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seismic_ADEPML_2D_viscoelastic_RK4_eighth_order.f90
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seismic_ADEPML_2D_viscoelastic_RK4_eighth_order.f90
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!
! 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 Ruiqi Shi and Youshan Liu, China.
!
! RK4 bug detected by Youshan Liu, China fixed by Quentin Brissaud, France and also Caltech (USA) in this version in March 2018.
!
! Ruiqi Shi, Department of Exploration Geophysics, China University of Petroleum, Beijing, China.
! Email: shiruiqi123 AT gmail DOT com
!
! This software is a computer program whose purpose is to solve
! the two-dimensional viscoelastic wave equation
! using a finite-difference method with Auxiliary Differential
! Equation Perfectly Matched Layer (ADE-PML) conditions.
!
! 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_ADEPML_2D_viscoelastic_RK4_eighth_order
! High order 2D explicit-semi implicit-implicit viscoelastic finite-difference code
! in velocity and stress formulation with Auxiliary Differential
! Equation Perfectly Matched Layer (ADE-PML) absorbing conditions for
! an SLS viscoelastic medium. It is fourth order Runge-Kutta (RK4) in time
! and 8th order in space using Holberg spatial discretization.
! Version 1.1.3
! by Roland Martin, University of Pau, France, Jan 2010
! with improvements by Ruiqi Shi and
! with a major bug fix in the Runge-Kutta implementation
! and also significant memory usage optimization by Youshan Liu, China, August 2015.
! based on seismic_CPML_2D_isotropic_second_order.f90
! by Dimitri Komatitsch and Roland Martin, University of Pau, France, 2007.
! *BEWARE* that the attenuation model implemented below is that of J. M. Carcione,
! Seismic modeling in viscoelastic media, Geophysics, vol. 58(1), p. 110-120 (1993), which is NON causal,
! i.e., waves speed up instead of slowing down when turning attenuation on.
! This comes from the fact that in that model the relaxed state at zero frequency is used as a reference instead of
! the unrelaxed state at infinite frequency. These days a causal model should be used instead,
! i.e. one using the unrelaxed state at infinite frequency as a reference.
! The 8th-order staggered-grid formulation of Holberg is used:
!
! ^ y
! |
! |
!
! +-------------------+
! | |
! | |
! | |
! | |
! | v_y |
! sigma_xy +---------+ |
! | | |
! | | |
! | | |
! | | |
! | | |
! +---------+---------+ ---> x
! v_x sigma_xx
! sigma_yy
!
! The ADE-PML implementation is based in part on formulas given in Roden and Gedney (2010)
!
! If you use this code for your own research, please cite some (or all) of these articles:
!
! @Article{BlKoChLoXi15,
! Title = {Positivity-preserving highly-accurate optimization of the {Z}ener viscoelastic model, with application
! to wave propagation in the presence of strong attenuation},
! Author = {\'Emilie Blanc and Dimitri Komatitsch and Emmanuel Chaljub and Bruno Lombard and Zhinan Xie},
! Journal = {Geophysical Journal International},
! Year = {2015},
! Note = {in press.}}
!
! @ARTICLE{MaKoGeBr10,
! author = {Roland Martin and Dimitri Komatitsch and Stephen D. Gedney and Emilien Bruthiaux},
! title = {A high-order time and space formulation of the unsplit perfectly matched layer
! for the seismic wave equation using {Auxiliary Differential Equations (ADE-PML)}},
! journal = {Comput. Model. Eng. Sci.},
! year = {2010},
! volume = {56},
! pages = {17-42},
! number = {1}}
!
! @ARTICLE{MaCo10,
! author = {Roland Martin and Carlos Couder-Casta{\~n}eda},
! title = {An improved unsplit and convolutional Perfectly Matched Layer
! absorbing technique for the Navier-Stokes equations using cut-off frequency shift},
! journal = {Comput. Model. Eng. Sci.},
! pages ={47-77}
! year = {2010},
! volume = {63},
! number = {1}}
!
! @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}}
!
! @ARTICLE{MaKoEz08,
! author = {Roland Martin and Dimitri Komatitsch and Abdelaaziz 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{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{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{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 2D results as color images, 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 "
!
! 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 = 141
integer, parameter :: NY = 621 ! NY = 800
! Explicit (epsn=1,epsn=0), implicit (epsn=0,epsn1=1), semi-implicit (epsn=0.5,epsn1=0.5)
integer, parameter :: iexpl=0
integer, parameter :: iimpl=0
integer, parameter :: isemiimpl=1
! size of a grid cell
double precision, parameter :: DELTAX = 5.d0, ONE_OVER_DELTAX = 1.d0 / DELTAX
double precision, parameter :: DELTAY = DELTAX
double precision, parameter :: ONE_OVER_DELTAY = ONE_OVER_DELTAX
double precision, parameter :: ONE=1.d0,TWO=2.d0, DIM=2.d0
! P-velocity, S-velocity and density
double precision, parameter :: cp_top = 3050.d0
double precision, parameter :: cs_top = 1950.d0
double precision, parameter :: rho_top = 2000.d0
double precision, parameter :: mu_top = rho_top*cs_top*cs_top
double precision, parameter :: lambda_top = rho_top*(cp_top*cp_top - 2.d0*cs_top*cs_top)
double precision, parameter :: lambdaplustwomu_top = rho_top*cp_top*cp_top
double precision, parameter :: cp_bottom = 2600.d0
double precision, parameter :: cs_bottom = 1500.d0
double precision, parameter :: rho_bottom = 1500.d0
double precision, parameter :: mu_bottom = rho_bottom*cs_bottom*cs_bottom
double precision, parameter :: lambda_bottom = rho_bottom*(cp_bottom*cp_bottom - 2.d0*cs_bottom*cs_bottom)
double precision, parameter :: lambdaplustwomu_bottom = rho_bottom*cp_bottom*cp_bottom
! total number of time steps
integer, parameter :: NSTEP = 5000
! time step in seconds
double precision, parameter :: DELTAT = 5.d-4
! parameters for the source
double precision, parameter :: f0 = 15.d0
double precision, parameter :: t0 = 1.20d0 / f0
double precision, parameter :: factor = 1.d5
! parameters for attenuation
! number of standard linear solids
integer, parameter :: N_SLS = 2
! Qp approximately equal to 13, Qkappa approximately to 20 and Qmu / Qs approximately to 10
double precision, parameter :: QKappa_att = 20.d0, QMu_att = 10.d0
double precision, parameter :: f0_attenuation = 16 ! in Hz
! flags to add PML layers to the edges of the grid
logical, parameter :: USE_PML_XMIN = .true.
logical, parameter :: USE_PML_XMAX = .true.
logical, parameter :: USE_PML_YMIN = .true.
logical, parameter :: USE_PML_YMAX = .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.
! source
! integer, parameter :: ISOURCE = NX - 2*NPOINTS_PML - 1
integer, parameter :: ISOURCE = NPOINTS_PML+11
integer, parameter :: JSOURCE = 2*NY / 3
double precision, parameter :: xsource = (ISOURCE) * DELTAX
double precision, parameter :: ysource = (JSOURCE) * DELTAY
double precision, parameter :: INTERFACE_HEIGHT = ysource - 125*DELTAY
integer, parameter:: JINTERFACE=INT(INTERFACE_HEIGHT/DELTAY)+1
! angle of source force clockwise with respect to vertical (Y) axis
double precision, parameter :: ANGLE_FORCE = 45.d0
! receivers
integer, parameter :: NREC = 3
double precision, parameter :: xdeb = xsource - 100.d0 ! first receiver x in meters
double precision, parameter :: ydeb = 2300.d0 ! first receiver y in meters
double precision, parameter :: xfin = xsource ! last receiver x in meters
double precision, parameter :: yfin = 300.d0 ! last receiver y in meters
! display information on the screen from time to time
integer, parameter :: IT_DISPLAY = 500
! 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
! Holberg (1987) coefficients, taken from
! @ARTICLE{Hol87,
! author = {O. Holberg},
! title = {Computational aspects of the choice of operator and sampling interval
! for numerical differentiation in large-scale simulation of wave phenomena},
! journal = {Geophysical Prospecting},
! year = {1987},
! volume = {35},
! pages = {629-655}}
double precision, parameter :: c1 = 1.231666d0
double precision, parameter :: c2 = -1.041182d-1
double precision, parameter :: c3 = 2.063707d-2
double precision, parameter :: c4 = -3.570998d-3
double precision, parameter :: coefficient_sum = abs(c1)+abs(c2)+abs(c3)+abs(c4)
! RK4 scheme coefficients, 2 per subloop, 8 in total
double precision, dimension(4) :: rk41, rk42
! power to compute d0 profile
double precision, parameter :: NPOWER = 2.d0
double precision, parameter :: NPOWER2 = 2.d0
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-11
!double precision, parameter :: K_MAX_PML = 7.d0
! double precision, parameter :: ALPHA_MAX_PML = 0.d0 ! from Festa and Vilotte
double precision, parameter :: ALPHA_MAX_PML_1 = 2.d0*PI*(f0/2.d0) ! from Festa and Vilotte
double precision K_MAX_PML_1
! double precision, parameter :: K_MAX_PML_2 = K_MAX_PML_1 / 15.d0
! double precision, parameter :: K_MAX_PML_2 = K_MAX_PML_1
! double precision, parameter :: ALPHA_MAX_PML_2 = ALPHA_MAX_PML_1 / 5.d0
! arrays for the memory variables
! could declare these arrays in PML only to save a lot of memory, but proof of concept only here
! We have as many memory variables as the number of frequency shift poles in the CPML
! Indices are 1 and 2 for the 2 frequency shift poles
double precision, dimension(4,-4:NX+4,-4:NY+4) :: &
memory_dvx_dx_1, &
memory_dvx_dy_1, &
memory_dvy_dx_1, &
memory_dvy_dy_1, &
memory_dsigmaxx_dx_1, &
memory_dsigmayy_dy_1, &
memory_dsigmaxy_dx_1, &
memory_dsigmaxy_dy_1
double precision, dimension(-4:NX+4,-4:NY+4) :: &
memory_vx_dx_1, &
memory_vx_dy_1, &
memory_vy_dx_1, &
memory_vy_dy_1, &
memory_sigmaxx_dx_1, &
memory_sigmayy_dy_1, &
memory_sigmaxy_dx_1, &
memory_sigmaxy_dy_1
double precision :: &
value_dvx_dx, &
value_dvx_dy, &
value_dvy_dx, &
value_dvy_dy, &
value_dsigmaxx_dx, &
value_dsigmayy_dy, &
value_dsigmaxy_dx, &
value_dsigmaxy_dy
double precision :: duxdx,duxdy,duydx,duydy,div
double precision :: epsn,epsn1,Sn
! 1D arrays for the damping profiles
double precision, dimension(-4:NX+4) :: d_x_1,K_x_1,alpha_prime_x_1,g_x_1,ksi_x
double precision, dimension(-4:NX+4) :: d_x_half_1,K_x_half_1,alpha_prime_x_half_1,g_x_half_1,ksi_x_half
double precision, dimension(-4:NY+4) :: d_y_1,K_y_1,alpha_prime_y_1,g_y_1,ksi_y
double precision, dimension(-4:NY+4) :: d_y_half_1,K_y_half_1,alpha_prime_y_half_1,g_y_half_1,ksi_y_half
! 1D arrays for the damping profiles
double precision, dimension(-4:NX+4) :: d_x_2,K_x_2,alpha_prime_x_2,g_x_2
double precision, dimension(-4:NX+4) :: d_x_half_2,K_x_half_2,alpha_prime_x_half_2,g_x_half_2
double precision, dimension(-4:NY+4) :: d_y_2,K_y_2,alpha_prime_y_2,g_y_2
double precision, dimension(-4:NY+4) :: d_y_half_2,K_y_half_2,alpha_prime_y_half_2,g_y_half_2
! coefficients that allow to reset the memory variables at each RK4 substep depend on the substepping and are then of dimension 4,
! 1D arrays for the damping profiles
double precision, dimension(4,-4:NX+4) :: a_x_1,b_x_1
double precision, dimension(4,-4:NX+4) :: a_x_half_1,b_x_half_1
double precision, dimension(4,-4:NY+4) :: a_y_1,b_y_1
double precision, dimension(4,-4:NY+4) :: a_y_half_1,b_y_half_1
double precision, dimension(-4:NX+4) :: r_x_1,s_x_1
double precision, dimension(-4:NX+4) :: r_x_half_1,s_x_half_1
double precision, dimension(-4:NY+4) :: r_y_1,s_y_1
double precision, dimension(-4:NY+4) :: r_y_half_1,s_y_half_1
! 1D arrays for the damping profiles
double precision, dimension(4,-4:NX+4) :: a_x_2
double precision, dimension(4,-4:NX+4) :: a_x_half_2
double precision, dimension(4,-4:NY+4) :: a_y_2
double precision, dimension(4,-4:NY+4) :: a_y_half_2
! PML
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, dimension(-4:NX+4,-4:NY+4) :: vx,vy,sigmaxx,sigmayy,sigmaxy
double precision, dimension(-4:NX+4,-4:NY+4) :: sigmaxx_R,sigmayy_R,sigmaxy_R
double precision, dimension(N_SLS,-4:NX+4,-4:NY+4) :: e1,e11,e22,e12
double precision, dimension(-4:NX+4,-4:NY+4) :: rho, mu,lambda,lambdaplustwomu
double precision rho_half_x_half_y
! variables are stored in four indices in the first dimension to implement RK4
! dv does not always indicate a derivative
double precision, dimension(4,-4:NX+4,-4:NY+4) :: dvx,dvy,dsigmaxx,dsigmayy,dsigmaxy
double precision, dimension(4,-4:NX+4,-4:NY+4) :: dsigmaxx_R,dsigmayy_R,dsigmaxy_R
double precision, dimension(N_SLS,4,-4:NX+4,-4:NY+4) :: de1,de11,de12
integer, parameter :: number_of_2Darrays = 2*8
integer, parameter :: number_of_3Darrays = 32
! for the source
double precision a,t,force_x,force_y,source_term
! for attenuation
double precision :: f_min_attenuation, f_max_attenuation
double precision, dimension(N_SLS) :: tau_epsilon_nu1,tau_sigma_nu1,tau_epsilon_nu2,tau_sigma_nu2
! for stability estimate
double precision :: c_max,c_min
! for receivers
double precision distval,dist
integer, dimension(NREC) :: ix_rec,iy_rec
double precision, dimension(NREC) :: xrec,yrec
! for seismograms
double precision, dimension(NSTEP,NREC) :: sisvx,sisvy
! max amplitude for color snapshots
double precision max_amplitudeVx
double precision max_amplitudeVy
! for evolution of total energy in the medium
double precision :: epsilon_xx,epsilon_yy,epsilon_xy
double precision, dimension(NSTEP) :: total_energy,total_energy_kinetic,total_energy_potential
double precision :: local_energy_kinetic,local_energy_potential
integer :: irec,inc
double precision :: mul_relaxed,lambdal_relaxed,lambdalplus2mul_relaxed
double precision :: mul_unrelaxed,lambdal_unrelaxed,lambdalplus2mul_unrelaxed
double precision :: Mu_nu1,Mu_nu2
double precision :: phi_nu1(N_SLS)
double precision :: phi_nu2(N_SLS)
double precision :: tauinv,inv_tau_sigma_nu1(N_SLS)
double precision :: taumin,taumax, tau1, tau2, tau3, tau4
double precision :: inv_tau_sigma_nu2(N_SLS)
integer :: i,j,it,it2
double precision :: Vsolidnorm
double precision Courant_number_bottom,Courant_number_top
double precision Dispersion_number_bottom,Dispersion_number_top
! timer to count elapsed time
character(len=8) datein
character(len=10) timein
character(len=5) :: zone
integer, dimension(8) :: time_values
integer ihours,iminutes,iseconds,int_tCPU
double precision :: time_start,time_end,tCPU
! names of the time stamp files
character(len=150) outputname
! main I/O file
integer, parameter :: IOUT = 41
!---
!--- the program starts here
!---
if (iexpl == 1) then
epsn = 1.d0
epsn1 = 0.d0
endif
if (iimpl == 1) then
epsn = 0.d0
epsn1 = 1.d0
endif
if (isemiimpl == 1) then
epsn = 0.5d0
epsn1 = 0.5d0
endif
! attenuation constants for standard linear solids
! nu1 is the dilatation/incompressibility mode (QKappa)
! nu2 is the shear mode (Qmu)
! array index (1) is the first standard linear solid, (2) is the second etc.
! from J. M. Carcione, Seismic modeling in viscoelastic media, Geophysics,
! vol. 58(1), p. 110-120 (1993) for two memory-variable mechanisms (page 112).
! Beware: these values implement specific values of the quality factors:
! Qp approximately equal to 13, Qkappa approximately to 20 and Qmu / Qs approximately to 10,
! which means very high attenuation, see that paper for details.
! tau_epsilon_nu1(1) = 0.0334d0
! tau_sigma_nu1(1) = 0.0303d0
! tau_epsilon_nu2(1) = 0.0352d0
! tau_sigma_nu2(1) = 0.0287d0
! tau_epsilon_nu1(2) = 0.0028d0
! tau_sigma_nu1(2) = 0.0025d0
! tau_epsilon_nu2(2) = 0.0029d0
! tau_sigma_nu2(2) = 0.0024d0
! from J. M. Carcione, D. Kosloff and R. Kosloff, Wave propagation simulation
! in a linear viscoelastic medium, Geophysical Journal International,
! vol. 95, p. 597-611 (1988) for two memory-variable mechanisms (page 604).
! Beware: these values implement specific values of the quality factors:
! Qkappa approximately to 27 and Qmu / Qs approximately to 20,
! which means very high attenuation, see that paper for details.
! tau_epsilon_nu1(1) = 0.0325305d0
! tau_sigma_nu1(1) = 0.0311465d0
! tau_epsilon_nu2(1) = 0.0332577d0
! tau_sigma_nu2(1) = 0.0304655d0
! tau_epsilon_nu1(2) = 0.0032530d0
! tau_sigma_nu1(2) = 0.0031146d0
! tau_epsilon_nu2(2) = 0.0033257d0
! tau_sigma_nu2(2) = 0.0030465d0
! f_min and f_max are computed as : f_max/f_min=12 and (log(f_min)+log(f_max))/2 = log(f0)
f_min_attenuation = exp(log(f0_attenuation)-log(12.d0)/2.d0)
f_max_attenuation = 12.d0 * f_min_attenuation
! use new SolvOpt nonlinear optimization with constraints from Emilie Blanc, Bruno Lombard and Dimitri Komatitsch
! to compute attenuation mechanisms
call compute_attenuation_coeffs(N_SLS,QKappa_att,f0_attenuation,f_min_attenuation,f_max_attenuation, &
tau_epsilon_nu1,tau_sigma_nu1)
call compute_attenuation_coeffs(N_SLS,QMu_att,f0_attenuation,f_min_attenuation,f_max_attenuation, &
tau_epsilon_nu2,tau_sigma_nu2)
print *
print *,'with new SolvOpt routine for attenuation:'
print *
print *,'N_SLS, QKappa_att, QMu_att = ',N_SLS, QKappa_att, QMu_att
print *,'f0_attenuation,f_min_attenuation,f_max_attenuation = ',f0_attenuation,f_min_attenuation,f_max_attenuation
print *,'tau_epsilon_nu1 = ',tau_epsilon_nu1
print *,'tau_sigma_nu1 = ',tau_sigma_nu1
print *,'tau_epsilon_nu2 = ',tau_epsilon_nu2
print *,'tau_sigma_nu2 = ',tau_sigma_nu2
print *
tau1 = tau_sigma_nu1(1)/tau_epsilon_nu1(1)
tau2 = tau_sigma_nu2(1)/tau_epsilon_nu2(1)
tau3 = tau_sigma_nu1(2)/tau_epsilon_nu1(2)
tau4 = tau_sigma_nu2(2)/tau_epsilon_nu2(2)
taumax = max(1.d0/tau1,1.d0/tau2,1.d0/tau3,1.d0/tau4)
taumin = min(1.d0/tau1,1.d0/tau2,1.d0/tau3,1.d0/tau4)
inv_tau_sigma_nu1(1) = ONE / tau_sigma_nu1(1)
inv_tau_sigma_nu2(1) = ONE / tau_sigma_nu2(1)
inv_tau_sigma_nu1(2) = ONE / tau_sigma_nu1(2)
inv_tau_sigma_nu2(2) = ONE / tau_sigma_nu2(2)
phi_nu1(1) = (ONE - tau_epsilon_nu1(1)/tau_sigma_nu1(1)) / tau_sigma_nu1(1)
phi_nu2(1) = (ONE - tau_epsilon_nu2(1)/tau_sigma_nu2(1)) / tau_sigma_nu2(1)
phi_nu1(2) = (ONE - tau_epsilon_nu1(2)/tau_sigma_nu1(2)) / tau_sigma_nu1(2)
phi_nu2(2) = (ONE - tau_epsilon_nu2(2)/tau_sigma_nu2(2)) / tau_sigma_nu2(2)
Mu_nu1 = ONE - (ONE - tau_epsilon_nu1(1)/tau_sigma_nu1(1)) - (ONE - tau_epsilon_nu1(2)/tau_sigma_nu1(2))
Mu_nu2 = ONE - (ONE - tau_epsilon_nu2(1)/tau_sigma_nu2(1)) - (ONE - tau_epsilon_nu2(2)/tau_sigma_nu2(2))
print *
print *,'2D visco-elastic FD code in velocity and stress formulation with ADE in 8th an RK4'
print *
! display size of the model
print *
print *,'NX = ',NX
print *,'NY = ',NY
print *
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 *,'Number of points of all the arrays = ',dble(NX+4*2+1)*dble(NY+4*2+1)*number_of_2Darrays + &
4*dble(NX+4*2+1)*dble(NY+4*2+1)*number_of_3Darrays
print *,'Size in GB of all the arrays = ',dble(NX+4*2+1)*dble(NY+4*2+1)*number_of_2Darrays*8.d0/(1024.d0*1024.d0*1024.d0) + &
4*dble(NX+4*2+1)*dble(NY+4*2+1)*number_of_3Darrays*8.d0/(1024.d0*1024.d0*1024.d0)
!--- 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 = 1.d-5
c_max = max(cp_bottom,cp_top)
c_min = min(cs_bottom,cs_top)
K_MAX_PML_1 = 1.d0
print *,'K_MAX_PML = ',K_MAX_PML_1
! 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) * c_max *dsqrt(taumax)* log(Rcoef) / (2.d0 * thickness_PML_x)
d0_y = - (NPOWER + 1) * c_max *dsqrt(taumax)* log(Rcoef) / (2.d0 * thickness_PML_y)
else
d0_x = - (NPOWER + 1) * cp_bottom *dsqrt(taumax)* log(Rcoef) / (2.d0 * thickness_PML_x)
d0_y = - (NPOWER + 1) * cp_bottom *dsqrt(taumax)* log(Rcoef) / (2.d0 * thickness_PML_y)
endif
print *
print *,'d0_x = ',d0_x
print *,'d0_y = ',d0_y
! parameters involved in RK4 time expansion
rk41(1) = ZERO
rk41(2) = 0.5d0
rk41(3) = 0.5d0
rk41(4) = 1.d0
rk42(1) = 1.d0 / 6.d0
rk42(2) = 2.d0 / 6.d0
rk42(3) = 2.d0 / 6.d0
rk42(4) = 1.d0 / 6.d0
ksi_x(:) = ZERO
ksi_x_half(:) = ZERO
d_x_1(:) = ZERO
d_x_half_1(:) = ZERO
K_x_1(:) = 1.d0
K_x_half_1(:) = 1.d0
alpha_prime_x_1(:) = ZERO
alpha_prime_x_half_1(:) = ZERO
a_x_1(:,:) = ZERO
a_x_half_1(:,:) = ZERO
g_x_1(:) = 5.d-1
g_x_half_1(:) = 5.d-1
ksi_y(:) = ZERO
ksi_y_half(:) = ZERO
d_y_1(:) = ZERO
d_y_half_1(:) = ZERO
K_y_1(:) = 1.d0
K_y_half_1(:) = 1.d0
alpha_prime_y_1(:) = ZERO
alpha_prime_y_half_1(:) = ZERO
a_y_1(:,:) = ZERO
a_y_half_1(:,:) = ZERO
g_y_1(:) = 1.d0
g_y_half_1(:) = 1.d0
d_x_2(:) = ZERO
d_x_half_2(:) = ZERO
K_x_2(:) = 1.d0
K_x_half_2(:) = 1.d0
alpha_prime_x_2(:) = ZERO
alpha_prime_x_half_2(:) = ZERO
a_x_2(:,:) = ZERO
a_x_half_2(:,:) = ZERO
g_x_2(:) = 1.d0
g_x_half_2(:) = 1.d0
d_y_2(:) = ZERO
d_y_half_2(:) = ZERO
K_y_2(:) = 1.d0
K_y_half_2(:) = 1.d0
alpha_prime_y_2(:) = ZERO
alpha_prime_y_half_2(:) = ZERO
a_y_2(:,:) = ZERO
a_y_half_2(:,:) = ZERO
g_y_2(:) = 1.d0
g_y_half_2(:) =1.d0
r_x_1(:) = ZERO
s_x_1(:) = ZERO
r_x_half_1(:) = ZERO
s_x_half_1(:) = ZERO
r_y_1(:) = ZERO
s_y_1(:) = ZERO
r_y_half_1(:) = ZERO
s_y_half_1(:) = ZERO
! damping in the X direction
! 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 = -4,NX+4
! abscissa of current grid point along the damping profile
xval = DELTAX * dble(i-1)
!---------- left edge
if (USE_PML_XMIN) 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_1(i) = d0_x * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_x_1(i) = 1.d0 + (K_MAX_PML_1 - 1.d0) * abscissa_normalized**NPOWER2
alpha_prime_x_1(i) = ALPHA_MAX_PML_1 * (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_1(i) = d0_x * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_x_half_1(i) = 1.d0 + (K_MAX_PML_1 - 1.d0) * abscissa_normalized**NPOWER2
alpha_prime_x_half_1(i) = ALPHA_MAX_PML_1 * (1.d0 - abscissa_normalized)
endif
endif
!---------- right edge
if (USE_PML_XMAX) 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_1(i) = d0_x * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_x_1(i) = 1.d0 + (K_MAX_PML_1 - 1.d0) * abscissa_normalized**NPOWER2
alpha_prime_x_1(i) = ALPHA_MAX_PML_1 * (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_1(i) = d0_x * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_x_half_1(i) = 1.d0 + (K_MAX_PML_1 - 1.d0) * abscissa_normalized**NPOWER2
alpha_prime_x_half_1(i) = ALPHA_MAX_PML_1 * (1.d0 - abscissa_normalized)
endif
endif
! 1 pole
d_x_2(i) = 0.d0
d_x_half_2(i) = 0.d0
! just in case, for -5 at the end
if (alpha_prime_x_1(i) < ZERO) alpha_prime_x_1(i) = ZERO
if (alpha_prime_x_half_1(i) < ZERO) alpha_prime_x_half_1(i) = ZERO
! just in case, for -5 at the end
if (alpha_prime_x_2(i) < ZERO) alpha_prime_x_2(i) = ZERO
if (alpha_prime_x_half_2(i) < ZERO) alpha_prime_x_half_2(i) = ZERO
! CPML damping parameters for the 4 sub time steps of RK4 algorithm
do inc=1,4
b_x_1(inc,i) = (1.-epsn*DELTAT*rk41(inc)*(d_x_1(i)/K_x_1(i) + alpha_prime_x_1(i)))/&
(1.+epsn1*DELTAT*rk41(inc)*(d_x_1(i)/K_x_1(i) + alpha_prime_x_1(i)))
b_x_half_1(inc,i) = (1.-epsn*DELTAT*rk41(inc)*(d_x_half_1(i)/K_x_half_1(i) &
+ alpha_prime_x_half_1(i)))/(1. +epsn1*DELTAT*rk41(inc)*(d_x_half_1(i)/K_x_half_1(i) &
+ alpha_prime_x_half_1(i)))
! this to avoid division by zero outside the PML
if (abs(d_x_1(i)) > 1.d-6) a_x_1(inc,i) = - DELTAT*rk41(inc)*d_x_1(i) / (K_x_1(i)* K_x_1(i))/&
(1. +epsn1*DELTAT*rk41(inc)*(d_x_1(i)/K_x_1(i) + alpha_prime_x_1(i)))
if (abs(d_x_half_1(i)) > 1.d-6) a_x_half_1(inc,i) =-DELTAT*rk41(inc)*d_x_half_1(i)/&
(K_x_half_1(i)*K_x_half_1(i) )/&
(1. +epsn1*DELTAT*rk41(inc)*(d_x_half_1(i)/K_x_half_1(i)&
+ alpha_prime_x_half_1(i)))
r_x_1(i) = -(d_x_1(i)/K_x_1(i) + alpha_prime_x_1(i))
s_x_1(i) = - d_x_1(i)/K_x_1(i)/K_x_1(i)
r_x_half_1(i) = -(d_x_half_1(i)/K_x_half_1(i) + alpha_prime_x_half_1(i))
s_x_half_1(i) = - d_x_half_1(i)/K_x_half_1(i)/K_x_half_1(i)
enddo
enddo
! damping in the Y direction
! origin of the PML layer (position of right edge minus thickness, in meters)
yoriginbottom = thickness_PML_y
yorigintop = (NY-1)*DELTAY - thickness_PML_y
do j = -4,NY+4
! abscissa of current grid point along the damping profile
yval = DELTAY * dble(j-1)
!---------- bottom edge
if (USE_PML_YMIN) 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_1(j) = d0_y * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_y_1(j) = 1.d0 + (K_MAX_PML_1 - 1.d0) * abscissa_normalized**NPOWER2
alpha_prime_y_1(j) = ALPHA_MAX_PML_1 * (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_1(j) = d0_y * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_y_half_1(j) = 1.d0 + (K_MAX_PML_1 - 1.d0) * abscissa_normalized**NPOWER2
alpha_prime_y_half_1(j) = ALPHA_MAX_PML_1 * (1.d0 - abscissa_normalized)
endif
endif
!---------- top edge
if (USE_PML_YMAX) 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_1(j) = d0_y * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_y_1(j) = 1.d0 + (K_MAX_PML_1 - 1.d0) * abscissa_normalized**NPOWER2
alpha_prime_y_1(j) = ALPHA_MAX_PML_1 * (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_1(j) = d0_y * abscissa_normalized**NPOWER
! from Stephen Gedney's unpublished class notes for class EE699, lecture 8, slide 8-2
K_y_half_1(j) = 1.d0 + (K_MAX_PML_1 - 1.d0) * abscissa_normalized**NPOWER2
alpha_prime_y_half_1(j) = ALPHA_MAX_PML_1 * (1.d0 - abscissa_normalized)
endif
endif
! just in case, for -5 at the end
if (alpha_prime_y_1(j) < ZERO) alpha_prime_y_1(j) = ZERO
if (alpha_prime_y_half_1(j) < ZERO) alpha_prime_y_half_1(j) = ZERO
! CPML damping parameters for the 4 sub time steps of RK4 algorithm
do inc=1,4
b_y_1(inc,j) = (1.-epsn*DELTAT*rk41(inc)*(d_y_1(j)/K_y_1(j) + alpha_prime_y_1(j)))/&
(1.+epsn1*DELTAT*rk41(inc)*(d_y_1(j)/K_y_1(j) + alpha_prime_y_1(j)))
b_y_half_1(inc,j) = (1.-epsn*DELTAT*rk41(inc)*(d_y_half_1(j)/K_y_half_1(j) + &
alpha_prime_y_half_1(j)))/(1.+epsn1*DELTAT*rk41(inc)*(d_y_half_1(j)/K_y_half_1(j)&
+ alpha_prime_y_half_1(j)))
! this to avoid division by zero outside the PML
if (abs(d_y_1(j)) > 1.d-6) a_y_1(inc,j) = - DELTAT*rk41(inc)*d_y_1(j)&
/ (K_y_1(j)* K_y_1(j))/&
(1.+epsn1*DELTAT*rk41(inc)*(d_y_1(j)/K_y_1(j) + alpha_prime_y_1(j)))
if (abs(d_y_half_1(j)) > 1.d-6) a_y_half_1(inc,j) = -DELTAT*rk41(inc)*d_y_half_1(j) /&
(K_y_half_1(j) * K_y_half_1(j) )/&
(1.+epsn1*DELTAT*rk41(inc)*(d_y_half_1(j)/K_y_half_1(j) + alpha_prime_y_half_1(j)))
enddo
r_y_1(j) = -(d_y_1(j)/K_y_1(j) + alpha_prime_y_1(j))
s_y_1(j) = - d_y_1(j)/K_y_1(j)/K_y_1(j)
r_y_half_1(j) = -(d_y_half_1(j)/K_y_half_1(j) + alpha_prime_y_half_1(j))
s_y_half_1(j) = - d_y_half_1(j)/K_y_half_1(j)/K_y_half_1(j)
enddo
! compute the Lame parameters and density
do j = -4,NY+4
do i = -4,NX+4
if (HETEROGENEOUS_MODEL .and. DELTAY*dble(j-1) > INTERFACE_HEIGHT) then
rho(i,j)= rho_top
mu(i,j)= mu_top
lambda(i,j) = lambda_top
lambdaplustwomu(i,j) = lambdaplustwomu_top
else
rho(i,j)= rho_bottom
mu(i,j)= mu_bottom
lambda(i,j) = lambda_bottom
lambdaplustwomu(i,j) = lambdaplustwomu_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
xrec(1) = xsource
yrec(1) = ysource - 393*DELTAY
xrec(2) = xsource
yrec(2) = ysource + 191*DELTAY
xrec(3) = xsource + 101*DELTAX
yrec(3) = ysource
! 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) - xrec(irec))**2 + (DELTAY*dble(j) - 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 *dsqrt(taumax)* DELTAT*sqrt(1.d0/DELTAX**2 + 1.d0/DELTAY**2)
Dispersion_number_bottom=cs_bottom*dsqrt(taumin)/(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/coefficient_sum) stop 'time step is too large, simulation will be unstable'
if (HETEROGENEOUS_MODEL) then
Courant_number_top = cp_top *dsqrt(taumax) * DELTAT* sqrt(1.d0/DELTAX**2 + 1.d0/DELTAY**2 )
Dispersion_number_top= cs_top*dsqrt(taumin) /(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 > 1.d0/coefficient_sum) stop 'time step is too large, simulation will be unstable'
endif
! erase main arrays
vx(:,:) = ZERO
vy(:,:) = ZERO
sigmaxy(:,:) = ZERO
sigmayy(:,:) = ZERO
sigmaxx(:,:) = ZERO
sigmaxy_R(:,:) = ZERO
sigmayy_R(:,:) = ZERO
sigmaxx_R(:,:) = ZERO
dvx(:,:,:) = ZERO
dvy(:,:,:) = ZERO
dsigmaxy(:,:,:) = ZERO
dsigmayy(:,:,:) = ZERO
dsigmaxx(:,:,:) = ZERO
dsigmaxy_R(:,:,:) = ZERO
dsigmayy_R(:,:,:) = ZERO
dsigmaxx_R(:,:,:) = ZERO
e1(1,:,:)=ZERO
e1(2,:,:)=ZERO
e11(1,:,:)=ZERO
e11(2,:,:)=ZERO
e12(1,:,:)=ZERO
e12(2,:,:)=ZERO
e22(1,:,:)=ZERO
e22(2,:,:)=ZERO
de1(1,:,:,:)=ZERO
de1(2,:,:,:)=ZERO
de11(1,:,:,:)=ZERO
de11(2,:,:,:)=ZERO
de12(1,:,:,:)=ZERO
de12(2,:,:,:)=ZERO
! PML
memory_vx_dx_1(:,:) = ZERO
memory_vx_dy_1(:,:) = ZERO
memory_vy_dx_1(:,:) = ZERO
memory_vy_dy_1(:,:) = ZERO
memory_sigmaxx_dx_1(:,:) = ZERO
memory_sigmayy_dy_1(:,:) = ZERO
memory_sigmaxy_dx_1(:,:) = ZERO
memory_sigmaxy_dy_1(:,:) = ZERO
memory_dvx_dx_1(:,:,:) = ZERO
memory_dvx_dy_1(:,:,:) = ZERO
memory_dvy_dx_1(:,:,:) = ZERO
memory_dvy_dy_1(:,:,:) = ZERO
memory_dsigmaxx_dx_1(:,:,:) = ZERO
memory_dsigmayy_dy_1(:,:,:) = ZERO
memory_dsigmaxy_dx_1(:,:,:) = ZERO
memory_dsigmaxy_dy_1(:,:,:) = ZERO