FAMILIAR OPERATORS

The simplest and most fundamental linear operators arise when a matrix operator reduces to a simple row or a column.

A row    is a summation operation.

A column is an impulse response.

If the inner loop of a matrix multiply ranges within a

row,    the operator is called sum or pull.

column, the operator is called spray or push.

A basic aspect of adjointness is that the adjoint of a row matrix operator is a column matrix operator. For example, the row operator [a,b]

$$y \eq \left[ \ a \ b \ \right] \left[ \begin{array} {l} x_1 \\ x_2\end{array}\right] \eq a x_1 + b x_2$$ (1)

has an adjoint that is two assignments:

$$\left[ \begin{array} {l} \hat x_1 \\ \hat x_2 \end{array... ...q \left[ \begin{array} {l} a \\ b \end{array} \right] \ y$$ (2)

The adjoint of a sum of N terms is a collection of N assignments.

 

• Adjoint derivative
• Transient convolution
• Internal convolution
• Zero padding is the transpose of truncation
• Adjoints of products are reverse-ordered products of adjoints
• Nearest-neighbor coordinates
• Data-push binning
• Linear interpolation
• Spray and sum : scatter and gather
• Causal and leaky integration
• Backsolving, polynomial division and deconvolution
• The basic low-cut filter
• Nearest-neighbor normal moveout (NMO)
• Coding chains and arrays