The stationary phase analysis of the Kirchhoff-type demigrated field
Sun Jian-Guo1,2
1. College for Geoexploration Science and Technology, Jilin University, Changchun 130026, China;
2. Laboratory for Integrated Geophysical Interpretation Theory of Ministry for Land and Resourc-Laboratory for Wave Theory and Imaging Technology, Changchun, 130026, China.
Abstract：
In the literature, stationary phase analysis of Kirchhoff-type demigrated fields is carried out mainly under the following two conditions: (1) The considered isochrone and the target reflector are tangential to each other; (2) The spatial duration of the wavelet of the depth-migrated image is short. For the isochrones that are not tangential to the target reflector and for the depth-migrated images that have a large spatial duration, the published stationary phase equation for the demigrated field will become invalid. For performing the stationary phase analysis of the Kirchhoff-type demigrated field under the conditions that the considered isochrone and the target reflector are not tangential to each other and that the spatial duration of the wavelet of the depth-migrated image is not short (the general conditions), I derive the formulas for the factors appearing in the stationary phase formula in two dimensions, from which I find that for different isochrones the horizontal coordinates of the stationary point of the depth difference function are different. Also, the equation for the Kirchhoff-type demigrated field consists of two parts. One is the true-amplitude demigrated signal and the other is the amplitude distortion factor. From these facts the following two conclusions can be drawn: (1) A demigrated signal is composed of many depth-migrated images and one depth-migrated image trace provides only one sample to the demigrated signal; and (2) The amplitude distortion effect is an effect inherent in the Kirchhoff-type demigration and cannot be eliminated during demigration. If this effect should be eliminated, one should do an amplitude correction after demigration.
SUN Jian-Guo. The stationary phase analysis of the Kirchhoff-type demigrated field[J]. APPLIED GEOPHYSICS, 2010, 6(1): 18-30.
[1]
Aki, K., and Richards, P. G., 1980, Quantitative seismology theory and methods, Volume 1:W. H. Freeman and Co.
[2]
Ben-Menahem, A., and Beydoun, W. B., 1985, Range of validity of seismic ray and beam methods in general inhomogeneous media - I. General theory: Geophys. J. Roy. Astr. Soc., 82, 207 - 234.
[3]
Bleistein, N., 1984, Mathematical methods for wave phenomena: Academic Press Inc.
[4]
ervený, V., 2001, Seismic ray theory: Cambridge University Press.
[5]
Claerbout, J. F., 1985, Imaging the earth’s interior: Blackwell Scientific Publications.
[6]
Gardner, G. H. F., French, W. S., and Matzuk, T., 1974, Elements of migration and velocity analysis: Geophysics, 39, 811 - 825.
[7]
Guo, S.J, Li, Z.C., Sun, X.D., Ye, Y.M., Teng, H.H., and Li, F., 2008, Post stack reverse time migration using a finite difference method based on triangular grid: Applied Geophysics, 5(2), 115 - 120.
[8]
Hubral, P., Schleicher, J., and Tygel, M., 1996, A unified approach in 3-D seismic reflection imaging, Part I: Basic concepts: Geophysics, 61(3), 742 - 758.
[9]
Jaramillo, H., Schleicher, J., and Tygel, M., 1998, Discussion to “A unified approach to 3-D seismic reflection imaging, Part II: Theory” (M. Tygel, J. Schleicher, and P. Hubral, Geophysics, 61, 759 - 775): Geophysics, 63, 670 - 673.
[10]
Jaramillo, H., and Bleistein, N., 1999, The link of Kirchhoff migration and demigration to Kirchhoff and Born modeling: Geophysics, 64, 1793 - 1805.
[11]
Liu, G.F., Liu, H., Li, B., and Meng, X.H., 2009, Getting pre-stack time migration travel times from the single square root operator: Applied Geophysics, 6(2), 129 - 137.
[12]
Luneburg, R. K., 1964, Mathematical theory of optics: University of California Press.
[13]
Kravtsov, Yu. A., and Orlov, Yu. I., 1990, Geometrical optics of inhomogeneous media: Springer-Verlag.
[14]
—1993, Caustics, catastrophes, and wave fields: Springer-Verlag.
[15]
Santos, L. T., Schleicher, J., Tygel, M., and Hubral, P., 2000, Seismic modeling by demigration: Geophysics, 65, 1281 - 1289.
[16]
Schleicher, J., Tygel, M., and Hubral, P., 1993, 3-D true-amplitude finite-offset migration: Geophysics, 58(8), 1112 - 1126.
[17]
Stamnes, J. J., 1986, Waves in focal region: Adam Hilgr.
[18]
Sun, J.G., 1995, 3-D crosswell transmissions: Paraxial ray solutions and reciprocity paradox: Geophysics, 60(3), 810 - 820.
[19]
Sun, J.G., 1996, The relationship between the first Fresnel zone and the normalized geometrical spreading factor: Geophysical Prospecting, 44, 351-374.
Sun, J.G., 2002a, On some problems in 3-D isochrone-stack demigration: Journal of Jilin University, Earth Science Edition, 32(3), 273 - 278 (in Chinese).
[22]
Sun, J.G., 2002b, Kirchhoff-type true-amplitude migration and demigration: Progress in Exploration Geophysics, 6(6), 1 - 5 (in Chinese).
[23]
Sun, J.G., 2004, True-amplitude weight functions in 3D limited-aperture migration revisited: Geophysics, 69(4), 1025 - 1036.
[24]
Tygel, M., Schleicher, J., and Hubral, P., 1994, Pulse distortion in depth migration: Geophysics, 59(10), 1561 - 1569.
[25]
Tygel, M., Schleicher, J., and Hubral, P., 1995, Dualities involving reflectors and reflection-time surfaces: Journal of Seismic Exploration, 4, 123 - 150.
[26]
Tygel, M., Schleicher, J., and Hubral, P., 1996, A unified approach in 3-D seismic reflection imaging, Part II: Theory: Geophysics, 61(3), 759 - 775.
[27]
Ye, Y.M., Li, Z.C., Xu, X.G., Zhu,X.F., and Tong, Z.Q., 2009, Preserved amplitude migration based on the one way wave equation in the angle domain: Applied Geophysics, 6(1), 50 - 58.
2010, Vol. 6 
