Acoustic wavefield evolution as function of source location perturbation

Theory

timedsS Figure 2. Illustration of the relation between the initial source location and a perturbed version given by a single source and image point locations. This is equivalent to a shift in the velocity field laterally by $l$.

In this section, I derive partial differential equations that relate changes in the wavefield shape to perturbations in the source location. We will start by taking derivatives of the wave equation with respect to lateral perturbation and then use the Taylor's series expansion to predict the wavefield form at another source.

In 3-D media, the acoustic wavefield $u$ is described as a function of $x$, $y$ and depth $z$ and is governed by a partial differential equation as a function of time $t$ given by,

$$\frac{\partial^2 u}{\partial x^2}+ \frac{\partial^2 u}{\partial y... ...al^2 u}{\partial z^2} = w(x,y,z) \frac{\partial^2 u}{\partial t^2} +f(x,y,z,t),$$ (1)

where $w(x,y,z)$ is the sloth (slowness squared) as a function position. and should be slowly varying with respect to the wavelength for proper amplitude description. The source can be included as a function added to equation (1) 2given by $f(x,y,z,t)$, defined usually at a point, or represented by the wavefield $u(x,y,z,t)$ around time $t=0$ as an initial condition. A change in the source location along the surface,2while keeping its source function stationary, is equivalently represented, in the far field, by shifting the velocity field laterally by the same amount in the opposite direction and thus can be represented by the following wave equation form:

$$\frac{\partial^2 u}{\partial x^2}+ \frac{\partial^2 u}{\partial y... ...ial^2 u}{\partial z^2}=w(x-l,y,z) \frac{\partial^2 u}{\partial t^2}+f(x,y,z,t),$$ (2)

2where $f(x,y,z,t)$, in this case, is stationary and independent of $l$. A simple variable change of $x^{'}=x-l$ can demonstrate this assertion, where $x^{'}$ is replaced by $x$ to simplify notation. For simplicity, I use the symbol $u$ to describe the new wavefield, as well. Figure 2 shows the depicts the relation for a single source and image point combination with the velocity shift. To evaluate the wavefield response to lateral perturbations, we take the derivative of equation (2) with respect to $l$, where the wavefield is dependent on the source location as well [$u(x,y,z,l,t)$], which yields:

$$\frac{\partial^3 u}{\partial x^2 \partial l}+ \frac{\partial^3 u}... ...c{\partial^2 u}{\partial t^2} + w \frac{\partial^3 u}{\partial t^2 \partial l}.$$ (3)

Substituting $D_x=\frac{\partial u}{\partial l}$, where $l$ is the equivalent source shift (actual velocity shift) in the $x$-direction, into equation (3), 2and setting $l=0$, the location in which we evaluate the equation for the Taylor's series expansion yields:

$$\frac{\partial^2 D_x}{\partial x^2}+ \frac{\partial^2 D_x}{\parti... ...partial t^2} - \frac{\partial w}{\partial x} \frac{\partial^2 u}{\partial t^2},$$ (4)

which has the form of the wave equation with the last term on the right hand side acting as a source function. If this source function is zero given by, for example, no lateral velocity variation ( $\frac{\partial w}{\partial x}$=0), then $D_x$=0, and as expected there will be no change in the wavefield form with a change in source position.

Therefore, the wavefield for a source located at a distance $l$ from the source used to estimate the wavefield $u$ can be approximated using the following Taylor's series expansion:

$$u(x,y,z,l,t) \approx u(x,y,z,l=0,t) + D_x(x,y,z,t) l.$$ (5)

This result obviously has first-order accuracy represented by the first order Taylor's series expansion. For higher order accuracy, we take the derivative of equation (3) again with respect to $l$, which yields:

$$\frac{\partial^4 u}{\partial x^2 \partial l^2}+ \frac{\partial^4 ... ...}{\partial t^2 \partial l} + w \frac{\partial^4 u}{\partial t^2 \partial l^2} .$$ (6)

Again, by substituting $D_{xx}=\frac{\partial^2 u}{\partial l^2}$, as well as, $D_x$ into equation (6) , 2and setting $l=0$, the second order perturbation equation is given by:

$$\frac{\partial^2 D_{xx}}{\partial x^2}+ \frac{\partial^2 D_{xx}}{... ...ial t^2} +\frac{\partial^2 w}{\partial x^2} \frac{\partial^2 u}{\partial t^2}.$$ (7)

Now, the wavefield for a source located at a distance $l$ from the original source can be approximated using the following second-order Taylor's series expansion:

$$u(x,y,z,l,t) \approx u(x,y,z,l=0,t) + D_x(x,y,z,t) l + \frac{1}{2} D_{xx}(x,y,z,t) l^2.$$ (8)

Equations (4) and (7) can be written in many forms and in the next section I show a velocity-derivative independent version of them. 2Equations (4) and (7) can be solved in many ways and in the next section I show some of the features gained by using an integral formulation given by the Green's function.

Acoustic wavefield evolution as function of source location perturbation