Array design by inverse methods

The generalized inverse method of Bennett and McIntosh (1982) is used to assess the efficiency of instrument arrays with observe deterministic fields. The arrays considered are combinations of points at which bottom pressures are observed (Cartwright, 1978), plus paths along which averaged barotropic velocities are observed, by the acoustic tomography technique of reciprocal shooting (Munk and Wunsch, 1982a,b). The barotropic M₂ tide is used as an example of a field which is being observed by the array, and for which an interpolation or smoothing is required. The treatment of observations of individual inter-annual events would be similar.It is shown that the generalized inverse method for the objective analysis of deterministic fields is formally identical to the Gandin (1965) method for the objective analysis of random fields.Construction of the generalized inverse requires the inversion of an Hermitian positive definite matrix. Array efficiency is characterized by the number of significant eigenvalues of the matrix.The admission of errors in the observations and dynamics necessitates the choice of weights in a variational principle. The choice is made by prior estimation of the relative error variances. It is also necessary to choose locally singular weighting functions for the dynamics, in order to ensure non-singular interpolating or smoothing fields. The dominant array modes are defined and constructed. These are the interpolating fields which make the most stable contribution to the generalized inverse.An example of inversion is carried out using simulated data.