GEOPHYSICS, VOL. 62, NO. 1 (JANUARY-FEBRUARY 1997); P. 325–334, 7 FIGS., 1 TABLE. Point-source τ-p transform: A review and comparison of computational methods Yanghua Wang ∗ and Gregory A. Houseman ‡ ABSTRACT The τ - p transformation of reflection seismic data ex- cited by a point source requires a cylindrical slant stack, which includescompensation for the phase shift and geo- metrical spreading associated with cylindrical geometry. Using a simple test model, we review and compare the different computationalmethodsused for the cylindrical slant stack. The two major method types are the Bessel- Fourier transform (BFT) method and a set of meth- ods (TD2D, TDX, TDTC and TDTS) based on time- domain convolution. The stack integral in each of these approachescan be separated into two parts,with the part correspondingto the contribution ofincomingwavesbe- ing negligible for large ray parameter p. Neglecting the latter term for large p generally avoids a major problem caused by aliasing between time and space domains. The TDTS method introduced here has the advan- tage of being expressed in terms of a conventional 2-D slant stack (weighted by offset x ) plus a correction term. This formulation facilitates a quantitative comparison of the 3-D cylindrical slant stack with a 2-D slant stack. The TDTS method compares favorably with the TD2D method, in which the cylindrical slant stack is expressed as a weighted integral of the 2-D slant stacks. TDTS avoids the singular weighting function needed in the in- tegralfor the TD2D method,and therefore haslessprob- lemswith numericalnoise.For allofthe methodsconsid- ered, the accuracy of the map transformation is limited by the spatial and temporal resolution of the test data set. Some corrections to previously published methods are also provided. INTRODUCTION A τ - p mapping is a data transformation that transforms a seismic data set from the conventional time-offset distance ( t , x ) plane to the transform domain defined bythe intercept time-slowness plane ( τ, p). The process of deriving U ( τ, p) from the reflected seismic waves S( t , x ) observed at source- receiver separation x and time t can be viewed as a means of obtaining the plane-wave response of the layered earth. The τ - p transform clearly separates different reflection events that may be superposed in the t -x domain, as is illustrated in the example of Figures 1 and 2; thus the transform facilitates in- terpretation. Most of the examples of τ - p transformations in the literature (e.g., Schultz and Claerbout, 1978; McMechan and Ottolini, 1980; Stoffa et al., 1981; Zhou and Greenhalgh, 1994) are based on the 2-D slant stack or Radon transform, with the simple summation of the signal along lines of constant slope within the data and the possible addition of a semblance, or autocorrelation criterion to enhance noise rejection. This Manuscript received by the Editor May 22, 1995; revised manuscript received April 1, 1996. ∗ Department of Geology, Imperial College, Prince Consort Road, London SW7 2BP, England. ‡Department ofMathematicsand Australian GeodynamicsCooperative Research Centre,Monash University,Wellington Road,Clayton,Vic3168, Australia. c  1997 Society of Exploration Geophysicists. All rights reserved. method is a sensitive means of velocity filtering for selecting waves with apparent slowness p and intercept time τ . The 2-D slant stack does not, however, preserve phase and amplitude information for a point source. Analyses such as full wave inversion, amplitude variation with offset (AVO) analy- sis, and dispersive analysis, for surface data excited by a point source require the cylindricalimplementation ofthe τ - p trans- form (e.g., Chapman, 1981; Henry et al., 1981; Phinney et al., 1981; Treitel et al., 1982; Harding, 1985; Brysk and McCowan, 1986; Wilson, 1986; Benoliel et al., 1987; Kappus et al., 1990; Dietrich, 1990; Fokkema et al., 1992; Zhao et al., 1994). The cylindrical τ - p transform is derived from the solution of the wave equation for a point source in a 3-D vertically stratified medium. In this paper, we review several different computa- tionalmethodsand test these methodsusinga simple synthetic data set.Our purpose isto illustrate the performance and com- pare the resultant τ - p maps of the cylindrical slant stack ob- tained using different numerical computation strategies. 325