Formulation of the . . .
Formulation of the . . .
Mathematical and
Computational Aspects of a
Joint Inversion Paper by
M. Moorkamp, A. G. Jones,
and S. Fishwick
Anibal Sosa and Vladik Kreinovich
Cyber-ShARE Center
University of Texas at El Paso
500 W. University
El Paso, TX 79968, USA
[email protected], [email protected]
Main Idea of the Paper
Pareto Optimality
Genetic Algorithm: . . .
Selecting a Single Model
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1.
Formulation of the Geophysical Problem
• Problem: we are interested in some quantity q.
Formulation of the . . .
Formulation of the . . .
Main Idea of the Paper
• Example: we are interested in how the density ρ depends on the depth d: ρ = ρ(d).
• Situation: we have several types t of measurement results t, e.g., they use seismic data, resistivity, etc.
Pareto Optimality
Genetic Algorithm: . . .
Selecting a Single Model
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• Measurement results: for each type of data t, we have
measurement results mt,i , i = 1, . . . , nt .
• Measurement accuracy: for each measurement, we have
estimates σt,i of the accuracy of this measurement.
• Problem: sometimes, we only have a general accuracy
estimate σt for all measurements of type t.
• Solution: in this case, we take σt,i ≈ σt .
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2.
Formulation of the Geophysical Problem (cont-d)
Formulation of the . . .
Formulation of the . . .
• Reminder:
Main Idea of the Paper
– we are interested in a quantity q;
– we have measurement results mt,i of different types t;
– we know (approximately) the accuracies σt,i of different measurements.
• Forward models Mt enables us, given q, to predict the
corresponding measured values
mi,t ≈ Mt (i, q).
Pareto Optimality
Genetic Algorithm: . . .
Selecting a Single Model
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• Least Squares formulation: find q that minimizes
X
t
def
Φt , where Φt =
nt
X
(mt,i − Mt (i, q))2
i=1
2
σt,i
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• Problem: the accuracies σt,i are only approximately
known.
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3.
Main Idea of the Paper
• Ideal case: if we knew the exact accuracies σt,i , we
could apply the Least Squares approach.
Formulation of the . . .
Formulation of the . . .
Main Idea of the Paper
Pareto Optimality
• In practice: we only know approximate values of σt,i .
• Reason: for some t, we systematically overestimate the
measurement errors; for other t, we underestimate.
• Whether we over- or under-estimate depends on t.
• Natural idea: assume that the actual accuracies are
act
σt,i
= kt · σt,i .
• Resulting solution: for all possible combinations of the
correction coefficients kt , find q that minimizes
nt
X
X 1
(mt,i − Mt (i, q))2
.
2 · Φt , where Φt =
2
k
σ
t
t,i
t
i=1
• Selection of an appropriate solution (“model”) q is made
by a geophysicist.
Genetic Algorithm: . . .
Selecting a Single Model
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Pareto Optimality
Formulation of the . . .
• Reminder: for all possible combinations of the correction coefficients kt , find q that minimizes
Formulation of the . . .
Main Idea of the Paper
Pareto Optimality
X 1
2 · Φt , where Φt =
k
t
t
nt
X
i=1
(mt,i − Mt (i, q))2
.
2
σt,i
• Known: this is ⇔ finding all Pareto optimal solutions
q, i.e., q which are not worse than any other q 0 :
q worse than q 0 ⇔ (Φt (q) ≤ Φt (q 0 ) for all t and
Φt (q) < Φt (q 0 ) for some t).
• How they find it: use genetic algorithm, with the mindef
imized function O(q) = #{q 0 : q 0 worse than q}.
Genetic Algorithm: . . .
Selecting a Single Model
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5.
Genetic Algorithm: Brief Description
• Each q is a sequence of values: e.g., ρ(di ) at different
depths di .
Formulation of the . . .
Formulation of the . . .
Main Idea of the Paper
Pareto Optimality
• We start with several randomly generated sequences.
Genetic Algorithm: . . .
• At each step, we repeatedly
Selecting a Single Model
– select two sequences s1 and s2 – the smaller O(q),
the larger probability of selection;
– select random splitting locations, so si = si1 si2 si3 . . .,
where si1 is before the 1st location, etc.;
– combine s1 and s2 into a new sequence s11 s22 s13 s24 . . .;
– mutate, i.e., randomly change some elements of the
new sequence.
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• These new sequences form a new generation, with which
we deal on the next step.
• We repeat this procedure many (N 1) times.
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Selecting a Single Model
Formulation of the . . .
• Reminder: we find all the solutions which are Paretooptimal with respect to Φ = (Φ1 , Φ2 , . . .).
• Interesting case: when we have two types of measurements.
• In this case: we find all the solutions which are Paretooptimal with respect to Φ = (Φ1 , Φ2 ).
• Empirical fact:
– if we plot the dependence of ln(Φ1 ) on ln(Φ2 ), then
– at the geophysically most meaningful solution, the
corresponding curve has the largest curvature.
• Name: the corresponding curve is called an L-curve,
since it has a sharp corner – like a letter L.
• Resulting idea: look for the solution at which the curvature is the largest.
Formulation of the . . .
Main Idea of the Paper
Pareto Optimality
Genetic Algorithm: . . .
Selecting a Single Model
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