Seismic exploration using exactly invertible discrete transformation into tau-p space

A method of seismic exploration of subsurface formations comprising:

• deriving a seismogram of subsurface formations which is in discrete form in a selected space;
• transforming the seismogram through the use of a discrete form of a transform to generate a seismogram in tau-p space, in a process which comprises utilizing a transformation matrix having the property of being block circulant when represented in the spatial domain;
• filtering the seismogram in tau-p space to generate a filtered seismogram;
• inverse-transforming the filtered seismogram from tau-p space through the use of an exact discrete form of an inverse transform to thereby generate a filtered seismogram in the selected space;
• and
• providing a tangible representation of the filtered seismogram in the selected space.

A method of seismic exploration of subsurface formations comprising:

• deriving an original seismogram of selected subsurface formations which is represented as a set of traces each comprising discrete samples of a selected seismic parameter over time for a selected spatial parameter;
• transforming the original seismogram into tau-p space by transforming the original seismogram into the frequency domain, multiplying the frequency domain version of the original seismogram by selected transformation matrices for respective frequencies to derive respective product arrays, and inverse transforming the product arrays out of the frequency domain to derive a representation of the original seismogram in discrete form in tau-p space and allowing exact discrete inverse-transformation;
• and
• providing a tangible representation of the seismogram in tau-p space in form suitable for modifying or display in tau-p space.

A method comprising:

• deriving a two-dimensional array x.sub.l (n) of seismic traces in (l,n) space, where l is an integer index which identifies a respective one of (2L+l) traces which are rows in the array and ranges from -L to +L, L being an integer, each row of the array comprises discrete samples of a selected seismic parameter, each sample being identified by an index n=1, 2, . . . , N, and x.sub.l (n) is a periodic sequence with the period N, N being an integer;
• deriving transformation matrices R.sub.jl (k), where j is an integer index which identifies a respective one of (2J+l) rows of a transform matrix and ranges from -J to +J, l is an integer index which identifies a respective one of (2L+l) columns of a transformation matrix and ranges from -L to +L, and where k is a frequency index and J and L are integers;
• transforming the array x.sub.l (n) into the frequency domain to derive the array x.sub.l (k+1);
• matrix-multiplying the array x.sub.l (k+1) by the matrices R.sub.jl (k) to derive the array y.sub.j (k+1) allowing exact discrete inverse-transformation;
• and
• modifying the array y.sub.j (k+1) to derive a modified array.

A method comprising:

• deriving an original record of measurements of a selected parameter of energy propagating in a medium and sampled in time and in space;
• transforming the original record to a record in tau-p space through a discrete transformation process which permits exact discrete inverse-transformation and which in processing in the frequency domain can be completed frequency-by-frequency and in processing in the spatial domain can utilize a transformation matrix having a block-circulant structure;
• processing the tau-p space record to enhance desirable features thereof and effect an exact discrete inverse transformation from tau-p space to generate an improved record;
• and
• providing a tangible representation of the improved record.

A method comprising:

• deriving an original record of data for a selected parameter sampled with respect to two independent variables defining an original domain;
• transforming the original record to a record in a different domain in which each point results from a sum of values of said parameter taken over a corresponding pattern in said original domain, through a discrete transformation which permits exact discrete inverse-transformation and which in processing in a frequency domain corresponding to one of said independent variables can be completed frequency-by-frequency and in processing in said original domain with respect to the other of said independent variables can utilize a transformation matrix having a block-circulant structure;
• processing said different domain record to enhance desirable features thereof and effect an exact discrete inverse transformation from said different domain to generate an improved record;
• and
• providing a tangible representation of the improved record.