Penland, C., and P. D. Sardeshmukh, 1995: Error and sensitivity analysis of geophysical systems. J. Climate, 8, 1988-1998.
The first-order perturbation technique is reviewed as a tool for investigating the error and sensitivity of results obtained from the eigenanalysis of geophysical systems. Expressions are provided for the change in a system's eigenfunctions (e.g., normal modes) and their periods and growth rates associated with a small change δL in the system matrix L. In the context of data analysis, these expressions can be used to estimate changes or uncertainties in the eigenstructure of matrices involving the system's covariance statistics. Their application is illustrated in the problems of 1) updating a subset of the empirical orthogonal functions and their eigenvalues when more data become available, 2) estimating uncertainties in the growth rate and spatial structure of the singular vectors of a linear dynamical system, and 3) estimating uncertainties in the period, growth rate, and spatial structure of the normal modes of a linear dynamical system. The linear system considered in examples 2 and 3 is an empirical stochastic-dynamic model of tropical sea surface temperature (SST) evolution derived from 35 years of SST observations in the tropical Indo-Pacific basin. Thus, the system matrix L is empirically derived. Estimates of the uncertainty in L, required for estimating the uncertainties in the singular vectors and normal modes, are obtained from a long Monte Carlo simulation. The analysis suggests that the singular vectors, which represent optimal initial structures for SST anomaly growth, are more reliably determined from the 35 years of observed data than are the individual normal modes of the system.