Downward continuation

Given the values of the function 472#472, downward continuation consists of finding the values of 473#473. An expression describing this process lays at the end of the following proof: Obtain the Helmholtz equation by applying the Fourier transform defined by () to the wave equation () while taking into account the notation () and the property () and rearranging:  

 474#474 (198)

By derivating relation () with respect to x and z we obtain:  

 475#475 (199)

 

 476#476 (200)

 

 477#477 (201)

 

 478#478 (202)

By plugging into in () and eliminating the exponential, we get:  

 479#479 (203)

The second derivative with respect to z can be eliminated by derivating with respect to z, multiplying by 480#480, and adding the result to ():  

 481#481 (204)

Note that no approximation has been made between the wave equation () and this point. Eq. is simply the wave equation in a different coordinate system. Now Q_zzz is approximated by zero:  

 482#482 (205)

For the case of a homogenous medium, 483#483 and the equation turns into the familiar 429#429 equation:  

 484#484 (206)

The 444#444 equation is obtained by neglecting the Q_xxz term also:  

 485#485 (207)

Downward continuation proceeds by considering  

 486#486 (208)

then by using one of the equations , or to find the values of 487#487 and by finally finding P by undoing the variable change:  

 488#488 (209)