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seismic_CPML_2D_poroelastic_fourth_order.f90
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seismic_CPML_2D_poroelastic_fourth_order.f90
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!
! SEISMIC_CPML Version 1.1.1, November 2009.
!
! Copyright Universite de Pau et des Pays de l'Adour, CNRS and INRIA, France.
! Contributors: Roland Martin, roland DOT martin aT get DOT obs-mip DOT fr
! and Dimitri Komatitsch, komatitsch aT lma DOT cnrs-mrs DOT fr
!
! This software is a computer program whose purpose is to solve
! the poroelastic elastic wave equation
! using a finite-difference method with Convolutional Perfectly Matched
! Layer (C-PML) conditions and Biot model with and without viscous dissipation.
!
! This program is free software; you can redistribute it and/or modify
! it under the terms of the GNU General Public License as published by
! the Free Software Foundation; either version 3 of the License, or
! (at your option) any later version.
!
! This program is distributed in the hope that it will be useful,
! but WITHOUT ANY WARRANTY; without even the implied warranty of
! MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
! GNU General Public License for more details.
!
! You should have received a copy of the GNU General Public License along
! with this program; if not, write to the Free Software Foundation, Inc.,
! 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
!
! The full text of the license is available in file "LICENSE".
program seismic_CPML_2D_poroelastic_fourth
! 2D poroelastic finite-difference code in velocity and stress formulation
! with Convolution-PML (C-PML) absorbing conditions
! with and without viscous dissipation
! Roland Martin, University of Pau, France, October 2009.
! based on the elastic code of Komatitsch and Martin, 2007.
! The fourth-order staggered-grid formulation of Madariaga (1976) and Virieux (1986) is used:
!
! ^ y
! |
! |
!
! +-------------------+
! | |
! | |
! | |
! | |
! | v_y |
! sigma_xy +---------+ |
! | | |
! | | |
! | | |
! | | |
! | | |
! +---------+---------+ ---> x
! v_x sigma_xx
! sigma_yy
!
! The C-PML implementation is based in part on formulas given in Roden and Gedney (2000).
! If you use this code for your own research, please cite some (or all) of these
! articles:
!
! @ARTICLE{MaKoEz08,
! author = {Roland Martin and Dimitri Komatitsch and Abdela\^aziz Ezziani},
! title = {An unsplit convolutional perfectly matched layer improved at grazing
! incidence for seismic wave equation in poroelastic media},
! journal = {Geophysics},
! year = {2008},
! volume = {73},
! pages = {T51-T61},
! number = {4},
! doi = {10.1190/1.2939484}}
!
! @ARTICLE{MaKo09,
! author = {Roland Martin and Dimitri Komatitsch},
! title = {An unsplit convolutional perfectly matched layer technique improved
! at grazing incidence for the viscoelastic wave equation},
! journal = {Geophysical Journal International},
! year = {2009},
! volume = {179},
! pages = {333-344},
! number = {1},
! doi = {10.1111/j.1365-246X.2009.04278.x}}
!
! @ARTICLE{MaKoGe08,
! author = {Roland Martin and Dimitri Komatitsch and Stephen D. Gedney},
! title = {A variational formulation of a stabilized unsplit convolutional perfectly
! matched layer for the isotropic or anisotropic seismic wave equation},
! journal = {Computer Modeling in Engineering and Sciences},
! year = {2008},
! volume = {37},
! pages = {274-304},
! number = {3}}
!
! @ARTICLE{KoMa07,
! author = {Dimitri Komatitsch and Roland Martin},
! title = {An unsplit convolutional {P}erfectly {M}atched {L}ayer improved
! at grazing incidence for the seismic wave equation},
! journal = {Geophysics},
! year = {2007},
! volume = {72},
! number = {5},
! pages = {SM155-SM167},
! doi = {10.1190/1.2757586}}
!
! The original CPML technique for Maxwell's equations is described in:
!
! @ARTICLE{RoGe00,
! author = {J. A. Roden and S. D. Gedney},
! title = {Convolution {PML} ({CPML}): {A}n Efficient {FDTD} Implementation
! of the {CFS}-{PML} for Arbitrary Media},
! journal = {Microwave and Optical Technology Letters},
! year = {2000},
! volume = {27},
! number = {5},
! pages = {334-339},
! doi = {10.1002/1098-2760(20001205)27:5 < 334::AID-MOP14>3.0.CO;2-A}}
! To display the results as color images in the selected 2D cut plane, use:
!
! " display image*.gif " or " gimp image*.gif "
!
! or
!
! " montage -geometry +0+3 -rotate 90 -tile 1x21 image*Vx*.gif allfiles_Vx.gif
! "
! " montage -geometry +0+3 -rotate 90 -tile 1x21 image*Vy*.gif allfiles_Vy.gif
! "
! then " display allfiles_Vx.gif " or " gimp allfiles_Vx.gif "
! then " display allfiles_Vy.gif " or " gimp allfiles_Vy.gif "
! To display the 2D results as PostScript vector plots with small arrows, use:
!
! " gs vect*.ps "
!
! IMPORTANT : all our CPML codes work fine in single precision as well (which is significantly faster).
! If you want you can thus force automatic conversion to single precision at compile time
! or change all the declarations and constants in the code from double precision to single.
implicit none
! total number of grid points in each direction of the grid
integer, parameter :: NX = 140
integer, parameter :: NY = 620
! size of a grid cell
double precision, parameter :: DELTAX = 0.5D0
double precision, parameter :: DELTAY = DELTAX
! flags to add PML layers to the edges of the grid
logical, parameter :: USE_PML_LEFT = .true.
logical, parameter :: USE_PML_RIGHT = .true.
logical, parameter :: USE_PML_BOTTOM = .true.
logical, parameter :: USE_PML_TOP = .true.
! thickness of the PML layer in grid points
integer, parameter :: NPOINTS_PML = 10
! heterogeneous model and height of the interface
logical, parameter :: HETEROGENEOUS_MODEL = .true.
double precision, parameter :: INTERFACE_HEIGHT =105.D0+NPOINTS_PML*DELTAY
integer, parameter:: JINTERFACE=INT(INTERFACE_HEIGHT/DELTAY)+1
double precision :: co,c1,c2,vtemp
! model mud saturated with water, see article by Martin and Komatitsch
double precision, parameter :: etaokappa_bottom=0.d0
double precision, parameter :: rmu_bottom = 5.25D09
double precision, parameter :: phi_bottom =0.25d0
double precision, parameter :: a_bottom = 2.49d0
double precision, parameter :: rhos_bottom = 2588.d0
double precision, parameter :: rhof_bottom = 952.4d0
double precision, parameter :: rho_bottom =2179.1d0
double precision, parameter :: rsm_bottom =9486.d0
double precision, parameter :: alpha_bottom=0.89d0
double precision, parameter :: rbM_bottom =7.71d09
double precision, parameter :: rlambdao_bottom = 6.2D08
double precision, parameter :: rlambdac_bottom =rlambdao_bottom+alpha_bottom**2*rbM_bottom
double precision, parameter :: ro11_b=rho_bottom+phi_bottom*rhof_bottom*(a_bottom-2.d0)
double precision, parameter :: ro12_b=phi_bottom*rhof_bottom*(1.d0-a_bottom)
double precision, parameter :: ro22_b=a_bottom*phi_bottom*rhof_bottom
double precision, parameter :: lambda_b=rlambdao_bottom+rbM_bottom*(alpha_bottom-phi_bottom)**2
double precision, parameter :: R_b=rbM_bottom*phi_bottom**2
double precision, parameter :: ga_b=rbM_bottom*phi_bottom*(alpha_bottom-phi_bottom)
double precision, parameter :: S_b=lambda_b+2*rmu_bottom
double precision, parameter :: c1_b=S_b*R_b-ga_b**2
double precision, parameter :: b1_b=-S_b*ro22_b-R_b*ro11_b+2*ga_b*ro12_b
double precision, parameter :: a1_b=ro11_b*ro22_b-ro12_b**2
double precision, parameter :: delta_b=b1_b**2-4.d0*a1_b*c1_b
double precision:: cp_bottom
double precision:: cps_bottom
double precision:: cs_bottom
double precision, parameter :: etaokappa_top=3.33D06
double precision, parameter :: rmu_top = 2.4D09
double precision, parameter :: phi_top =0.1d0
double precision, parameter :: a_top = 2.42d0
double precision, parameter :: rhos_top = 2250.d0
double precision, parameter :: rhof_top = 1040.d0
double precision, parameter :: rho_top = 2129.d0
double precision, parameter :: rsm_top =25168.d0
double precision, parameter :: alpha_top=0.58d0
double precision, parameter :: rbM_top = 7.34d09
double precision, parameter :: rlambdao_top =6.D08
double precision, parameter :: rlambdac_top =rlambdao_top+alpha_top**2*rbM_top
double precision, parameter :: ro11_t=rho_top+phi_top*rhof_top*(a_top-2.d0)
double precision, parameter :: ro12_t=phi_top*rhof_top*(1.d0-a_top)
double precision, parameter :: ro22_t=a_top*phi_top*rhof_top
double precision, parameter :: lambda_t=rlambdao_top+rbM_top*(alpha_top-phi_top)**2
double precision, parameter :: R_t=rbM_top*phi_top**2
double precision, parameter :: ga_t=rbM_top*phi_top*(alpha_top-phi_top)
double precision, parameter :: S_t=lambda_t+2*rmu_top
double precision, parameter :: c1_t=S_t*R_t-ga_t**2
double precision, parameter :: b1_t=-S_t*ro22_t-R_t*ro11_t+2*ga_t*ro12_t
double precision, parameter :: a1_t=ro11_t*ro22_t-ro12_t**2
double precision, parameter :: delta_t=b1_t**2-4.d0*a1_t*c1_t
double precision:: cp_top
double precision:: cps_top
double precision:: cs_top
! total number of time steps
integer, parameter :: NSTEP = 100000
! time step in seconds
double precision, parameter :: DELTAT = 1.d-04
! parameters for the source
double precision, parameter :: f0 = 80.d0
double precision, parameter :: t0 = 1.d0/f0
double precision, parameter :: factor =1.d02
! source
integer, parameter :: ISOURCE = NX/2+1
integer, parameter :: JSOURCE = NY/2 +1
integer, parameter :: IDEB = NX / 2 + 1
integer, parameter :: JDEB = NY / 2 + 1
double precision, parameter :: xsource = DELTAX * ISOURCE
double precision, parameter :: ysource = DELTAY * JSOURCE
! angle of source force clockwise with respect to vertical (Y) axis
double precision, parameter :: ANGLE_FORCE = 0.d0
! receivers
integer, parameter :: NREC = 2
double precision, parameter :: ydeb = NPOINTS_PML*DELTAY+10.D0 ! first receiver x in meters
double precision, parameter :: yfin = NY*DELTAY-NPOINTS_PML*DELTAY-10.d0 ! first receiver x in meters
double precision, parameter :: xdeb =xsource ! first receiver y in meters
double precision, parameter :: xfin =xdeb ! first receiver y in meters
! display information on the screen from time to time
integer, parameter :: IT_DISPLAY = 200
! value of PI
double precision, parameter :: PI = 3.141592653589793238462643d0
! conversion from degrees to radians
double precision, parameter :: DEGREES_TO_RADIANS = PI / 180.d0
! zero
double precision, parameter :: ZERO = 0.d0
! large value for maximum
double precision, parameter :: HUGEVAL = 1.d+30
! velocity threshold above which we consider that the code became unstable
double precision, parameter :: STABILITY_THRESHOLD = 1.d+25
! main arrays
double precision, dimension(0:NX+1,0:NY+1) :: vx,vy,sigmaxx,sigma2,alp_sigma2,sigmayy,sigmaxy,vnorm
double precision, dimension(0:NX+1,0:NY+1) :: vxf,vyf
double precision, dimension(0:NX+1,0:NY+1) :: rho,rhof,rsm,rmu,rlambdac,rbM,alpha,etaokappa,rlambdao
! to interpolate material parameters at the right location in the staggered grid cell
double precision rho_half_x_half_y,rhof_half_x_half_y,rsm_half_x_half_y,etaokappa_half_x_half_y
! for evolution of total energy in the medium
double precision epsilon_xx,epsilon_yy,epsilon_xy
double precision, dimension(NSTEP) :: total_energy_kinetic,total_energy_potential
double precision c33_half_y
! power to compute d0 profile
double precision, parameter :: NPOWER = 2.d0
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-11
double precision, parameter :: K_MAX_PML = 1.d0
double precision, parameter :: ALPHA_MAX_PML = 2.d0*PI*(f0/2.d0) ! from Festa and Vilotte
! 2D arrays for the memory variables
double precision, dimension(0:NX+1,0:NY+1) :: gamma11,gamma22
double precision, dimension(0:NX+1,0:NY+1) :: gamma12_1
double precision, dimension(0:NX+1,0:NY+1) :: xi_1,xi_2
double precision, dimension(0:NX+1,0:NY+1) :: &
memory_dx_vx1,memory_dx_vx2,memory_dy_vx,memory_dx_vy,memory_dy_vy1,memory_dy_vy2, &
memory_dx_sigmaxx,memory_dx_sigmayy,memory_dx_sigmaxy, &
memory_dx_sigma2vx,memory_dx_sigma2vxf,memory_dy_sigma2vy,memory_dy_sigma2vyf, &
memory_dy_sigmaxx,memory_dy_sigmayy,memory_dy_sigmaxy
! 1D arrays for the damping profiles
double precision, dimension(NX) :: d_x,K_x,alpha_x,a_x,b_x,d_x_half_x,K_x_half_x,alpha_x_half_x,a_x_half_x,b_x_half_x
double precision, dimension(NY) :: d_y,K_y,alpha_y,a_y,b_y,d_y_half_y,K_y_half_y,alpha_y_half_y,a_y_half_y,b_y_half_y
double precision thickness_PML_x,thickness_PML_y,xoriginleft,xoriginright,yoriginbottom,yorigintop
double precision Rcoef,d0_x,d0_y,xval,yval,abscissa_in_PML,abscissa_normalized
double precision value_dx_vx1,value_dx_vx2,value_dx_vy,value_dx_sigmaxx,value_dx_sigmaxy
double precision value_dy_vy1,value_dy_vy2,value_dy_vx,value_dy_sigmaxx,value_dy_sigmaxy
double precision value_dx_sigma2vxf,value_dy_sigma2vyf
! for the source
double precision a,t,source_term
! for receivers
double precision xspacerec,yspacerec,distval,dist
integer, dimension(NREC) :: ix_rec,iy_rec
double precision, dimension(NREC) :: xrec,yrec
! for seismograms
double precision, dimension(NSTEP,NREC) :: sisvx,sisvy,sisp
integer i,j,it,irec
double precision Courant_number_bottom,Courant_number_top,velocnorm_all,max_amplitude
double precision Dispersion_number_bottom,Dispersion_number_top
!---
!--- program starts here
!---
cp_bottom=(-b1_b+sqrt(delta_b))/(2.d0*a1_b);
cps_bottom=(-b1_b-sqrt(delta_b))/(2.d0*a1_b);
cp_bottom=sqrt(cp_bottom)
cps_bottom=sqrt(cps_bottom)
cs_bottom=sqrt(rmu_bottom/(ro11_b-ro12_b**2/ro22_b))
cp_top=(-b1_t+sqrt(delta_t))/(2.d0*a1_t);
cps_top=(-b1_t-sqrt(delta_t))/(2.d0*a1_t);
cp_top=sqrt(cp_top)
cps_top=sqrt(cps_top)
cs_top=sqrt(rmu_top/(ro11_t-ro12_t**2/ro22_t))
print *,'cp_bottom= ',cp_bottom
print *,'cps_bottom=',cps_bottom
print *,'cs_bottom= ',cs_bottom
print *,'cp_top= ',cp_top
print *,'cps_top=',cps_top
print *,'cs_top= ',cs_top
print *,'rho_bottom= ',rho_bottom
print *,'rsm_bottom= ',rsm_bottom
print *,'rho_top= ',rho_top
print *,'rsm_top= ',rsm_top
print *,'rmu_bottom= ',rmu_bottom
print *,'rlambdac_bottom= ',rlambdac_bottom
print *,'rlambdao_bottom= ',rlambdao_bottom
print *,'alpha_bottom= ',alpha_bottom
print *,'rbM_bottom= ',rbM_bottom
print *,'etaokappa_bottom= ',etaokappa_bottom
print *,'rmu_top= ',rmu_top
print *,'rlambdac_top= ',rlambdac_top
print *,'rlambdao_top= ',rlambdao_top
print *,'alpha_top= ',alpha_top
print *,'rbM_top= ',rbM_top
print *,'etaokappa_top= ',etaokappa_top
print *, 'DELTAT CPML=', DELTAT
print *,'2D poroelastic finite-difference code in velocity and stress formulation with C-PML'
print *
! display size of the model
print *
print *,'NX = ',NX
print *,'NY = ',NY
print *
print *,'size of the model along X = ',(NX - 1) * DELTAX
print *,'size of the model along Y = ',(NY - 1) * DELTAY
print *
print *,'Total number of grid points = ',NX * NY
print *
!--- define profile of absorption in PML region
! thickness of the PML layer in meters
thickness_PML_x = NPOINTS_PML * DELTAX
thickness_PML_y = NPOINTS_PML * DELTAY
! reflection coefficient (INRIA report section 6.1) http://hal.inria.fr/docs/00/07/32/19/PDF/RR-3471.pdf
Rcoef = 0.001d0
! check that NPOWER is okay
if (NPOWER < 1) stop 'NPOWER must be greater than 1'
! compute d0 from INRIA report section 6.1 http://hal.inria.fr/docs/00/07/32/19/PDF/RR-3471.pdf
if (HETEROGENEOUS_MODEL) then
d0_x = - (NPOWER + 1) * max(cp_bottom,cp_top) * log(Rcoef) / (2.d0 * thickness_PML_x)
d0_y = - (NPOWER + 1) * max(cp_bottom,cp_top) * log(Rcoef) / (2.d0 * thickness_PML_y)
else
d0_x = - (NPOWER + 1) * cp_bottom * log(Rcoef) / (2.d0 * thickness_PML_x)
d0_y = - (NPOWER + 1) * cp_bottom * log(Rcoef) / (2.d0 * thickness_PML_y)
endif
print *,'d0_x = ',d0_x
print *,'d0_y = ',d0_y
d_x(:) = ZERO
d_x_half_x(:) = ZERO
d_y(:) = ZERO
d_y_half_y(:) = ZERO
K_x(:) = 1.d0
K_x_half_x(:) = 1.d0
K_y(:) = 1.d0
K_y_half_y(:) = 1.d0
alpha_x(:) = ZERO
alpha_x_half_x(:) = ZERO
alpha_y(:) = ZERO
alpha_y_half_y(:) = ZERO
a_x(:) = ZERO
a_x_half_x(:) = ZERO
a_y(:) = ZERO
a_y_half_y(:) = ZERO
! origin of the PML layer (position of right edge minus thickness, in meters)
xoriginleft = thickness_PML_x
xoriginright = (NX-1)*DELTAX - thickness_PML_x
do i = 1,NX
! abscissa of current grid point along the damping profile
xval = DELTAX * dble(i-1)
!!!! ---------- left edge
if (USE_PML_LEFT) then
! define damping profile at the grid points
abscissa_in_PML = xoriginleft - xval
if (abscissa_in_PML >= ZERO) then
abscissa_normalized = abscissa_in_PML / thickness_PML_x
d_x(i) = d0_x * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_x(i) = 1.d0 + (K_MAX_PML - 1.d0) * abscissa_normalized**NPOWER
alpha_x(i) = ALPHA_MAX_PML * (1.d0 - abscissa_normalized)
endif
! define damping profile at half the grid points
abscissa_in_PML = xoriginleft - (xval + DELTAX/2.d0)
if (abscissa_in_PML >= ZERO) then
abscissa_normalized = abscissa_in_PML / thickness_PML_x
d_x_half_x(i) = d0_x * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_x_half_x(i) = 1.d0 + (K_MAX_PML - 1.d0) * abscissa_normalized**NPOWER
alpha_x_half_x(i) = ALPHA_MAX_PML * (1.d0 - abscissa_normalized)
endif
endif
!!!! ---------- right edge
if (USE_PML_RIGHT) then
! define damping profile at the grid points
abscissa_in_PML = xval - xoriginright
if (abscissa_in_PML >= ZERO) then
abscissa_normalized = abscissa_in_PML / thickness_PML_x
d_x(i) = d0_x * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_x(i) = 1.d0 + (K_MAX_PML - 1.d0) * abscissa_normalized**NPOWER
alpha_x(i) = ALPHA_MAX_PML * (1.d0 - abscissa_normalized)
endif
! define damping profile at half the grid points
abscissa_in_PML = xval + DELTAX/2.d0 - xoriginright
if (abscissa_in_PML >= ZERO) then
abscissa_normalized = abscissa_in_PML / thickness_PML_x
d_x_half_x(i) = d0_x * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_x_half_x(i) = 1.d0 + (K_MAX_PML - 1.d0) * abscissa_normalized**NPOWER
alpha_x_half_x(i) = ALPHA_MAX_PML * (1.d0 - abscissa_normalized)
endif
endif
! just in case, for -5 at the end
if (alpha_x(i) < ZERO) alpha_x(i) = ZERO
if (alpha_x_half_x(i) < ZERO) alpha_x_half_x(i) = ZERO
b_x(i) = exp(- (d_x(i) / K_x(i) + alpha_x(i)) * DELTAT)
b_x_half_x(i) = exp(- (d_x_half_x(i) / K_x_half_x(i) + alpha_x_half_x(i)) * DELTAT)
! this to avoid division by zero outside the PML
if (abs(d_x(i)) > 1.d-6) a_x(i) = d_x(i) * (b_x(i) - 1.d0) /&
(K_x(i) * (d_x(i) + K_x(i) * alpha_x(i)))
if (abs(d_x_half_x(i)) > 1.d-6) a_x_half_x(i) = d_x_half_x(i)&
* (b_x_half_x(i) - 1.d0) / (K_x_half_x(i) * (d_x_half_x(i) + K_x_half_x(i) * alpha_x_half_x(i)))
enddo
!!!!!!!!!!!!! added Y damping profile
! origin of the PML layer (position of right edge minus thickness, in meters)
yoriginbottom = thickness_PML_y
yorigintop = NY*DELTAY - thickness_PML_y
do j = 1,NY
! abscissa of current grid point along the damping profile
yval = DELTAY * dble(j-1)
!!!! ---------- bottom edge
if (USE_PML_BOTTOM) then
! define damping profile at the grid points
abscissa_in_PML = yoriginbottom - yval
if (abscissa_in_PML >= ZERO) then
abscissa_normalized = abscissa_in_PML / thickness_PML_y
d_y(j) = d0_y * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_y(j) = 1.d0 + (K_MAX_PML - 1.d0) * abscissa_normalized**NPOWER
alpha_y(j) = ALPHA_MAX_PML * (1.d0 - abscissa_normalized)
endif
! define damping profile at half the grid points
abscissa_in_PML = yoriginbottom - (yval + DELTAY/2.d0)
if (abscissa_in_PML >= ZERO) then
abscissa_normalized = abscissa_in_PML / thickness_PML_y
d_y_half_y(j) = d0_y * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_y_half_y(j) = 1.d0 + (K_MAX_PML - 1.d0) * abscissa_normalized**NPOWER
alpha_y_half_y(j) = ALPHA_MAX_PML * (1.d0 - abscissa_normalized)
endif
endif
!!!! ---------- top edge
if (USE_PML_TOP) then
! define damping profile at the grid points
abscissa_in_PML = yval - yorigintop
if (abscissa_in_PML >= ZERO) then
abscissa_normalized = abscissa_in_PML / thickness_PML_y
d_y(j) = d0_y * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_y(j) = 1.d0 + (K_MAX_PML - 1.d0) * abscissa_normalized**NPOWER
alpha_y(j) = ALPHA_MAX_PML * (1.d0 - abscissa_normalized)
endif
! define damping profile at half the grid points
abscissa_in_PML = yval + DELTAY/2.d0 - yorigintop
if (abscissa_in_PML >= ZERO) then
abscissa_normalized = abscissa_in_PML / thickness_PML_y
d_y_half_y(j) = d0_y * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_y_half_y(j) = 1.d0 + (K_MAX_PML - 1.d0) * abscissa_normalized**NPOWER
alpha_y_half_y(j) = ALPHA_MAX_PML * (1.d0 - abscissa_normalized)
endif
endif
! just in case, for -5 at the end
! if (alpha_y(j) < ZERO) alpha_y(j) = ZERO
! if (alpha_y_half_y(j) < ZERO) alpha_y_half_y(j) = ZERO
b_y(j) = exp(- (d_y(j) / K_y(j) + alpha_y(j)) * DELTAT)
b_y_half_y(j) = exp(- (d_y_half_y(j) / K_y_half_y(j) + alpha_y_half_y(j)) * DELTAT)
! this to avoid division by zero outside the PML
if (abs(d_y(j)) > 1.d-6) a_y(j) = d_y(j) * (b_y(j) - 1.d0) &
/ (K_y(j) * (d_y(j) + K_y(j) * alpha_y(j)))
if (abs(d_y_half_y(j)) > 1.d-6) a_y_half_y(j) = d_y_half_y(j)&
* (b_y_half_y(j) - 1.d0) / (K_y_half_y(j) * (d_y_half_y(j) + K_y_half_y(j) * alpha_y_half_y(j)))
enddo
! compute the Lame parameters and density
do j = 0,NY+1
do i = 0,NX+1
if (HETEROGENEOUS_MODEL .and. DELTAY*dble(j-1) > INTERFACE_HEIGHT) then
rho(i,j)= rho_top
rhof(i,j) = rhof_top
rsm(i,j) = rsm_top
rmu(i,j)= rmu_top
rlambdac(i,j) = rlambdac_top
rbM(i,j) = rbM_top
alpha(i,j)=alpha_top
etaokappa(i,j)=etaokappa_top
rlambdao(i,j) = rlambdao_top
else
rho(i,j)= rho_bottom
rhof(i,j) = rhof_bottom
rsm(i,j) = rsm_bottom
rmu(i,j)= rmu_bottom
rlambdac(i,j) = rlambdac_bottom
rbM(i,j) = rbM_bottom
alpha(i,j)=alpha_bottom
etaokappa(i,j)=etaokappa_bottom
rlambdao(i,j) = rlambdao_bottom
endif
enddo
enddo
! print position of the source
print *
print *,'Position of the source:'
print *
print *,'x = ',xsource
print *,'y = ',ysource
print *
! define location of receivers
print *
print *,'There are ',nrec,' receivers'
print *
xspacerec = (xfin-xdeb) / dble(NREC-1)
yspacerec = (yfin-ydeb) / dble(NREC-1)
do irec=1,nrec
xrec(irec) = xdeb + dble(irec-1)*xspacerec
yrec(irec) = ydeb + dble(irec-1)*yspacerec
enddo
! find closest grid point for each receiver
do irec=1,nrec
dist = HUGEVAL
do j = 1,NY
do i = 1,NX
distval = sqrt((DELTAX*dble(i-1) - xrec(irec))**2 + (DELTAY*dble(j-1) - yrec(irec))**2)
if (distval < dist) then
dist = distval
ix_rec(irec) = i
iy_rec(irec) = j
endif
enddo
enddo
print *,'receiver ',irec,' x_target,y_target = ',xrec(irec),yrec(irec)
print *,'closest grid point found at distance ',dist,' in i,j = ',ix_rec(irec),iy_rec(irec)
print *
enddo
! check the Courant stability condition for the explicit time scheme
! R. Courant et K. O. Friedrichs et H. Lewy (1928)
Courant_number_bottom = cp_bottom * DELTAT / min(DELTAX,DELTAY)
Dispersion_number_bottom=min(cs_bottom,cps_bottom)/(2.5d0*f0*max(DELTAX,DELTAY))
print *,'Courant number at the bottom is ',Courant_number_bottom
print *,'Dispersion number at the bottom is ',Dispersion_number_bottom
print *
if (Courant_number_bottom > 1.d0/sqrt(2.d0)) stop 'time step is too large, simulation will be unstable'
if (HETEROGENEOUS_MODEL) then
Courant_number_top = max(cp_top,cp_bottom) * DELTAT / min(DELTAX,DELTAY)
Dispersion_number_top=min(cs_top,cs_bottom,cps_bottom,cps_top)/(2.5d0*f0*max(DELTAX,DELTAY))
print *,'Courant number at the top is ',Courant_number_top
print *
print *,'Dispersion number at the top is ',Dispersion_number_top
if (Courant_number_top > 6.d0/7.d0/sqrt(2.d0)) stop 'time step is too large, simulation will be unstable'
endif
! suppress old files
! call system('rm -f Vx_*.dat Vy_*.dat vect*.ps image*.pnm image*.gif')
! initialize arrays
vx(:,:) = ZERO
vy(:,:) = ZERO
sigmaxx(:,:) = ZERO
sigmayy(:,:) = ZERO
sigmaxy(:,:) = ZERO
sigma2(:,:) = ZERO
alp_sigma2(:,:) = ZERO
gamma11(:,:)=0.d0
gamma22(:,:)=0.d0
gamma12_1(:,:)=0.d0
gamma12_1(:,:)=0.d0
xi_1(:,:)=0.d0
xi_2(:,:)=0.d0
vxf(:,:) = ZERO
vyf(:,:) = ZERO
memory_dx_vx1(:,:)=0.d0
memory_dx_vx2(:,:)=0.d0
memory_dy_vx(:,:)=0.d0
memory_dx_vy(:,:)=0.d0
memory_dy_vy1(:,:)=0.d0
memory_dy_vy2(:,:)=0.d0
memory_dx_sigmaxx(:,:)=0.d0
memory_dx_sigmayy(:,:)=0.d0
memory_dx_sigmaxy(:,:)=0.d0
memory_dx_sigma2vx(:,:)=0.d0
memory_dx_sigma2vxf(:,:)=0.d0
memory_dy_sigmaxx(:,:)=0.d0
memory_dy_sigmayy(:,:)=0.d0
memory_dy_sigmaxy(:,:)=0.d0
memory_dy_sigma2vy(:,:)=0.d0
memory_dy_sigma2vyf(:,:)=0.d0
! initialize seismograms
sisvx(:,:) = ZERO
sisvy(:,:) = ZERO
sisp(:,:) = ZERO
! initialize total energy
total_energy_kinetic(:) = ZERO
total_energy_potential(:) = ZERO
!---
!--- beginning of time loop
!---
do it = 1,NSTEP
!----------------------
! compute stress sigma
!----------------------
!-----------------------------------
! update memory variables for C-PML
!-----------------------------------
do j = 2,NY
do i = 1,NX-1
! memory of sigmaxx
value_dx_sigmaxx =(27.d0*vx(i+1,j)-27.d0*vx(i,j)-vx(i+2,j)+vx(i-1,j))/DELTAX/24.D0
value_dy_sigmaxx =(27.d0*vy(i,j)-27.d0*vy(i,j-1)-vy(i,j+1)+vy(i,j-2))/DELTAY/24.D0
memory_dx_sigmaxx(i,j) = b_x_half_x(i) * memory_dx_sigmaxx(i,j) + a_x_half_x(i) * value_dx_sigmaxx
memory_dy_sigmaxx(i,j) = b_y(j) * memory_dy_sigmaxx(i,j) + a_y(j) * value_dy_sigmaxx
gamma11(i,j) = gamma11(i,j)+DELTAT*(value_dx_sigmaxx / K_x_half_x(i) + memory_dx_sigmaxx(i,j))
gamma22(i,j) = gamma22(i,j)+DELTAT*(value_dy_sigmaxx / K_y(j) + memory_dy_sigmaxx(i,j))
! sigma2
value_dx_sigma2vxf=(27.d0*vxf(i+1,j)-27.d0* vxf(i,j)-vxf(i+2,j)+vxf(i-1,j)) / DELTAX/24.D0
value_dy_sigma2vyf=(27.d0*vyf(i,j)-27.d0*vyf(i,j-1)-vyf(i,j+1)+vyf(i,j-2)) / DELTAY/24.D0
memory_dx_sigma2vxf(i,j) = b_x_half_x(i) * memory_dx_sigma2vxf(i,j) + a_x_half_x(i) * value_dx_sigma2vxf
memory_dy_sigma2vyf(i,j) = b_y(j) * memory_dy_sigma2vyf(i,j) + a_y(j) * value_dy_sigma2vyf
xi_1(i,j) = xi_1(i,j) -(value_dx_sigma2vxf/ K_x_half_x(i) + memory_dx_sigma2vxf(i,j))*DELTAT
xi_2(i,j) = xi_2(i,j) -(value_dy_sigma2vyf/K_y(j)+memory_dy_sigma2vyf(i,j))*DELTAT
sigma2(i,j)=-alpha(i,j)*rbM(i,j)*(gamma11(i,j)+gamma22(i,j))+rbM(i,j)*(xi_1(i,j)+xi_2(i,j))
enddo
enddo
! add the source (point source located at a given grid point)
a = pi*pi*f0*f0
t = dble(it-1)*DELTAT
! Gaussian
source_term = factor * exp(-a*(t-t0)**2)/(-2.d0*a)
! first derivative of a Gaussian
! source_term = factor * 2.d0*a*(t-t0)*exp(-a*(t-t0)**2)
! source_term = factor *(t-t0)*exp(-a*(t-t0)**2)
! Ricker source time function (second derivative of a Gaussian)
! source_term = factor * (1.d0 - 2.d0*a*(t-t0)**2)*exp(-a*(t-t0)**2)
! define location of the source
i = ISOURCE
j = JSOURCE
! add the source term
sigma2(i,j) = sigma2(i,j) + source_term*rbM(i,j)
do j = 1,NY-1
do i = 2,NX
! interpolate material parameters at the right location in the staggered grid cell
c33_half_y = 2.d0/(1.d0/rmu(i,j)+1.d0/rmu(i,j+1))
c33_half_y = rmu(i,j+1)
value_dx_sigmaxy = (27.d0*vy(i,j) - 27.d0*vy(i-1,j)-vy(i+1,j)+vy(i-2,j)) / DELTAX/24.D0
value_dy_sigmaxy = (27.d0*vx(i,j+1) - 27.d0*vx(i,j)-vx(i,j+2)+vx(i,j-1)) / DELTAY/24.D0
memory_dx_sigmaxy(i,j) = b_x(i) * memory_dx_sigmaxy(i,j) + a_x(i) * value_dx_sigmaxy
memory_dy_sigmaxy(i,j) = b_y_half_y(j) * memory_dy_sigmaxy(i,j) + a_y_half_y(j) * value_dy_sigmaxy
sigmaxy(i,j) = sigmaxy(i,j) + &
c33_half_y/1.d0 * (value_dx_sigmaxy / K_x(i) + memory_dx_sigmaxy(i,j) + &
value_dy_sigmaxy / K_y(j) + memory_dy_sigmaxy(i,j)) * DELTAT
enddo
enddo
do j = 2,NY
do i = 1,NX-1
sigmaxx(i,j)=(rlambdao(i,j)+2.d0*rmu(i,j))*gamma11(i,j)+rlambdao(i,j)*gamma22(i,j) -alpha(i,j)*sigma2(i,j)
sigmayy(i,j)=rlambdao(i,j)*gamma11(i,j)+(rlambdao(i,j)+2.d0*rmu(i,j))*gamma22(i,j) -alpha(i,j)*sigma2(i,j)
enddo
enddo
!------------------
! compute velocity
!------------------
!-----------------------------------
! update memory variables for C-PML
!-----------------------------------
do j = 2,NY
do i = 2,NX
co=(rho(i,j)*rsm(i,j)-rhof(i,j)*rhof(i,j))/DELTAT
c1=co+rho(i,j)*etaokappa(i,j)*0.5d0
c2=co-rho(i,j)*etaokappa(i,j)*0.5d0
vtemp=vxf(i,j)
value_dx_vx1 = (27.d0*sigmaxx(i,j) - 27.d0*sigmaxx(i-1,j)&
-sigmaxx(i+1,j)+sigmaxx(i-2,j)) / DELTAX/24.D0
value_dx_vx2 = (27.d0*sigma2(i,j) - 27.d0*sigma2(i-1,j)-sigma2(i+1,j)+sigma2(i-2,j)) / DELTAX/24.D0
value_dy_vx = (27.d0*sigmaxy(i,j) - 27.d0*sigmaxy(i,j-1)-sigmaxy(i,j+1)+sigmaxy(i,j-2)) / DELTAY/24.D0
memory_dx_vx1(i,j) = b_x(i) * memory_dx_vx1(i,j) + a_x(i) * value_dx_vx1
memory_dx_vx2(i,j) = b_x(i) * memory_dx_vx2(i,j) + a_x(i) * value_dx_vx2
memory_dy_vx(i,j) = b_y(j) * memory_dy_vx(i,j) + a_y(j) * value_dy_vx
vxf(i,j) = (c2*vxf(i,j) + &
(-rhof(i,j)*(value_dx_vx1/ K_x(i) + memory_dx_vx1(i,j) &
+ value_dy_vx / K_y(j) + memory_dy_vx(i,j)) &
-rho(i,j)*(value_dx_vx2/ K_x(i) + memory_dx_vx2(i,j)) &
)) /c1
vtemp=(vtemp+vxf(i,j))*0.5d0
vx(i,j) = vx(i,j) + &
(rsm(i,j)*(value_dx_vx1/ K_x(i) + memory_dx_vx1(i,j)+ &
value_dy_vx / K_y(j) + memory_dy_vx(i,j))+&
rhof(i,j)*(value_dx_vx2/ K_x(i) + memory_dx_vx2(i,j)) + &
rhof(i,j)*etaokappa(i,j)*vtemp)&
/co
enddo
enddo
do j = 1,NY-1
do i = 1,NX-1
rho_half_x_half_y = rho(i,j+1)
rsm_half_x_half_y = rsm(i,j+1)
rhof_half_x_half_y = rhof(i,j+1)
etaokappa_half_x_half_y = etaokappa(i,j+1)
co=(rho_half_x_half_y*rsm_half_x_half_y-rhof_half_x_half_y**2)/DELTAT
c1=co+rho_half_x_half_y*etaokappa_half_x_half_y*0.5d0
c2=co-rho_half_x_half_y*etaokappa_half_x_half_y*0.5d0
vtemp=vyf(i,j)
value_dx_vy = (27.d0*sigmaxy(i+1,j) - 27.d0*sigmaxy(i,j)-sigmaxy(i+2,j)+sigmaxy(i-1,j)) / DELTAX/24.D0
value_dy_vy1 = (27.d0*sigmayy(i,j+1)- 27.d0*sigmayy(i,j)&
-sigmayy(i,j+2)+sigmayy(i,j-1)) / DELTAY/24.D0
value_dy_vy2 = (27.d0*sigma2(i,j+1) - 27.d0*sigma2(i,j)-sigma2(i,j+2)+sigma2(i,j-1)) / DELTAY/24.D0
memory_dx_vy(i,j) = b_x_half_x(i) * memory_dx_vy(i,j) + a_x_half_x(i) * value_dx_vy
memory_dy_vy1(i,j) = b_y_half_y(j) * memory_dy_vy1(i,j) + a_y_half_y(j) * value_dy_vy1
memory_dy_vy2(i,j) = b_y_half_y(j) * memory_dy_vy2(i,j) + a_y_half_y(j) * value_dy_vy2
vyf(i,j) = (c2*vyf(i,j) + &
(-rhof_half_x_half_y*(value_dx_vy / K_x_half_x(i) + memory_dx_vy(i,j) &
+value_dy_vy1 / K_y_half_y(j) + memory_dy_vy1(i,j))&
-rho_half_x_half_y*(value_dy_vy2 / K_y_half_y(j) + memory_dy_vy2(i,j)))&
) /c1
vtemp=(vtemp+vyf(i,j))*0.5d0
vy(i,j) = vy(i,j) + &
(rsm_half_x_half_y*(value_dx_vy / K_x_half_x(i) + memory_dx_vy(i,j)&
+ value_dy_vy1 / K_y_half_y(j) + memory_dy_vy1(i,j))&
+ rhof_half_x_half_y*(value_dy_vy2 / K_y_half_y(j) + memory_dy_vy2(i,j))&
+ rhof_half_x_half_y*etaokappa_half_x_half_y*vtemp)&
/co
enddo
enddo
! Dirichlet conditions (rigid boundaries) on the edges or at the bottom of the PML layers
vx(1,:) = ZERO
vx(NX,:) = ZERO
vx(:,1) = ZERO
vx(:,NY) = ZERO
vy(1,:) = ZERO
vy(NX,:) = ZERO
vy(:,1) = ZERO
vy(:,NY) = ZERO
vxf(1,:) = ZERO
vxf(NX,:) = ZERO
vxf(:,1) = ZERO
vxf(:,NY) = ZERO
vyf(1,:) = ZERO
vyf(NX,:) = ZERO
vyf(:,1) = ZERO
vyf(:,NY) = ZERO
! store seismograms
do irec = 1,NREC
! x component
sisvx(it,irec) = vx(ix_rec(irec),iy_rec(irec))
! y component
sisvy(it,irec) = vy(ix_rec(irec),iy_rec(irec))
! fluid pressure
sisp(it,irec) = sigma2(ix_rec(irec),iy_rec(irec))
enddo
! compute total energy
! compute kinetic energy first, defined as 1/2 rho ||v||^2
! in principle we should use rho_half_x_half_y instead of rho for vy
! in order to interpolate density at the right location in the staggered grid cell
! but in a homogeneous medium we can safely ignore it
total_energy_kinetic(it) = 0.5d0 * &
sum(rho(NPOINTS_PML:NX-NPOINTS_PML+1,NPOINTS_PML:NY-NPOINTS_PML+1)&
*(vx(NPOINTS_PML:NX-NPOINTS_PML+1,NPOINTS_PML:NY-NPOINTS_PML+1)**2&
+vy(NPOINTS_PML:NX-NPOINTS_PML+1,NPOINTS_PML:NY-NPOINTS_PML+1)**2))&
*DELTAX * DELTAY+&
0.5d0*sum(rsm(NPOINTS_PML:NX-NPOINTS_PML+1,NPOINTS_PML:NY-NPOINTS_PML+1)&
*(vxf(NPOINTS_PML:NX-NPOINTS_PML+1,NPOINTS_PML:NY-NPOINTS_PML+1)**2&
+vyf(NPOINTS_PML:NX-NPOINTS_PML+1,NPOINTS_PML:NY-NPOINTS_PML+1)**2))&
*DELTAX*DELTAY
! add potential energy, defined as 1/2 epsilon_ij sigma_ij
! in principle we should interpolate the medium parameters at the right location
! in the staggered grid cell but in a homogeneous medium we can safely ignore it
total_energy_potential(it) = ZERO
do j = NPOINTS_PML,NY-NPOINTS_PML+1
do i = NPOINTS_PML,NX-NPOINTS_PML+1
epsilon_xx = ((rlambdao(i,j) + 2.d0*rmu(i,j)) * sigmaxx(i,j) - rlambdao(i,j) * sigmayy(i,j)) / &
(4.d0 * rmu(i,j) * (rlambdao(i,j) + rmu(i,j)))
epsilon_yy = ((rlambdao(i,j) + 2.d0*rmu(i,j)) * sigmayy(i,j) - rlambdao(i,j) * sigmaxx(i,j)) / &
(4.d0 * rmu(i,j) * (rlambdao(i,j) + rmu(i,j)))
epsilon_xy = sigmaxy(i,j) / (2.d0 * rmu(i,j))
total_energy_potential(it) = total_energy_potential(it) + &
0.5d0 * (epsilon_xx * sigmaxx(i,j) + epsilon_yy * sigmayy(i,j) + 2.d0 * epsilon_xy * sigmaxy(i,j)&
+sigma2(i,j)**2/rbM(i,j)&
+2.d0*rhof(i,j)*(vx(i,j)*vxf(i,j)+vy(i,j)*vyf(i,j)))*DELTAX * DELTAY
enddo
enddo
! output information
if (mod(it,IT_DISPLAY) == 0 .or. it == 5) then
! print maximum of norm of velocity
velocnorm_all = maxval(sqrt(vx(:,:)**2 + vy(:,:)**2))
print *,'Time step # ',it,' out of ',NSTEP
print *,'Time: ',sngl((it-1)*DELTAT),' seconds'
print *,'Max norm velocity vector V (m/s) = ',velocnorm_all
print *,'total energy = ',total_energy_kinetic(it) + total_energy_potential(it)
print *
! check stability of the code, exit if unstable
if (velocnorm_all > STABILITY_THRESHOLD) stop 'code became unstable and blew up'
vnorm(:,:)=sqrt(vx(:,:)**2+vy(:,:)**2)
call create_color_image(vx,NX+2,NY+2,it,ISOURCE,JSOURCE,ix_rec,iy_rec,nrec, &
NPOINTS_PML,USE_PML_LEFT,USE_PML_RIGHT,USE_PML_BOTTOM, &
USE_PML_TOP,1,max_amplitude,JINTERFACE)
call create_color_image(vy,NX+2,NY+2,it,ISOURCE,JSOURCE,ix_rec,iy_rec,nrec, &
NPOINTS_PML,USE_PML_LEFT,USE_PML_RIGHT,USE_PML_BOTTOM, &
USE_PML_TOP,2,max_amplitude,JINTERFACE)
! save temporary partial seismograms to monitor the behavior of the simulation
! while it is running
call write_seismograms(sisvx,sisvy,sisp,NSTEP,NREC,DELTAT,t0)
endif
enddo ! end of time loop
! save seismograms
call write_seismograms(sisvx,sisvy,sisp,NSTEP,NREC,DELTAT,t0)
! save total energy
open(unit=20,file='energy.dat',status='unknown')
do it = 1,NSTEP
write(20,*) sngl(dble(it-1)*DELTAT), sngl(total_energy_kinetic(it) + total_energy_potential(it))
enddo
close(20)
! create script for Gnuplot for total energy
open(unit=20,file='plot_energy',status='unknown')
write(20,*) 'set term x11'
write(20,*) '# set term postscript landscape monochrome dashed "Helvetica" 22'
write(20,*) '# set xrange [0:7]'
write(20,*) '# set yrange [-4:4.5]'
write(20,*)
write(20,*) 'set xlabel "Time (s)"'