Development of an Image De-Noising Method in Preparation for the Surface Water and Ocean Topography Satellite Mission

In a near future, the Surface Water Ocean Topography (SWOT) mission will provide images of altimetric data at kilometric resolution. This unprecedented 2-dimensional data structure will allow the estimation of geostrophy-related quantities that are essential for studying the ocean surface dynamics and for data assimilation uses. To estimate these quantities, i.e. compute spatial derivatives of the Sea Surface Height (SSH) measurements, the small-scale noise expected to aﬀect the SWOT data must be smoothed out while minimizing the loss of relevant, physical SSH information. This paper introduces a new technique for de-noising the future SWOT SSH images. The de-noising model is formulated as a regularized least-square problem with a Tikhonov regularization based on the ﬁrst, second, and third-order derivatives of SSH. The method is implemented and compared to other, convolution-based ﬁltering methods with boxcar and Gaussian kernels. This is performed using a large set of pseudo-SWOT data generated in the Western Mediterranean Sea, from a 1/60 ◦ simulation and the SWOT simulator. Based on Root Mean Square Error and spectral diagnostics, our de-noising method shows a better performance than the convolution-based methods. We ﬁnd the optimal parametrization to be when only the second-order SSH derivative is penalized. This de-noising reduces the spatial scale resolved by SWOT by a factor of 2, and at 10 km wavelengths the noise level is reduced by 10 4 and 10 3 for Summer and Winter respectively. This is encouraging for the processing of the future SWOT data.


1
The Surface Water Ocean Topography (SWOT) [1] mission will provide an unprecedented 2 two-dimensional view of ocean surface topography at a pixel resolution of 2 km. The launch 3 is scheduled for 2021. SWOT's wide-swath altimeter, based upon SAR interferometry tech-most relevant conclusions, discusses them, and suggests possible future research paths.
2. Variational de-noising of SWOT images with penalization of derivatives 68 2.1. Formulation of the image de-noising problem 69 The primary purpose of image de-noising here is to allow the computation of first and second-70 order SSH spatial derivatives of SWOT data as accurately as possible. The two reasons, already 71 mentioned in the introduction, are: (i) these quantities represent geostrophic velocities and rel-72 ative vorticity, respectively, whose estimation is central to the success of SWOT mission; and 73 (ii) these quantities can be needed to draw maximum benefits from the assimilation of SWOT 74 data into ocean circulation models [14,15]. We therefore propose a method that explicitly 75 constrains these derivatives.

77
The proposed de-noising model is formulated as a regularized least-square problem with a Tikhonov regularization. The de-noised SWOT image h is searched for by minimizing the following cost function: where represents the L 2 -norm, h obs is the original noisy image (i.e., our observation, the 78 pseudo-SWOT data) , ∇ = (∂/∂x, ∂/∂y) is the gradient operator, and ∆ = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 79 is the Laplacian operator. Letter m and sign • represent a mask and the entrywise matrix 80 product, respectively. They can be ignored for the present and the next sub-section: their role 81 is discussed in Section 2.3 below. The regularization terms impose regularity constraints on 82 geostrophic velocity, vorticity, and vorticity gradient, respectively. Parameters λ 1 λ 2 and λ 3 83 must be prescribed. The search for their optimal values is reported in Section 3.3. The variational problem displayed in eq. 1 is solved using a gradient descent method [23]. The gradient of J is written: so that the solution can be reached after convergence of the following iterations: Stability of iterations is guaranteed if τ ≤ (1 + 8λ 1 + 64λ 2 + 512λ 3 ) −1 . In practice, it is 87 taken equal to this value. Two improvements on the method's implementation accelerate the  The Laplacian operator is discretized with finite differences using the five-point stencils of where N is the number of pixels and i a pixel index. Single image RMSEs are then averaged out the test of a de-noising method with a specific set of parameters results in 9 RMSE values.

196
To evaluate the improvement after the application of the different de-noising techniques and 197 parameters, we also calculate the percentage of the initial RMSE left. We calculate this RMSE residual (RM SE r ) as: where h is the de-noised field and h obs the original noisy field (SSH obs K or SSH obs).

201
The spatial spectra of the de-noised SWOT SSH are compared with the spectra of the noise- MSR is computed as:

210
where N k is the number of wavelengths considered; P SD j (h true ) and P SD j (h) are the power 211 spectral density values at wavelength j for the original, noise-free SWOT field and the de-212 noised SWOT field, respectively. The considered wavelengths span the interval from 9 km, the 213 approximate effective resolution of NAtl60, to 200 km, the size of images along-track. MSR is 214 defined above so that the best score is 0.  The orders of magnitude of the terms ∇h 2 , ∆h 2 and ∇∆h 2 composing the cost func-230 tion (eq. 1) are estimated to coarsely scale the parameters λ 1 , λ 2 and λ 3 . The rationale is, for 231 one of these terms (with its weight) to have some impact on the solution, it must be of an order to be in the range 1 to 100 in the same region. This has been estimated using the noise-free field. with the first order term only as the λ 1 -method. We similarly refer to the λ 2 -method and to 251 the (λ 1 + λ 2 )-method when the first two penalization terms are considered, and so on.

253
For each scenario, a two-step procedure is implemented to identify an optimal set of pa-

Optimal de-noising method
In this Section, the optimal de-noising method is searched for based on the RMSE and MSR 262 scores described in Section 3. We investigate the KaRIn-noise-only scenario, then the all noises 263 scenario, and finally have a closer look at the method identified as optimal. As it becomes  which shows that other methods do not beat it clearly, but also by the fact that it is much 295 easier to parametrize a single-parameter method rather than a two or three-parameter method.  (table 2). This is obviously due to the spatially 299 correlated component of the noise (see Figure 2), which is not filtered out by any of the methods 300 used here. Other approaches must be used to remove the correlated noise in order to obtain 301 more accurate estimates. In terms of MSRs, the methods involving λ 2 perform significantly better than the others, 315 including the λ 3 -and the Gaussian methods. These last two exhibit MSR larger than the others 316 by factors of 1.5 to 4. In Winter, the λ 2 -method is a little less effective than the multi-parameter 317 methods, with a MSR twice as large.      5. Retrieved SWOT fields and spatial spectra      To remove the small-scale SWOT noise, we propose a de-noising method that performs 419 better than conventional convolution-based methods both in terms of RMSE (physical space 420 diagnostic) and spectra. The method, which originates from image processing applications, is  [41] to significantly reduce the impact of the geometrically structured, highly correlated SWOT 475 errors (roll, phase, timing, and baseline errors).

554
Laplacian are computed using finite differences, following the method proposed by [22]. We note h the image of size N x × N y . In a first step, the two components of the gradient are computed as (i = 1, ..., N x ; j = 1, ..., N y ): In a second step, Laplacian is computed as the divergence of the gradient. Divergence of vector a = (a x , a y ) is computed as: div(a) = b h k+1 = h k + τ m • (h obs − y k ) + λ 1 ∆y k −λ 2 ∆∆y k + λ 3 ∆∆∆y k t k+1 = (1 + 1 + 4t 2 k )/2 Appendix C. Calculation of spatial spectra 595 The spatial spectra used as one of the scores for the de-noising parameterizations are cal-596 culated as follows: Gaussian method, the laplacian is less noisy than with our method, but the gradient is over-