Three-layer case

Let us now turn to the right plot of Figure 15, which depicts the situation where we have a three-layered medium and the total traveltime is given by

$$t = t_0(\mathbf{x_0},\mathbf{x_1}(h)) + t_1(\mathbf{x_1}(h),\mathbf{x_2}(h)) + t_2(\mathbf{x_2}(h),\mathbf{x_3}(h)) \quad h = x_3-x_0~.$$ (44)

We can proceed along the same line of argument as in the two-layer case and differentiating the total time with respect to $h$, which leads to

$$\frac{\partial t}{\partial h} = \frac{\partial t_2}{\partial h} + \sum^2_{k=1} \frac{\partial t}{\partial x_k} \frac{\partial x_k}{\partial h}~.$$ (45)

Again, $\partial t / \partial x_k = 0$ due to the Fermat's principle, we then have

$$\frac{\partial t}{\partial h} = \frac{\partial t_2}{\partial h}~.$$ (46)

We further differentiate the expression with respect to $h$ and arrive at

$$\frac{\partial^2 t}{\partial h^2} = \frac{\partial^2 t_2}{\partia... ...frac{\partial^2 t_2}{\partial x_2 \partial h}\left( \frac{d x_2}{d h}\right) ~.$$ (47)

To find $d x_2/ d h$, we differentiate the Fermat's conditions ( $\partial t / \partial x_k = 0$ for $k \in \{1,2\}$) in a similar manner as in the two-layer case.

1. At the first interface ($k=1$), the condition $\partial t / \partial x_1 = \partial ( t_0+t_1) / \partial x_1 = 0$ leads to

   $$\frac{\partial^2 (t_0 + t_1)}{\partial x_1^2} \left( \frac{d x_1}... ...{\partial^2 t_1}{\partial x_1 \partial x_2} \left( \frac{d x_2}{d h}\right) = 0$$ (48)

2. At the second interface ($k=2$), the condition $\partial t / \partial x_2 = \partial ( t_1+t_2) / \partial x_2 = 0$ leads to

   $$\frac{\partial^2 t_1}{\partial x_1 \partial x_2} \left( \frac{d x... ...( \frac{d x_2}{d h}\right) + \frac{\partial^2 t_2}{\partial x_2 \partial h} = 0$$ (49)

At this stage, we can see that equations (48) and (49) contain two unknown variables $d x_1/d h$ and $d x_2/ d h$ to be solved for. We propose to look at this problem in a specific way that will facilitate an extension to the general multi-layer case. We first rearrange equation (48) into the following form

$$r = \left. \left( \frac{d x_1}{d h}\right) \middle/ \left( \frac{... ...\middle/ \left( \frac{\partial^2 (t_0 + t_1)}{\partial x_1^2} \right)\right. ~,$$ (50)

which is reminiscent of equation (42) for the two-layer case when $x_2 = h +$   const. Substituting equation (50) into equation (49), we arrive at

$$\frac{dx_2}{dh} = \left. -\left( \frac{\partial^2 t_2}{\partial x... ...\partial x_2} + \frac{\partial^2 (t_1 + t_2)}{\partial x_2^2} \right)\right. ~,$$ (51)

which can be substituted into equation (47) to evaluate the desired second-order total traveltime derivative.