Least-Squares Deconvolution Tests

I constructed a simple, yet relevant synthetic test case for the TLS algorithm: deconvolution. The known model is a sequence of spikes of random amplitude and placement. To create data, the known model was convolved with a Ricker wavelet. Gaussian-distributed noise with a variance of 1 was added to the data, and also to the filter used in the deconvolution.

Figures - compare the standard least-squares (LS), the TLS, and DLS solutions to the problem. The LS solution is undoubtedly poor. In the ``quiet'' zones of the model, where the known model is zero-valued, the estimated LS model has almost as much energy as where the spikes are. Still, the modeled data appears to fit the input data quite well.

 

decon.ls.noisy
Figure 1 Top to bottom: 1) Known filter plus noise, 2) Known model, 3) Estimated standard least-squares model overlaying known model, 4) Noisy data, 5) Modeled data, 6) Residual error.

 

decon.tls.noisy
Figure 2 Top to bottom: 1) Known filter plus noise, 2) Known model, 3) Estimated total least-squares model overlaying known model, 4) Noisy data, 5) Modeled data, 6) Residual error.

 

decon.dls.noisy
Figure 3 Top to bottom: 1) Known filter plus noise, 2) Known model, 3) Estimated damped least-squares model overlaying known model, 4) Noisy data, 5) Modeled data, 6) Residual error.

The TLS and DLS solutions appear somewhat similar. Both approaches seem to suppress unwanted noise in the estimated model in the quiet regions. However, the TLS model seems to have better resolution of the true spikes. Also, the TLS method's residual error appears better balanced than the DLS's. Both TLS and DLS have higher residual error energy than the LS solution.