The propagator of a linear model plays a central role in empirical normal mode and finite-time instability problems. Its estimation will affect whether the linear stability characteristics of the corresponding dynamic system can be properly extracted. In this study, we introduce two alternative methods for estimating the linear propagator and finite-time growth rates from data. The first is the generalized singular value decomposition (GSVD) and the second is the singular value decomposition combined with the cosine-sine decomposition (SVD-CSD). Both methods linearize the relation between the predictor and the predictand by decomposing them to have a common evolution structure and then make the estimations. Thus, the linear propagator and the associated singular vectors can be simultaneously derived. The GSVD clearly reveals the connection between the finite-time amplitude growth rates and the singular values of the propagator. However, it can only be applied in situations when given data have more state variables than observations. Furthermore, it generally encounters an over-fitting problem when data are contaminated by noise. To fix these two drawbacks, the GSVD is generalized to the SVD-CSD to include data filtering capability. Therefore, it has more flexibility in dealing with general data situations. These two methods as well as the Yule-Walker equation were applied to two synthetic datasets and the Kaplan's sea surface temperature anomalies (SSTA) to evaluate their performance. The results show that, because of linearization and flexible filtering capabilities, the propagator and its associated properties could be more accurately estimated with the SVD-CSD than other methods.
- Finite-time instability
- Generalized singular value decomposition