3:30 PM - 5:00 PM
[S01P-10] Symplectic numerical integration method for normal mode computation of the spherically symmetric Earth
地震や地滑り火山噴火などの励起源が生じる永久変位を包括する波動現象（地震・津波・大気波動）は、重力弾性結合系の地球自由振動解を用いて、変位（と時間・空間微分）の理論波形が簡便高速に計算される。近年報告されている、P波到達前の重力場変動計算では、純粋な弾性地震波動場の粗密に起因する重力場を時間領域で計算や、励起項に相当する非斉次項を有する重力摂動を含む周波数領域の運動方程式の直接積分による手法とられてきた。重力摂動を含む重力弾性結合系の地球自由振動解を用いるとP波到着前の重力場変動が高速簡便に計算されると期待されるが、こらまで、高周波（〜１Hz）領域まで安定的に数値計算することが困難であった。 周波数領域で変数分離した後の動径方向を独立変数とする微分方程式は、重力摂動を含めると６変数連立１次微分方程式となる。安定的に数値積分を行うためにいくつかの手法が提案されてきたが(斎藤2009, Woodhouse 1988)、数値不安定のため高周波まで計算ができない。その連立微分方程式は、変位と重力ポテンシャルの固有関数を一般化座標ベクトルq、歪みと重力ポテンシャルの微分の固有関数を一般化運動量ベクトルpとする、６次元位相空間のハミルトニアン形式d(q,p)dr=J H(q,p)と書くことができる。このハミルトニアンH(r)は一般化座標qと一般化運動量pの2次形式となっており、H(q,p)＝U(q)+T(p)のように分離できないため安定的数値積分法として知られているシンプレクティック（Symplectic）数値計算法を用いることができなかった。今回発見したQ=Q(q,p), P=P(q,p)とする座標変換によりハミルトニアンHがH(Q,P)=U(Q)+T(P)分離できた。さらに、一般化座標として動径rと一般化運動量としてH(r)に相当するE(r)加えた８次元位相空間の新たハミルトニアンH(Q,P)-Eは座標rに関して一定となり、安定的に数値計算できることが判明した。
The normal mode method has been used for wave phenomena including permanent deformation, tsunami and atmospheric waves when we include the solid earth and ocean and atmosphere as a single gravitationally coupled elastic body. Once the normal mode set have been obtained, the computation time is very short, compared with other numerical simulation methods. Recently pre-P signal in seismograms induced by gravity potential change has been reported and confirmed by numerical simulations. We try to shorten the computation time by using the normal mode method. After the separation of variables of the differential equation for a symmetric Earth model in the frequency domain, 6 dependent variables as functions of independent variable r compose a 1 st order simultaneous differential equation. The numerical solution methods for the simultaneous differential has been proposed, however , it has been difficult to compute the solutions at high frequencies, ~1Hz. We extend the Hamiltonian formulation of the differential equation d(q,p)dr=J H(q,p). H(q,p) is a quadratic function of q and p and cannot be separated into H(q,p)＝U(q)+T(p). We found a coordinate transformation Q=Q(q,p) and P=P(q,p) so that the new Hamiltonian H(Q,P)= U(Q)+T(P) is separable. So a numerically stable symplectic integration method is applicable for the differential equations. Further, we extended the dimension by adding a new generalized coordinate r and a new generalized momentum E and compose a new Hamiltonian H(Q,P)-E. The new Hamiltonian can be integrated stably.
The normal mode method has been used for wave phenomena including permanent deformation, tsunami and atmospheric waves when we include the solid earth and ocean and atmosphere as a single gravitationally coupled elastic body. Once the normal mode set have been obtained, the computation time is very short, compared with other numerical simulation methods. Recently pre-P signal in seismograms induced by gravity potential change has been reported and confirmed by numerical simulations. We try to shorten the computation time by using the normal mode method. After the separation of variables of the differential equation for a symmetric Earth model in the frequency domain, 6 dependent variables as functions of independent variable r compose a 1 st order simultaneous differential equation. The numerical solution methods for the simultaneous differential has been proposed, however , it has been difficult to compute the solutions at high frequencies, ~1Hz. We extend the Hamiltonian formulation of the differential equation d(q,p)dr=J H(q,p). H(q,p) is a quadratic function of q and p and cannot be separated into H(q,p)＝U(q)+T(p). We found a coordinate transformation Q=Q(q,p) and P=P(q,p) so that the new Hamiltonian H(Q,P)= U(Q)+T(P) is separable. So a numerically stable symplectic integration method is applicable for the differential equations. Further, we extended the dimension by adding a new generalized coordinate r and a new generalized momentum E and compose a new Hamiltonian H(Q,P)-E. The new Hamiltonian can be integrated stably.