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## Statistics and Physical Oceanography (1993) Commission on Physical Sciences, Mathematics, and Applications (CPSMA)

### Citation Manager

. "6: Interpolation, Nonlinear Smoothing, Filtering, and Prediction." Statistics and Physical Oceanography. Washington, DC: The National Academies Press, 1993.

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Statistics and Physical Oceanography

by the data. The problem is very difficult, and only a few attempts have thus far been made to address the issue (Wunsch, 1989; Schlax and Chelton, 1992).

Satellite instruments, such as AVHRR, that work in the visible and infrared range of the electromagnetic spectrum provide ocean observations only in the absence of clouds. Hence, maps based on these observations have gaps. One way of achieving full coverage of a specific ocean area is by creating composite images that combine data from different time periods (cf., NRC, 1992a). However, since the fields (for example, sea surface temperature) are time dependent, the composite images represent only some average picture of sea surface temperature distribution for the period covered. Therefore, it is important to know how this picture and its statistical properties are connected with the statistical properties of cloud fields, and how representative the composite image is with respect to the ensemble average of the temperature field (see, e.g., Chelton and Schlax, 1991).

#### Mapping Satellite Data: Motivation and Methods

For most applications, satellite data must be represented on a regular grid. The most common method of mapping satellite observations onto a geographic grid is by interpolating the data from nearby points at the satellite measurement locations. Given the complicated statistical geometry of oceanographic fields (see Chapter 2), such gridding may lead to considerable distortion. Therefore, it is important to study effects of intermittent and rare events, as well as effects of statistical anisotropy and inhomogeneity of oceanographic fields, on the gridding process.

Each interpolated value is typically computed from the 10 to 1000 closest data points, selected from the millions of points typically found in satellite data sets. Common non-trivial methods of interpolating include natural or smoothing spline fits, successive corrections, statistical interpolation, and fitting analytical basis functions such as spherical harmonics. In all cases the interpolated values are linear functions of some judiciously chosen subset of the data.

Applications of natural splines and smoothing splines to interpolate irregularly spaced data are as common in oceanography as they are in most other fields of science and engineering. The methods have been well documented in the literature (e.g., Press et al., 1986; Silverman, 1985).

Successive corrections (Bratseth, 1986; Tripoli and Krishnamurti, 1975) is an iterative scheme, with one iteration per spatial and temporal scale starting with the larger ones. The interpolating weights are a function only of the scale and an associated quantity, the search radius (e.g., Gaussian of given width arbitrarily set to zero for distances greater than the search radius). This scheme is computationally very fast and adapts reasonably well to irregular data distributions, but does not usually provide a formal error estimate of the interpolated field, although it is straightforward to add one. Somewhat related is an iterative scheme that solves the differential equation for minimum curvature (Swain, 1976) of the interpolated surfaces with predetermined stiffness parameter, akin to cubic splines; however, the extension to three-dimensional data is not commonly available.

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 Front Matter (R1-R10) 1: Overview (1-16) 2: Statistical Isues in the Multiple-Scale Variability of Oceanographic Fields (17-22) 3: Lagrangian and Eulerian Data and Models (23-26) 4: Feature Identification (27-32) 5: Visualization (33-36) 6: Interpolation, Nonlinear Smoothing, Filtering, and Prediction (37-44) 7: Model and Data Comparisons (45-46) 8: Non-Gaussian Random Fields (47-50) 9: Encouraging Collaboration Between Statisticians and Oceanographers (51-54) Bibliography (55-62)