Blocky velocity inversion by hybrid norm

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# Dix inversion as an -optimization problem

The Dix equation can be made linear by relating the square of interval velocity to the square of RMS velocity ,

 (1)

where is the two-way traveltime. By defining and , we can set up the Dix inversion problem in an sense as follows:

 (2)

where is a weight function proportional to the pick strength in the velocity scan divided by , is the causal integration operator, and and are vectors containing all the values of and , respectively. The division by reduces the strength of the later events to balance the data fitting strength along the time axis.

The hybrid norm above defines the cost function as follows:

 (3)

where is the residual and is a threshold which defines a smooth transition between the and norms (Claerbout, 2009).

Fitting goal (2) is not enough to fully constrain the inversion, because it has a large null space (Li and Maysami, 2009). Moreover, picking errors can lead to incorrect RMS velocities and unreasonable interval velocities. Therefore, a second fitting goal (i.e. a regularization term) is required to constrain this inversion. The regularization term can be written as follows:

 (4)

where is typically a roughening operator, and is a scalar to balance the two fitting goals.

Notice that the norm in fitting goal (2) has a different effect than the norm in fitting goal (4). Using the hybrid norm in data fitting makes the inversion less sensitive to outliers. On the other hand, using the hybrid norm in model styling affects the general shape of the estimated model, which is the goal of this paper.

Li and Maysami (2009) successfully produced blockiness in 1D when using the first derivative as a regularization operator. In the following sections, we will try different regularization operators to achieve the same goals in 2D.

 Blocky velocity inversion by hybrid norm

Next: Regularization by the Laplacian Up: Almomin: Blocky velocity inversion Previous: Introduction

2010-05-19