Non-linear problems

Non-linear problems are much more difficult to solve than linear problems. A method that is often successful is to make some approximations that will turn the non-linear problem into a linear problem. We can make this conversion by doing a Taylor expansion (ignoring second and higher order terms) around the initial guess at the slowness field, where $\bf s_0$ is the initial guess at slowness, is the Frechet derivative, a linear approximation of $\bf T_{nl}$ at $\bf s_0$,$\bf \Delta s$ is the change in slowness, $\bf t_0$ are the modeled travel times by applying to $\bf s_0$,$\bf \Delta t$ are the differences between the modeled travel times, , and the measured travel times, .Reflection seismic tomography problems are too large to do direct matrix inversion so iterative methods, such as conjugate gradient, are used instead. Once we have used an iterative method to converge to an acceptable $\bf \Delta s$, we update the slowness estimate,

$$\bf s_1 = \bf s_{0} + \bf \Delta s.$$ (3)

We can then re-linearize around this new model ($\bf s_1$), constructing a new tomography operator $\bf T_{1}$ and repeating the process.

Unfortunately, the linearization does not share many of the properties of a truly linear problem. First, the step length calculation that is valid in the linear problem is usually not valid in the non-linear problem. Instead, we must take into account the linearization point when calculating the step size. Second, and more importantly, the problem isn't guaranteed to converge. To see why, let's return to the simple example. The operator is based on the initial slowness estimate, which is in error. Explained in terms of rays we are back projecting along the guess at the ray trajectories, not their true trajectories. As a result we will end up back projecting the slowness changes to the wrong portions of the model space.

Figure 3 shows the velocity estimate after applying linearized tomography. Generally, we have moved towards the correct velocity model (Figure 1) but we can also see the effects of the linear approximation. In the upper right portion of Figure 3 the velocity has been incorrectly increased. If we look at the ray connecting (z=1.6,x=0) and (z=0,x=9.5) in Figures 2 and 1 we can begin to see why. The guess at the raypath indicates that 30% of the ray length is spent in the lower layer while in fact 50% is traveling through this high velocity layer. To account for the traveltime discrepancy, the inversion has increased the velocity in the in the upper portion of the model.

 

first.tomo Figure 3 The tomography result using the slowness model and raypaths of Figure 2 as the initial estimates.

A second inconsistency is that equation (2) is an example of transmission tomography rather than reflection tomography. Reflection tomography attempts to invert the slowness field from raypaths that go from a known source, to a unknown reflection point in the subsurface, to a known receiver. Not knowing the reflection point introduces a whole new set of unknowns into the inversion. Figure 4 shows a synthetic anticline model. Overlaid in solid lines are the correct reflector positions and the correct raypaths to these reflector positions. The dashed lines are the reflector positions and raypaths that would be estimated using a s(z) initial slowness model. Note that the guess at the reflector position is in significant error, making the guess at the raypaths, and the corresponding tomography operator, in greater error than its transmission tomography counterpart. In addition, traveltime error can be accounted for by shifting the reflector, changing the slowness field, or some combination of the two. The result is that reflection tomography is much more likely to get stuck in local minima and maxima than its transmission counterpart.

 

intro-rays
Figure 4 The solid lines show raypaths through the correct velocity and the correct reflector position. The dashed lines are raypaths through the initial model and the initial reflector positions. Note that the estimated raypaths have significant error, therefore the tomography operator will have significant error.

The possibility of converging to a local minima is well known. The general solution is to use as accurate an initial model as possible (to move the initial point as close the global minima as possible). This is usually done by first applying conventional velocity analysis methods such as semblance analysis and interval velocity estimation by Dix () or some derivation on Toldi's approach. In addition, smaller step sizes and smoother slowness changes at early iterations are used to avoid going outside the valid range of the linear approximation. Both of these approaches have potential negative consequences. A smaller step size requires a greater number of expensive non-linear iterations. In addition, conventional velocity estimation often does not provide a good initial estimate and might in fact lead us to a local maxima.