1.
- Crossref
- Search Google Scholar
- Export Citation
Heimbach, P., S. Hasselmann, and K. Hasselmann, 1998: Statistical analysis and intercomparison of WAM model data with global ERS-1 SAR wave mode spectral retrievals over 3 years. J. Geophys. Res., 103, 7931–7977, https://doi.org/10.1029/97JC03203.
- Crossref
- Search Google Scholar
- Export Citation
Holt, B., A. K. Liu, D. W. Wang, A. Gnanadesikan, and H. S. Chen, 1998: Tracking storm-generated waves in the northeast Pacific ocean with ERS-1 synthetic aperture radar imagery and buoy. J. Geophys. Res., 103, 7917–7929, https://doi.org/10.1029/97JC02567.
- Crossref
- Search Google Scholar
- Export Citation
Krogstad, H. E., and S. F. Barstow, 1999: Satellite wave measurement for coastal engineering applications. Coastal Eng., 37, 283–307, https://doi.org/10.1016/S0378-3839(99)00030-7.
- Crossref
- Search Google Scholar
- Export Citation
Kuo, Y. Y., L. G. Leu, and I. L. Kao, 1999: Directional spectrum analysis and statistics obtained from ERS-1 SAR wave images. Ocean Eng., 26, 1125–1144, https://doi.org/10.1016/S0029-8018(98)00058-4.
- Crossref
- Search Google Scholar
- Export Citation
Li, X. F., W. Pichel, M. He, S. Wu, K. S. Friedman, P. Clemente-Colon, and C. Zhao, 2002: Observation of hurricane generated ocean swell refraction at the Gulf Stream North Wall with the RADARSAT-1 synthetic aperture radar. IEEE Trans. Geosci. Remote Sens., 40, 2131–2142, https://doi.org/10.1109/TGRS.2002.802474.
- Crossref
- Search Google Scholar
- Export Citation
Lyzenga, D. R., 1986: Numerical simulation of synthetic aperture radar image spectra for ocean waves. IEEE Trans. Geosci. Remote Sens., GE-24, 863–872, https://doi.org/10.1109/TGRS.1986.289701.
- Crossref
- Search Google Scholar
- Export Citation
Melsheimer, C., M. Bao, and W. Alpers, 1998: Imaging of ocean waves on both sides of an atmospheric front by the SIR-C/X-SAR multifrequency synthetic aperture radar. J. Geophys. Res., 1031, 18 839–18 849, https://doi.org/10.1029/98JC00457.
- Crossref
- Search Google Scholar
- Export Citation
Monaldo, F. M., and D. R. Lyzenga, 1988: Comparison of Shuttle Imaging Radar-B ocean wave image spectra with linear model predictions based on aircraft measurements. J. Geophys. Res., 93, 15 374–15 388, https://doi.org/10.1029/JC093iC12p15374.
- Crossref
- Search Google Scholar
- Export Citation
Monaldo, F. M., and R. C. Beal, 1998: Comparison of SIR-C SAR wavenumber spectra with WAM model predictions. J. Geophys. Res., 103, 18 815–18 825, https://doi.org/10.1029/98JC01457.
- Crossref
- Search Google Scholar
- Export Citation
Monaldo, F. M., T. G. Gerling, and D. Tilley, 1993: Comparison of SIR-B SAR wave image spectra with wave model predictions: Implications on the SAR modulation transfer function. IEEE Trans. Geosci. Remote Sens., 31, 1199–1209, https://doi.org/10.1109/36.317443.
- Crossref
- Search Google Scholar
- Export Citation
Mouche, A., and B. Chapron, 2015: Global C-Band Envisat, RADARSAT-2 and Sentinel-1 SAR measurements in copolarization and cross-polarization. J. Geophys. Res. Oceans, 120, 7195–7207, https://doi.org/10.1002/2015JC011149.
- Crossref
- Search Google Scholar
- Export Citation
Parzen, E., 1962: Stochastic Processes. Holden-Day, 324 pp.
- Search Google Scholar
- Export Citation
Schuler, D. L., J. S. Lee, D. Kasilingam, and E. Pottier, 2004: Measurement of ocean surface slopes and wave spectra using polarimetric SAR image data. Remote Sens. Environ., 91, 198–211, https://doi.org/10.1016/j.rse.2004.03.008.
- Crossref
- Search Google Scholar
- Export Citation
Schulz-Stellenfleth, J., S. Lehner, and D. Hoja, 2005: A parametric scheme for the retrieval of two-dimensional ocean wave spectra from synthetic aperture radar look cross spectra. J. Geophys. Res., 110, C05004, https://doi.org/10.1029/2004JC002822.
- Crossref
- Search Google Scholar
- Export Citation
Schulz-Stellenfleth, J., T. Konig, and S. Lehner, 2007: An empirical approach for the retrieval of integral ocean wave parameters from synthetic aperture radar data. J. Geophys. Res., 112, C03019, https://doi.org/10.1029/2006JC003970.
- Crossref
- Search Google Scholar
- Export Citation
Tilley, D. G., and R. C. Beal, 1994: ERS-1 and Almaz estimates of directional ocean wave spectra conditioned by simultaneous aircraft SAR and buoy measurements. Atmos.–Ocean, 32, 113–142, https://doi.org/10.1080/07055900.1994.9649492.
- Crossref
- Search Google Scholar
- Export Citation
Wright, J. W., 1968: A new model for sea clutter. IEEE Trans. Antennas Propag., 16, 217–223, https://doi.org/10.1109/TAP.1968.1139147.
- Crossref
- Search Google Scholar
- Export Citation
Zhang, B., W. Perrie, and Y. He, 2010: Validation of RADARSAT-2 fully polarimetric SAR measurements of ocean surface waves. J. Geophys. Res., 115, C06031, https://doi.org/10.1029/2009JC005887.
- Search Google Scholar
- Export Citation

Share Link

Copy this link, or click below to email it to a friend

Email this content

or copy the link directly:

https://journals.ametsoc.org/view/journals/phoc/50/11/JPO-D-20-0045.1.xml

Link copied successfully

Journal of Physical Oceanography

Abstract
1. Introduction
2. Method
3. The tilt MTFs
4. The retrieval algorithm
5. The retrieved results and discussions
6. Conclusions
The Detailed Derivation of Eq. (9)

ProCite

RefWorks

Reference Manager

BibTeX

Zotero

EndNote

View raw image

Fig. 1.

The values of (a) $P_{k hhvv}^{S_{1}}$ , (b) $P_{k hhvv}^{S_{2}}$ , and (c) $P_{k hhvv}^{S_{3}}$ . Here, their values have been normalized with F_k.

View raw image

Fig. 2.

A comparison between the empirical and theoretical tilt MTFs.

View raw image

Fig. 3.

The empirical tilt MTFs for different wind speeds for (left) hh and (right) vv polarization.

View raw image

Fig. 4.

A comparison of the empirical $| T_{k hhvv}^{t_e} (u, θ_{i}, ϕ_{i}) |$ and $| T_{k hhvv}^{t} |$ for different wind speeds and wind directions.

View raw image

Fig. 5.

(a) Vertically (vv) polarized image of the SAR ID 1 data in Table 1; (b) attitude information of the SAR image.

View raw image

Fig. 6.

A comparison between (a) the first-guess spectrum and (b) the retrieved spectrum from the SAR ID 1 data.

View raw image

Fig. 7.

The corresponding 512 × 512 pixel size images selected from SAR ID 1 image: (a) hh-polarized NRCS, (b) vv-polarized NRCS, and (c) the new polarimetric image.

View raw image

Fig. 8.

View raw image

Fig. 9.

As in Fig. 5, but for the SAR ID 2 image.

View raw image

Fig. 10.

As in Fig. 7, but for the SAR ID 2 image.

View raw image

Fig. 11.

As in Fig. 8, but for the SAR ID 2 image.

View raw image

Fig. 12.

As in Fig. 5, but for the SAR ID 3 image.

View raw image

Fig. 13.

As in Fig. 7, but for the SAR ID 3 image.

View raw image

Fig. 14.

As in Fig. 8, but for the SAR ID 3 image.

	All Time	Past Year	Past 30 Days
Abstract Views	230	0	0
Full Text Views	427	198	21
PDF Downloads	438	166	23

Numerical Simulations of the 2013 Alberta Flood: Dynamics, Thermodynamics, and the Role of Orography

Authors:

Shawn M. Milrad

Kelly Lombardo

Eyad H. Atallah

, and

John R. Gyakum

Assessment of Vertical Mesh Refinement in Concurrently Nested Large-Eddy Simulations Using the Weather Research and Forecasting Model

Authors:

Jeffrey D. Mirocha

and

Katherine A. Lundquist

Diagnosing the Horizontal Propagation of Rossby Wave Packets along the Midlatitude Waveguide

Authors:

Gabriel Wolf

and

Volkmar Wirth

The Structure, Evolution, and Dynamics of a Nocturnal Convective System Simulated Using the WRF-ARW Model

Authors:

Benjamin T. Blake

David B. Parsons

Kevin R. Haghi

, and

Stephen G. Castleberry

Toward 1-km Ensemble Forecasts over Large Domains

Authors:

Craig S. Schwartz

Glen S. Romine

Kathryn R. Fossell

Ryan A. Sobash

, and

Morris L. Weisman

On the Nonlinear Mapping of an Ocean Wave Spectrum into a New Polarimetric SAR Image Spectrum

Abstract

Abstract

1. Introduction

2. Method

3. The tilt MTFs

4. The retrieval algorithm

5. The retrieved results and discussions

6. Conclusions

Acknowledgments

APPENDIX

The Detailed Derivation of Eq. (9)

REFERENCES

Share Link

AMS Publications

Get Involved with AMS

Affiliate Sites

Follow Us

Contact Us

Sections

References

Figures

Cited By

Metrics

Related Content

On the Nonlinear Mapping of an Ocean Wave Spectrum into a New Polarimetric SAR Image Spectrum

Abstract

Abstract

1. Introduction

2. Method

3. The tilt MTFs

4. The retrieval algorithm

5. The retrieved results and discussions

6. Conclusions

Acknowledgments

APPENDIX

The Detailed Derivation of Eq. (9)

REFERENCES

Share Link

Headings

References

Figures

Cited By

Metrics

Related Content

AMS Publications

Get Involved with AMS

Affiliate Sites

Follow Us

Contact Us