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Statistics and Physical Oceanography 4 FEATURE IDENTIFICATION A fundamental problem in oceanographic data analysis is the identification of features in image data: their shape, size, and motion. The data used in identification are typically satellite images, e.g., infrared or visible images from the NOAA polar-orbiting satellites or from synthetic aperture radar (SAR). Features are identified in order to quantify their statistics (e.g., ring size and frequency, front locations), to understand the evolution of the fields (e.g., ice leads and floes), and in successive images to infer motion in the field (e.g., sea surface temperature (SST)). Statistics of the features can be used to determine the accuracy of numerical models that describe the physics of the process. Feature identification can also be used to generate realistic fields from data with numerous gaps for assimilation into numerical models for prediction. Feature identification is usually complicated by the presence of instrument noise or geophysical (e.g., clouds) noise. Automation of feature identification using statistical measures is a primary issue; to date, few automated techniques have matched the success of a skilled analyst. TRACKING OF FRONTS AND RINGS The locations of major current systems and the location, tracks, diameters, and lifetimes of rings have been studied using infrared images from the Advanced Very High Resolution Radiometer (AVHRR) sensor on the NOAA polar-orbiting satellites. Brown et al. (1986) characterized the warm-core rings in the Gulf Stream system using 10 years of AVHRR data; a histogram of ring lifetimes showed two distinct peaks at 54 days and 229 days. Auer (1987) analyzed rings as well as the “north wall” of the Gulf Stream, defined subjectively as the location of the maximum SST gradient, using analysis charts derived from AVHRR images. Among other findings, Auer found that the position of the north wall had an annual signal, and that its interannual variability in position was comparable to its annual variability. Cornillon (1986) examined variations in the Gulf Stream position upstream and downstream of the New England Seamounts, again locating the north wall subjectively, and found that the meander envelope did not increase due to the seamounts, but that the mean path length did increase. Cornillon and Watts (1987) found that subjective identification of the north wall was more accurate than that enabled by any “conventional algorithm,” such as the location of the maximum SST gradient, and found that the root-mean-square difference between the AVHRR-derived location and a traditional definition based on in situ temperature measurements was less than 15 km. Ring motion is generally determined by the ring displacement over periods of tens of days, but there may be substantial changes in ring structure and motion over these time periods. Cornillon et al. (1989), in an attempt to determine the motion of warm-core rings relative to the motion of the Gulf Stream slope water, confined their analyses to pairs of observations separated by 36 hours or less. The ring outline was determined from AVHRR images, again by subjective methods, and the ring center was found by the best fit to an
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Statistics and Physical Oceanography ellipse. This fit to the ellipses was found to be better than both a center-of-mass estimate or the intersection of perpendicular bisectors from the ring edge. Absolute velocity estimates were derived from adjacent pairs of ring centers. The velocity of the slope water was determined by a subjective tracking of small SST features in pairs of images (horizontal velocity estimation is discussed in more detail below), and the difference between the velocity estimates was the desired result. The uncertainties in all of the motion estimates were quite large. A related problem is the determination of the ring characteristics and frequency of occurrence based on a series of line samples (as from a radar altimeter subtrack), where the spacing between tracks is as large as a ring diameter and the time between successive tracks is comparable to the time required to move to another track (an “aliasing” problem). Mariano (1990) developed a method for combining different types of data to produce a map of a field that preserves typical feature shapes, rather than smearing them out as in an optimal estimate. Optimal estimates (generally known as “objective” maps in oceanography) minimize the expected squared error of the field value; Mariano’s contour analysis produces instead an optimal estimate of the location of each contour of the field values. Thus it preserves the typical magnitudes of the field gradients; i.e., it preserves the shapes and sizes of rings and ocean fronts. Because the gradients affect the dynamics of the field in the simulation, the analyzed contour fields give more realistic input for assimilation into numerical simulation models. Mariano’s method requires a pattern recognition algorithm to first delineate the contours in each type of data, before the optimal estimate of the final contour location can be made. All of these statistical characterizations using images have in common the problem of detecting features in the presence of extensive cloud contamination or instrument noise; subjective methods have probably been most successful because the human eye can compensate for slight changes in the values of the field and locate a feature by its shape. The problem with subjective methods is that they tend to be labor intensive. A successful automated technique is highly desirable, especially for the case of analyzing large quantities of data (e.g., satellite observations or numerical model output). Ring studies have the additional problem of isolating an elliptically shaped feature that has numerous streamers and smaller eddies attached to it. The delineation of fronts is similar to a contouring problem: a single line must be designated in a noisy field, and the presence of closed contours must be determined to distinguish a ring from the front. SEA ICE TRACKING There are several problems in feature identification in sea ice for which good statistical estimators are needed. Some examples are given here. The motion of pack ice, using a feature-tracking method to determine velocities from a sequence of images, is similar to that of cloud motion or movement of water parcels (e.g., Ninnis et al., 1986). This problem is closely related to ocean velocity estimation, which is discussed below. Feature identification algorithms are needed to characterize ice floes (Banfield and Raftery, 1991;
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Statistics and Physical Oceanography see also Chapter 3 of NRC, 1991b) and leads (the open water between the ice floes): floe size distribution, and lead direction, spacing, and width distributions. If one considers a set of markers on sea ice, their subsequent changes in position can be decomposed into four components: a translation, a rotation, an isotropic scaling, and a change in shape. An alternative decomposition would be into rigid body motion and deformation, and the deformation may be further decomposed into affine and nonaffine components. Shape statistics, concerned with the analysis of shapes such as these, includes the examination of a series of shapes evolving over time. In the context of polar oceanography, the emphasis is not so much on the shape itself—as it might be in biology where much of shape statistics originates—but rather on the motion and deformation of the shapes. The deformations and motions of various shapes must be reconciled with each other to establish the evolution of the entire field, and to infer something about the field dynamics. A combination of feature identification and feature tracking is used to estimate the opening and closing of sea ice leads, which is necessary for models that estimate sea ice thickness (e.g., Fily and Rothrock, 1990). The object of this analysis is to produce an estimate of the fractional increase or decrease in size of sea ice leads from a pair of sequential SAR images. The first step in the estimation requires the designation of tie points between the same features in sequential images, which are determined by cross-correlations between subsets of the images. This procedure is quite similar to that required for estimation of ice motion. The next step requires the classification of the entire image into ice or lead, which is a statistical problem by itself, similar to that of flagging AVHRR images for cloud cover, or classifying AVHRR images by cloud type. The net increase or decrease in the area covered by the leads based on a comparison of the two classified images gives the required estimate. ESTIMATION OF HORIZONTAL VELOCITIES FROM IMAGE SEQUENCES Another oceanographic problem that might benefit from the application of advanced statistical methods is the estimation of horizontal ocean velocities using pairs of satellite images. One method of estimating these velocities is to track identifiable features in a tracer field, usually the sea surface temperature (SST; Emery et al., 1986). Other methods use the heat advection equation (1.4) (Kelly, 1989) or an assumption of geostrophic balance (Kouzai and Tsuchiya, 1990) to relate observed SST to the velocity field. SST images from the AVHRR have a horizontal resolution of approximately 1.1 km, with temporal separations of 4 to 8 hours. While clouds often obscure much of the ocean, there are occasionally periods of 1 to 3 days with relatively few clouds during which 4 to 12 images can be collected. Most of the velocity estimates assume that changes in SST are due to horizontal advection; however, other processes also change the SST seen by AVHRR: contamination by undetected clouds and fog, heating and cooling by the sun and air, vertical mixing and vertical motion, and changes in the top “skin” of the ocean (less than 1 mm thick). In the absence of these complications, the problem of estimating velocities would be one of mapping the location of all pixels in the first image onto the second image. It has been suggested that other statistical methods, such as simulated annealing (see, e.g., Chapter 2 in
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Statistics and Physical Oceanography NRC, 1992c), might produce such a mapping of individual pixels, but this has not been attempted to date. The feature tracking method has been automated using a maximum cross-correlation (MCC) method, first applied by Emery et al. (1986) and derived from the methods used to track the motion of pack ice. The procedure is to cross-correlate a subregion of an initial image with the same-sized subregion in a subsequent image, searching for the location in the second image that gives the maximum cross-correlation coefficient. The size of the region searched in the second image depends on the maximum displacement that could be caused by reasonable velocities in the surface ocean. There is a trade-off between the spatial resolution of the velocity estimates and the statistical reliability of the cross-correlation. The small-scale features can be enhanced by the calculation of gradients or by high-pass filtering. It has been suggested that wavelet transforms might provide another way of first correlating larger-scale features and then smaller-scale features, but this has not been tried. Further references to the MCC method include Collins and Emery (1988), Kamachi (1989), Garcia and Robinson (1989), Tokmakian et al. (1990), and Emery et al. (1992). Identifying features in consecutive images is not the most difficult problem in velocity estimation, although there is room for improvement here. Two related unresolved issues are ring motion (or rotation) and inferring velocity along isolines of the tracer field or in regions of small gradients. These flows produce only small changes in the tracer field, but the magnitudes of the velocities may be larger than those of the velocities that produce large changes in the tracer field. The MCC method can be modified to accommodate rotation of the features. Besides simply displacing the initial search region and calculating displacement, the initial region can be rotated through a reasonable range of angles (Kamachi, 1989; Tokmakian et al., 1990). However, the additional searches increase the chance of random high correlations, and the benefit is questionable. Emery et al. (1992) have investigated an alternate method of following rotation in closed rings and eddies, also noting that the basic method, without rotation, produces similar results. Another method, which addresses the latter problem, solves the heat advection equation using inverse methods to find the velocity field most consistent with the change in SST fields observed in the two images (Kelly, 1989). The heat equation used, based on equation (1.4), is Tt+uTx+vTy-m(x,y)=S(x,y), (4.1) where u, v are the horizontal velocity components, Tx, Ty are the horizontal derivatives of SST, Tt is the temporal derivative of SST, S(x, y) is a term that describes SST fluctuations with relatively large spatial scales (which are not due to advection), and m(x,y) is the misfit. As in the MCC method, there is an optimal temporal lag d between images for the inversion: approximately 12 hours, compared to values of 4 to 6 hours preferred for the MCC method. Velocity fields that include the along-isoline velocity component can be obtained by adding constraints on the velocity solution, notably the minimization of horizontal divergence, with a weighting factor a relative to the heat equation (4.1), that is,
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Statistics and Physical Oceanography a(ux+vy=0). (4.2) Two-dimensional biharmonic splines were used as basis functions for the velocity fields in the inversion to give a continuous solution, unlike the feature-tracking methods, which give estimates at discrete grid points (Kelly and Strub, 1992). The spatial resolution of the solution depends on a parameter that sets the number of data per knot in the spline, and on the size of the subregion used to compute the SST gradients. A statistical challenge in this inverse problem is determining the best solution as a trade-off between the fit to the heat advection equation and the constraints. Although inverse theory methods exist to solve this problem more rigorously, it has not yet been done. The horizontal velocity problem has been examined by many scientists and engineers. Other methods include the use of a single image in conjunction with the thermal wind equation, which relates horizontal SST gradients to vertical velocity shear (Kouzai and Tsuchiya, 1990). This method neglects salinity effects and requires an empirical relation between SST gradients and velocity from field data. Wahl and Simpson (1991) explored a variety of artificial intelligence methods for modifying the basic feature-tracking method and improving the cross-isoline solution. These methods have not been evaluated using field measurements. The MCC and heat advection inverse methods have been compared by Kelly and Strub (1992) to in situ velocities from surface drifters and acoustic Doppler current profilers (ADCP), and to geostrophic velocities from the Geosat altimeter. They found that both methods produce velocity fields that captured the main features of the horizontal velocity field in a region of the coastal ocean approximately 500 km square. Both methods also underestimated the maximum velocities in the most energetic jets (velocities over 1 ms-1). Detailed examination of the SST fields showed that in some cases the MCC method was not underestimating the displacements of identifiable features within the jet. Rather, drifters at 15-m depth within the jet were moving to locations beyond the SST feature in the second image. Thus, substantial errors in both methods occur because some of the largest velocities in the ocean do not produce observable SST changes. Although further modifications of these two methods or entirely new techniques might improve the estimates, these errors suggest that even a perfect mapping of SST fields would not give an accurate velocity field in regions with energetic jets. One promising approach is to incorporate independent velocity measurements into the estimate, either from radar altimeters or from drifters. PROSPECTIVE DIRECTIONS FOR RESEARCH Identifying features through the analysis of oceanographic data presents many opportunities for statistical research to contribute to progress on important physical oceanographic issues. The following particular issues exemplify some of the challenges for which statistical advances that improve on current approaches would be valuable:
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Statistics and Physical Oceanography Detection of SST fronts and rings (maximum gradients) in the presence of noise with a variety of spatial scales; Characterization of rings or eddies by shape, frequency, and motion in a series of images or from a series of line samples, which may lead to aliasing of the feature motion; Characterization of the evolution of ice floes and leads, using a time series of images. The emphasis is on inference of the dynamics of the field from the feature evolution and statistics; and Estimation of oceanic velocity using a time series of tracer fields, where the relationship between the velocity field and the tracer is not unique and the velocity field is subject to some dynamical constraints.
Representative terms from entire chapter: