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Abstract

The Spectral Broadening SFM provides a number of methods to broaden the spectra of the input seismic data. The methods are Stochastic Deconvolution and two types of Sparse Spike processing.

NOTICE Copyright protection as an unpublished work is claimed by WesternGeco. The work was created in 2008. Should publication of the work occur, the following notice shall apply. " 2008 Westerngeco". This work contains valuable tradesecrets; disclosure without written authorization is prohibited.

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Contents

1.0 Technical Discussion 1.1 1.2 1.3 Introduction 1.1.1 Assumptions Stochastic Deconvolution 1.2.1 Parameters for Stochastic Deconvolution Sparse Spike 1.3.1 Minimum Entropy Deconvolution with Frequency-Domain Constraints (FMED) 1.3.2 Linear Programming Method 2.0 3.0 References Inputs and Outputs 3.1 3.2 4.0 Inputs Outputs

Literal Summary 4.1 Inputs 4.1.1 INPUT_TRACES Port 4.1.2 INPUT_WAVELET Port 4.2 Outputs 4.2.1 OUTPUT_TRACES Port 4.2.2 OUTPUT_CUBE_WAVELETS Port 4.2.3 OUTPUT_GLOBAL_WAVELET Port

5.0 6.0

Parameter Set Summary Setup Parameters 6.1 6.2 Stochastic Deconvolution Sparse Spike

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1.0 Technical Discussion

1.0 Technical Discussion

1.1 Introduction

The Spectral Broadening SFM provides three ways of broadening the spectra of the input seismic data. Only the first method (stochastic deconvolution) is a true deconvolution, the other two try to compute a reflectivity series consisting of sparse spike reflectors. The methods are described below in detail.

1.1.1 Assumptions

For Sparse Spike processing the input data should be pre-processed

so that the spectrum within the given frequency band is flat. For Stochastic Deconvolution the input seismic data should have reflectors which are continuous in the horizontal direction. In both cases the vertical sequences should be sparse.

1.2 Stochastic Deconvolution

The Stochastic Deconvolution method is designed to simultaneously estimate a wavelet and deconvolved seismic reflection signal. It is based on a convolution model. i.e. the seismic signal s(t,x,y) is given by Equation 1: s(t,x,y) = w(t,x,y) * r(t,x,y) where r refers to the reflectivity and w refers to the wavelet. The time/depth dimension is denoted by t, and the lateral dimensions are denoted by x and y respectively. Noise is neglected in this equation. The major a priori assumptions are: The vertical sequences are sparse Reflectivity tends to be continuous in the horizontal direction The wavelet is common to several traces

Based on these assumptions the algorithm recovers a non-minimum phase wavelet and resolved closely spaced reflectors. This is not a unique problem. The wavelet is estimated using the least square method using no model assumption; the phase is estimated as well. Commonly only a few iterations are needed. November 2008 - WesternGeco SPECTRAL_BROADENING 3

1.0 Technical Discussion

The process starts with an initial guess of the reflectivity and uses it together with the seismic to find a wavelet estimate. In the next step it assumes the wavelet to be the correct one and estimates a new reflectivity. This reflectivity is better than the initial guess, it iterates the procedure alternatively improving the wavelet and reflectivity until the process converges. It is possible to output either a single global wavelet operator or a cube of operators (one for each trace). These can then be input into the SFM and used directly. The operators outputted are the ones calculated in the final iteration of the process. The iterative nature of the process means that the results produced from inputting a wavelet cube will not exactly match the original output from the job that produced the wavelet cube.

1.2.1 Parameters for Stochastic Deconvolution

The absolute density factor controls the density of reflectors to be estimated. A small value gives fewer reflectors. If the resulting reflectivity estimate has large gaps between typically strong reflectors this parameter should be increased. On the other hand, if a strong reflector splits up into several reflectors this factor should be decreased. The dynamic density factor allows the process to vary density along the seismic trace. Certain data may require a high-density factor to prevent splitting but this may miss weak reflectivity parts.

In this case the absolute density factor should be chosen to avoid splitting and the dynamic density factor should be increased to allow weak reflectors to show up. Neighborhood processing type determines the quality and speed of the estimation. 2D neighborhood processing is fast but gives a limited quality of reflectivity estimate. 3D neighborhood processing gives improved reflectivity quality and is suitable for volume processing. Both types provide a good wavelet estimation. The Wavelet Length should be long enough to allow the decaying of the side-lobes without allowing the amplitudes at the head and tail of the wavelet to approach too close to zero.

1.3 Sparse Spike

Both of the sparse spike methods seek to compute a reflectivity series based on the assumption that the data consists of sparse (not frequent or regular) reflectors.

1.3.1 Minimum Entropy Deconvolution with Frequency-Domain Constraints (FMED)

The FMED method aims to reconstruct the reflectivity spectum of the input data using the minimum entropy criterion. The keyword here is reconstruction; the spectrum out of the user-specified frequency band is computed from the input data, while the data inside the frequency band is preserved untouched. This is not a true deconvolution since no inverse filter is computed and

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1.0 Technical Discussion

applied to the data. This technique seeks the smallest number of large spikes that are consistent with the input data. Ideally, the wavelet in the frequency band will have been replaced during pre-processing by a zero-phase wavelet of constant spectral amplitude. This is because the data in this frequency band will not be touched by this process. The method is described by Sacchi et al (1994).

FMED Algorithm

The algorithm is an iterative process. Users should experiment with the number of iterations during testing, but it has been found that a small iteration count (3?C4) is sufficient. The process can work on the entire trace or on user-specified time gates. Loop over j=1,????,niter. 1. Compute normalized squared amplitude trace qj(i) as Equation 2.

Where n is the number of iterations, yj the input seismic trace and N the number of samples. The normalization sum is calculated only once. 2. Compute entropy norm Vj as in Equation 3.

Where lqj is the logarithm of the normalized squared amplitude trace.

3. Compute sample by sample trace weights wj as in Equation 4.

4. Compute updated enhanced trace by applying the weights as in Equation 5. bj(i)=wj(i)yj(i), i=1,????,N 5. Transform the updated trace to the Fourier domain and merge updated trace spectrum with original trace spectrum while retaining the original spectrum within the frequency band as in Equation 6.

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1.0 Technical Discussion

6. Transform merged trace back to time domain and proceed with next iteration cycle.

1.3.2 Linear Programming Method

The Linear Programming method computes and outputs a sparse spike reflectivity series RS(i), with the minimum number of non-zero values. This is achieved using a simplex optimization method to minimize the absolute value of RS(1). The details of this method are described by Oldenburg et al (1983) . The fast simplex algorithm used to find the solution is taken from Barrodale and Roberts (1980). The sparse spike algorithm is accelerated further by dividing the data into time gates of 25 samples.

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2.0 References

2.0 References

Barrodale, I., and Roberts. F. D. K., 1980, Algorithm 552, Solution of the constrained l1 linear approximation problem: ACM Transactions on Mathematical Software, 6, no. 2, 231?C235. Oldenburg, D.W., Scheuer, T., and Levy, L. L., 1983, Recovery of acoustic impedance from reflection seismograms: Geophysics 48, no. 10, 1318?C1337. Sacchi, M.D., D.R. Velis and A.H. Cominguez, 1994. Minimum entropy deconvolution with frequency-domain constraints. Geophysics 59, p. 938?C945.

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3.0 Inputs and Outputs

3.0 Inputs and Outputs

3.1 Inputs

Standard Input (Required) Seismic traces in any sort order, after merging the data with its geometry. INPUT_TRACES

Wavelet Input (Optional) INPUT_WAVELET Wavelet operator, either one global operator or a cube of operators one for each seismic trace. This is only read if the Stochastic Deconvolution Parameter Set is used. If this input port is used, the supplied operations will be used to deconvolve the seismic input data. If this port is not connected, stochastic decon will be performed, resulting in deconvolved output traces and output of a global wavelet and cube of wavelets from their respective output ports.

3.2 Outputs

Standard Output (Required) Processed seismic traces. OUTPUT_TRACES

Cube Wavelets Output (Required) OUTPUT_CUBE_WAVELETS Only output when using Stochastic Deconvolution method. Cube of de-convolution operator wavelets, one for each input trace. Global Wavelet Output (Optional) OUTPUT_GLOBAL_WAVELET Only output when using Stochastic Deconvolution method. One global wavelet operator.

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4.0 Literal Summary

4.0 Literal Summary

4.1 Inputs

4.1.1 INPUT_TRACES Port

Limited identification header literals are required, TIME_SHIFT_ALIGNMENT is only required when Sparse Spike processing is selected. Processing grid position trace header literals are needed to identify the relative position of the traces for the Stochastic Deconvolution method.

LITERAL DATA_DESC EARLIEST_TIME MAX_GATHER_MULT SAMP_INT DESCRIPTION Data Set Description Earliest TIME_SHIFT_ALIGNMENT Maximum Gather Multiplicity Sampling Interval

LITERAL 3DT_PRIM_INDEX DESCRIPTION Trace Primary Index. This literal is used in the Stochastic Deconvolution method to internally identify inline (column) position of the trace Trace Secondary Index. This literal is used in the Stochastic Deconvolution method to internally identify cross line (row) position of the trace. Trace Type. This literal is used to internally reject any trace that has a STACK_WORD of zero. Length of the Trace in Samples

3DT_SEC_INDEX

STACK_WORD LTRSAM

4.1.2 INPUT_WAVELET Port

This is only used if the Stochastic Deconvolution Parameter Set is

used.

LITERAL DATA_DESC MAX_GATHER_MULT DESCRIPTION Data Set Description Maximum Gather Multiplicity

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Spectral Broadening LITERAL SAMP_INT DESCRIPTION Sampling Interval

4.0 Literal Summary

LITERAL 3DT_PRIM_INDEX DESCRIPTION Trace Primary Index. This literal is used in the Stochastic Deconvolution method to internally identify inline (column) position of the trace Trace Secondary Index. This literal is used in the Stochastic Deconvolution method to internally identify cross line (row) position of the trace. Trace Type. This literal is used to internally reject any trace that has a STACK_WORD of zero. Length of the Trace in Samples

3DT_SEC_INDEX

STACK_WORD LTRSAM

4.2 Outputs

4.2.1 OUTPUT_TRACES Port

No literals are updated.

No literals are updated.

4.2.2 OUTPUT_CUBE_WAVELETS Port

Trace headers output that match data from INPUT_TRACES port.

LITERAL EARLIEST_TIME MAX_REFLECT_TIME DESCRIPTION Earliest TIME_SHIFT_ALIGNMENT Maximum Reflection Time

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4.0 Literal Summary

No literals are updated.

4.2.3 OUTPUT_GLOBAL_WAVELET Port

LITERAL EARLIEST_TIME MAX_REFLECT_TIME DESCRIPTION Earliest TIME_SHIFT_ALIGNMENT Maximum Reflection Time

No literals are updated.

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5.0 Parameter Set Summary

5.0 Parameter Set Summary

Parameter Set Name Geophysical Language STOCHASTIC_DECON OP_LENGTH ITERATIONS ABS_DENSITY DYN_DENSITY NEIGH_PROC TIME_VAR_EST SPARSE_SPIKE METHOD LOW_FREQ HIGH_FREQ ITERATIONS WINDOW_LENGTH Parameter Set Title Parameter Title Stochastic Deconvolution Operator Length Number of Iterations Absolute Density Factor Dynamic Density Factor Neighborhood Processing Type Estimate Time Variant Wavelet Sparse Spike Sparse Spike Decon Method Lower Frequency Boundary Higher Frequency Boundary Number of Iterations for FMED method Window Length for FMED Algorithm Status Optional

Optional

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6.0 Setup Parameters

6.0 Setup Parameters

6.1 Stochastic Deconvolution

General Information

Parameters for performing Stochastic Deconvolution

STOCHASTIC_DECON

(Status: Optional, Type: standard)

PARAMETERS:

OP_LENGTH Operator Length This parameter specifies the operator length in samples, it is only used if an operator trace is not supplied via an input port. Optional: Type: Trace-varying: Multi-valued: Constraint: Default: ITERATIONS Number of Iterations This parameter specifies the number of iterations Optional: Type: Trace-varying: Multi-valued: Constraint: Default: No integer No No param() >= 1 and param() <= 50 5 No integer No No param() >= 1 20

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Spectral Broadening ABS_DENSITY Absolute Density Factor This parameter specifies the absolute density factor. It must be a value greater than 0 and less than 1.0 Optional: Type: Trace-varying: Multi-valued: Constraint: Default: DYN_DENSITY Dynamic Density Factor This parameter specifies the dynamic density factor. Optional: Type: Trace-varying: Multi-valued: Constraint: Default: NEIGH_PROC

Neighborhood Processing Type Type of neighborhood processing. Optional: Type: Trace-varying: Multi-valued: Options: 'NONE' '2D' '2D_EXTENDED' '3D' '3D_EXTENDED' Default: No option No No No number No No param() >= 0 and param() < 1.0 0.5 No number No No param() > 0 and param() < 1.0 0.1

6.0 Setup Parameters

Wiener-Levinson spiking deconvolution. Calculate minimum phase from amplitude spectra using Hilbert transform. predictive (gapped) deconvolution predictive (gapped) deconvolution predictive (gapped) deconvolution 3D

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Spectral Broadening TIME_VAR_EST Estimate Time Variant Wavelet This parameter specifies whether to estimate time variant wavelets. Optional: Type: Trace-varying: Multi-valued: Default: No boolean No No false

6.0 Setup Parameters

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6.0 Setup Parameters

6.2 Sparse Spike

General Information

SPARSE_SPIKE

(Status: Optional, Type: standard)

Parameters for performing sparse spike deconvolution, either by FMED or LP methods If this Parameter set is used only the INPUT_TRACES port and the OUTPUT_TRACES port should be used.

PARAMETERS:

METHOD Sparse Spike Decon Method This parameter determines which algorithm to use to generate the sparse spike sequence. FMED is Frequency Constrained Minimum Entropy Deconvolution. LP is a Linear Programming method. Optional: Type: Trace-varying: Multi-valued: Options: 'FMED' 'LP' Default: LOW_FREQ Lower Frequency Boundary This parameter specifies the lower frequency limit. Frequencies between the lower and higher frequency boundaries will not be modified by this SFM. Optional: Type: Trace-varying: Multi-valued: Constraint: No number No No param() >0.0 No string No No Use the FMED algorithm. This uses the minimum entropy deconvolution technique with frequency-domain constraints. Uses the Linear Programming technique. FMED

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Spectral Broadening HIGH_FREQ Higher Frequency Boundary This parameter specifies the higher frequency limit.

6.0 Setup Parameters

If this is set to a frequency higher than the Nyquist Frequency, the Nyquist Frequency is used instead. Frequencies between the lower and higher frequency boundaries will not be modified by this SFM Optional: Type: Trace-varying: Multi-valued: Constraint: No number No No param() >0.0

ITERATIONS Number of Iterations for FMED method This parameter specifies the number of iterations used in the FMED processing. Values from 3 to 5 have been show to give good results. This parameter is ignored if 'Sparse Spike Decon Method' is LP. Optional: Type: Trace-varying: Multi-valued: Constraint: Default: No integer No No param() >0 3

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Spectral Broadening WINDOW_LENGTH Window Length for FMED Algorithm

6.0 Setup Parameters

This parameter specifies the window length to use in the FMED algorithm. If the parameter is left as COMPUTED a single window is used with a length of the entire trace. This may not be satisfactory for the data. If the Sparse Spike Decon Method is LP this parameter is ignored. Optional: Type: Trace-varying: Multi-valued: Constraint: Default: No number No No param() >0.0 COMPUTED

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