Measure-Theoretic parameter estimation and prediction for contaminant transport and coastal ocean modeling

Transcription

1 Measure-Theoretic parameter estimation and prediction for contaminant transport and coastal ocean modeling Steven Mattis and Lindley Graham RMSWUQ 2015 July 17, 2015

2 Collaborators 2/45 The University of Texas at Austin Clint Dawson Lindley Graham Jiachuan He The University of Colorado Denver Troy Butler Scott Walsh Colorado State University Don Estep Los Alamos National Laboratory Monty Vesselinov Daniel O Malley Support from DoE DiaMonD Center

3 Table of Contents 3/45 1 Groundwater Contamination Problem Algorithm Preliminary Results 6

4 Table of Contents 4/45 1 Groundwater Contamination Problem Algorithm Preliminary Results 6

5 Motivation: LANL Chromium Site 5/45 54,000 kg of Cr6+ released in Sandia Canyon between 1956 and 1972 (uncertainties). Cr6+ detected above MCL (50 ppb; NM standard) in 4 monitoring wells in the regional aquifer beneath LANL. Cr6+ plume size is about 2 km 2 (region above MCL). Cr6+ plume is located near LANL site boundary and water-supply wells are located nearby. Contaminant mass distribution in the subsurface is unknown. Contaminant source location and mass flux at the top of the regional aquifer are unknown due to complex 3D pathways through the vadose zone. Limited remedial options due to aquifer depth ( 300 m below the ground surface) and complexities in the subsurface flow. Goal: Quantify existing uncertainties/unknowns.

6 Generating a numerical model A Simplified Transport Model A simple advection-diffusion-reaction system for homogenous porous media: U t + v U (D U) + λu = I /φ, where U is the contaminant concentration, v is the flow velocity, D is the dispersion tensor, λ is the reactive decay constant, I is a source, and φ is the porosity. Initial simplifying assumptions 6/45 Semi-infinitive aquifer thickness Box-shaped contaminant source Homogeneous contaminant flux within source Constant porosity and flow angle/velocity Constant dispersion tensor

7 Uncertain Parameters and Data Model Parameters Porosity φ, [ ] Pore velocity, [4 60 m/y] Flow angle θ, [ ] Reaction constant λ, [ /y] D x, [ m] D y, [1 14 m] D z, [ m] Source Parameters Source location/size Source flux, [ kg/y] Time source is active, [t 0 t F ], t 0 [0 40 y] Quantities of Interest Concentration in well at a given time. Mean concentration in a region. 7/45

8 Numerical Model 8/45 Model Evaluation For fast exploration of sample space and data space an approximate analytical solution to the tranport problem is evaluated. Utilize open source LANL code MADS: Model Analyses and Decision Support. Real World Concerns Gathering data is difficult and expensive. Space of parameters is generally higher-dimensional than space of independent quantities of interest. Little or no prior information is known.

9 Table of Contents 9/45 1 Groundwater Contamination Problem Algorithm Preliminary Results 6

10 Groundwater Contamination Problem Inverse problem: Set-valued solutions The simplest inverse problem. Given a value of Q, find all values of λ Λ with Q(λ) = Q. Q(λ) Λ The solution is generally a set of values. 10/45

11 Groundwater Contamination Problem Contour manifolds Theorem There exist smooth lower dimensional contour manifolds in Λ representing set-valued solutions. D D Λ L Let L be the set of these manifolds, defining an equivalence class for Λ. 11/45 Λ

12 Groundwater Contamination Problem Inverse Sensitivity Inverse stochastic sensitivity analysis A probability measure PD is imposed on (D, BD ). The bijection between L and D induces a σ algebra BL and a volume measure µl on L. Theorem A probability measure PD on (D, BD ) corresponds to a unique probability measure PL on (L, BL ). L 12/45 Λ

13 Structure of measures on Λ 13/45 A probability measure on Λ can be deconstructed into that transverse to and along the generalized countrous. Recall that the probability P L is induced by the model after the specification of P D. Theorem Specifying P l on generalized contours corresponding to l L determines a unique probability measure on (Λ, B Λ ). We assume that the probability measure is uniform with respect to the volume measure along the generalized countours.

14 Table of Contents 14/45 1 Groundwater Contamination Problem Algorithm Preliminary Results 6

15 Algorithmic outline 15/45 Start with: parameter space Λ, map Q : Λ D, and probability measure P D on the data space. Step 1: Define (implicit) Voronoi tessellation {V j } N j=1 of Λ used to approximate measurable sets in both Λ and L. Step 2: Define approximation to P D (or its density) on (D, B D ) by computation of P D (I i ) for tessellation {I i } M i=1 of D. Step 3: Use forward model solves and Monte Carlo integration to determine P Λ (V j ). Step 4: Compute P Λ (A) for events of interest A B Λ using P Λ (V j ).

16 Determining useful QoI 16/45 We want as many QoI as possible that give us new information. Multiple QoI are geometrically distinct (GD) at a given parameter if their set-valued inverses through this parameter are not parallel. QoI that are GD reduce the dimension of the generalized contours defined by the set-valued inverses. Each QoI corresponds to an expensive and time consuming experimental observation so we want to avoid wasting experiments on QoI that do not add new information.

17 Geometrically Indistinct QoIs 17/ Concentration in two wells wr.t initial source time and source flux.

18 QoI Not Sensitive to Data At All 18/ Concentration in two wells w.r.t dispersivity and pore velocity.

19 Geometrically Distinct! 19/ Concentrations in two wells have different sensitivities w.r.t. dispersivity and pore velocity.

20 Parameters Groundwater Contamination Problem Unknown Parameters Minimum Maximum Source x-coordinate x [m] Source y-coordinate y [m] Contaminant source flux f [kg/y] Porosity n [-] Flow angle θ [degrees] Pore velocity u [m/y] 6 60 Dispersivity x α x [m] Time Scale Dispersivity [-] /45 Known Parameters Value Source size x x s [m] 250 Source size y y s [m] 250 Source location z z 0 [m] 0 Source size z z 1 [m] 1 Reactive decay λ [1/y] 0.01 Dispersivity y α y [m] α x /10 Dispersivity z α z [m] α x /100 Initial source time t 0 [y] 0 End source time t 1 [y] 20

21 21/45 Data Groundwater Contamination Problem y [m] w7 w w10 w8 w1 w3 w w9 w2 w x [m] Well name x [m] y [m] Conc. [mg/kg] w w w w w w w E-05 w n/a w n/a w n/a Let Q map from space of unknown parameters to concentrations in wells with measured concentrations. Let P D be uniform in a hyperbox around the measured concentrations.

22 Marginal Probabilities: Choice of QoI 22/ Source x coordinate [m] Source y coordinate [m] Source x coordinate [m] ρ ρ ρ Source x coordinate [m] Source y coordinate [m] Source x coordinate [m] (q 1(λ), q 3(λ)) (q 1(λ), q 7(λ)) Source y coordinate [m] Source y coordinate [m] ρ Contaminant flux [kg/y] Contaminant flux [kg/y] ρ Contaminant flux [kg/y] Contaminant flux [kg/y] ρ

23 High-Probability Regions 23/45 Most of the probability is concentrated in very small regions of the parameter space. P(A Λ) Number of samples in A µ(a Λ)/µ(Λ)

24 Table of Contents 24/45 1 Groundwater Contamination Problem Algorithm Preliminary Results 6

25 Remediation Strategies 25/45 1 Strategy 1: Set t 1 to 1y. 2 Strategy 2: Set t 1 to 5y. 3 Strategy 3: Increase λ to 0.1 1/y. 4 Strategy 4: Increase λ to 1.0 1/y. 5 Strategy 5: Increase λ to /y. Maximum Concentration Level is 25 mg/kg.

26 Effect of Remediation on Probable Parameter Set No Remediation t = 5y w4 w1 w3 w5 w w7 w4 w10 w8 w1 w3 w5 w9 w2 t = 10y t = 20y 25.0 w7 w4 w10 w8 w1 w3 w5 w9 w w7 w10 w8 w9 w6 w6 w6 50 Strategy 1 w4 w1 w3 w5 w w7 w7 w7 w4 w4 w10 w10 w10 w8 w1 w8 w1 w8 w3 w3 w5 w5 w9 w2 w9 w2 w9 w6 w6 w Strategy 3 Strategy 2 w4 w1 w3 w5 w2 w4 w1 w3 w5 w2 w7 w4 w10 w8 w1 w3 w5 w9 w2 w6 w7 w w10 w8 w1 w3 w9 w2 w w w7 w4 w10 w8 w1 w3 w5 w9 w2 w6 w7 w4 w8 w5 w9 w w10 w8 w9 w6 w7 w10 w10 w1 w8 w3 w5 w2 w9 w Concentration [mg/kg] 26/45 Strategy 5 Strategy 4 w1 w3 w2 w7 w7 w7 w4 w4 w4 w10 w8 w1 w3 w5 w9 w2 w w10 w8 w1 w5 w9 w2 w w w5 w9 w6 w6 w6 w6 w10 w8 w7 w7 w7 w4 w4 w4 w10 w10 w10 w1 w8 w1 w8 w1 w8 w3 w3 w3 w5 w5 w5 w2 w9 w2 w9 w2 w

27 PDFs of Concentration in well w8 at t=10y 27/45 Strategy 1 Strategy R1-P R1-U None R2-P R2-U None PDF [-] PDF [-] Concentration [mg/kg] Concentration [mg/kg]

28 PDFs of Concentration in well w8 at t=10y Strategy 3 Strategy R3-P R3-U None R4-P R4-U None PDF [-] PDF [-] Concentration [mg/kg] Concentration [mg/kg] Strategy 5 R5-P R5-U None 10-2 PDF [-] / Concentration [mg/kg]

29 Probability of concentration being over MCL (failure) 29/45 Well time [y] Remediation Strategy None R1 R2 R3 R4 R5 w w w w w w

30 30/45 Table of Contents Groundwater Contamination Problem Algorithm Preliminary Results 1 Groundwater Contamination Problem Algorithm Preliminary Results 6

31 31/45 Groundwater Contamination Problem Estimating Probability, P Λ (A) Algorithm Preliminary Results Sources of Error approximation of events error in numerical computation of forward model

32 32/45 Groundwater Contamination Problem Algorithm Preliminary Results Motivations: Reduce error in approximation of events Approximate rare events All of the above in high-dimensions

33 33/45 Groundwater Contamination Problem Algorithm Preliminary Results Goal Adaptively generate samples within regions of interest in the data space. Specify regions of interest {D k } D and choose a probability on D p k (q) = ρ D (q) 1 Dk (q) ρ D dµ D D k

34 34/45 Groundwater Contamination Problem Algorithm Preliminary Results Choose a set of initial samples { λ (0,j)} M (LHS, i.i.d., regular grid). j=1 Set p 0,k = 1 Dk (Q(λ (0,j) )) D k.

35 34/45 Groundwater Contamination Problem Algorithm Preliminary Results Generate next M parameter samples { λ (1,j)} M j=1 using P t(λ (0,j), B(s 0 )). Set p 1,k = 1 Dk (Q(λ (1,j) )) D k.

36 34/45 Groundwater Contamination Problem Algorithm Preliminary Results Generate next M parameter samples { λ (i,j)} M j=1 using P t(λ (i 1,j), B(s j )). Alter step size used to determine B(s) = R(s) Λ if p i,j > (1 + γ)p i 1,j then s j = s decrease s j else if p i,j < γp i 1,j then s j = s increase s j elses j = s j end if Determine remaining samples { λ (i,j) } N i=2.

37 Groundwater Contamination Problem Algorithm Preliminary Results (i,j) λ 35/45 Q(λ(i,j) ) D

38 36/45 Preliminary Results Groundwater Contamination Problem Algorithm Preliminary Results Regular Grid Adaptive Total Forward Solves Forward Solves in D k Fraction in D k λ λ λ The approximation of ρ Λ for Q = (q 1, q 6) where λ reference = (λ 1, λ 2) = (0.0781, 0.168). Left: The approximation of ρ Λ using a regular grid of samples. Right: The approximation of ρ Λ using the adaptive sampling Algorithm. λ1

39 36/45 Convergence Groundwater Contamination Problem Algorithm Preliminary Results Definition Given a Voronoi tessellation of Λ denoted by {V j } N j=1 Λ we say that A N is the Voronoi coverage of A B Λ and µ k (A) is its volume defined by A N := λ (j) A,1 j N V(λ (j) ), and µ k (A) := µ Λ (A k ) = N µ Λ (V(λ (j) ))1 λ (j) A (1) j=1 Definition A rule for defining any N samples { λ (j)} N j=1 Λ is called B Λ-consistent if µ Λ (A A k ) 0 as N, A B Λ, s.t. µ Λ ( A) = 0. (2)

40 37/45 Groundwater Contamination Problem Convergence, Q = (q 1, q 2 ) Algorithm Preliminary Results

41 38/45 Groundwater Contamination Problem Convergence, Q = (q 1, q 6 ) Algorithm Preliminary Results

42 39/45 Groundwater Contamination Problem Convergence, Q = (q 1, q 2 ) Algorithm Preliminary Results x = µ Λ(Q 1 (E)) µ Λ (Λ) e = µ Λ(Q 1 (E)) µ Λ (Q 1 (E)) µ Λ (Q 1 (E)) Table : Statistics of adaptive volume estimates for the idealized inlet for (q 1, q 2) for 20 sets of samples. Sample type Relative volume Relative error x σ x ē σ e 9680 uniform 1.126e e e e uniform with emulation 1.125e e e e Adaptive 1.124e e e e uniform 1.122e e e e-03

43 40/45 Groundwater Contamination Problem Convergence, Q = (q 1, q 6 ) Algorithm Preliminary Results Table : Statistics of adaptive volume estimates for the idealized inlet for (q 1, q 6) for 20 sets of samples. Sample type Relative volume Relative error x σ x ē σ e 9680 uniform 2.034e e e e uniform with emulation 2.048e e e e Adaptive 2.055e e e e uniform 2.050e e e e-03

44 Table of Contents 41/45 1 Groundwater Contamination Problem Algorithm Preliminary Results 6

45 Improving Sampling 42/45 Possible Tools: Surrogate Models Gradient Information A posteriori error estimates

46 Adding complexity 43/45 More complicated models Analytical models provide a great deal of data cheaply but require major simplifications. Want to allow complex permeability fields, flow fields, geometries, and sources. A FEM is used to approximate the solution and the Quantity of Interest (QoI) for more general cases. Using adjoint based a posteriori error analysis we can estimate the numerical error in the computed QoI. We employ numerical and exact solutions as available in the UQ methodology.

47 Reliable error estimates for transport model - example Test Problem: Manufactured Solution Solution is highly oscillatory. Effectivity Ratio = Error Estimate True Error Error estimates can be used to form better QoI estimates. 44/45 QoI True Error Error Estimate Effectivity Ratio U(0.85, 0.5; 0.1) U(0.35, 0.855; 0.1) U(0.347, 0.235; 0.1) U(0.86, 0.785; 0.3) U(0.86, 0.785; 1.0)

48 45/45 Improving Inverse Solutions Goal-Oriented Adaptive Sampling Automated Calculation of Optimal QoIs A priori Error Estimates A posteriori Error Estimates (balancing sampling error and numerical error in forward solution) More complicated models (FEM) to handle more realistic physics. BET Software An open source software package for measure-theoretic inverses Demo after lunch.