Regularization
KirchMig.ConvMap — Function.

ConvMap([T,] wavelet, n...) -> W

Construct a convolution/correlation operator W which will act on an AbstractVector.

Parameters

  • T : Type, optional

Type of argument of W. Defaults to Float64.

  • wavelet

One-dimensional wavelet.

  • n...

Sequence of dimensions of the data.

Usage

  • Forward map and Adjoint maps

The forward map W convolves the last dimension of the input with wavelet. The adjoint map W correlates the last dimension of the input with wavelet.

KirchMig.LaplacianMap — Function.

LaplacianMap([T,] n...) -> Δ

Construct a discretized Laplacian operator Δ which will act on an AbstractVector.

Parameters

  • T : Type, optional

Type of argument of Δ. Defaults to Float64.

  • n...

Sequence of spatial dimensions of Δ.

Usage

  • Forward map and Adjoint maps

The forward map Δ multiplies a model vector of size nz × nx × ny × ... its second order derivative. It is symmetric.

Description

The forward and adjoint maps computes the following operation

\[Δm = \sum_{i} δ_i m\]

where

\[δ_l m_{i,j,k,...} = - m_{...,l+1,...} + 2m_{...,l,...} - m_{...,l-1,...}\]
KirchMig.DiffZMap — Function.

DiffZMap([T,] n...) -> δz

Construct a discretized z-derivative operator δz which will act on an AbstractVector.

Parameters

  • T : Type, optional

Type of argument of δz. Defaults to Float64.

  • n...

Sequence of spatial dimensions of δz.

Usage

  • Forward map and Adjoint maps

The forward map δz multiplies a model vector of size nz × nx × ny × ... its first order z-derivative. The adjoint map is minus the forward map.

Description

The forward map computes the following operation

\[δ_z m_{i,j,k,...} = (m_{l+1,...} - m_{l-1,...})/2\]

and the adjoint map computes -δz.

KirchMig.DiffXMap — Function.

DiffXMap([T,] n...) -> δx

Construct a discretized x-derivative operator δx which will act on an AbstractVector.

Parameters

  • T : Type, optional

Type of argument of δx. Defaults to Float64.

  • n...

Sequence of spatial dimensions of δx.

Usage

  • Forward map and Adjoint maps

The forward map δx multiplies a model vector of size nz × nx × ny × ... its first order x-derivative. The adjoint map is minus the forward map.

Description

The forward map computes the following operation

\[δ_x m_{i,j,k,...} = (m_{l+1,...} - m_{l-1,...})/2\]

and the adjoint map computes -δx.

KirchMig.GradDivMap — Function.

GradDivMap([T,] n...) -> GD

Construct a discretized gradient operator GD which will act on an AbstractVector.

Parameters

  • T : Type, optional

Type of argument of GD. Defaults to Float64.

  • n...

Sequence of spatial dimensions of GD.

Usage

  • Forward map

Calculates the discrete gradient of a nz × nx × ny × ... using first order central differences.

\[∇m_{i,j,k,...} = [δ_x m_{i,j,k,...}, ..., δ_z m_{i,j,k,...}]\]
  • Adjoint map

Calculates the discrete negative divergence of a nz × nx × ny × ... using first order central differences.

\[∇\cdot m_{i,j,k,...} = δ_x m_{i,j,k,...} + ... + δ_z m_{i,j,k,...}\]