Data Processing Flows with SCons
Jim Jennings
Houston, Texas
July, 2010
Outline
•
•
•
Introduction to data processing flows with SCons
two simple examples
Random Fields with Madagascar
what is stochastic simulation?
how to make random field with FFTs
implementation in Madagascar
examples
Variograms with Madagascar
what is a variogram?
how to compute a variogram with FFTs
implementation in Madagascar
examples
2
Stochastic Simulation
• Unconditional stochastic simulation in geostatistics is
•
the process of generating a random field with a
specified variogram model.
Conditional stochastic simulation makes random
fields that have a specified variogram and have
specified values at given control points.
3
Stochastic Gaussian Simulation
v x   rxhwh
h
moving average method
V  RW
C  VV *
 RWR *W *
 RR *WW *
 WW *
 W2
symmetric weight function
4
Stochastic Simulation with FFTs
w x  F1
C
v x   rxhwh
moving average method
h
 
v x  F1 R C
FFT moving average
5
Implementation in Madagascar
• A recipe for stochastic simulation of unconditional
•
Gaussian random fields in 1, 2, & 3 dimensions is in
$RSFSRC/book/Recipes/rfield.py
An example SConstruct using the recipe is in
$RSFSRC/book/geostats/simulate/rfield
6
Deep-Water Channels with Background Noise
7
8
9
10
11
12
Variogram Array
13
Variogram Computation with FFTs
The trick is to think of an FFT not as an approximation
to the Fourier integral transform, but as a tool for exact
and efficient computation of the discrete product sum:


AB
1
a
b

F
F(a)F(b)
 x xh
x
 F1
… for all possible values of the discrete lag vector h.
14
Variogram Computation with FFTs
Then, expand the variogram definition into a collection
of product sums:


2
1
 (h) 
v x  v xh

2N(h) x
1
2
2
v

v
  v x v xh

x
xh
2 x
x

N(h)


15
16
Variogram Computation with FFTs
Then, expand the variogram definition into a collection
of product sums:


2
1
 (h) 
v x  v xh

2N(h) x
1
2
2
v

v
  v x v xh

x
xh
2 x
x

N(h)
1
2
2 
v x i xh   i x v xh    v x v xh


 x
2 x
x

 i x i xh


x
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Variogram Computation with FFTs
… that can be computed efficiently with FFTs:

 (h) 

 V 2 I*  IV 2*

F1 
 VV * 
2



F1 II*
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Variogram Computation with FFTs
… that can be computed efficiently with FFTs:

 (h) 

 V 2 I*  IV 2*

F1 
 VV * 
2



F1 II*
Marcotte, D., 1996, Fast variogram computation with FFT, Computers &
Geosciences, v 22, n 10, pp. 1175–1186.
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Implementation in Madagascar
• A recipe for computation of variograms and other
•
spatial statistics from array data is in
$RSFSRC/book/Recipes/spatial_stats.py
An example SConstruct using the recipe is in
$RSFSRC/book/geostats/spatial_stats/variogram
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