A platform for Kirchhoff data mapping in scalar models of data acquisition

Kirchhoff data mapping (KDM) is a procedure for transforming data from a gi ven input source/receiver configuration and background earth model to data corresponding to a different output source/receiver configuration and backg round model. The generalization of NMO/DMO, datuming and offset continuatio n are three examples of KDM applications. This paper describes a 'platform' for KDM for scalar wavefields. The word, platform, indicates that no calcu lations are carried out in this paper that would adapt the derived formula to any one of a list of KDMs that are presented in the text. Platform formu lae are presented in 3D and in 2.5D. For the latter, the validity of the pl atform equation is verified - within the constraints of high-frequency asym ptotics - by applying it to a Kirchhoff approximate representation of the u pward scattered data from a single reflector and for an arbitrary source/re ceiver configuration. The KDM formalism is shown to map this Kirchhoff mode l data in the input source/receiver configuration to Kirchhoff data in the output source/receiver configuration, with one exception. The method does n ot map the reflection coefficient. Thus, we verify that, asymptotically, th e ray theoretical geometrical spreading effects due to propagation and refl ection (including reflector curvature) are mapped by this formalism, consis tent with the input and output modelling parameters, while the input reflec tion coefficient is preserved. In this sense, this is a 'true-amplitude' fo rmalism. As with earlier Kirchhoff inversion, a slight modification of the kernel of KDM provides alternative integral operators for estimating the sp ecular reflection angle, both in the input configuration and in the output configuration, thereby providing a basis for amplitude-versus-angle analysi s of the data.