Fast time-to-depth conversion and interval velocity estimation in the case of weak lateral variations

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:

$$w (x,z) \approx w_r(z) + \Delta w(x,z)~,$$ (8)

$$x_0 (x,z) \approx x + \Delta x_0(x,z)~,$$ (9)

$$t_0 (x,z) \approx \int_0^z \frac{1}{\sqrt{w_r(z)}} dz + \Delta t_0(x,z)~.$$ (10)

The first terms on the right-hand side of equations 8-10 correspond to the correct values of the velocity squared $w_r$ , $x_0$ , and $t_0$ for the reference laterally homogeneous background. Our objective is to seek for $\Delta w$ , $\Delta x_0$ , and $\Delta t_0$ 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

$$w_d (x,z) = w_r (z) \left(1+ 2\frac{\partial \Delta x_0 }{\partial x}(x,z) \right) + \Delta w(x,z)~,$$ (11)

$$\frac{\partial \Delta t_0 }{\partial x}(x,z) = -\frac{1}{\sqrt{w_r(z)}} \left(\frac{\partial \Delta x_0 }{\partial z}(x,z)\right)~,$$ (12)

$$\Delta w(x,z) = -2 w_r(z)\sqrt{w_r(z)} \left(\frac{\partial \Delta t_0 }{\partial z}(x,z)\right)~,$$ (13)

which is a considerably simpler system to solve than the original one. However, implementing the proposed system requires the knowledge of $w_d (x,z)$ , which is unavailable from migration velocity analysis because the Dix velocity squared $w_d (x_0,t_0)$ is still expressed in the time domain. In the same spirit as before, we propose to consider instead a linearized approximation given by,

$$w_d (x_0(x,z),t_0(x,z)) = w_d \left(x + \Delta x_0,\int_0^z \frac{1}{\sqrt{w_r}}~ dz + \Delta t_0 \right)~,$$

$$\approx w_d\left(x,\int_0^z \frac{1}{\sqrt{w_r}}~ dz\right) + \Delta x_0(x,z)\left(\frac{\partial w_d}{\partial x_0}(x_0(x,z),t_0(x,z)) \right) +$$

$$\Delta t_0(x,z)\left(\frac{\partial w_d}{\partial t_0}(x_0(x,z),t_0(x,z)) \right)~.$$ (14)

Following the similar procedure and retaining only the first-order terms, we can approximate $\frac{\partial w_d}{\partial x_0}(x_0(x,z),t_0(x,z))$ and $\frac{\partial w_d}{\partial t_0}(x_0(x,z),t_0(x,z))$ , which results in

$$w_d (x,z) \approx w_{dr}(x,z) + \Delta x_0 (x,z) \left(\frac{\par... ...right) + \Delta t_0 (x,z)\left(\frac{\partial w_d}{\partial t_0}(x,z) \right)~,$$ (15)

where the reference $w_{dr}(x,z)$ denotes the $w_d (x_0,t_0)$ converted to depth based on the laterally homogeneous background assumption, and the two derivatives are evaluated first in the original $(x_0,t_0)$ coordinates followed by similar conversion. Substituting equation 15 into equation 11 leads to the following first-order linear system:

$$\frac{\partial \mathbf{u} }{\partial z} = \mathbf{A} \frac{\parti... ... x} -\frac{1}{2 w_r\sqrt{w_r}}\left(\mathbf{B} \mathbf{u} + \mathbf{f}\right)~,$$ (16)

where $\mathbf{u} = [\Delta t_0, \Delta x_0]^T$ ,

$$\mathbf{A} = \begin{bmatrix} 0 & 1/\sqrt{w_r} \\ -\sqrt{w_r} & 0 \end{bmatrix}~,$$ (17)

$$\mathbf{B} = \begin{bmatrix} \frac{\partial w_d}{\partial t_0}& \frac{\partial w_d}{\partial x_0} \\ 0 & 0 \end{bmatrix}~,$$ (18)

$$\mathbf{f} = \begin{bmatrix} w_{dr}-w_r \\ 0 \end{bmatrix}~.$$ (19)

Equation 16 can be solved by stepping in the depth $z$ direction given the following initial conditions at the surface $z=0$ :

$$\Delta t_0(x,0) = 0 \quad \Delta x_0(x,0) = 0~.$$ (20)

In our numerical experiments, we adopt the following procedure to solve system 16:

1. Provided the $w_d (x_0,t_0)$ from migration velocity analysis, we compute the $\frac{\partial w_d}{\partial x_0}$ and $\frac{\partial w_d}{\partial t_0}$ using smooth differentiation.
2. Convert the velocity and both derivatives to depth based on the assumption of laterally homogeneous media to obtain $w_{dr}(x,z)$ , $\frac{\partial w_d}{\partial x_0}(x,z)$ , and $\frac{\partial w_d}{\partial t_0}(x,z)$ .
3. Choose a reference laterally homogeneous background $w_r(z)$ from $w_{dr}(x,z)$ .
4. Given the initial condition 20 on $\mathbf{u}$ and other known parameters from the previous steps, we compute $\frac{\partial \mathbf{u}}{\partial x}$ 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

   $$\frac{\partial \mathbf{u} }{\partial z} \approx \frac{\mathbf{u}_{k+1}-\mathbf{u}_{k}}{\Delta z}~,$$ (21)

   
   
   where $\Delta z$ represents the depth increment of the model, the current layer is denoted by $k$ and the next layer is denoted by $k+1$ .
6. Repeat steps 4 and 5 for the next layer til the final layer.
7. Compute $\Delta w$ from the estimated $\mathbf{u}$ using equation 13.

Fast time-to-depth conversion and interval velocity estimation in the case of weak lateral variations