System and method for estimating one-way propagation operators

1. A method, comprising:

obtaining, from a seismic survey, a seismic data set and a multi-dimensional seismic wave propagation velocity model;

using a seismic processing system:

approximating the seismic wave propagation velocity model as a plurality of layers each bounded by a first bounding depth and a second bounding depth,

for each of the plurality of layers:

simulating propagation of a seismic wave from the first bounding depth to the second bounding depth using a two-way seismic wave propagation simulator;

forming an over-determined system of linear equations relating at least one mono-frequency component of the seismic wave at the first bounding depth to at least one mono-frequency component of the seismic wave at the second bounding depth;

determining a plurality of one-way seismic wave propagation operators by inverting the over-determined system of linear equations; and

verifying that the plurality of one-way seismic wave propagation operators has been determined for each layer of the plurality of layers,

processing the seismic data set using at least one of the one-way seismic wave propagation operators for at least one of layer;

using a computer processor:

determining a presence and a location of a hydrocarbon reservoir based, at least in part, on the processed seismic data set, and

determining a borehole trajectory through a subterranean region of interest based, at least in part, on the presence of the hydrocarbon reservoir; and

drilling, using a drilling system, a borehole guided by the borehole trajectory.

2. The method of claim 1 , wherein the seismic wave propagation velocity model comprises:

a seismic P-wave propagation velocity model,

a seismic S-wave propagation velocity model, and

a density model.

3. The method of claim 1 , wherein inverting the over-determined system of linear equations comprises inverting a plurality of independent systems of linear equations,

wherein each of the independent systems of linear equations comprises equating a first vector to a matrix-vector product of a matrix and a second vector.

4. The method of claim 3 , wherein the first vector represents an amplitude and a phase of a mono-frequency component of the seismic wave at a plurality of locations at the second bounding depth, the second matrix represents an amplitude and a phase of a plurality of one-way seismic wave propagators, and the second vector represents an amplitude and a phase of a mono-frequency component of plurality of seismic waves at a plurality of locations at the first bounding depth.

5. The method of claim 1 , wherein inverting a plurality of independent systems is performed using at least one of a biconjugate gradient method, a biconjugate gradient stabilized method, a generalized minimal residual method, a quasi-minimal residual method, genetic algorithm, and a simulated annealing algorithm.

6. The method of claim 1 , wherein the two-way seismic wave propagation simulator is a two-way frequency-domain seismic wave propagation simulator.

7. A system, comprising:

a seismic source to emit radiated seismic P-waves;

a plurality of seismic receivers for detecting and recording a seismic data set generated by the radiated seismic P-waves;

a seismic processor, configured to:

receive a seismic data set and a multi-dimensional seismic wave propagation velocity model,

approximate the seismic wave propagation velocity model as a plurality of layers each bounded by a first bounding depth and a second bounding depth,

for each of the plurality of layers:

simulate propagation of a seismic wave from the first bounding depth to the second bounding depth using a two-way seismic wave propagation simulator;

form an over-determined system of linear equations relating at least one mono-frequency component of the seismic wave at the first bounding depth to at least one mono-frequency component of the seismic wave at the second bounding depth; and

determine a plurality of one-way seismic wave propagation operators by inverting the over-determined system of linear equations;

verify that the plurality of one-way seismic wave propagation operators has been determined for each layer of the plurality of layers, and

process the seismic data set using at least one of the one-way seismic wave propagation operators for at least one of layers;

a computer processor, configured to:

determine a presence and a location of a hydrocarbon reservoir based, at least in part, on the processed seismic data set, and

determine a borehole trajectory through a subterranean region of interest based, at least in part, on the presence of the hydrocarbon reservoir; and

a drilling system, configured to drill a borehole guided by-the borehole trajectory.

8. The system of claim 7 , wherein the seismic wave propagation velocity model comprises:

a seismic P-wave propagation velocity model,

a seismic S-wave propagation velocity model, and

a density model.

9. The system of claim 7 , wherein inverting the over-determined system of linear equations comprises inverting a plurality of independent systems,

wherein each of the independent systems comprise equating a first vector to a matrix-vector product of a matrix and a second vector.

10. The system of claim 9 , wherein the first vector represents an amplitude and a phase of a mono-frequency component of a seismic wave at a plurality of locations at the second bounding depth, the second matrix represents an amplitude and a phase of a plurality of one-way seismic wave propagator, and the matrix representing an amplitude and a phase of a mono-frequency component of a plurality of seismic waves at a plurality of locations at the first bounding depth.

11. The system of claim 7 , wherein inverting a plurality of independent systems is performed using at least one of a biconjugate gradient method, a biconjugate gradient stabilized method, a generalized minimal residual method, a quasi-minimal residual method, genetic algorithm, and a simulated annealing algorithm.

12. The system of claim 7 , wherein the two-way seismic wave propagation simulator is a two-way frequency-domain seismic wave propagation simulator.