Aspects of sequential and simultaneous assimilation May 29, 2013 - - PowerPoint PPT Presentation
Aspects of sequential and simultaneous assimilation May 29, 2013 Motivation There exist several algorithms for conditioning a model to data, s.a. EnKF, RML, ES, EnRML, MDA,... All methods generate approximate samples from the same
◮ There exist several algorithms for conditioning a model to data, s.a.
◮ All methods generate approximate samples from the same
◮ Methods sample correctly for linear problems ◮ Methods give different results for non-linear problems
◮ Differences between methods are defined by some key characteristics ◮ Focus on: Sequential vs. simultaneous assimilation of data for
◮ Formal Bayesian expression
◮ Seq. data assimilation = sim. data assimilation
◮ Formal Bayesian expression
◮ Seq. data assimilation = sim. data assimilation
◮ Approximate methods
◮ Linear forward models: seq. data assimilation = sim. data
◮ Formal Bayesian expression
◮ Seq. data assimilation = sim. data assimilation
◮ Approximate methods
◮ Linear forward models: seq. data assimilation = sim. data
◮ Non-linear forward models: seq. data assimilation = sim. data
◮ Formal Bayesian expression
◮ Seq. data assimilation = sim. data assimilation
◮ Approximate methods
◮ Linear forward models: seq. data assimilation = sim. data
◮ Non-linear forward models: seq. data assimilation = sim. data
5 10 15 5 10 15 20 25 0.05 0.1 0.15 0.2
5 10 15 5 10 15 20 25 0.05 0.1 0.15 0.2
◮ Note: Analytical result exist for seq. vs sim. RML with linear data
◮ Note: Analytical result exist for seq. vs sim. RML with linear data ◮ Strategy:
◮ Note: Analytical result exist for seq. vs sim. RML with linear data ◮ Strategy:
◮ Define comparable variants of seq./sim. RML and EnKF/ES
◮ Note: Analytical result exist for seq. vs sim. RML with linear data ◮ Strategy:
◮ Define comparable variants of seq./sim. RML and EnKF/ES ◮ Analyze differences between the methods
◮ Note: Analytical result exist for seq. vs sim. RML with linear data ◮ Strategy:
◮ Define comparable variants of seq./sim. RML and EnKF/ES ◮ Analyze differences between the methods ◮ Extend linear RML result to new RML variants for combination of
◮ Note: Analytical result exist for seq. vs sim. RML with linear data ◮ Strategy:
◮ Define comparable variants of seq./sim. RML and EnKF/ES ◮ Analyze differences between the methods ◮ Extend linear RML result to new RML variants for combination of
◮ Extend linear RML result for variants of EnKF/ES
◮ Note: Analytical result exist for seq. vs sim. RML with linear data ◮ Strategy:
◮ Define comparable variants of seq./sim. RML and EnKF/ES ◮ Analyze differences between the methods ◮ Extend linear RML result to new RML variants for combination of
◮ Extend linear RML result for variants of EnKF/ES
◮ Define variants of EnKF and the RML method ◮ Remove impact of other characteristics than seq./sim. by ensuring
◮ Choose versions of RML and EnKF honoring 1-3
◮ EnKF honors
◮ Point 1: Updates based on ensemble ◮ Point 2: Perform one complete run
◮ EnKF honors
◮ Point 1: Updates based on ensemble ◮ Point 2: Perform one complete run
◮ EnKF does not honor
◮ Point 3: Focus on static parameters
◮ EnKF honors
◮ Point 1: Updates based on ensemble ◮ Point 2: Perform one complete run
◮ EnKF does not honor
◮ Point 3: Focus on static parameters
◮ Solution
◮ Restart from initial time after each assimilation
◮ EnKF honors
◮ Point 1: Updates based on ensemble ◮ Point 2: Perform one complete run
◮ EnKF does not honor
◮ Point 3: Focus on static parameters
◮ Solution
◮ Restart from initial time after each assimilation ◮ EnKF → Half-iterative EnKS (Hi-EnKS)
◮ EnKF honors
◮ Point 1: Updates based on ensemble ◮ Point 2: Perform one complete run
◮ EnKF does not honor
◮ Point 3: Focus on static parameters
◮ Solution
◮ Restart from initial time after each assimilation ◮ EnKF → Half-iterative EnKS (Hi-EnKS)
◮ If data are assimilated simultaneously: Hi-EnKS → ES
◮ EnKF honors
◮ Point 1: Updates based on ensemble ◮ Point 2: Perform one complete run
◮ EnKF does not honor
◮ Point 3: Focus on static parameters
◮ Solution
◮ Restart from initial time after each assimilation ◮ EnKF → Half-iterative EnKS (Hi-EnKS)
◮ If data are assimilated simultaneously: Hi-EnKS → ES ◮ Hi-EnKS: sequential scheme honoring 1-3 ◮ ES: simultaneous scheme honoring 1-3
◮ RML honors
◮ Point 3: Focus on static parameters
◮ RML honors
◮ Point 3: Focus on static parameters
◮ RML does not honor
◮ Point 1: Updates based on ensemble
◮ RML honors
◮ Point 3: Focus on static parameters
◮ RML does not honor
◮ Point 1: Updates based on ensemble
◮ Solution
◮ EnRML updates using an ensemble approximation to gradient: RML
◮ RML honors
◮ Point 3: Focus on static parameters
◮ RML does not honor
◮ Point 1: Updates based on ensemble ◮ Point 2: Perform one complete run
◮ Solution
◮ EnRML updates using an ensemble approximation to gradient: RML
◮ RML honors
◮ Point 3: Focus on static parameters
◮ RML does not honor
◮ Point 1: Updates based on ensemble ◮ Point 2: Perform one complete run
◮ Solution
◮ EnRML updates using an ensemble approximation to gradient: RML
◮ Minimize utilizing one full Gauss-Newton step: EnRML
◮ RML honors
◮ Point 3: Focus on static parameters
◮ RML does not honor
◮ Point 1: Updates based on ensemble ◮ Point 2: Perform one complete run
◮ Solution
◮ EnRML updates using an ensemble approximation to gradient: RML
◮ Minimize utilizing one full Gauss-Newton step: EnRML
◮ Sim. GN-EnRML: Simultaneous scheme honoring 1-3 ◮ Seq. GN-EnRML: Sequential scheme honoring 1-3
◮ Note: Analytical result exist for seq. vs sim. RML with linear data ◮ Strategy:
◮ Define comparable variants of seq./sim. RML and EnKF/ES ◮ Analyze differences between the methods ◮ Extend linear RML result to new RML variants for combination of
◮ Extend linear RML result for variants of EnKF/ES
◮ GN-EnRML update:
j = mf j + Cm ˜
j
j = mf j + ˜
j
◮ GN-EnRML update:
j = mf j + Cm ˜
j
j = mf j + ˜
j
◮ Cm ˜
◮ ˜
◮ Ensemble gradient given by the pseudo inverse
◮ Ensemble covariance
◮ Ensemble gradient given by the pseudo inverse
◮ Ensemble covariance
◮ Rewriting
◮ Inserting for ˜
p ∆dT
◮ Inserting for ˜
p ∆dT ◮ Ne ≤ Nm =
p = I =
1 Ne−1∆d∆dT = ˜
◮ Inserting for ˜
p ∆dT ◮ Ne ≤ Nm =
p = I =
1 Ne−1∆d∆dT = ˜
1 Ne−1∆m∆mT
◮ Inserting for ˜
p ∆dT ◮ Ne ≤ Nm =
p = I =
1 Ne−1∆d∆dT = ˜
1 Ne−1∆m∆mT
◮ Same result when comparing ES & sim. GN-EnRML
◮ For Ne > Nm the difference between GN-EnRML and Hi-EnKS
p ◮ ∆dt: True predicted data pertubation ◮ ∆dp: Predicted linearized data pertubation
p
◮ For Ne > Nm the difference between GN-EnRML and Hi-EnKS
p ◮ ∆dt: True predicted data pertubation ◮ ∆dp: Predicted linearized data pertubation
p ◮ Difference depends on the non-linearity of the data:
p
◮ ej: Truncation error
◮ Note: Analytical result exist for seq. vs sim. RML with linear data ◮ Strategy:
◮ Define comparable variants of seq./sim. RML and EnKF/ES ◮ Analyze differences between the methods ◮ Extend linear RML result to new RML variants for combination of
◮ Extend linear RML result for variants of EnKF/ES
◮ Utilizing seq./sim. GN-EnRML extend linear RML result for
◮ Choice of covariance update for seq. GN-EnRML
m = C f m − C f mG T
mG T + Cd
m
m = 1 Ne−1∆m∆mT
◮ Utilizing seq./sim. GN-EnRML extend linear RML result for
◮ Choice of covariance update for seq. GN-EnRML
m = C f m − C f mG T
mG T + Cd
m
m = 1 Ne−1∆m∆mT ◮ We choose 1 for seq. GN-EnRML
◮ Compare sim. assimilation with seq. assimilation of two data groups
◮ Compare sim. assimilation with seq. assimilation of two data groups
j
j
◮ Compare sim. assimilation with seq. assimilation of two data groups
j
j
j
j
◮ Compare sim. assimilation with seq. assimilation of two data groups
j
j
j
j ◮ GN-EnRML: seq. = sim. for non-linear → linear ◮ GN-EnRML: seq. = sim. for linear → non-linear
◮ Note: Analytical result exist for seq. vs sim. RML with linear data ◮ Strategy:
◮ Define comparable variants of seq./sim. RML and EnKF/ES ◮ Analyze differences between the methods ◮ Extend linear RML result to new RML variants for combination of
◮ Extend linear RML result for variants of EnKF/ES
◮ Remember:
1 Ne−1∆m∆mT
◮ Remember:
1 Ne−1∆m∆mT
◮ Seq. GN-EnRML updated covariance as:
m = C f m − C f mG T
mG T + Cd
m
◮ Remember:
1 Ne−1∆m∆mT
◮ Seq. GN-EnRML updated covariance as:
m = C f m − C f mG T
mG T + Cd
m ◮ Need: Ne → ∞ for C a m = ˜
◮ GN-EnRML = Hi-EnKS
◮ Remember:
1 Ne−1∆m∆mT
◮ Seq. GN-EnRML updated covariance as:
m = C f m − C f mG T
mG T + Cd
m ◮ Need: Ne → ∞ for C a m = ˜
◮ GN-EnRML = Hi-EnKS
◮ Difference between methods depend on non-linearity
◮ Perform numerical studies for Hi-EnKS with weakly non-linear data
◮ Difference between Hi-EnKS and GN-EnRML varies with ∆e ◮ Numerical study to investigate:
◮ Optimal assimilation strategy ◮ GN-EnRML result valid for Hi-EnKS
◮ Difference between Hi-EnKS and GN-EnRML varies with ∆e ◮ Numerical study to investigate:
◮ Optimal assimilation strategy ◮ GN-EnRML result valid for Hi-EnKS
◮ Numerical experiments
◮ Univariate: ◮ Simple forward model ◮ One linear data group ◮ One non-linear data group ◮ Multivariate ◮ 1D Reservoir ◮ One weakly non-linear data group ◮ One data group with stronger non-linearity
◮ Difference between Hi-EnKS and GN-EnRML varies with ∆e ◮ Numerical study to investigate:
◮ Optimal assimilation strategy ◮ GN-EnRML result valid for Hi-EnKS
◮ Numerical experiments
◮ Univariate: ◮ Simple forward model ◮ One linear data group ◮ One non-linear data group ◮ Multivariate ◮ 1D Reservoir ◮ One weakly non-linear data group ◮ One data group with stronger non-linearity
◮ Assess quality of Hi-EnKS/ES by Kullback-Leibler Divergence (KLD)
◮ Nearest Neighbor kernel density estimator
◮ Simple forward model
◮ Assimilate d1 → d2 and d2 → d1 with Hi-EnKS, and assimilate
d1/2
2 3 4 5 6 2 4 6 8 10
2 3 4 5 6 2 4 6 8 10
2 3 4 5 6 2 4 6 8 10
2 3 4 5 6 2 4 6 8 10
2 3 4 5 6 2 4 6 8 10
2 3 4 5 6 2 4 6 8 10
2 3 4 5 6 2 4 6 8 10
2 3 4 5 6 2 4 6 8 10
◮ 1D reservoir consisting of 31 unknown parameters ◮ Two data groups
◮ d1 : 6 measurements of log(perm) field. Measurements made at wells
◮ d2 : 6 pressure observations, made at a single well marked Pres obs
1 Inj Hard obs 2 · · · 6 7 Hard obs 8 · · · 12 13 Hard obs 14 15 16 Pres obs 17 18 19 Hard obs 20 · · · 24 25 Hard obs 26 · · · 30 31 Prod Hard obs
Hi-EnKS d1 → d2 Hi-EnKS d2 → d1 ES
◮ GN-EnRML: seq. = sim. for non-linear data before linear data ◮ For Ne ≤ Nm ∧ ˜
1 Ne−1∆m∆mT: GN-EnRML = Hi-EnKS ◮ For Ne > Nm: (GN-EnRML - Hi-EnKS) ∝ ∆e
◮ GN-EnRML: seq. = sim. for non-linear data before linear data ◮ For Ne ≤ Nm ∧ ˜
1 Ne−1∆m∆mT: GN-EnRML = Hi-EnKS ◮ For Ne > Nm: (GN-EnRML - Hi-EnKS) ∝ ∆e
◮ Hi-EnKS: seq. = sim. non-linear data before linear data ◮ Data with weakest non-linearity first:
◮ Best univariate and multivariate mean ◮ Best univariate and multivariate KLD