Summary of operators

All wave-equation migration velocity analysis operators described in the preceding sections are similar in that they relate perturbations of the image with perturbations of the (slowness) model. In all cases, this velocity estimation procedure takes advantage of features of migrated images which indicate incorrect imaging. The imaging inaccuracies can have several causes, i.e. incorrect downward continuation, irregular illumination, limited acquisition aperture etc., but the wave-equation MVA operators translate all inaccuracies in model updates. This feature, however, is a fundamental limitation of all migration velocity analysis techniques and we do not expand on this topic further.

Since the migration velocity analysis operators link image perturbations with slowness perturbations, they are all composed of several common parts, but with implementations that are specific for each imaging configuration. Thus, a wave-equation MVA operator is composed of an extrapolation operator (for wavefield reconstruction from recorded data), an imaging operator (for image construction from reconstructed wavefields) and a scattering operator (for relating wavefield perturbations to slowness perturbations). The following table summarizes the wave-equation MVA operator components in different imaging configurations, as described in detail in the preceding sections.

	extrapolation operator			imaging operator	scattering operator
	equation	velocity	extrapolation	type	equation
zero-offset	SSR	half	backward	zero time	linearized SSR
	equations 1, 2			equation 3	equations 8, 11
survey-sinking	DSR	full	backward	zero time	linearized DSR
				implicit cross-corr.
	equations 12, 15			equation 16	equations 20, 23,24
shot-record	SSR	full	forward	zero time	linearized SSR
			backward	explicit cross-corr.
	equations 25,26,27			equation 28	equations 31,32,36,37