Stochastic simulation of a heterogeneous glacial structure using transition probabilities
and Markov models (TProGS).
Julian Kochab, Xin Hea, Jens C. Refsgaarda, Karsten H. Jensenb
a Geological
Survey of Denmark and Greenland (GEUS), Copenhagen, Denmark
b Department of Geography and Geology, University of Copenhagen, Copenhagen, Denmark
Main Objectives:
1. Translate geophysical observations (resistivity) into soft data (facies
probability).
2. Define a suitable Markov model that represents the measured transition
probabilities of a two facies (sand and clay) system accordingly.
Results:
Top layers:
Transition probabilities and the defined Markov model:
The simulated uncertainty of the top 10m is of special interest when the
predictive uncertainty of e.g. shallow particle tracking is addressed.
Markov parameters: Sand proportion: 23% and correlation length:
500m and 5m for the horizontal and vertical direction, respectively
3. Generate a set of realizations using TProGS.
Measured P(Sand)
Simulated P(Sand)
Out of the top 5
cells; How many
belong to class:
Class 1: 0 – 20%
Sand Probability
4. Validate the set of realizations
Study Site and Data:
The target area is a delineated glacial
structure in the Norsminde catchment,
eastern Jutland, Denmark.
112 boreholes are classified into sand and
clay and are assigned a trust score that
allows soft conditioning.
Methodology:
Airborne based geophysical survey
(SkyTEM) with 100.000 sounding points
at 2000 flight lines. The data is gridded by
a spatially constrained inversion algorithm
Class 2: 40 – 60%
Sand Probability
Probability maps:
Show the intra-variability of a set of realizations (25 in this study).
Overconditioning (spatially correlated data) causes the simulation of a
very deterministic image  Out thin the conditioning datasat
Conditioning
Data
SkyTEM
P(Sand)
Class 3: 80 – 100%
Sand Probability
Validation criteria:
Histogram approach:
The categorical data from the boreholes are
paired with the resistivity values .The
resistivity data is then grouped in bins
of 10 Ωm and a facies percentage gets
calculated for each bin. Curve fitting allows
a direct reading of sand probability.
Validation Criteria
SkyTEM 20 m
P(Sand)
SkyTEM 100 m
P(Sand)
SkyTEM 500 m
P(Sand)
 46Ωm cut off value (50% sand probability)
20m soft data
200m rotational soft data
1. Simulated sand proportion
(Deviation)
+2%
+6.3%
2. Simulated correlation length
(X / Y)
-21% / -20%
-37% / -37%
3. Simulated geobody
Connectivity (θ / Г)
-2.1% / -1.1%
-2.8% / -1.4%
4. Simulated uncertainty
distribution
TProGS:
Poor (approx. 70% cells with zero Satisfying (approx. 15% cells with
change)
zero change)
5. RMSE – sim. and meas.
p(Sand)
The transition probability tjk (h) is a measure of spatial variability:
𝑡𝑗𝑘 ℎ = 𝑃 𝑘 𝑜𝑐𝑐𝑢𝑟𝑠 𝑎𝑡 𝑥 + ℎ 𝑗 𝑜𝑐𝑐𝑢𝑟𝑠 𝑎𝑡 𝑥
k and j are categories, x a spatial location vector and h a separation vector (lag).
’Given that a facies j is present at location x, what is the probability that another
(or the same) facies occurs at location x+h.’
The Markov model represents the transition
probability for each possible transition at each
specified lag (h) in direction Φ:
 t1,1h  t1,K h 


T (h )   

 
t K ,1h  t K ,K h 
simulated uncertainty distribution:
simulated uncertainty allocation:
0.20
0.06
Conclusions:
• Out thinning of the conditioning dataset can work around the problem of
overconditioning
• 200m spaced soft data as conditioning yields the best simulation
performance
• Validation criteria are very fruitful
Julian Koch
[email protected]