Gradient of travel time in layered media?

[ytht100]

I have a problem of the following picture. x_0, y_0, z_0, V_0, and V_1 are fixed.

http://postimg.org/image/6r0ogcx3f/

The travel time is obviously $$t = \frac{1}{{{V_1}}}{[{({x_1} - {x_c})^2} + D_1^2]^{1/2}} + \frac{1}{{{V_0}}}{[{({x_c} - {x_0})^2} + D_0^2]^{1/2}}$$

According to a high-profile publication, the derivative of traveltime with regard to x1 is $$\frac{{\partial t}}{{\partial {x_1}}} = \frac{1}{{{V_1}}}\frac{{({x_1} - {x_c})}}{{\sqrt {{{({x_1} - {x_c})}^2} + D_1^2} }}$$. Here, it seems to ignore dx_c/dx_1, is it correct?

I guess it is wrong.

Thanks a lot!

 

[ytht100]

Please, someone give me a hint.

 

[nasu]

What is that publication? Can you give a reference to the article?