In this report, I use least squares Kirchhoff (LSK) and reverse time migrations (LSRTM) to analyze how least squares techniques behave in the presence of mismatches between the physics that produce the data and the operators we use to predict them. By implementing both approaches with similar algorithms I show that noise is added during the inversion because of contributions to the residuals from poor operator approximations. When applying LSRTM with exact operators the method works really well, and we don’t observe any noise. However, when the operators differ, errors start to accumulate very quickly from inversion. Examples for Kirchhoff migration with different types of traveltime tables show that noise in LSK is produced by multipathing and shadow zones and inaccurate traveltime calculations. I interpret this noise as a consequence of calculating the gradient of the inversion as a mapping from a wrong residual space to the model space. I interpret this noise as a consequence of assuming that the prediction operator is linear and accurate when is not. Finally, I analyze two well-known mechanisms based on data and model weights to control this problem: regularization by adaptive residual (data space) and image (model space) filtering.
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