Formulation of the . . . Formulation of the . . . Mathematical and Main Idea of the Paper Pareto Optimality Computational Aspects of a Genetic Algorithm: . . . Selecting a Single Model Joint Inversion Paper by M. Moorkamp, A. G. Jones, Title Page ◭◭ ◮◮ and S. Fishwick ◭ ◮ Anibal Sosa and Vladik Kreinovich Page 1 of 7 Go Back Cyber-ShARE Center University of Texas at El Paso Full Screen 500 W. University El Paso, TX 79968, USA Close usosaaguirre@miners.utep.edu, vladik@utep.edu Quit

1. Formulation of the Geophysical Problem Formulation of the . . . Formulation of the . . . • Problem: we are interested in some quantity q . Main Idea of the Paper • Example: we are interested in how the density ρ de- Pareto Optimality pends on the depth d : ρ = ρ ( d ). Genetic Algorithm: . . . • Situation: we have several types t of measurement re- Selecting a Single Model sults t , e.g., they use seismic data, resistivity, etc. Title Page • Measurement results: for each type of data t , we have ◭◭ ◮◮ measurement results m t,i , i = 1 , . . . , n t . ◭ ◮ • Measurement accuracy: for each measurement, we have estimates σ t,i of the accuracy of this measurement. Page 2 of 7 • Problem: sometimes, we only have a general accuracy Go Back estimate σ t for all measurements of type t . Full Screen • Solution: in this case, we take σ t,i ≈ σ t . Close Quit

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 ; Pareto Optimality – we have measurement results m t,i of different types t ; Genetic Algorithm: . . . – we know (approximately) the accuracies σ t,i of dif- Selecting a Single Model ferent measurements. Title Page • Forward models M t enables us, given q , to predict the corresponding measured values ◭◭ ◮◮ ◭ ◮ m i,t ≈ M t ( i, q ) . Page 3 of 7 • Least Squares formulation: find q that minimizes Go Back n t ( m t,i − M t ( i, q )) 2 def � � Φ t , where Φ t = . Full Screen σ 2 t,i t i =1 Close • Problem: the accuracies σ t,i are only approximately Quit known.

3. Main Idea of the Paper Formulation of the . . . Formulation of the . . . • Ideal case: if we knew the exact accuracies σ t,i , we Main Idea of the Paper could apply the Least Squares approach. Pareto Optimality • In practice: we only know approximate values of σ t,i . Genetic Algorithm: . . . • Reason: for some t , we systematically overestimate the Selecting a Single Model measurement errors; for other t , we underestimate. Title Page • Whether we over- or under-estimate depends on t . ◭◭ ◮◮ • Natural idea: assume that the actual accuracies are σ act t,i = k t · σ t,i . ◭ ◮ • Resulting solution: for all possible combinations of the Page 4 of 7 correction coefficients k t , find q that minimizes Go Back n t ( m t,i − M t ( i, q )) 2 1 � � Full Screen · Φ t , where Φ t = . k 2 σ 2 t t,i t i =1 Close • Selection of an appropriate solution (“model”) q is made Quit by a geophysicist.

4. Pareto Optimality Formulation of the . . . Formulation of the . . . • Reminder: for all possible combinations of the correc- Main Idea of the Paper tion coefficients k t , find q that minimizes Pareto Optimality n t ( m t,i − M t ( i, q )) 2 1 Genetic Algorithm: . . . � � · Φ t , where Φ t = . k 2 σ 2 Selecting a Single Model t t,i t i =1 • Known: this is ⇔ finding all Pareto optimal solutions Title Page q , i.e., q which are not worse than any other q ′ : ◭◭ ◮◮ q worse than q ′ ⇔ (Φ t ( q ) ≤ Φ t ( q ′ ) for all t and ◭ ◮ Φ t ( q ) < Φ t ( q ′ ) for some t ) . Page 5 of 7 • How they find it: use genetic algorithm, with the min- Go Back = # { q ′ : q ′ worse than q } . def imized function O ( q ) Full Screen Close Quit

5. Genetic Algorithm: Brief Description Formulation of the . . . Formulation of the . . . • Each q is a sequence of values: e.g., ρ ( d i ) at different Main Idea of the Paper depths d i . Pareto Optimality • We start with several randomly generated sequences. Genetic Algorithm: . . . • At each step, we repeatedly Selecting a Single Model – select two sequences s 1 and s 2 – the smaller O ( q ), Title Page the larger probability of selection; ◭◭ ◮◮ – select random splitting locations, so s i = s i 1 s i 2 s i 3 . . . , where s i 1 is before the 1st location, etc.; ◭ ◮ – combine s 1 and s 2 into a new sequence s 11 s 22 s 13 s 24 . . . ; Page 6 of 7 – mutate, i.e., randomly change some elements of the Go Back new sequence. Full Screen • These new sequences form a new generation, with which Close we deal on the next step. Quit • We repeat this procedure many ( N ≫ 1) times.

6. Selecting a Single Model Formulation of the . . . • Reminder: we find all the solutions which are Pareto- Formulation of the . . . Main Idea of the Paper optimal with respect to Φ = (Φ 1 , Φ 2 , . . . ). Pareto Optimality Genetic Algorithm: . . . • Interesting case: when we have two types of measure- Selecting a Single Model ments. • In this case: we find all the solutions which are Pareto- optimal with respect to Φ = (Φ 1 , Φ 2 ). Title Page ◭◭ ◮◮ • Empirical fact: ◭ ◮ – if we plot the dependence of ln(Φ 1 ) on ln(Φ 2 ), then Page 7 of 7 – at the geophysically most meaningful solution, the corresponding curve has the largest curvature. Go Back • Name: the corresponding curve is called an L-curve , Full Screen since it has a sharp corner – like a letter L. Close • Resulting idea: look for the solution at which the cur- Quit vature is the largest.

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