Geoinformatika 2014; 4(52) : 48-53 (in Ukrainian)
A METHOD OF INCREASING THE NUMERICAL STABILITY OF THE WAVE FIELD CALCULATION BASED ON THE MATRIX METHOD OF THOMSON–HASKELL
R.M. Pak
Army Academy named after Hetman Petro Sahaydachnyi, Gvardiyska Str., 32, Lviv 79012, Ukraine,
e-mail: rpak@email.ua
Purpose. It is known that the area of application of the Thomson–Haskell calculation scheme is in principle limited during the direct computer realization. The purpose of this work is the development of approaches to fix this problem for the case of arbitrary dipole effective-point source which is located in any layer of horizontal-layered half-space.
Design/methodology/approach. The suggested approach is based on the method of transition from image of the wave field using characteristic matrices of the 4th order (method of Thomson–Haskell) to image, using characteristic matrices of the 6th order.
Findings. The result of this work is the wave field presentation which does not include the characteristic matrices of the 4th order. Each minor of these matrices is replaced with the corresponding element of characteristic matrices of the 6th order. There is a doubling of columns and rows in the resulting matrices of the 6th order. Because of this, the order of characteristic matrices is lowered to five by reducing them to a quasi diagonal form. This will save computer time during calculation of interference fields.
Practical value/implications. The developed approach increases the numerical stability during the calculation of wave fields. As a result, the computation of synthetic seismograms can be done for very thick layers of medium and higher wave frequencies.
Keywords: mathematical modelling, seismic wave-field, matrix method, vertically heterogeneous medium, tensor of seismic moment, synthetic seismogram.
- Aki K., Richards P. Kolichestvennaya seysmologiya [Quantitative Seismology]. Moscow, Mir, 1983, 519 p.
- Molotkov L.A. Matrichnyy metod v teorii rasprostraneniya voln v sloistykh, uprugikh i zhidkikh sredakh [Matrix method in the theory of wave propagation in layered, elastic and liquid media]. Leningrad, Nauka, 1984, 201 p.
- Molotkov L.A. O rasprostranenii uprugikh voln v sredakh, soderzhashchikh tonkie ploskoparallel’nye sloi [Propagation of elastic waves in media containing thin plane-parallel layers]. Voprosy dinamicheskoy teorii rasprostraneniya seysmicheskikh voln [Questions dynamic theory of seismic wave propagation]. Leningrad, Publishing house of Leningrad University, 1961, no. 5, pp.240-280.
- Pak R.M. Modeliuvannia khvylovoho polia, zbudzhenoho hlybynnym dzherelom u vertykalno-neodnoridnomu seredovyshchi [Modeling of wave-fields excited by deep source in a vertically-heterogeneous medium]. Geofizicheskij zhurnal [Geophysical Journal], 2005, vol. 27, no. 5, pp. 887-894.
- Pak R.M. Khvylove pole v odnoridnomu seredovyshchi dlia dzherela u vyhliadi odynarnoi syly abo podviinoi pary syl [Wave field calculation for rupture with displacement along inner surface]. Geoinformatika [Geoinformatics (Ukraine)], 2004, no. 1, pp.36-44.
- Chapman C.H. Yet another elastic plane-wave, layer-matrix algorithm. Geophysical Journal International, 2003, vol. 154, pp.212-223.
- Dunkin I.W. Computation of modal solution in layered elastic media at high frequencies. Bulletin of the Seismological Society of America, 1965, vol. 55, pp. 335-358.
- Knopoff L.A. Matrix method for elastic wave problems. Bulletin of the Seismological Society of America, 1964, vol. 54, pp.431-438.
- Thrower E.N. The computation of the dispersion of elastic waves in layered media. Journal of Sound and Vibration, 1965, vol.2, pp. 210-226.
- Wang R. A simple orthonormalization method for stable and efficient computation of Green’s functions. Bulletin of the Seismological Society of America, 1999, vol. 89, pp. 733-741.
- Watson T.N. A note on fast computation of Rayleigh wave dispersion in the multilayered halfspace. Bulletin of the Seismological Society of America, 1970, vol. 60, pp. 161-166.
//