Introduction

fwi (Tarantola, 2005,1984; Lailly, 1984), rtm (McMechan, 1983; Whitmore, 1983; Baysal et al., 1983), lsm (Nemeth et al., 1999; Tang, 2009; Dai et al., 2012; Duquet et al., 2000) and wave-equation traveltime inversion (Luo, 1991; Woodward, 1992; De Hoop and van Der Hilst, 2005; Woodward, 1989; Luo and Schuster, 1991) are important tools for imaging seismic data at the engineering (Buddensiek et al., 2008), exploration (Pica et al., 1990; Mora, 1988; Shin and Cha, 2008; Zhou et al., 1995; Virieux and Operto, 2009; Mora, 1989; Pratt and Goulty, 1991; Krebs et al., 2009) and earthquake (Tong et al., 1998; Fichtner and Trampert, 2011b; Marquering et al., 1999; Van Der Hilst and Maarten, 2005; Fichtner et al., 2009; Tape et al., 2009; Fichtner, 2011; De Hoop and van Der Hilst, 2005; Fichtner and Trampert, 2011a) scales. In all of the above methods, the wave equation is solved to estimate the model that minimizes, in some sense, the difference between the predicted and observed data. The main value of these wave equation-based imaging methods is that they overcome the high-frequency assumption of ray-based methods and use many, if not all, of the arrivals to reconstruct a finely detailed earth model. The hope is to find models with spatial resolution of one-half wavelength, and perhaps even better if evanescent energy can be exploited (Schuster et al., 2012; de Fornel, 2001; Fink, 2008). The main limitations of wave equation imaging are computational cost, extensive preprocessing of the data, and extensive trial-and-error testing compared to the ray-based methods.

To optimize the use of wave equation imaging one must understand its limits of spatial resolution. Without this understanding, models can be over parameterized and lead to solutions that honor the data but violate the wavelength-based resolution limits of wave propagation. Such models should be avoided in our attempts to understand the earth.

In the last 30 years there has been much progress in mathematically defining the resolution limits of seismic images. The two most important categories of seismic imaging and their resolution limits are for traveltime tomography and reflection imaging.