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Green’s function retrieval by iterative substitution or inversion (
Green’s function retrieval by iterative substitution or inversion (?) of the Marchenko equation Joost van der Neut (Delft University of Technology)
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Acknowledgements Satyan Singh, Jyoti Behura, Roel Snieder
Filippo Broggini, Dirk-Jan van Manen Kees Wapenaar, Evert Slob Jan Thorbecke Ivan Vasconcelos Bowen Guo, Jerry Schuster Andrey Bakulin
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Introduction
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Input for Marchenko Redatuming
^ ^ ^ ^ ^ 1 EPSI output Data Multiple-free Wavelet-free Ghost-free Data Source Function Surface Reflectivity Data (Van Groenestijn & Verschuur, 2009; Lin and Herrmann, 2013)
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Input for Marchenko Redatuming
^ ^ ^ ^ ^ 1 EPSI output Data Multiple-free Wavelet-free Ghost-free Data Source Function Surface Reflectivity Data (Van Groenestijn & Verschuur, 2009; Lin and Herrmann, 2013) 2 Background Green’s function Focal point
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The aim of Marchenko redatuming
X gdown Marchenko redatuming g0 gup
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Redatuming below a complex overburden
= * gup Xdatum gdown Retrieve e.g. by inversion
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Examples
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Example 1 T0 R Conventional image Marchenko image
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Example 2
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Example 2 – Conventional image
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Example 2 – Marchenko image
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Example 3 - Conventional
Model Target area Image
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(with adaptive subtraction)
Example 3 – Marchenko (with adaptive subtraction) Model Target area Image
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Example 4 - Conventional
Velocity Density Image Red = without multiples Yellow with multiples Yellow Red 7000 m/s 4000 kg / m3
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Example 4 - Marchenko Velocity Density Image Red = without multiples
Yellow Marchenko result Yellow Red 7000 m/s 4000 kg / m3
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Example 4 with erroneous (constant) velocity - Conventional
Density Image Red = without multiples Yellow with multiples Yellow Red 7000 m/s 4000 kg / m3
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Example 4 with erroneous (constant) velocity – Marchenko
Density Image Red = without multiples Yellow Marchenko result Yellow Red 7000 m/s 4000 kg / m3
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Example 5 - Sigsbee Marchenko Conventional Behura et al. (2014)
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Theory
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The focusing function (Wapenaar et al., 2014)
Focal point Response Earth Focusing function
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The focusing function (Wapenaar et al., 2014)
Heterogeneous Focal point Response Earth Focusing function
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The focusing function (Wapenaar et al., 2014)
Heterogeneous Focal point Response Earth Focusing function
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Data acting on the focusing function
Heterogeneous Focal point Heterogeneous Time-reversed Focusing function & Green’s function Reflection response Focusing function
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Representation in matrix-vector notation
Time-reversal Focusing function Green’s function Convolution with data Focusing function
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Representation in matrix-vector notation
Time-reversal Focusing function Green’s function Convolution with data Focusing function
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Representation in matrix-vector notation
Time-reversal Focusing function Green’s function Convolution with data Focusing function Unknown Unknown
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Exploiting causality Time-reversed Focusing function Green’s function
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1 Exploiting causality Window function Time-reversed Focusing function
Green’s function
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Exploiting causality 1 Window function Time reversal Focusing function
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Exploiting causality 1 Window function Green’s function
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1 1 Exploiting causality Window function Green’s function Background
Window function Green’s function Background Green’s function
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Green’s function representation
Apply Intial Green’s function (model) Window Time Reversal Convolution with data Focusing function
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Green’s function representation
Apply Intial Green’s function (model) Window Time Reversal Convolution with data Focusing function
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Retrieved Green’s functions - 1D results from spgl1
Velocity Density Iterative solution (50 iterations) spgl1 inversion (100 iterations) 7000 m/s 4000 kg / m3
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Applications for Least-Squares Migration?
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Least-Squares Migration
Data = perturbation
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Born approximation Data = perturbation
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Using Marchenko-based Green’s functions?
Data = perturbation Marchenko mapping:
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Discussion
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Revisiting the problem
1. EPSI (inversion): 2. Focusing function Retrieval (inversion): 3. Green’s function Retrieval (forward): 4. Least-Squares Migration (inversion): δ δ
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