Near-surface ocean velocity from infrared images: Global Optimal Solution to an inverse model

We address the problem of obtaining ocean surface velocities from sequences of thermal (AVHRR) space-borne images by inverting the heat conservation equation (including sources of surface heat fluxes and vertical entrainment). We demonstrate the utility of the technique by deriving surface velocities from (1) The motion of a synthetic surface tracer in a numerical model and (2) a sequence of five actual AVHRR images from 1 day. Typical formulations of this tracer inversion problem yield too few equations at each pixel, which is often remedied by imposing additional constraints (e. g., horizontal divergence, vorticity, and energy). In contrast, we propose an alternate strategy to convert the underdetermined equation set to an overdetermined one. We divide the image scene into many subarrays and define velocities and sources within each subarray using bilinear expressions in terms of the corner points (called knots). In turn, all velocities and sources on the knots can be determined by seeking an optimum solution to these linear equations over the large scale, which we call the Global Optimal Solution (GOS). We test the accuracy of the GOS by contaminating the model output with up to 10% white noise but find that filtering the data with a Gaussian convolution filter yields velocities nearly indistinguishable from those without the added noise. We compare the GOS velocity fields with those from the numerical model and from the Maximum Cross Correlation (MCC) technique. A histogram of the difference between GOS and numerical model velocities is narrower and more peaked than the similar comparison with MCC, irrespective of the time interval (Delta t = 2 or 4 h) between images. The calculation of the root mean square error difference between the GOS (and MCC) results and the model velocities indicates that the GOS/model error is only half that of the MCC/model error irrespective of the time interval (Dt = 2 or 4 h) between images. Finally, the application of the technique to a sequence of five NOAA AVHRR images yields a velocity field, which we compare with that from a Coastal Ocean Dynamics Radar (CODAR) array. We find that the GOS velocities generally agree more closely with those from the CODAR than they do with those from the MCC. Specifically, the root mean square error obtained by differencing GOS and CODAR velocities is smaller than that from the similar calculation with MCC velocities. The magnitude of the complex correlation between GOS and CODAR is larger than that between MCC and CODAR. The phase of the complex correlation indicates that both MCC and GOS on average yield velocity vectors biased in the clockwise direction relative to the CODAR vectors for the period examined.