    Fast time-to-depth conversion and interval velocity estimation in the case of weak lateral variations  Next: Examples Up: Theory Previous: Theory

## Taking weak lateral variations into account

Instead of attempting to solve equations 5-7 directly, we assume that the lateral variations are mild and the parameters can be approximated with respect to the laterally homogeneous background up to the first-order linearization as follows:   (8)   (9)   (10)

The first terms on the right-hand side of equations 8-10 correspond to the correct values of the velocity squared , , and for the reference laterally homogeneous background. Our objective is to seek for , , and that quantify the first-order effects from lateral heterogenity and satisfy the system of PDEs in equations 5-7. Substituting equations 8-10 into equations 5-7 and restrict our consideration only up to the first-order perturbations, we can derive   (11)   (12)   (13)

which is a considerably simpler system to solve than the original one. However, implementing the proposed system requires the knowledge of , which is unavailable from migration velocity analysis because the Dix velocity squared is still expressed in the time domain. In the same spirit as before, we propose to consider instead a linearized approximation given by,      (14)

Following the similar procedure and retaining only the first-order terms, we can approximate and , which results in (15)

where the reference denotes the converted to depth based on the laterally homogeneous background assumption, and the two derivatives are evaluated first in the original coordinates followed by similar conversion. Substituting equation 15 into equation 11 leads to the following first-order linear system: (16)

where ,   (17)   (18)   (19)

Equation 16 can be solved by stepping in the depth direction given the following initial conditions at the surface : and (20)

In our numerical experiments, we adopt the following procedure to solve system 16:
1. Provided the from migration velocity analysis, we compute the and using smooth differentiation.
2. Convert the velocity and both derivatives to depth based on the assumption of laterally homogeneous media to obtain , , and .
3. Choose a reference laterally homogeneous background from .
4. Given the initial condition 20 on and other known parameters from the previous steps, we compute for the topmost layer using a derivative filter followed by smoothing which helps alleviate the effects from sharp contrasts and their corresponding numerical artifacts that may get carried on to the next depth step.
5. We make a step in depth based on (21)

where represents the depth increment of the model, the current layer is denoted by and the next layer is denoted by .
6. Repeat steps 4 and 5 for the next layer til the final layer.
7. Compute from the estimated using equation 13.    Fast time-to-depth conversion and interval velocity estimation in the case of weak lateral variations  Next: Examples Up: Theory Previous: Theory

2018-11-16