Kirchhoff Map
KirchMig.KirchMap — Function.

KirchMap{eltype(t)}(t, trav_r, [trav_s]; parallel_threaded_serial="serial")

Construct a Kirchhoff LinearMap object, which can perform Kirchhoff modeling (or more accurately, demigration) as a forward operator and Kirchhoff migration as an adjoint operation.

Parameters

  • t : (nt,), AbstractVector{<:Real}

Contains time samples which correspond to the data domain. KirchMap(t, trav)*m will create data whose time axis is given by t. Its type defines the type of KirchMap.

  • trav_r : (nz, [nx, ny, ...], nr), AbstractArray{<:Real, M}

Contains traveltimes between each model parameter and receiver location. The first M-1 dimensions are model dimensions (z, x, y, ...), and the last dimension corresponds to receivers.

  • trav_s : (nz, [nx, ny, ...], ns), AbstractArray{<:Real, M}, optional

Like trav_r but for sources. If omitted, defaults to trav_r, i.e., sources and receivers are assumed colocated.

  • parallel_threaded_serial : String

Defines which Kirchhoff methods to use. Can be the default, parallel, which uses Julia's distributed computing to parallelize over receivers; threaded which uses multi-threading to parallelize over receivers; or serial. It is highly recommended to not use the serial version, even when only using a single worker or thread.

Usage

  • Forward map

The forward map L multiplies a model vector of size nz × nx × ny × ... to create a data vector of size nr × ns × nt.

  • Adjoint map

The adjoint map L' multiplies a data vector of size nr × ns × nt to create a model vector of size nz × nx × ny × ...

Description

The forward map computes the discretized version of the following operation

\[d(r, s, t) = \int m(x)\,\delta(t - \tau_{sx} - \tau_{xr})\,\mathrm{d}x.\]

and the adjoint map computes

\[m(x) = \int d(t, r, s)\,\delta(t - \tau_{sx} - \tau_{xr})\,\mathrm{d}r\,\mathrm{d}s\,\mathrm{d}t.\]

In both computations, the differential elements (which only affect amplitude) are neglected.