Inverse methods for assessing ship‐of‐opportunity networks and estimating circulation and winds from tropical expendable bathythermograph data
Authors: Bennett Andrew F. 
Published: February 2008

This is a study of inverse methods for estimating the surface circulation of the equatorial Pacific, by combining a linear, reduced‐gravity shallow‐water model with the Tropical Ocean‐Global Atmosphere ship‐of‐opportunity expendable bathythermograph (TOGA SOP XBT) observing program. The circulation of interest has large spatial scales and interannual time scales. The temporal structure is parameterized with a few frequencies, which provide a sufficiently accurate description over the 2‐year data‐smoothing interval. No predictive skill is implied, beyond the smoothing interval. A circulation is sought as the weighted least squares best fit to the dynamics and the data; thus the approach is formally that of an inverse problem in which forcing fields are estimated from noisy observations of the circulation. These observations are not the original temperature data, which are randomly scattered along the ship tracks. Instead, an orderly data set is defined by projecting the data onto Hermite functions of latitude. The projections could be summed and sampled at regular intervals along the tracks, but the projections themselves suffice as smoothed yet still somewhat noisy data. The number of measurements becomes the number of tracks (13) multiplied by the number of Hermite projections or moments (6), that is, 78 measurements at each of three frequencies (0.25 cpy, 0.5 cpy, 1.0 cpy). Each measurement may be described mathematically by a linear functional. The best fit minimizes a quadratic cost functional and is expressed as a sum of the influence functions or representers associated with the cost functional (which makes roughness expensive) and the measurement functionals. The sum of representers is finite, as there is one representer per measurement. Unobservable circulations are discarded by the minimization algorithm. It is shown that the representer expansion is closely related to Gauss‐Markov smoothing or “objective analysis,” and also to the inverse method of Backus and Gilbert (1970). The analysis of the inverse using representers yields an Hermitian nonnegative matrix which characterizes the efficiency of the XBT observing program or “array.” Spectral decomposition of the matrix also yields modes for the circulation and for the forcing (solution array modes (SAMs) and residual array modes (RAMs), respectively), which are linear combinations of the representers and their dynamical residuals, respectively, and are the patterns most stably estimated by the inverse. The inverse assumes that a prior estimate has been made for the forcing, corresponding to which there is a prior estimate of circulation. The data may be regarded as prior estimates of measurements of the true circulation. The inverse also assumes that error covariances for these prior estimates are available. The results of the inverse calculation include posterior estimates for the errors in the prior forcing estimates, for the prior circulation estimates, and for the measurements. The representers and their residuals, or equivalentIy the SAMs and RAMs, yield the error covariances for these posterior estimates. Finally the reduction of the cost functional may be expressed in terms of the trace of the Hermitian representer matrix. The dynamics and the measurement functionals are linear while the cost functional is quadratic. Consequently, the structure described above does not depend on any realization of the circulation and may be determined once and for all. The structure is computed here for various choices of oceanic phase speed, forcing error statistics and measurement error statistics. The robustness of the inverse is also tested with synthetic forcing and data which variously are or are not consistent with the assumed prior statistics. The approach taken here is placed in the context of other variational assimilation schemes.