SUFDMOD2_PML - Finite-Difference MODeling (2nd order) for acoustic wave
    equation with PML absorbing boundary conditions.			
 Caveat: experimental PML absorbing boundary condition version,	
may be buggy!								

 sufdmod2_pml <vfile >wfile nx= nz= tmax= xs= zs= [optional parameters]

 Required Parameters:							
 <vfile		file containing velocity[nx][nz]		
 >wfile		file containing waves[nx][nz] for time steps	
 nx=			number of x samples (2nd dimension)		
 nz=			number of z samples (1st dimension)		
 xs=			x coordinates of source				
 zs=			z coordinates of source				
 tmax=			maximum time					

 Optional Parameters:							
 nt=1+tmax/dt		number of time samples (dt determined for stability)
 mt=1			number of time steps (dt) per output time step	

 dx=1.0		x sampling interval				
 fx=0.0		first x sample					
 dz=1.0		z sampling interval				
 fz=0.0		first z sample					

 fmax = vmin/(10.0*h)	maximum frequency in source wavelet		
 fpeak=0.5*fmax	peak frequency in ricker wavelet		

 dfile=		input file containing density[nx][nz]		
 vsx=			x coordinate of vertical line of seismograms	
 hsz=			z coordinate of horizontal line of seismograms	
 vsfile=		output file for vertical line of seismograms[nz][nt]
 hsfile=		output file for horizontal line of seismograms[nx][nt]
 ssfile=		output file for source point seismograms[nt]	
 verbose=0		=1 for diagnostic messages, =2 for more		

 abs=1,1,1,1		Absorbing boundary conditions on top,left,bottom,right
 			sides of the model. 				
 		=0,1,1,1 for free surface condition on the top		

 ...PML parameters....                                                 
 pml_max=1000.0        PML absorption parameter                        
 pml_thick=0           half-thickness of pml layer (0 = do not use PML)

 Notes:								
 This program uses the traditional explicit second order differencing	
 method. 								

 Two different absorbing boundary condition schemes are available. The 
 first is a traditional absorbing boundary condition scheme created by 
 Hale, 1990. The second is based on the perfectly matched layer (PML)	
 method of Berenger, 1995.						



 Authors:  CWP:Dave Hale
           CWP:modified for SU by John Stockwell, 1993.
           CWP:added frequency specification of wavelet: Craig Artley, 1993
           TAMU:added PML absorbing boundary condition: 
               Michael Holzrichter, 1998

 References: (Hale's absobing boundary conditions)
 Clayton, R. W., and Engquist, B., 1977, Absorbing boundary conditions
 for acoustic and elastic wave equations, Bull. Seism. Soc. Am., 6,
	1529-1540. 

 Clayton, R. W., and Engquist, B., 1980, Absorbing boundary conditions
 for wave equation migration, Geophysics, 45, 895-904.

 Hale, D.,  1990, Adaptive absorbing boundaries for finite-difference
 modeling of the wave equation migration, unpublished report from the
 Center for Wave Phenomena, Colorado School of Mines.

 Richtmyer, R. D., and Morton, K. W., 1967, Difference methods for
 initial-value problems, John Wiley & Sons, Inc, New York.

 Thomee, V., 1962, A stable difference scheme for the mixed boundary problem
 for a hyperbolic, first-order system in two dimensions, J. Soc. Indust.
 Appl. Math., 10, 229-245.

 Toldi, J. L., and Hale, D., 1982, Data-dependent absorbing side boundaries,
 Stanford Exploration Project Report SEP-30, 111-121.

 References: (PML boundary conditions)
 Jean-Pierre Berenger, ``A Perfectly Matched Layer for the Absorption of
 Electromagnetic Waves,''  Journal of Computational Physics, vol. 114,
 pp. 185-200.

 Hastings, Schneider, and Broschat, ``Application of the perfectly
 matched layer (PML) absorbing boundary condition to elastic wave
 propogation,''  Journal of the Accoustical Society of America,
 November, 1996.

 Allen Taflove, ``Electromagnetic Modeling:  Finite Difference Time
 Domain Methods'', Baltimore, Maryland: Johns Hopkins University Press,
 1995, chap. 7, pp. 181-195.


 Trace header fields set: ns, delrt, tracl, tracr, offset, d1, d2,
                          sdepth, trid