Introduction

Usually, multiples in seismic data have been considered as noise for the imaging of the primaries Berkhout and Verschuur (1997). This is because it is difficult to put the multiples onto their scattering points, since the commonly used imaging conditions can not correctly and simultaneously pick up both the focused primaries and multiples. Schuster et al. (2003) proposed that if the source below the surface is unknown, the autocorrelation of each trace can be used to determine a pseudo source on the surface, since the autocorrelation of the direct wave is t=0 time delay and the direct wave is thus eliminated in the autocorrelogram. The autocorrelogram can be thought to be acquired with the pseudo shot-receiver pair at the surface. Therefore, conventional prestack depth migration can focus and image ghost wave, or the first-order multiple. However, the disadvantages of the method are that the autocorrelogram does not satisfy the wave equation, and the traveltime of the direct wave can not be correctly estimated and cancelled, which makes the travel time calculation in the integral migration not match the travel time in the autocorrelogram and the imaging noises occur. The crosstalk in the autocorrelogram also causes the imaging noise. On the other hand, the ghost wave is the first-order multiple and the imaging of higher-order multiples is ignored.

The imaging condition proposed by Claerbout (1971) should be modified if primaries and multiples are simultaneously imaged, whether the source position is known or unknown. The imaging condition I propose states that the radius of curvature of the wavefront equals zero. This is called the depth-focusing imaging condition. MacKay and Abma (1993) use depth focusing to carry out velocity analysis. If the migration velocity is larger than the medium velocity, then the focusing depth is less than the reflection depth, and the imaging depth is larger than the reflection depth; on the other hand, if the migration velocity is less than the medium velocity, then the focusing depth is greater than the reflection depth, and the imaging depth is less than the reflection depth. The real reflection depth lies at the mid-point between the focusing depth and the imaging depth. In that paper, the authors proposed a method for estimating the radius of the curvature of the wavefront. However the formula is suitable only for imaging the primaries. For a given scattering point, the primary and multiple scattering from it are simultaneously focused at the same depth in the model space and at different times in the data space with the downward wavefield continuation. The "focusing" means that the received scattered wavefield is collapsed into the scattering point, and the radius of curvature of the wavefront diminishes to zero. With the depth-focusing imaging condition, the focused imaging values of the primary and multiples can be simultaneously picked up from the depth-extrapolated wavefield, which is expressed in the time domain. The following are some advantages of depth-focusing imaging. The primaries and multiples (including the higher-order multiples) can be simultaneously imaged; the source position can be known (for the primaries) or unknown (for the multiples); all of the scattered wavefield can be added together, and computation efficiency can be improved. The disadvantage is that the depth-focusing imaging condition is difficult to use, especially for data with a lot of noise.