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## The pressure operator

The pressure operator from physics from the equation of state'' that the time rate of change of the pressure is an external source s(x,y,t) minus the divergence of the velocity times the incompressibility K.
 (11)

In the heat flow equation the field variable is a scalar (the temperature). For wave propagation in two dimensions, the wavefield variable has three components, pressure and two components of material velocity. The pressure operator P transforms ptx,y to pt+1x,y but it is an identity matrix for the two components of velocity (which will actually be transformed in a later process). We distribute the wavefield variables throughout space with this tiling:
 (12)
We define the basic tile for this discrete mesh as
 (13)
Computationally we refer to this basic tile on the inputs as and on the outputs as although the three components in the tile are at slightly different physical locations from one another. Expressing the pressure equation for these tiles we have
 (14)
In subroutine pressure() below the input pressure and velocity field is q(x,y,*) and the output is r(x,y,*). Equation (14) is expressed in the subroutine on the line beginning with r(x,y,p)=.

# operator of pressure change from divergence of velocity
#
real                           kappa(nx,ny),q(nx,ny,3), r(nx,ny,3)
p=1; u=2; v=3
do x= 2, nx-1 {
do y= 2, ny-1 {
r(x,y,p)= r(x,y,p) + q(x,y,p) - (q(x,y,u) - q(x-1,y  ,u)) * kappa(x,y) -
(q(x,y,v) - q(x  ,y-1,v)) * kappa(x,y)
r(x,y,u)= r(x,y,u) + q(x,y,u)
r(x,y,v)= r(x,y,v) + q(x,y,v)
} else {
q(x  ,y  ,p)= q(x  ,y  ,p) + r(x,y,p)
q(x  ,y  ,u)= q(x  ,y  ,u) - r(x,y,p) * kappa(x,y)  + r(x,y,u)
q(x-1,y  ,u)= q(x-1,y  ,u) + r(x,y,p) * kappa(x,y)
q(x  ,y  ,v)= q(x  ,y  ,v) - r(x,y,p) * kappa(x,y)  + r(x,y,v)
q(x  ,y-1,v)= q(x  ,y-1,v) + r(x,y,p) * kappa(x,y)
}
}}
return; end


Notice that the identity matrix pass through'' of the velocity components shows up as two extra assignments in the forward operator and thus as two added terms in the adjoint operator.

 frog Figure 1 A few frogs hopped into the pond. Chirup!

Next: The velocity operator Up: WAVE PROCESS CHAIN Previous: WAVE PROCESS CHAIN
Stanford Exploration Project
11/12/1997