Downward continuation

Given the values of the function $P\left( {\forall \vec x,z,\forall \omega } \right)$, downward continuation consists of finding the values of $P\left( {\forall \vec x,z + \delta z,\forall \omega } \right)$. An expression describing this process lays at the end of the following proof:

Obtain the Helmholtz equation by applying the Fourier transform defined by (24) to the wave equation (22) while taking into account the notation (26) and the property (25) and rearranging:

 

$$\Delta P + P_{zz} + m^2 P = 0.$$ (29)

By derivating relation (28) with respect to x and z we obtain:

 

$$P_{x_i } = Q_{x_i } e^{i\bar mz} ,$$ (30)

 

$$\Delta P = \left( {\Delta Q} \right)e^{i\bar mz} ,$$ (31)

 

$$P_z = (Q_z + i\bar mQ)e^{i\bar mz} ,$$ (32)

 

$$P_{zz} = (Q_{zz} + 2i\bar mQ_z - \bar m^2 Q)e^{i\bar mz} .$$ (33)

By plugging into in (29) and eliminating the exponential, we get:

 

$$\Delta Q + Q_{zz} + 2i\bar mQ_z + \left( {m^2 - \bar m^2 } \right)Q = 0.$$ (34)

The second derivative with respect to z can be eliminated by derivating with respect to z, multiplying by $\frac{i}{{2\bar m}}$, and adding the result to (34):  

$$\frac{i}{{2\bar m}}Q_{zzz} + \frac{i}{{2\bar m}}\left( {\Del... ... + \frac{{im}}{{\bar m}}\frac{{\partial m}}{{\partial z}}Q = 0.$$ (35)

Note that no approximation has been made between the wave equation (22) and this point. Eq. 35 is simply the wave equation in a different coordinate system. Now Q_zzz is approximated by zero:  

$$\frac{i}{{2\bar m}}\left( {\Delta Q} \right)_z + \Delta Q + ... ... + \frac{{im}}{{\bar m}}\frac{{\partial m}}{{\partial z}}Q = 0.$$ (36)

For the case of a homogenous medium, $\bar m = m$ and the equation turns into the familiar $45^\circ$ equation:

 

$$\frac{i}{{2\bar m}}\left( {\Delta Q} \right)_z + \Delta Q + 2i\bar mQ_z = 0 .$$ (37)

The $15^\circ$ equation is obtained by neglecting the Q_xxz term also:

 

$$\Delta Q + 2i\bar mQ_z = 0 .$$ (38)

Downward continuation proceeds by considering

 

$$Q\left( {\forall \vec x,z,\forall \omega } \right) = P\left( {\forall \vec x,z,\forall \omega } \right)$$ (39)

then by using one of the equations 36, 37 or 38 to find the values of $Q\left( {\forall \vec x,z + \delta z,\forall \omega } \right)$ and by finally finding P by undoing the variable change:

 

$$P\left( {\forall \vec x,z + \delta z,\forall \omega } \right... ... \omega } \right)e^{im_{\left( {\vec x,z} \right)} \delta z} .$$ (40)