recfunk_svd.f version of recfunk with SVD in the inner loop

click the title of post to obtain the code recfunk_svd.f in

From unpublished manuscript on RF estimation:

Receiver Functions from Multiple-Taper Spectral Correlation:
Statistics, Single-Station Migration, and Jackknife Uncertainty
J. Park and V. Levin, submitted to GJI

RF-computation suggests that the coherence between P and SH-converted energy in the P coda is comparable to the coherence between P and SV-converted energy. Because aggregate coherence C(f)< 0.5, one can question whether the SV and SH converted phases are mutually correlated. The alternative hypothesis is that SV and SH scattering arise from statistically distinct portions of the incoming P wave. Poor correlation between SV and SH scattering can be tested by simultaneous cross-correlation of all three particle-motion components, computed via the singular-value decomposition (SVD). Define a K ×3 matrix A(f) of eigenspectral estimates in the P-SV -SH coordinate system (in our application K = 3):

For noise-free, perfectly correlated data, the matrix A(f) = u (f) v(f), a single dyad in which the 3-component vector v(f) contains the correlation coefficients between components and the K-component vector u(f) describes the spectral signature of the correlated signal in a narrow frequency band about f. The receiver functions HSV (f),HSH(f) correspond to the component ratios vSV /vP and vSH/vP , where v(f) = [vP , vSV , vSH]. For real noisy data, we associate the SVD receiver function with the singular vectors of A that correspond to the largest singular value, similar to the ”pure state” filtering algorithm (Samson, 1983). We SVD decompose A(f). The uncertainty of the SVD receiver function can be estimated by analogy to uncertainties of RF computed from complex cross-correlation and coherence.

In tests with real data, the added sophistication of the SVD receiver function algorithm leads only to small changes in the estimated RFs. We infer that P-SV conversion and P-SH conversion in the P coda are co-dependent, in a statistical sense. It should therefore be feasible to estimate HSV (f) and HSH(f) jointly. On the other hand, our experiments suggest that the ordinary pairwise MTC RF estimator is adequate in practice. The SVD-based RF estimator may have utility in complex situations, such as the estimation of S receiver functions for S-to-P converted energy.