Linear Operators

JUDI is building on JOLI.jl to implement matrix-free linear operators. These operators represent the discretized wave-equations and sensitivit (Jacobian) for different acquisition schemes.

• Linear Operators
• judiModeling
• judiJacobian
• judiProjection
• judiLRWF

judiModeling

Seismic modeling operator for solving a wave equation for a given right-hand-side.

judiModeling(model; options=Options())
judiModeling(model, src_geometry, rec_geometry; options=Options())

Construction:

• Construct a modeling operator without source/receiver projections:

F = judiModeling(model; options=opt)

• Construct a modeling operator with source/receiver projections:

F = judiModeling(model, src_geometry, rec_geometry)

• Construct a modeling operator from an existing operator without geometries and projection operators:

F = Pr*F*Ps'

where Ps and Pr are source/receiver projection operators of type judiProjection.

• Construct a modeling operator for extended source modeling:

F = Pr*F*Pw'

where Pw is a judiLRWF (low-rank-wavefield) projection operator.

Accessible fields:

# Model structure
F.model

# Source injection (if available) and geometry
F.qInjection
F.qInjection.geometry

# Receiver interpolation (if available) and geometry
F.rInterpolation
F.rInterpolation.geometry

# Options structure
F.options

Usage:

# Forward modeling (F w/ geometries)
d_obs = F*q

# Adjoint modeling (F w/ geometries)
q_ad = F'*d_obs

# Forward modeling (F w/o geometries)
d_obs = Pr*F*Ps'*q

# Adjoint modelng (F w/o geometries)
q_ad = Ps*F'*Pr'*d_obs

# Extended source modeling (F w/o geometries)
d_obs  = Pr*F*Pw'*w

# Adjoint extended source modeling (F w/o geometries)
w_ad = Pw*F'*Pr'*d_obs

# Forward modeling and return full wavefield (F w/o geometries)
u = F*Ps'*q

# Adjoint modelnig and return wavefield (F w/o geometries)
v = F'*Pr'*d_obs

# Forward modeling with full wavefield as source (F w/o geometries)
d_obs = Pr*F*u

# Adjoint modeling with full wavefield as source (F w/o geometries)
q_ad = Ps*F*v

judiJacobian

Jacobian of a non-linear forward modeling operator. Corresponds to linearized Born modeling (forward mode) and reverse-time migration (adjoint mode).

judiJacobian(F,q)

Construction:

• A judiJacobian operator can be create from an exisiting forward modeling operator and a source vector:

J = judiJacobian(F, q)  # F w/ geometries

J = judiJacobian(Pr*F*Ps', q)   # F w/o geometries

where Ps and Pr are source/receiver projection operators of type judiProjection.

• A Jacobian can also be created for an extended source modeling operator:

J = judiJacobian(Pr*F*Pw', w)

where Pw is a judiLRWF operator and w is a judiWeights vector (or 2D/3D Julia array).

Accessible fields::

# Model structure
J.model

# Underlying propagator
J.F

# Source injection (if available) and geometry throughpropagator
J.F.qInjection
J.F.qInjection.geometry

# Receiver interpolation (if available) and geometry through propagator
J.F.rInterpolation
J.F.rInterpolation.geometry

# Source term, can be a judiWeights, judiWavefield, or a judiVector
J.q

# Options structure
J.options

Usage:

# Linearized modeilng
d_lin = J*dm

# RTM
rtm = J'*d_lin

# Matrix-free normal operator
H = J'*J

judiProjection

Abstract linear operator for source/receiver projections. A (transposed) judiProjection operator symbolically injects the data with which it is multiplied during modeling. If multiplied with a forward modeling operator, it samples the wavefield at the specified source/receiver locations.

judiProjection(geometry)

Accessible fields:

# Source/receiver geometry
P.geometry

Usage:

# Multiply with judiVector to create a judiRHS
rhs1 = Pr'*d_obs
rhs2 = Ps'*q

# Sample wavefield at source/receiver location during modeling
d_obs = Pr*F*Ps'*q
q_ad = Ps*F*Pr'*d_obs

judiLRWF

Abstract linear operator for sampling a seismic wavefield as a sum over all time steps, weighted by a time-varying wavelet. Its transpose injects a time-varying wavelet at every grid point in the model.

judiWavelet(dt, wavelet)

Accessible fields:

# Wavelet of i-th source location
P.wavelet[i]

Usage:

# Multiply with a judiWeight vector to create a judiExtendedSource
ex_src = Pw'*w

# Sample wavefield as a sum over time, weighted by the source
u_ex = Pw*F'*Pr'*d_obs