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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.

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** Up:** Brown: Total least-squares
** Previous:** Relation of TLS to
Stanford Exploration Project

11/11/2002