Acoustic Waveform Inversion and its Numerical Implementation: Waveform tomography inverts seismic traces for the earth’s elastic parameters. Unlike traveltime tomography, it utilizes both the amplitude and phase information of all arrivals rather than only the traveltimes of selected events. Although elegant in theory, waveform inversion is not always used in exploration geophysics because it is computationally expensive (often more than an order of magnitude more expensive than migration), it requires an expert user to get it to work well, and sometimes it does not work at all. The most challenging problem is that the iterative nonlinear inversions tend to get stuck in a local minimum. Fortunately, the increasing speed of computers, more effective inversion strategies, and a greater demand for accurate lithology information will increase its use in the future.