Internal wave parameters retrieval from space-borne SAR image

Abstract

Based on oceanic internal wave SAR imaging mechanism and the microwave scattering imaging model for oceanic surface features, we developed a new method to extract internal wave parameters from SAR imagery. Firstly, the initial wind fields are derived from NCEP reanalysis data, the sea water density and oceanic internal wave pycnocline depth are estimated from the Levites data, the surface currents induced by the internal wave are calculated according to the KDV equation. The NRCS profile is then simulated by solving the action balance equation and using the sea surface radar backscatter model. Both the winds and internal wave pycnocline depth are adjusted by using the dichotomy method step by step to make the simulated data approach the SAR image. Then, the wind speed, pycnocline depth, the phase speed, the group velocity and the amplitude of internal wave can be retrieved from SAR imagery when a best fit between simulated signals and the SAR image appears. The method is tested on one scene SAR image near Dongsha Island, in the South China Sea, results show that the simulated oceanic internal wave NRCS profile is in good agreement with that on the SAR image with the correlation coefficient as high as 90%, and the amplitude of oceanic internal wave retrieved from the SAR imagery is comparable with the SODA data. Besides, the phase speeds retrieved from other 16 scene SAR images in the South China Sea are in good agreement with the empirical formula which describes the relations between internal wave phase speed and water depths, both the root mean square and relative error are less than 0.11 m·s⁻¹ and 7%, respectively, indicating that SAR images are useful for internal wave parameters retrieval and the method developed in this paper is convergent and applicable.

Keywords

synthetic aperture radar / internal wave / retrieval

Introduction

Oceanic internal waves are major dynamic features which travel within the interior of the ocean and play an important role in ocean mixing, acoustic signal propagation, sediment re-suspension, and cross-shore pollutant transport ( Fan, 2002; Cai et al., 2003). Oceanic internal wave fields have been measured by instruments deployed in the ocean, like temperature and salinity sensors or current meters, or by acoustic instruments like sonar. However, information on internal wave fields can also be extracted from their sea surface manifestations ( Zhao, 2004).

Recently, Synthetic Aperture Radar (SAR) has demonstrated the potential to obtain high-resolution ocean surface images from which internal wave features could be identified, and SAR technique for internal wave parameters retrieval has been investigated for many years ( Alpers, 1985; Liu et al., 1998; Lai, 1999; Li et al., 2000; Yang et al., 2003; Fan et al., 2010).

At present, the distribution and the wavelength of internal wave could be directly extracted from SAR images ( Rodenas and Garello, 1998, Alpers et al., 2005; Zheng et al., 2007), and also the internal wave parameters, such as pycnocline depth, phase speed, group velocity and amplitude could also be retrieved from SAR images ( Porter and Thompson 1999; Lai, 1999; Zhao et al., 2004; Le Caillec, 2006). This ability to provide quantitative estimates of internal wave parameters makes SAR images especially significant for ocean internal wave research.

By assuming that the local semidiurnal tide period is the generating source for these waves, and by measuring the distance between the wave packets, Li et al. (2000) calculated the group velocity of the internal waves directly from SAR images and derived both the mixed-layer depth and the amplitude under the assumption that the ocean is a two-layer finite depth model ( Li et al., 2000; Yang et al., 2003). Also, Zhao et al. ( 2004) extracted upper layer depth and density difference from polarity conversion of internal waves observed in one RADARSAT-1 SAR image taken over the northeastern South China Sea. Porter and Thompson ( 1999) estimated the pycnocline depth and the density of the surface layer from the dispersion relation obtained from a two-layer fluid model.

However, the above inversion method’s retrieval is constrained by the accuracy of the velocity of the internal waves, and in the case when the local semidiurnal tide period is not exactly 12.4 hours, CTD et al. data will be needed to calculate internal wave speed ( Gan et al., 2007). Also, it is impossible to use the above SAR techniques to retrieve internal wave parameters from just one train of internal waves within one SAR image without the signature of polarity conversions between depression and elevation internal waves, or from one SAR image containing several trains of internal waves, but the local semidiurnal tide period is not the generating source for these waves. Besides, owing to the lack of a fully perfect microwave scattering imaging model for oceanic surface features, the above methods have not used the information of the profiles of internal waves.

Brandt et al. ( 1999) have studied the feasibility of retrieving ocean internal wave parameters from the internal wave profile in the SAR images based on the microwave scattering imaging model for oceanic surface features. Lai ( 1999) has extracted the ocean internal wave peak surface currents information through the simulated signals and the normalized radar crossing section (NRCS) from SAR images, and the peak surface currents estimates are within 15% of the observed in situ data. Le Caillec ( 2006) have also tried to estimate the pycnocline depth from profiles of the SAR signature of internal waves.

The above methods could be applied without the assumption that the local semidiurnal tide period is the generating source for these waves, but have the crucial need for the radar microwave imaging model for ocean surface. So the study of the internal wave parameters retrieval from SAR images by combining both the radar microwave imaging model and the internal waves profile NRCS in an SAR image is still in the exploratory stage.

Based on SAR imaging of internal waves, we developed a new method for internal wave parameters retrieval from space-borne SAR imagery through iteratively fitting the simulated NRCS with SAR image data. Then the wind speed, pycnocline depth, phase speed, group velocity and amplitude of the internal wave can be retrieved from SAR imagery when a best fit between simulated signals and the SAR image appears. This method is applied to several case studies in the South China Sea. The results show that the internal wave parameters, calculated from the SAR images, are in good agreement with the SODA data or the empirical formula of oceanic internal wave.

SAR imaging mechanism of oceanic internal wave

The SAR cannot directly observe ocean internal waves, as radar waves can penetrate seawater only a few centimeters. However, the convergent and divergent areas over the surface modulated by the oceanic internal wave will result in surface roughness variation which can be captured in the SAR image. This surface roughness variation reflects the internal wave characteristics, and therefore, we can detect the rich information about internal waves from SAR images under favorable meteorological and hydrodynamic conditions (wind conditions 2 −10 m·s⁻¹) ( Alpers, 1985; Thompson and Gasparovic, 1986).

Generally, imaging of oceanic internal waves by SAR can be described by hydrodynamics modulation theory, which consists of three physical processes ( Alpers and Hennings, 1984; Alpers, 1985; Thompson and Gasparovic, 1986). Firstly, propagation of ocean internal waves results in the modulation of surface flow velocity. Secondly, the modulation of surface flow velocity induces the variations of the surface wave spectrum. Finally, the variation of the surface spectrum causes modulation in radar backscattering. Based on above three processes, oceanic internal wave can be imaged using SAR ( Jackson and Apel, 2004).

SAR microwave scattering imaging ocean surface

A few imaging models for simulating radar microwave scattering from the ocean surface were developed based on the hydrodynamic theory and radar microwave scattering models, such as the Model for Fully Two-dimensional Simulation (M4S), Radar Imaging Model (RIM), Weak Hydrodynamic Interaction Theory (WHIT) and ERIM Ocean Model (EOM) ( Lyzenga, 2003; Romeiser, 2005). In this paper, we use the microwave scattering imaging model for oceanic surface features of M4S developed by Dr. R. Romeiser, a software toolbox for numerical simulations of oceanic surface features with two components: wave-current interaction and radar microwave scattering ( Romeiser et al., 1994; Romeiser and Alpers, 1997b). For the wave-current interaction, the surface wave spectra modulated by sea surface winds and surface currents are computed from the wave action balance equation by the ray tracing method ( Romeiser and Alpers, 1997a). The radar microwave backscatter model is an improved composed-surface model, which can calculate the NRCS using the modulated surface wave spectra and radar parameters etc.

Wave action balance equation

The wave action balance equation describes the evolution of the energy of a wave packet traveling through a slowly varying surface current field ( Alpers and Hennings 1984) and is expressed as

(1) {\displaystyle \frac{dA}{dt}}={\displaystyle \frac{\partial \mathrm{\Omega }}{\partial k_x}}{\displaystyle \frac{\partial A}{\partial x}}+{\displaystyle \frac{\partial \mathrm{\Omega }}{\partial k_y}}{\displaystyle \frac{\partial A}{\partial y}}-{\displaystyle \frac{\partial \mathrm{\Omega }}{\partial x}}{\displaystyle \frac{\partial A}{\partial k_x}}-{\displaystyle \frac{\partial \mathrm{\Omega }}{\partial y}}{\displaystyle \frac{\partial A}{\partial k_y}}=S,

where A\left(\overrightarrow{r},\overrightarrow{k},t\right) is the action spectral density of the wave packet, \mathrm{\Omega }\left(\overrightarrow{r},\overrightarrow{k},t\right) the apparent frequency in the moving medium, \overrightarrow{r}=(x,y) the parameters of the space, \overrightarrow{k}=(k_x,k_y) the wave number vectors of the wave packet, and S\left(\overrightarrow{r},\overrightarrow{k},t\right) a source function.

In the microwave scattering imaging model, the spatially varying wave spectra were further computed by numerical integration of the wave action balance equation through the ray-tracing method.

Radar microwave backscatter model

An improved composite surface model for calculation of the ocean surface NRCS was developed in the 1990s ( Romeiser et al., 1994; Romeiser and Alpers, 1997b). This kind of radar microwave backscatter model is also called a “three-scale model”, because in the model, the NRCS has three components: Bragg scattering, longer-wave scattering, and short-wave scattering which is modulated by oceanic internal wave-induced surface current velocity gradients. The model can reflect the effects of surface current on the NRCS, and its results are in good agreement with the observed data ( Romeiser and Alpers, 1997a). It is appropriate to use the improved composite surface model to retrieve internal wave parameters or shallow water depths from SAR images ( Romeiser and Alpers, 1997a).

In the microwave scattering imaging model, radar backscattering in the presence of spatially varying surface currents induced by an oceanic internal wave is calculated by using the wave spectra from the wave action balance equation and the radar parameters.

Method for internal wave parameters retrieval by SAR

The evolution of nonlinear solitary internal wave trains on a continental shelf has been formulated and the wave amplitude η can be expressed by the following Korteweg-de Vries (KdV) equation ( Ostrovsky and Stepanyants, 1989; Liu et a1., l998; Fan, 2002)

(2) {\displaystyle \frac{\partial \eta }{\partial t}}+(C_{\mathrm{0}}+\alpha \eta +{\alpha }_{\mathrm{1}}{\eta }^{\mathrm{2}}){\displaystyle \frac{\partial \eta }{\partial x}}+\beta {\displaystyle \frac{{\partial }^{\mathrm{3}}\eta }{\partial x^{\mathrm{3}}}}+\kappa \eta -{\displaystyle \frac{\epsilon }{\mathrm{2}}}{\displaystyle \frac{\partial {\eta }^{\mathrm{2}}}{\partial x^{\mathrm{2}}}}=\mathrm{0},

where the parameters C₀, α, α₁, β, κ and ε are the coefficients for the linear, nonlinear, high-order nonlinear, dispersion, shoaling, and dissipation effects. In a two-layer system with mixed layer depth (h₁) and bottom layer thickness (h₂), mixed layer density (ρ₁) and bottom layer density (ρ₂), the above coefficients are

(3) \alpha ={\displaystyle \frac{\mathrm{3}{C}_{\mathrm{0}}(h_{\mathrm{1}}-h_{\mathrm{2}})}{\mathrm{2}{h}_{\mathrm{1}}{h}_{\mathrm{2}}}},

(4) \beta ={\displaystyle \frac{C_{\mathrm{0}}{h}_{\mathrm{1}}{h}_{\mathrm{2}}}{\mathrm{6}}},

(5) C_{\mathrm{0}}=\sqrt{{\displaystyle \frac{g\mathrm{\Delta }\rho h_{\mathrm{1}}{h}_{\mathrm{2}}}{\rho (h_{\mathrm{1}}+h_{\mathrm{2}})}}},

where g is the gravity acceleration, ρ is the density of the water, and Δρ=ρ₂−ρ₁.

When the nonlinear effect is in equilibrium with the dispersion effect, the steady-state independent wave solution for Eq. (2) is

(6) \eta (x,t)={\eta }_{\mathrm{0}}{\mathrm{sech}}^2\left({\displaystyle \frac{x-C_{p}t}{l}}\right),

(7) U_x=\pm \left|{\displaystyle \frac{C_{\mathrm{0}}{\eta }_{\mathrm{0}}}{h_1}}\right|{\mathrm{sech}}^2\left({\displaystyle \frac{x-C_{p}t}{l}}\right),

where η₀ is the maximum wave amplitude, C_p is the phase speed, l is the half-width of the solitary wave. U_x is the velocity of the surface current induced by the internal wave in the x direction, and the positive and negative are taken for the depression and elevation internal waves respectively. Both C_p and l are described as

(8) C_p=C_{\mathrm{0}}\left[\mathrm{1}+{\displaystyle \frac{{\eta }_{\mathrm{0}}(h_{\mathrm{2}}-h_{\mathrm{1}})}{\mathrm{2}{h}_{\mathrm{1}}{h}_{\mathrm{2}}}}\right],

(9) l={\displaystyle \frac{\mathrm{2}{h}_{\mathrm{1}}{h}_{\mathrm{2}}}{\sqrt{\mathrm{3}{\eta }_{\mathrm{0}}\left|h_{\mathrm{2}}-h_{\mathrm{1}}\right|}}}.

For the KDV equation which satisfies shallow water stereotypes of the solitary internal wave, the surface currents of internal waves will be the function of both the pycnocline depth and the amplitude of the internal wave according to Eq. (7). The amplitude could also be expressed by the pycnocline depth and the half-width of the solitary wave. Then based on the Eqs. (5), (7), (8), and (9), the formula which describes the relations between the surface currents and the pycnocline depth of the internal wave is given by,

(10) U_x=\pm \left|{\displaystyle \frac{\mathrm{4}{h}_{\mathrm{1}}{h}_{\mathrm{2}}^{\mathrm{2}}}{\mathrm{3}{l}^{\mathrm{2}}(h_{\mathrm{2}}-h_{\mathrm{1}})}}\sqrt{{\displaystyle \frac{g\mathrm{\Delta }\rho }{\rho (h_{\mathrm{1}}+h_{\mathrm{2}})}}}\right|{\mathrm{sech}}^2\left({\displaystyle \frac{x^{\prime }}{l}}\right).

Depth h can be obtained from terrain data or bathymetric survey data, and Eq. (11) will be obtained by inserting the equation h₂=h−h₁ into the above equation,

(11) U_x=\pm \left|{\displaystyle \frac{\mathrm{4}{h}_{\mathrm{1}}{(h-h_{\mathrm{1}})}^{\mathrm{2}}}{\mathrm{3}{l}^{\mathrm{2}}(h-\mathrm{2}{h}_{\mathrm{1}})}}\sqrt{{\displaystyle \frac{g\mathrm{\Delta }\rho }{\rho h}}}\right|{\mathrm{sech}}^2\left({\displaystyle \frac{x^{\prime }}{l}}\right).

The density of the water ρ will be estimated from the Levites data, especially for ρ₁, ρ₂ and Δρ will also be calculated by

(12) {\rho }_{\mathrm{1}}={\displaystyle \frac{\mathrm{1}}{h_{\mathrm{1}}}}{\displaystyle {\int }_{\mathrm{0}}^{h_{\mathrm{1}}}\rho (z)}dz,

(13) {\rho }_{\mathrm{2}}={\displaystyle \frac{\mathrm{1}}{h_{\mathrm{2}}}}{\displaystyle {\int }_{h_{\mathrm{1}}}^h\rho (z)}dz,

(14) {\displaystyle \frac{\mathrm{\Delta }\rho }{\rho }}={\displaystyle \frac{{\rho }_{\mathrm{2}}-{\rho }_{\mathrm{1}}}{({\rho }_{\mathrm{2}}+{\rho }_{\mathrm{1}})/\mathrm{2}}}.

The half-width of the solitary wave l could be directly measured by the distance D between the center of the bright band and that of the dark band in SAR imagery ( Lin et al., 2010).

(15) l=D/\mathrm{1.32}.

Eq. (11) suggests that the surface currents will be determined by h_1, and then the relative NRCS (Δσ⁰ (dB)) of internal wave SAR images is proportional to the internal wave pycnocline depth h_{\mathrm{1}} , where \mathrm{\Delta }{\sigma }^{\mathrm{0}}={\sigma }^{\mathrm{0}}-{\sigma }_{\mathrm{0}}^{\mathrm{0}},{\sigma }^{\mathrm{0}} is the ocean surface NRCS for the internal wave, {\sigma }_{\mathrm{0}}^{\mathrm{0}} is the surface background NRCS.

Alpers (1985) has also shown that the relative NRCS is zero at the location of surface current velocity peaks. In other words, the backscattering is equal to the surface background NRCS ( {\sigma }_{\mathrm{0}}^{\mathrm{0}} ). So it is mainly affected by sea surface winds, and the surface wind speed could be estimated by the NRCS at the peak location ( {\sigma }_M^{\mathrm{0}} (dB) or {\sigma }_{\mathrm{0}}^{\mathrm{0}} (dB)) ( Alpers, 1985). In our study, sea surface wind directions are derived from the National Centers for Environmental Prediction (NCEP) reanalysis data or from in-situ wind information.

So the inversion process starts with both initial guess parameters of h₁ and wind speeds, with radar parameters and wind direction etc. that are previously known and fixed into the microwave scattering imaging model for oceanic surface features and both wind speeds and the pycnocline depth adjusted. When all of this information is inputted into the microwave scattering imaging model for oceanic surface features, the signals induced by the internal wave can be simulated.

The appropriate initial wind speed and the pycnocline depth h₁ will speed up the convergence of simulation. In this paper, the first guessed pycnocline depth h₁ could be estimated from the Levites data and the initial wind speed is estimated by COMD-4 model ( Lehner et al., 2000; Portabella and Stoffelen, 2002), where the wind direction comes from the NCEP reanalysis wind data.

During the iterative retrieval procedure, the wind speeds and the pycnocline depth are adjusted using the dichotomy method at every iteration until the best match is achieved between SAR observations and simulated data. We also define three criteria for the best match, i.e., the correlation coefficients analysis between SAR observations and the simulated signal, the difference ε analysis of {\sigma }_M^{\mathrm{0}} in SAR images and {\sigma }_{SM}^{\mathrm{0}} in simulated images, and another difference ε analysis of relative NRCS of the elongated bright and dark bands (R_BD (dB)) in SAR profile data and S_BD in the simulated profile data. Finally, the winds and the pycnocline depth h₁ could be retrieved, and are the optimum estimations.

Figure 1 gives the flow chart of internal wave parameters retrieval by SAR.

Results of internal wave parameters retrieval by SAR

The method for retrieving internal wave parameters from SAR images described above has been applied to one scene of SAR imagery near Dongsha Island, in the South China Sea. Figure 2 shows the ERS-2 SAR image with oceanic internal wave characteristics of elongated bright and dark features in the test site, representing the internal solitary wave packets in each train.

In order to remove the speckle noise and get the real signal of the internal wave modulation, both the Lee and mean filter with 5 × 5 filter windows ( Fan et al., 2010) were used during the SAR image pre-processing. The SAR data along section A→B from the image (Fig. 2) represents the NRCS profile and the internal wave parameters would be calculated along the section A→B. The SAR image was acquired at 14:10 UTC on 23 June, 1998, the pixel size of the SAR image is about 12.5 m, and the SAR is on the ascending track with working wavelength of 5.66 cm and incidence angle of radar beam of 21.4°.

The local wind direction over the SAR imaging area is obtained from NCEP reanalysis data by Kriging interpolation, indicating south-easterlies at 12:00 UTC on June 23, 1998. During the internal wave parameters retrieval process, the initial wind speed is 2.81 m·s⁻¹ estimated using CMOD-4 model and the pycnocline depth h₁, the density of the upper and lower layer sea water ρ₁, ρ₂, and Δρ estimated from the Levites data are listed in Table 1, these initial parameters can speed up the convergent rate of iteration. The water depth in the test area is 319 m based on bathymetric survey data. And the internal wave surface currents could be estimated by Eq. (11)

After several iteration steps shown on the flowchart given in Fig. 1, the final estimate of wind speed is 1.41 m·s⁻¹, which is comparable with the wind speed obtained from NCEP reanalysis data 1.78 m·s⁻¹, and the pycnocline depth h₁ is 97 m. Table 2 and Table 3 give the iterative process of adjusting the sea surface wind speeds and the pycnocline depth h₁. Both tables also give the difference ε analysis between {\sigma }_M^{\mathrm{0}} (or {\sigma }_{\mathrm{0}}^{\mathrm{0}} ) and {\sigma }_{SM}^{\mathrm{0}} , R_BD and S_BD.

Figure 3 gives both the NRCS profile of the internal wave SAR image (the solid line), and the simulated NRCS for the internal wave (the dashed line). They have a correlation coefficient as high as 96.6%. Combining with the above difference \epsilon analysis, these indicate that the simulated NRCS and the observations are in excellent agreement and the iterative retrieval method is convergent and applicable.

Although the amplitude has not been retrieved from the in-situ measurements or reanalysis data to validate the retrieved results, the amplitude of 23.9 m estimated from the SAR image is within the range of prior research reports of 20−50 m. Figure 4 presents the temperature and salinity profiles at 116.5°E, 21.0°N on 23 June 1998, which is in the vicinity of the internal waves observed in the SAR image, from the data of CARTON-GIESE SODA Version 2.0.2-4. These data show that the pycnocline depth h₁ is about 95 m, which suggests that the pycnocline depth h₁ calculated from the SAR image is consistent with the SODA data assimilation reanalysis.

In this paper, we also demonstrated the retrieval results from another 16 SAR images at 19 different research sites in the South China Sea. Following the iterative procedure given in Fig. 1, the internal wave parameters could be calculated from the SAR image (Table 4).

Zhang (2010) proposed an empirical formula which describes the relations between internal wave phase speed and water depths and is expressed as,

(16) C_p=\mathrm{0.08}*\sqrt{H}.

To validate the results retrieved by SAR, we compare the internal wave parameters C_p retrieved from SAR image with the phase speed calculated from the empirical formula (Fig. 5). The error of the root mean square is 0.09 m·s⁻¹ and the relative error is 6.9%, which shows that the retrieval results agree well the empirical formula.

Discussion and conclusions

A new method for estimating internal wave parameters, including wind speed, pycnocline depth, phase speed, group velocity and amplitude, using space-borne SAR imagery is introduced based on SAR internal wave imaging mechanism and the radar microwave scattering imaging model. The convergence and applicability of this method is tested by difference ε analysis of three criteria between the simulated NRCS profile and that from the SAR image, such as correlation coefficient etc., and also the comparison demonstrates that the internal wave parameters retrieved from SAR images are comparable with the SODA data assimilation reanalysis and are in good agreement with the empirical formula.

Taking advantage of the iterative procedure, we do not need to assume the dynamic model of the ocean internal waves when doing the retrieval. Thus, this method can be applied to any internal wave case under any stratification and any water depth. Also, since the microwave scattering imaging model is applied during the iterative process, this will reduce the speckle noise in SAR image. However, this method employs just a one dimensional model and works well only under the condition in which the pycnocline depth is not close to half of the water depth.

Applications of the method to SAR images in the South China Sea indicate that the simulated internal wave NRCS is consistent with the SAR data. Their correlation coefficient is as high as 90%. The pycnocline depths h₁ calculated from SAR images are consistent with the SODA data assimilation reanalysis. Also, when the phase speeds C_p retrieved from SAR images are compared with the empirical formula, the root mean square is within 0.11 m·s⁻¹ with relative errors less than 7%, indicating the method introduced in this paper is reliable and practical. There are several sources of error inherent in the difference comparison. The numerical models used in this study are not perfect and thus introduce some errors. There are also errors associated with the calculation procedure, the initial conditions, the radar microwave scattering imaging model and the SAR image quality.

Although the new method for internal wave parameters retrieval by space-borne SAR imagery introduced in this paper may not be precisely accurate, such a technique is the best way to retrieving the internal wave parameters in remote coastal areas where the internal wave group velocity could not be easily detected or where just one internal solitary wave existed. In the future, in-situ measurements and studies are required to address the validation of the method to overcome the limitations of the present method. These could be operated in most coastal areas day and night.

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