frechet.c

su 的源代码库

/* Copyright (c) Colorado School of Mines, 2006.*/
/* All rights reserved.                       */

/*                                                              */
/*  Function RmFrechet()                                        */
/*                                                              */
/*  Computing reflectivity matrices for a given model at a      */
/*  specific slowness and temporal frequency and its            */
/*  correspondent Frechet derivatives 	                        */
/*                                                              */
/*  Input parameters:                                           */
/*  alpha..................p-wave velocities of the model       */
/*                         global variable                      */
/*  beta...................s-wave velocities of the model       */
/*                         global variable                      */
/*  rho....................densities of the elastic model       */
/*                         global variable                      */
/*  thick..................thicknesses of the model             */ 
/*                         global variable                      */
/*  wC.....................complex frequency                    */
/*                         global variable                      */
/*  uC.....................complex slowness                     */
/*                         global variable                      */
/*                                                              */
/*  Output parameters:                                          */
/*  rm.....................the reflectivity matrix              */
/*  DmB....................Frechet derivatives of the           */
/*                         reflectivity matrix                  */
/*  note the DmB data structure:                                */
/*  DmB[# of Layers][derivatives from the layer (or lim[0])     */
/*                   to the layer (or lim[1])][4 elements]      */
/*                                                              */
/*                                              Wences Gouveia  */
/*                                              September 1995  */
#include "posteriori.h"
void RmFrechet()
{
   /* declaration of variables */
   int i, iL, iDer;             /* counter */
   complex aux1;                /* auxiliar quantities */
   complex am, amI, bm, bmI;    /* vertical slownesses for P and S waves */
                                /* amI = am * I, bmI = bm * I */
   complex wThick;              /* wC * thickness */
   complex E[2][2];             /* phase-shift matrix */
   complex mT[2][2];            /* reflectivity matrix at top of layer */
   complex mTtD[2][2];          /* mt * tD */
   complex tUinv[2][2];         /* tU * inv */
   complex mB[2][2];            /* reflectivity matrix at bottom of layer */
   complex rD[2][2], tD[2][2];  /* reflec. and transm. coefficients for */
                                /* downgoing waves */
   complex rU[2][2], tU[2][2];  /* reflec. and transm. coefficients for */
                                /* upgoing waves */
   complex mAux[2][2];          /* auxiliar matrix */
   complex inv[2][2];           /* inv = (I - mT * rU)^-1 */
   
   /* initializing the reflectivity matrix at the bottom of half space */
   mT[0][0] = zeroC; mT[0][1] = zeroC;
   mT[1][0] = zeroC; mT[1][1] = zeroC;
   
   /* initializing the reflectivity matrix at the bottom */
   /* of layer nL - 1 */
   RTd(nL - 1, nL);

   mB[0][0] = coeffD[0]; mB[0][1] = coeffD[1]; 
   mB[1][0] = coeffD[2]; mB[1][1] = coeffD[3];

   /* checking depth limits */
   for (iDer = 0; iDer < numberPar; iDer++)
   {
      /* Vp -> Vs -> rho, if all active */
      if (lim[0] <= nL - 1 && nL - 1 < lim[1])
      {
	 /* Indexes: */
	 /*      [ 0  1 ]       */
	 /*      [ 2  3 ]       */
	 frechetRTd(nL - 1, nL, nL - 1);

	 DmB[nL - 1][limRange * iDer + 0][0] = coeffDFr[iDer][0];
	 DmB[nL - 1][limRange * iDer + 0][1] = coeffDFr[iDer][1];
	 DmB[nL - 1][limRange * iDer + 0][2] = coeffDFr[iDer][2];
	 DmB[nL - 1][limRange * iDer + 0][3] = coeffDFr[iDer][3];	       
      }
      if (lim[0] <= nL && nL < lim[1])
      {
	 frechetRTd(nL - 1, nL, nL);

 
	 DmB[nL - 1][limRange * iDer + 1][0] = coeffDFr[iDer][0];
	 DmB[nL - 1][limRange * iDer + 1][1] = coeffDFr[iDer][1];
	 DmB[nL - 1][limRange * iDer + 1][2] = coeffDFr[iDer][2];
	 DmB[nL - 1][limRange * iDer + 1][3] = coeffDFr[iDer][3];
      }
   }

   /* main loop over the nL layers */
   for (iL = nL - 1; iL >= 1; iL--)
   {
      /* square-root of PSlowness^2 - uuC */    
      auxm1 = PSlowness[iL][0].r - uuC.r;
      auxm2 = PSlowness[iL][0].i - uuC.i;
      auxm3 = sqrt(auxm1 * auxm1 + auxm2 * auxm2);
      auxm3 = sqrt(auxm3);
      angle = atan2(auxm2, auxm1) / 2;
      am.r = auxm3 * cos(angle);
      am.i = auxm3 * sin(angle);

      /* am * I */
      amI.r = -am.i;
      amI.i = am.r;

      /* square-root of SSlowness^2 - uuC */    
      auxm1 = SSlowness[iL][0].r - uuC.r;
      auxm2 = SSlowness[iL][0].i - uuC.i;
      auxm3 = sqrt(auxm1 * auxm1 + auxm2 * auxm2);
      auxm3 = sqrt(auxm3);
      angle = atan2(auxm2, auxm1) / 2;
      bm.r = auxm3 * cos(angle);
      bm.i = auxm3 * sin(angle);

      /* bm * I */
      bmI.r = -bm.i;
      bmI.i = bm.r;

      /* computing phase-shift matrix */
      wThick.r = wC.r * (-2 * thick[iL]);
      wThick.i = wC.i * (-2 * thick[iL]);

      /* cexp (amI * wThick) */
      auxm1 = amI.r * wThick.r - amI.i * wThick.i;
      auxm2 = amI.r * wThick.i + amI.i * wThick.r;
      E[0][0].r = exp(auxm1) * cos(auxm2);
      E[0][0].i = exp(auxm1) * sin(auxm2);

      /* cexp((amI + bmI) * (wThick * .5)) */
      auxm1 = amI.r + bmI.r;
      auxm2 = amI.i + bmI.i;
      auxm3 = .5 * (auxm1 * wThick.r - auxm2 * wThick.i);
      auxm4 = .5 * (auxm1 * wThick.i + auxm2 * wThick.r);
      E[0][1].r = exp(auxm3) * cos(auxm4);
      E[0][1].i = exp(auxm3) * sin(auxm4);

      E[1][0] = E[0][1];

      /* cexp (bmI * wThick) */
      auxm1 = bmI.r * wThick.r - bmI.i * wThick.i;
      auxm2 = bmI.r * wThick.i + bmI.i * wThick.r;
      E[1][1].r = exp(auxm1) * cos(auxm2);
      E[1][1].i = exp(auxm1) * sin(auxm2);
	
      /* applying phase-shift */
      mT[0][0].r = mB[0][0].r * E[0][0].r - mB[0][0].i * E[0][0].i;
      mT[0][0].i = mB[0][0].r * E[0][0].i + mB[0][0].i * E[0][0].r;
      mT[0][1].r = mB[0][1].r * E[0][1].r - mB[0][1].i * E[0][1].i;
      mT[0][1].i = mB[0][1].r * E[0][1].i + mB[0][1].i * E[0][1].r;
      mT[1][0].r = mB[1][0].r * E[1][0].r - mB[1][0].i * E[1][0].i;
      mT[1][0].i = mB[1][0].r * E[1][0].i + mB[1][0].i * E[1][0].r;
      mT[1][1].r = mB[1][1].r * E[1][1].r - mB[1][1].i * E[1][1].i;
      mT[1][1].i = mB[1][1].r * E[1][1].i + mB[1][1].i * E[1][1].r;

      /* bottom-layer matrix - need a sequence of Ref and TRANS coeff. */
      RTd(iL - 1, iL);

      rD[0][0] = coeffD[0]; rD[0][1] = coeffD[1]; 
      rD[1][0] = coeffD[2]; rD[1][1] = coeffD[3];
      
      tD[0][0] = coeffD[4]; tD[0][1] = coeffD[5]; 
      tD[1][0] = coeffD[6]; tD[1][1] = coeffD[7];

      /* computing mT * tD */
      mTtD[0][0].r = mT[0][0].r * tD[0][0].r - mT[0][0].i * tD[0][0].i
	           + mT[0][1].r * tD[1][0].r - mT[0][1].i * tD[1][0].i;
      mTtD[0][0].i = mT[0][0].r * tD[0][0].i + mT[0][0].i * tD[0][0].r
	           + mT[0][1].r * tD[1][0].i + mT[0][1].i * tD[1][0].r;
      mTtD[0][1].r = mT[0][0].r * tD[0][1].r - mT[0][0].i * tD[0][1].i
	           + mT[0][1].r * tD[1][1].r - mT[0][1].i * tD[1][1].i;
      mTtD[0][1].i = mT[0][0].r * tD[0][1].i + mT[0][0].i * tD[0][1].r
	           + mT[0][1].r * tD[1][1].i + mT[0][1].i * tD[1][1].r;
      mTtD[1][0].r = mT[1][0].r * tD[0][0].r - mT[1][0].i * tD[0][0].i
                   + mT[1][1].r * tD[1][0].r - mT[1][1].i * tD[1][0].i;
      mTtD[1][0].i = mT[1][0].r * tD[0][0].i + mT[1][0].i * tD[0][0].r
      	           + mT[1][1].r * tD[1][0].i + mT[1][1].i * tD[1][0].r;
      mTtD[1][1].r = mT[1][0].r * tD[0][1].r - mT[1][0].i * tD[0][1].i
      	           + mT[1][1].r * tD[1][1].r - mT[1][1].i * tD[1][1].i;
      mTtD[1][1].i = mT[1][0].r * tD[0][1].i + mT[1][0].i * tD[0][1].r
      	           + mT[1][1].r * tD[1][1].i + mT[1][1].i * tD[1][1].r;

      RTu(iL - 1, iL);

      rU[0][0] = coeffU[0]; rU[0][1] = coeffU[1]; 
      rU[1][0] = coeffU[2]; rU[1][1] = coeffU[3];
      
      tU[0][0] = coeffU[4]; tU[0][1] = coeffU[5]; 
      tU[1][0] = coeffU[6]; tU[1][1] = coeffU[7];

      /* inv = (I - mT * rU)^-1 */
      auxm1 = mT[0][0].r * rU[0][0].r - mT[0][0].i * rU[0][0].i;
      auxm2 = mT[0][0].r * rU[0][0].i + mT[0][0].i * rU[0][0].r;
      auxm3 = mT[0][1].r * rU[1][0].r - mT[0][1].i * rU[1][0].i;
      auxm4 = mT[0][1].r * rU[1][0].i + mT[0][1].i * rU[1][0].r;
      mAux[0][0].r = 1 - (auxm1 + auxm3);
      mAux[0][0].i = - (auxm2 + auxm4);

      auxm1 = mT[0][0].r * rU[0][1].r - mT[0][0].i * rU[0][1].i;
      auxm2 = mT[0][0].r * rU[0][1].i + mT[0][0].i * rU[0][1].r;
      auxm3 = mT[0][1].r * rU[1][1].r - mT[0][1].i * rU[1][1].i;
      auxm4 = mT[0][1].r * rU[1][1].i + mT[0][1].i * rU[1][1].r;
      mAux[0][1].r = - (auxm1 + auxm3);
      mAux[0][1].i = - (auxm2 + auxm4);

      auxm1 = mT[1][0].r * rU[0][0].r - mT[1][0].i * rU[0][0].i;
      auxm2 = mT[1][0].r * rU[0][0].i + mT[1][0].i * rU[0][0].r;
      auxm3 = mT[1][1].r * rU[1][0].r - mT[1][1].i * rU[1][0].i;
      auxm4 = mT[1][1].r * rU[1][0].i + mT[1][1].i * rU[1][0].r;
      mAux[1][0].r = - (auxm1 + auxm3);
      mAux[1][0].i = - (auxm2 + auxm4);

      auxm1 = mT[1][0].r * rU[0][1].r - mT[1][0].i * rU[0][1].i;
      auxm2 = mT[1][0].r * rU[0][1].i + mT[1][0].i * rU[0][1].r;
      auxm3 = mT[1][1].r * rU[1][1].r - mT[1][1].i * rU[1][1].i;
      auxm4 = mT[1][1].r * rU[1][1].i + mT[1][1].i * rU[1][1].r;
      mAux[1][1].r = 1 - (auxm1 + auxm3);
      mAux[1][1].i = - (auxm2 + auxm4);

      /* inverting */
      auxm1 = mAux[0][0].r * mAux[1][1].r - mAux[0][0].i * mAux[1][1].i;
      auxm2 = mAux[0][0].r * mAux[1][1].i + mAux[0][0].i * mAux[1][1].r;
      auxm3 = mAux[0][1].r * mAux[1][0].r - mAux[0][1].i * mAux[1][0].i;
      auxm4 = mAux[0][1].r * mAux[1][0].i + mAux[0][1].i * mAux[1][0].r;
      aux1.r = auxm1 - auxm3;
      aux1.i = auxm2 - auxm4;

      /* 1 / aux1 */
      aux = aux1.r * aux1.r + aux1.i * aux1.i;
      aux1.r = aux1.r / aux;
      aux1.i = -aux1.i / aux;

      inv[0][0].r = mAux[1][1].r * aux1.r - mAux[1][1].i * aux1.i;
      inv[0][0].i = mAux[1][1].r * aux1.i + mAux[1][1].i * aux1.r;
      inv[0][1].r = -1 * (mAux[0][1].r * aux1.r - mAux[0][1].i * aux1.i);
      inv[0][1].i = -1 * (mAux[0][1].r * aux1.i + mAux[0][1].i * aux1.r);
      inv[1][0].r = -1 * (mAux[1][0].r * aux1.r - mAux[1][0].i * aux1.i);
      inv[1][0].i = -1 * (mAux[1][0].r * aux1.i + mAux[1][0].i * aux1.r);
      inv[1][1].r = mAux[0][0].r * aux1.r - mAux[0][0].i * aux1.i;
      inv[1][1].i = mAux[0][0].r * aux1.i + mAux[0][0].i * aux1.r;
      

      /* computing tU * inv */
      tUinv[0][0].r = tU[0][0].r * inv[0][0].r - tU[0][0].i * inv[0][0].i
  	            + tU[0][1].r * inv[1][0].r - tU[0][1].i * inv[1][0].i;
      tUinv[0][0].i = tU[0][0].r * inv[0][0].i + tU[0][0].i * inv[0][0].r
	            + tU[0][1].r * inv[1][0].i + tU[0][1].i * inv[1][0].r;
      tUinv[0][1].r = tU[0][0].r * inv[0][1].r - tU[0][0].i * inv[0][1].i
	            + tU[0][1].r * inv[1][1].r - tU[0][1].i * inv[1][1].i;
      tUinv[0][1].i = tU[0][0].r * inv[0][1].i + tU[0][0].i * inv[0][1].r
	            + tU[0][1].r * inv[1][1].i + tU[0][1].i * inv[1][1].r;
      tUinv[1][0].r = tU[1][0].r * inv[0][0].r - tU[1][0].i * inv[0][0].i
                    + tU[1][1].r * inv[1][0].r - tU[1][1].i * inv[1][0].i;
      tUinv[1][0].i = tU[1][0].r * inv[0][0].i + tU[1][0].i * inv[0][0].r
      	            + tU[1][1].r * inv[1][0].i + tU[1][1].i * inv[1][0].r;
      tUinv[1][1].r = tU[1][0].r * inv[0][1].r - tU[1][0].i * inv[0][1].i
      	            + tU[1][1].r * inv[1][1].r - tU[1][1].i * inv[1][1].i;
      tUinv[1][1].i = tU[1][0].r * inv[0][1].i + tU[1][0].i * inv[0][1].r
      	            + tU[1][1].r * inv[1][1].i + tU[1][1].i * inv[1][1].r;
      
      /* finally the matrix */
      auxm1 = mTtD[0][0].r * tUinv[0][0].r - mTtD[0][0].i * tUinv[0][0].i;
      auxm2 = mTtD[0][0].r * tUinv[0][0].i + mTtD[0][0].i * tUinv[0][0].r;
      auxm3 = mTtD[1][0].r * tUinv[0][1].r - mTtD[1][0].i * tUinv[0][1].i;
      auxm4 = mTtD[1][0].r * tUinv[0][1].i + mTtD[1][0].i * tUinv[0][1].r;
      mB[0][0].r = rD[0][0].r + (auxm1 + auxm3);
      mB[0][0].i = rD[0][0].i + (auxm2 + auxm4);

      auxm1 = mTtD[0][1].r * tUinv[0][0].r - mTtD[0][1].i * tUinv[0][0].i;
      auxm2 = mTtD[0][1].r * tUinv[0][0].i + mTtD[0][1].i * tUinv[0][0].r;
      auxm3 = mTtD[1][1].r * tUinv[0][1].r - mTtD[1][1].i * tUinv[0][1].i;
      auxm4 = mTtD[1][1].r * tUinv[0][1].i + mTtD[1][1].i * tUinv[0][1].r;
      mB[0][1].r = rD[0][1].r + (auxm1 + auxm3);
      mB[0][1].i = rD[0][1].i + (auxm2 + auxm4);

      auxm1 = mTtD[0][0].r * tUinv[1][0].r - mTtD[0][0].i * tUinv[1][0].i;
      auxm2 = mTtD[0][0].r * tUinv[1][0].i + mTtD[0][0].i * tUinv[1][0].r;
      auxm3 = mTtD[1][0].r * tUinv[1][1].r - mTtD[1][0].i * tUinv[1][1].i;
      auxm4 = mTtD[1][0].r * tUinv[1][1].i + mTtD[1][0].i * tUinv[1][1].r;
      mB[1][0].r = rD[1][0].r + (auxm1 + auxm3);
      mB[1][0].i = rD[1][0].i + (auxm2 + auxm4);

      auxm1 = mTtD[0][1].r * tUinv[1][0].r - mTtD[0][1].i * tUinv[1][0].i;
      auxm2 = mTtD[0][1].r * tUinv[1][0].i + mTtD[0][1].i * tUinv[1][0].r;
      auxm3 = mTtD[1][1].r * tUinv[1][1].r - mTtD[1][1].i * tUinv[1][1].i;
      auxm4 = mTtD[1][1].r * tUinv[1][1].i + mTtD[1][1].i * tUinv[1][1].r;
      mB[1][1].r = rD[1][1].r + (auxm1 + auxm3);
      mB[1][1].i = rD[1][1].i + (auxm2 + auxm4);
      
      /* computing Frechet derivatives */
      frechetRm(E, tD, tUinv, mTtD, mT, rU, inv, wThick, am, bm, iL);
   }
   
   /* computing final phase-shift matrix */
   /* square-root of Pslowness^2 - uuC */    
   auxm1 = PSlowness[0][0].r - uuC.r;
   auxm2 = PSlowness[0][0].i - uuC.i;
   auxm3 = sqrt(auxm1 * auxm1 + auxm2 * auxm2);
   auxm3 = sqrt(auxm3);
   angle = atan2(auxm2, auxm1) / 2;
   am.r = auxm3 * cos(angle);
   am.i = auxm3 * sin(angle);

   /* am * I */
   amI.r = -am.i;
   amI.i = am.r;

   /* square-root of SSlowness^2 - uuC */    
   auxm1 = SSlowness[0][0].r - uuC.r;
   auxm2 = SSlowness[0][0].i - uuC.i;
   auxm3 = sqrt(auxm1 * auxm1 + auxm2 * auxm2);
   auxm3 = sqrt(auxm3);
   angle = atan2(auxm2, auxm1) / 2;