Introduction

Waveform tomography is the inversion of the seismic traces for the earth's velocity (and sometimes density) distribution. Instead of just inverting the, e.g. transmission traveltimes, waveform tomography inverts both the amplitude and phase information of all arrivals. Although elegant in theory, waveform inversion has not enjoyed much use in exploration geophysics partly because it is computationally expensive (more than an order of magnitude more expensive than migration) and partly because it requires expert user intervention to get it to work. However, faster computers, better inversion strategies and a greater demand for accurate lithology information should increase its use in the future. Books by Tarantola (1987), Lines (1988) and Nolet (1987) contain more detailed information about this subject, and some examples with field data are contained in Mora (1987), Crase et al. (1992), Tarantola (1986), Gauthier et al. (1986), Beydoun and Mendes (1989), Zhou et al. (1995) and Zhou et al. (1997).

There are two reasons why waveform tomography provides more detailed information about the earth than provided by traveltime tomography:

1.

Compared to traveltime data, the waveform amplitudes can be more sensitive to model parameter changes. For example, replacement of brine by gas in a sedimentary rock does not significantly change the propagation velocity or the associated traveltimes of reflected waves, but it can significantly change the amplitude of reflections to produce "bright spots" (see Nur and Wang, 1988). Bright spot technology has been widely used in the Gulf of Mexico to pinpoint hydrocarbon reserves that would otherwise be hidden from seismic view.

2.

There is a much larger quantity of information contained in the waveform data than contained in the corresponding traveltime data. For example, a seismogram will contain one transmission traveltime, but will also contain N_t samples of waveform data where N_t is the number of samples in the trace. Therefore, there might be quite a few slowness models that will fit the sparse traveltime data but there will likely be many fewer models that can explain the waveform data.

The overabundance of information in waveform data is both a blessing and a curse. It is a blessing because it is so rich in information, but it is a curse because it usually leads to a highly non-linear relationship between the model and the data. This means that the misfit function contains many local minima or long twisting valleys that can lure the gradient optimizer to an incorrect model (Jannane et al., 1989; Tarantola et al., 1989; Sneider et al., 1989). The next set of examples demonstrate the pseudo-linear nature of traveltime misfit functions and the non-linear nature of waveform misfit functions.

Example 1   Pseudo-linear Traveltime Misfit Function: Seismograms are generated for a 2 layer medium with a rigid surface at the top, where the first layer has a thickness d=0.25 km and velocity V=1.0 km/s, respectively; and the second layer velocity is very large so that strong multiples are generated. Figure 3.1a shows a shot gather with a 10 Hz Ricker wavelet source. The direct wave traveltimes t^obs(x)at the x offset are picked and used to generate a traveltime misfit function. The predicted traveltimes t_pred(x) are for a hypothetical velocity model with thickness equal to 0.25 km. This traveltime misfit function is plotted in Figure 3.1d and shows that the misfit function monotonically decreases as the hypothetical velocity V approaches the correct velocity 1.0 km/s. Thus a gradient optimization method will quickly converge to the global minimum.

  

Figure: (a) CSG seismograms p_obs(x,t) for a 2-layer earth model with a rigid surface. Note that the PP primary reflection energy is followed by 5 multiple arrivals, and the direct wave has been muted. (b). Associated PP traveltime t_obs(x) curve. (c). Waveform misfit function $\epsilon =\sum _t \sum _x[p_{pred}(x,t)-p_{obs} (x,t)]^2$ plotted against different values of the hypothetical velocity V of the first layer. The correct value of V is 1.0  km/s, where the misfit functions have a value of zero and the subscript pred denotes "data predicted from the hypothetical velocity model". (d). Traveltime misfit function $\epsilon =\sum _x [t_{pred}(x)-t_{obs} (x)]^2$ . A gradient method is likely to get stuck in a local minima for waveform inversion, compared to quickly moving to the global minima for traveltime inversion.

The next example demonstrates a highly non-linear misfit function.

Example 2   Highly Non-linear Waveform Misfit Function: In contrast to the traveltime misfit function, the waveform misfit function in Figure 3.1c has many local minima between the correct velocity of 1.0  km/sand the starting model velocity of 1.5  km/s. Hence a gradient optimization method will get stuck in a local minima well before it reaches the global minima. The reason for so many local minima is that the traces are quite wiggly w/r to time, and slight changes in the hypothetical value of V will time shift the seismograms. Thus a velocity change that shifts the wiggly seismograms by about a period or two will yield a similar looking seismogram and a misfit function with about the same value as before the shift. Hence the numerous local minima. Note that the waveform misfit curve becomes smoother for larger hypothetical velocities because large velocities are associated with longer wavelengths in the data.

How does one avoid creating misfit function functions that are dense in local minima or subject to flat valleys or plains? There is no general remedy to avoiding local minima, but some tricks can be attempted. One trick is to low-pass filter the data or model and then use a gradient search to find a model near the global minima. Using this model as the new starting model, begin a gradient search on the unfiltered data (see Nemeth et al., 1997). The next example demonstrates the above remedy by low pass filtering of the Figure 3.1 seismograms.

Example 3   Mildly Non-linear Waveform Misfit Function: The Figure  3.1a seismograms are low=pass filtered to give the seismograms shown in Figure 3.2a. The associated waveform misfit function in Figure 3.2c shows many fewer local minima compared to the misfit function constructed from the unfiltered data. Thus, a gradient search method will make good progress in getting to the global minimum.

  

Figure 3.2: Same as Figure 1 except the data are low-pass filtered by a 0-2 Hz filter. The wrinkles in the waveform misfit function have been mostly ironed out.

The goal of this chapter is to explain the basics of waveform inversion, concentrating on its mathematical derivation and the intuitive interpretation the equations. For simplicity of exposition, the case of an acoustic medium will be treated.