Improving Surface Geostrophic Current from a GOCE-Derived Mean Dynamic Topography Using Edge-Enhancing Diffusion Filtering

Abstract

With increased geoid resolution provided by the gravity and steady-state ocean circulation explorer (GOCE) mission, the ocean’s mean dynamic topography (MDT) can be now estimated with an accuracy not available prior to using geodetic methods. However, an altimetric-derived MDT still needs filtering in order to remove short wavelength noise unless integrated methods are used in which the three quantities are determined simultaneously using appropriate covariance functions. We studied nonlinear anisotropic diffusive filtering applied to the ocean´s MDT and a new approach based on edge-enhancing diffusion (EED) filtering is presented. EED filters enable controlling the direction and magnitude of the filtering, with subsequent enhancement of computations of the associated surface geostrophic currents (SGCs). Applying this method to a smooth MDT and to a noisy MDT, both for a region in the Northwestern Pacific Ocean, we found that EED filtering provides similar estimation of the current velocities in both cases, whereas a non-linear isotropic filter (the Perona and Malik filter) returns results influenced by local residual noise when a difficult case is tested. We found that EED filtering preserves all the advantages that the Perona and Malik filter have over the standard linear isotropic Gaussian filters. Moreover, EED is shown to be more stable and less influenced by outliers. This suggests that the EED filtering strategy would be preferred given its capabilities in controlling/preserving the SGCs.

Appendix: Practical Implementation of the Filters

1.1 1.1 Gaussian Filter

The Gaussian filter was applied in the spectral domain as proposed by Jekeli (1981). This is by filtering the spherical harmonic coefficients W _n by,

$$W_{n + 1} = - \frac{2l - 1}{b}W_{n} + W_{n - 1}$$

(7)

where n denotes the harmonic degree; W ₀  = 1/2π, W ₁  = W ₀ (1 + exp(−2b))/(1 − exp(−2b)) − W ₀ /b, and b = log2/(1 − cos(r/a)), r being the half wavelength and a the mean equatorial radius of the Earth.

1.2 1.2 Perona and Malik Filter

Following the same notation in Sect. 2, the PMF sets (2) as

$$I_{t} = {\text{div}}(g(\left| {\nabla I} \right|^{2} )\cdot\nabla I)$$

(8)

where g(.) was defined in (3). A simple discretization of (8) is provided by Perona and Malik (1990):

$$I_{i,j}^{t + 1} = I_{i,j}^{t} + {\text{d}}t[c_{N} \varDelta_{N} I_{i,j} + c_{S} \varDelta_{S} I_{i,j} + c_{E} \varDelta_{E} I_{i,j} + c_{W} \varDelta_{W} I_{i,j} ]^{t}$$

(9)

which defines the iterative process to filter the original grid I ₀  = MDT for any location (i, j). In (9), $\varDelta_{m}$ indicates the nearest neighbor differences for m = N,S,E,W (north, south, east, and west; e.g. $\varDelta_{N} = I_{i + 1,j} - I_{i,j}$). The conduction coefficients cm are estimated from (3), being $c_{m} = g(\varDelta_{m} )$. Here the time-step is required to be 0 < dt ≤ 1/4 for the numerical scheme to be stable. It was set to dt = 0.1 as for the EED.

1.3 1.3 Edge Enhancing Diffusion

To perform the EED filter we first determine the structure tensor J ₀ and convolve it with a Gaussian filter K _σ to obtain the regularized structure tensor J _σ . In this case, the pre-filtering implied by K _σ is carried out in the geographical domain and applied to MDTs determined as described in data section. Weighting the eigenvalues from J _σ by (3) the diffusion tensor D is determined as described in Sect. 2. Then, if

$$D = \left( {\begin{array}{*{20}c} a & b \\ b & c \\ \end{array} } \right),$$

(10)

the diffusion Eq. (2) can be rewritten as

$$I_{i,j}^{t + 1} = I_{i,j}^{t} + {\text{d}}t[\partial_{x} (a\partial_{x} I_{i,j} ) + \partial_{x} (b\partial_{y} I_{i,j} ) + \partial_{y} (b\partial_{x} I_{i,j} ) + \partial_{y} (c\partial_{y} I_{i,j} )]^{t}$$

(11)

which defines the iterative process to filter the original grid I ₀ = MDT for any location (i, j). Partial differential equations in (11) were discretized as following central differences. That is,

$$\partial_{x} (a\partial_{x} I_{i,j} ) = \frac{1}{2}\left[ {a_{i + 1,j} \frac{{I_{i + 1,j} - I_{i,j} }}{2} + a_{i - 1,j} \frac{{I_{i - 1,j} - I_{i,j} }}{2}} \right]$$

(12)

$$\partial_{x} (b\partial_{y} I_{i,j} ) = \frac{1}{4}\left[ {b_{i + 1,j} \frac{{I_{i + 1,j + 1} - I_{i + 1,j - 1} }}{4} + b_{i - 1,j} \frac{{I_{i - 1,j + 1} - I_{i - 1,j - 1} }}{4}} \right]$$

(13)

$$\partial_{y} (b\partial_{x} I) = \frac{1}{4}\left[ {b_{i,j + 1} \frac{{I_{i + 1,j + 1} - I_{i - 1,j + 1} }}{4} + b_{i,j - 1} \frac{{I_{i + 1,j - 1} - I_{i - 1,j - 1} }}{4}} \right]$$

(14)

$$\partial_{y} (c\partial_{y} I) = \frac{1}{2}\left[ {c_{i,j + 1} \frac{{I_{i,j + 1} - I_{i,j} }}{2} + c_{i,j - 1} \frac{{I_{i,j - 1} - I_{i,j} }}{2}} \right]$$

(15)

The time-step dt in (11) is required to be ${\text{d}}t < \left( {\sum\nolimits_{l} {{1 \mathord{\left/ {\vphantom {1 {{\text{d}}_{l}^{2} }}} \right. \kern-0pt} {{\text{d}}_{l}^{2} }}} } \right)^{ - 1}$ where d_l denotes distance from I _i,j to any location I _l involved in its filtering process (Weicker and Benhamouda 1997). As we are applying EED to filter I _i,j from its eight nearest neighbors, dt < 1/6. We set dt = 0.1 for the PMF.