Analysis of the impact of model nonlinearities in inverse problem - PDF document

Analysis of the impact of model nonlinearities in inverse problem solving Tomislava Vukicevic University of Colorado Collaboration Derek Posselt University of Michigan � Inverse problem formulation � Diagnostic numerical analysis – Impact of • model nonlinearity • modeling errors • observations • prior information Examples of parameter, initial condition and state estimation � Conclusions and Discussion 1

Inverse Problem Formulation after Mosegaard and Tarantola (2002) (International Handbook of Earthquake & Engineering Seismology, Academic Press) •Conjunction of information in a joint space of observations and parameters (includes state) •Information is represented by probability density functions Parameters are quantities which we wish to estimate, hereafter denoted [ ] m ∈ = 1 K M : m ( m ,...., m ), i 1 , I i i Observations are in space [ ] ∈ = 1 N D : y ( y ,...., y ), l 1 , L l l Mapping or forward model is y = * f ( m ) Conjunction of pdfs D × Joint space M p ( m , y ) p ( m , y ) ∫ × γ = 1 p ( m , y ) p ( m , y ) 1 2 = 1 2 p ( m , y ) , ν ( m , y ) γ ν D M ( m , y ) p ( m , y ) Joint pdf given information from model only 1 Joint pdf given observations and prior parameters (no p ( m , y ) 2 model involved) D × ν Joint homogenous pdf on space M ( m , y ) pdf of unit volume in the space; constant if space is linear Joint posterior pdf p ( m , y ) 2

Breakout of contributing pdfs Model only Marginal pdf for parameters is homogenous = ν p ( m , y ) p ( y / m ) ( m ) Information about existence only 1 1 Prior and Observations = Observations and prior assumed independent p ( m , y ) p ( m ) p ( y ) 2 2 2 because before particular observations are used there is no dependence Prior Observations Homogenous Independent for the same reason ν = ν ν ( m , y ) ( m ) ( y ) Inverse problem solution ∫ = Marginal of joint posterior pdf p ( m ) p ( m , y ) dy m D [ ] 1 ∫ = p ( m ) p ( m ) p ( y ) p ( y / m ) dy γ m 2 2 1 * D homogenous pdf folded into the constant 3

Familiar case ⎛ ⎞ 1 1 = − − − − ⎜ ⎟ T 1 p ( y ) exp ( y y ) C ( y y )) Observations 2 ⎝ y ⎠ 1 2 π ( 2 ) det C 2 y ⎛ ⎞ 1 1 = − − − − ⎜ T 1 ⎟ Model p ( y / m ) exp ( y f ( m )) C ( y f ( m )) 1 1 ⎝ s ⎠ 2 π ( 2 ) det C 2 s ⎛ ⎞ Convolving two 1 = − − − − = + ⎜ T 1 ⎟ C C C p ( m ) kp ( m ) exp ( f ( m ) y ) C ( f ( m ) y ) Gaussians , , m 2 ⎝ D ⎠ D y s 2 ⎛ ⎞ 1 1 = − − − − ⎜ ⎟ T 1 p ( m ) exp ( m m ) C ( m m )) Prior 2 1 ⎝ prior m prior ⎠ 2 π ( 2 ) det C 2 m ( ) = − p ( m ) const exp J ( m ) CONJUCTED m [ ] ) ( ) ( ) 1 ( ) ( = − − − + − T − − T 1 1 J ( m ) f ( m ) y C f ( m ) y m m C m m D prior m prior 2 Linear familiar case ≡ f ( m ) Hm Posterior marginal pdf is Gaussian with first and second moments, respectively ( ) − 1 = + + − T T m m C H HC H C ( y Hm ) prior m m D prior = − + − − T 1 1 1 C ( H C H C ) D m First moment or mean is also minimum of cost function [ ] ) ( ) ( ) 1 ( ) ( = − − − + − T − − T 1 1 J ( m ) f ( m ) y C f ( m ) y m m C m m D prior m prior 2 4

Nonlinear case � Explicit numerical evaluation of posterior joint and marginal pdfs for a nonlinear model given observations and priors � When this is possible could evaluate Impact of •model nonlinearity •model uncertainties •Gaussian prior •observation uncertainties Discretization [ ] 1 ∫ = p ( m ) p ( m ) p ( y ) p ( y / m ) dy γ m 2 2 1 * D Combination of function mapping on a discrete multidimensional grid in phase space and Monte Carlo sampling of known parametric pdfs Not a data assimilation algorithm ! 5

Example 1 Example 1 Damped harmonic oscillations χ χ 2 d d = Α − λ + η τ + Α − λ − η τ + α + ωχ = ( ) ( ) f ( m ) e e 0 τ τ 1 2 2 d d •Model error is assumed Gaussian at each solution y=f(m), for discrete set of m values within interval of permissible values; • Each m unit volume has the same probability (homogenous pdf for marginal in m) ⎛ ⎞ 1 1 = − − − − ⎜ ⎟ T 1 p ( y / m ) exp ( y f ( m )) C ( y f ( m )) s 1 ⎝ ⎠ 2 π ( 2 ) det C 2 s m is initial condition m is natural frequency y y m m Example 1, continued Observation: Gaussian with mean from true reference at an arbitrary time point ⎛ ⎞ 1 1 = − − − − ⎜ T 1 ⎟ p ( y ) exp ( y y ) C ( y y )) y 1 ⎝ ⎠ 2 π ( 2 ) det C 2 y pdf y Prior: uniform 6

Result of conjunction initial condition, linear model Natural frequency, exponential model p m ( m ) p m ( m ) p ( m , y ) p m ( m ) p ( m , y ) p m ( m ) Uniform prior Resulting posterior is Resulting posterior is Gaussian approximately Log-Normal Example 2 Parameterized dry convection model ( Lorenz, 1963 ) dX = − − a ( X Y ) τ d Only X component is dY observed 4 times = − − rX Y XZ τ d dZ = − XY bZ τ d 7

Inverse problems with Lorenz model � Coefficients � Initial conditions � State estimation pdfs at individual observation instances for coefficient a solution time 8

Cumulative influence of observations still inverse problem for coefficient a Posterior after first observation time; prior for the second sequential Posterior after second observation time Final marginal for the given observation set is asymmetrical, but unimodal pdfs at individual observation instances for initial condition X 9

Multidimensional pdfs � 3 coefficients or initial conditions � State estimation - 3 state components Experiments •Varying amplitude of model error •Gaussian prior update at each observation input 3 coefficients / negligible model error True pdf Final posterior pdf using all 4 observation times Intermediate times Well constrained problem with 4 observatios 10

Impact of model error and Gaussian prior estimation of coefficients Small model error Large model error Gaussian prior update Impact of model error and Gaussian prior initial condition problem Small model error Large model error Gaussian prior update 11

State estimation Full pdf or Gaussian prior update “true pdf” Gaussian time X is observed Example of joint pdf in state full Gaussian 12

Conclusions and discussion � Nonmonotonic forward model gives rise to the potential for a multimodal posterior pdf, the realization of which depends on the information content of the observations, and on observation and model uncertainties � The presence of model error greatly increases the possibility of capturing multiple modes in the posterior pdf � Cumulative effect of observations, over time, space or both, could render unimodal final posterior pdf even with the nonmonotonic forward model � A greater number of independent observations are needed to constrain the solution in the case of a nonmonotonic nonlinear model than for a monotonic model for same number of degrees of freedom in control parameter space � Gaussian prior update has a similar effect to an increase in model error, which indicates there is the potential for inaccurate estimate (relative to truth) even when observations and model are unbiased. 13