Example

The simplest case to study analytically is that of vertically-incident waves in laterally homogeneous media. In this case, all operators become functions of the scalar variable s (unknown layer slowness). If, for a particular temporal frequency 2#2 and the layer thickness 175#175, we measure the slowness in units of 176#176, the wave continuation operator is simply the phase shift  

 177#177 (76)

and the fundamental nonlinear equation takes the form  

 178#178 (77)

Noting that  

 179#179 (78)

where 180#180 is the true slowness, and that the convergence of Newton's method does not depend on scaling the equation by a constant, we can modify equation () to the simpler form  

 181#181 (79)

where 182#182. The obvious solution of Equation () is 183#183. Our task is to find the convergence limits and their dependence on 165#165.

After a number of algebraic and trigonometric simplifications, the operator G from equation () takes the form of the function  

 184#184 (80)

Its derivative is  

 185#185 (81)

The method will converge in the region around 183#183, where the absolute value of 186#186 is strictly smaller than one. This region (as a function of 141#141 and 165#165) is plotted in Figure . We can see that the convergence region has a finite extent. Its width is the same for 161#161, 162#162, and 174#174. Indeed,  

 187#187 (82)

and  

 188#188 (83)

In both cases, the absolute value of the derivative is smaller than one if 189#189. If we take 190#190 and 191#191, then the convergence radius is 192#192. At small 175#175, 

 193#193 (84)

and  

 194#194 (85)

The convergence rate is of the same order (cubic) but faster in the case of the bilinear method (174#174), because of the twice smaller constant. Here is an example of iterations starting with s₀=2 and converging to 195#195. The Born iteration: The bilinear iteration: A faster convergence can be achieved at some other values of 165#165.Examining the Taylor series of 186#186 around 183#183: 

 198#198 (86)

we find that the order of convergence is optimized for 199#199. In this case,  

 200#200 (87)

and the convergence is fifth order! The example iterations with the optimal value of 165#165 are: The radius of convergence with the optimal value of 165#165 is 202#202.

 

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Figure 3 Convergence region for the Newton-Gauss method in the vertical plane-wave example. Left: 3-D projection. Right: contours. The non-white region on the right plot corresponds to the convergence area. Horizontal axis: 141#141. Vertical axis: 165#165.

Of course, this analysis does not apply directly to the case of non-vertical wave propagation and laterally inhomogenous slowness fields. For reflection wavefields at multiple offsets, the symmetry between downward and upward continuation is broken, as is clear from the experimental results of this paper. However, the simple analysis points to the potential benefits of modifying the Born approximation in the wave-equation velocity estimation.