Introduction

The main objective of migration imaging is to generate an image of the reflectors, that is, to position reflection points and scattering points at their true subsurface locations. The methodology is to downward continuate the observed wavefield to the reflection points or scattering points using a known macro-velocity model with appropriate propagators, and to pick out the focused wavefield with an imaging condition. The focused wavefield displays the image of reflectors or scatterers. Therefore, I give the definition for migration imaging: Based on some assumptions about the geological medium and with the help of mathematical models, the observed seismic wavefield is extrapolated to the subsurface reflectors using a macro-velocity model with a propagator, and the imaging amplitudes are extracted with an imaging condition. Generally, the geological medium is assumed to be an acoustic medium, and the mathematical model is either the one-way wave equation or the Kirchhoff integral operator. However, migration imaging has not completely satisfied the needs of oil and gas exploration, since many reservoirs found recently are controlled not only by their geological structures but also by their lithology. Therefore, the lithological parameters are increasingly important to oil and gas exploration. Lithological parameter estimation is typically an inverse problem. In essence, migration imaging is an inverse problem, and it is also ill-posed. However, migration imaging is changed into an apparently well-posed problem by splitting it into two processes: wavefield extrapolation and macro-velocity analysis. The main objective of inversion imaging is to estimate lithological parameters or their disturbances, including reflectivity, P-wave velocity, S-wave velocity, and the density. There are linearized and non-linear inversions. The basis of linearized inversion is to linearize the formula characterizing the scattering wavefield with the Born approximation. The Born approximation is a "physical" approximation, with which only the primaries are modeled. The analytical (for constant background) or formal (for variable background) inversion formulas can be derived from the linearized forward-modeling formulas. This is a non-iterative linearized inversion. Based on least-squares theory, an iterative linearized inversion approach can be derived from linearized forward modeling. For the non-linear waveform inversion, only the wave propagator is linearized at a point in the model space. With the propagator, all of the wave phenomena are characterized. We call this linearization as a "mathematical" approximation, with which both primaries and multiples are simulated. This is the main difference between the two inversion approaches. Theoretically, the non-linear inversion Mora (1987); Tarantola (1984) is superior to the linearized inversion Bleistein et al. (1987); Bleistein (1987). In practice, it is very difficult to recover all wavenumber components of the lithological parameters, since the seismic data is frequency-band-limited and aperture-limited and polluted with non-Gaussian noise. Therefore, the linearized migration/inversion is becoming more and more important, especially the iterative linearized migration/inversion approach. Stolt and Weglein (1985) discussed the relation between the migration and the linearized inversion. Gray (1997) gave a comparison of three different examples of true-amplitude imaging. So-called true-amplitude imaging tries to recover the reflectivity of the reflectors.

In this paper, I compare non-iterative linearized migration/inversion imaging, iterative linearized migration/inversion imaging, and nonlinear waveform inversion. All of these imaging methods can be considered as back-projection and backscattering imaging. From backscattering imaging, we know that seismic wave illumination has a key influence on so-called true-amplitude imaging, and I give an analysis for the possibility of relative true-amplitude imaging. I also analyze the factors which affect the image quality. Finally, I point out that the Born approximation is not a good approximation for linearized migration/inversion imaging, and that the De Wolf approximation is a better choice.