High performance finite difference (FD) schemes for computation of synthetic seismograms are developed and tested following mathematical analysis and evaluations of their cost-effectiveness (as quantified by the computation time required to obtain a given level of accuracy). On the basis of these tests we find that optimally accurate O(2,2) (second order in space and time) FD schemes are preferable for practical computations. We then present calculations for a standard 2-D test model.

　A general criterion for optimally accurate numerical operators was derived by Geller & Takeuchi (1995) and, based on this criterion, an O(2,2) optimally accurate FD scheme (second order in both time and space) was derived by Geller & Takeuchi (1998) for 1-D cases and by Takeuchi & Geller (2000) for 2-D and 3-D cases.Following a similar procedure, we derive two new optimally accurate schemes for the 1-D case: an O(2,4) optimally accurate FD scheme (second order in time and fourth order in space) and an optimally accurate scheme using the spectral element method (SEM). We then compare the various optimally accurate schemes for 1-D heterogeneous and homogeneous models. All are broadly similar in cost-performance ratios for solution errors of around 1%, which is the accuracy range commonly required for practical applications. However, due to ease of programming, the O(2,2) optimally accurate FD method seems preferable in practice.However, if extremely high accuracy (solution errors of, say, less than 0.01%) were required, then SEM approaches might be preferable, but the difficulty of grid generation for complex structures is a significant problem. We show that all of the optimally accurate schemes are superior to all of the conventional schemes (schemes which do not satisfy the criterion for optimally accurate operators). We also show that staggered grid (SG) schemes, which are widely used, can be transformed to conventional FD schemes which use displacement as the only dependent variable and that such schemes have no advantage in computational accuracy and efficiency over other conventional FD schemes. A major advantage of the FD schemes considered here (both optimally accurate and conventional schemes) is that they can stably handle external free surface boundaries, as they are based on the weak form of the equation of motion.

　Previous optimally accurate O(2,2) schemes handled internal lithological discontinuities by treating each such boundary as a potential external free surface, and then "overlapping" the operators for the respective regions. This approach can be used for simple models, but is impractical for the complex heterogeneous models used in exploration seismology. In order to extend the optimally accurate O(2,2) operators to such complex models, we developed an optimally accurate heterogeneous method, following similar approaches using conventional FD operators. A theoretical analysis supports the use of this method at internal boundaries, and computational examples demonstrate its accuracy. We also developed a method for computing synthetics in combined fluid-solid media. Finally we apply the new heterogeneous scheme to the "Marmousi model," a standard test model used in exploration seismology, and demonstrate that the new scheme is well suited for application to actual problems. The Marmousi model was originally presented as an acoustic model. We use a Poisson's ratio of 0.25 and also make calculations for an elastic Marmousi model.

　All of the calculations in this thesis are for 1-D or 2-D cases, but the heterogeneous O(2,2) scheme presented here can be applied to the 3-D case.

Figure 1: Discontinuous two-layered model with free surface boundary conditions (left) and relative r.m.s. error versus CPU Time (right). Compared schemes are: OPT2 and OPT4 (O(2,2) and O(2,4) optimally accurate scheme), SEM-OPT (optimally accurate SEM) and CONV2 and CONV4 (conventional second and fourth order scheme). Each line color is displayed by Courant number that each scheme shows the best performance with it. Relations between the color and Courant number are red: 0.1, blue: 0.3 and magenta: 0.8.

Figure 2: Snapshots of a computation in combined fluid-solid mdeia. The upper and lower layer are fluid and solid respectively. The fluid-solid interface exists in the middle of the model and a source of a point force is located in the fluid layer. Snapshots at t = 0.1875 (s) of the x-(left) and z-(right) components of the displacement with the pressure change. Propagation of P wave in the solid layer can be seen to be faster than in the fluid layer and S-wave can be also seen in the solid layer. In the fluid layer reflected P wave from the boundary is seen.

Figure 3: a 2-D numerical example for validation of the new scheme. Snapshot of x (left) displacement at 0.25 (s) computed by OPT2 and used as a numerical solution to compute relative RMS errors and relative RMS errors versus elements per wavelength of x displacement (right). The log scale used for the vertical axis and results by OPT2 and CONV2 are shown in red and green respectively.

Figure 4: P velocity of Marmousi model (top) and snapshots of x displacement of P-SV problem from 0.4 to 1.6s by 0.4s interval (from second to bottom). A point force is located at x=2800 and z=6000m. The free surface condition is used for the top boundary and an absorbing boundary is used for other external boundaries. These figures indicate aplicability of the new scheme to actual problems arise in the geophysical seismology.

　地球浅部の3次元速度構造探査では，観測地震波形と比較する理論波形計算が必要となる．理論波形計算手法として現在FDM(差分法)，SEM(スペクトル有限要素法)，PSM(Pseudoスペクトル法)などが広く用いられている．本論文は，Geller and Takeuchi(1995)が導入した最適演算子を用いたFDM，SEMと，最適演算子を用いない従来のFDMについて，精度・計算量・プログラミングの容易さなどの観点から優劣を評価した．また，誤差解析を行いStaggered grid法が過大に評価されていることを指摘した．更に，不均質媒質中の内部境界の表現法として，従来の差分法において用いられているheterogeneous methodが最適演算子計算においても適用可能であることを示した．以上を踏まえ，単純な不均質媒質から現実的なMarmousiモデルまでの様々なモデルについて波形計算を行い，最適演算子を用いる高精度計算手法の優位性を確認し，物理探査への応用が可能であることを示した．

　本論文は7章からなる．

　第1章はイントロダクションである．既存の計算スキームであるPSM，SEM，FDM等の紹介，Geller and Takeuchi(1995)に始まる最適演算子を用いた計算スキームの発展，本論文の目的・構成が述べられている．

　第2章では，先行研究であるGeller and Takeuchi(1998)に示された時間2次・空間2次精度の最適演算子O(2,2)を基に，時間2次・空間4次精度の最適演算子O(2,4)を導出した．また，Mizutani et al.(2000)によるPSMの最適演算子の導出に基づき，SEMの最適演算子を導出した．

　最適演算子を用いない従来の2次精度FDM(CONV-2)，4次精度FDM(CONV4-4)と最適演算子O(2,2)やO(2,4)を用いたFDM(OPT-2，OPT-4)および最適演算子を用いたSEM(SEM-OPT)について，1次元の様々なモデルを用いて優劣を比較した．その結果，最適演算子を用いたOPT-2，OPT-4，SEM-OPTがCONV-2，CONV-4に比して圧倒的なコストパフォーマンスを持つことを示した．一方，OPT-4は均質媒質ではOPT-2に大きく勝るが，強い不連続を含む媒質ではOPT-2に対する優位性は著しく低下することが明らかになった．また，誤差0.01%以下の非常に高い精度を必要とする場合はSEM-OPTが卓越するが，現実的に必要とされる精度である0.1-1%の誤差範囲ではOPT-4，OPT-2とSEM-OPTの精度は同程度であった．これらの数値実験結果から，現実的な応用においてはプログラミングの容易なOPT-2が優れていると結論付けた．

　第3章では，staggered grid(SG)を用いた差分スキームが従来型のスキームCONV-2・CONV-4と同等かあるいは同等以下の精度しかないことを示し，安定化条件や分散関係を用いて定量的な比較を行った．

　第4章では，"heterogeneous method"の誤差解析を行い，2次スキームとして十分な精度を持つことを示した．これにより，複雑な形状の内部境界を容易に処理できることを示した．また，液体一固体境界を扱う時間領域演算子を導いた．

　第5章では，単純な2次元モデル用いてCONV-2とOPT-2を比較し，OPT-2の圧倒的な優位性を確認した．

　第6章では，物理探査手法の比較評価に用いられる2DのMarmousiモデルに対する最適演算子計算を行い，OPT-2が物理探査への応用に耐える段階にあることを示した．

　第7章は全体のまとめである．

　本論文は先行研究であるGeller&Takeuchi(1995)，Geller & Takeuchi(1998)，Takeuchi & Geller(2000)等を踏まえ，先行研究には無いO(2,4)演算子とSEMに対する最適演算子を新たに導出した．更に，様々な演算子を定量的に評価し，実際の応用においてはO(2,2)演算子スキームが最も優れていることを初めて明らかにした．また，O(2,2)スキームを物理探査のスキーム比較に用いられるMarmousiモデルに適用し，最適演算子を用いたスキームが物理探査への現実的応用に耐える段階にあること示した．

　なお，本論文の第2章，第3章の一部はRobert J. Geller・水谷宏光との共著であるが，O(2,4)やSEMの最適演算子の導出，計算の実行，誤差評価，安定性評価，分散関係の導出等は論文提出者が主体となって行っており，論文提出者の寄与が十分に大きいと判断できる．

　従って，博士(理学)の学位を授与できるものと認める．