Post on 09-Jul-2020
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Kalman Filtering in Climate Models
Kalman Filtering in Climate Models
George Hu
Brown University
December 2019
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Kalman Filtering in Climate Models
Table of Contents
1 Climate and StatisticsIntroductionData Assimilation
2 Kalman FilteringIntuition and ProofEnergy Balance ApplicationEnsemble Kalman Filter
3 Conclusion
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Kalman Filtering in Climate Models
Climate and Statistics
Introduction
Climate Data and Statistics
Our knowledge about Earth’s climate relies on data at itscore.
”How is the Earth’s climate changing?” is inherently astatistical question.
What is an ”anomaly” when we have only 1 test subject(the Earth)?How can climate scientists account for less reliablemeasurements and proxies for past data?How do climate models properly incorporate uncertainty?
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Kalman Filtering in Climate Models
Climate and Statistics
Introduction
Menard and Errera, International Space Science Institute
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Kalman Filtering in Climate Models
Climate and Statistics
Data Assimilation
Data Assimilation
Main Focus
Data is integrated into mathematical processes consistent withthe physical sciences through Data Assimilation.
Applications:
CO2 Proxy Data (ice cores, sediment analysis, etc.)Temperature Data (sea surface, land, troposphere)Forecasting Atmospheric Dynamics
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Kalman Filtering in Climate Models
Climate and Statistics
Data Assimilation
Data Assimilation Abstraction
Discrete Spatio-Temporal Process:
State Variables {Xi : i = 1, 2, . . .}, Xi ∈ Rn
Observation Variables {Yi : i = 2, 3, . . .}, Yi ∈ Rm
Transition Model f : Rn → Rn, Xi+1 = f (Xi)
Observation Model g : Rn → Rm, Yi = g(Xi)
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Kalman Filtering in Climate Models
Climate and Statistics
Data Assimilation
Facets of Data Assimilation
Tasks in data assimilation include:
Finding some Xk given all observations Yi is the task ofreanalysis.
Finding some Xk given observations until the present, Yi ,i ≤ k , is the task of filtering.
Finding some Xk given past observations, Yi , i < k , is thetask of forecasting.
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Kalman Filtering in Climate Models
Climate and Statistics
Data Assimilation
Facets of Data Assimilation
Data Assimilation can also be divided into:
Variational methods, in which the problem is framed asan optimization problem
minx
f (x)
Sequential methods, in which the problem is framed asusing conditioning
E[f (x) | some stuff ]
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Kalman Filtering in Climate Models
Kalman Filtering
Kalman Filtering
Sequential Filtering
The Kalman Filter, proposed by Rudolf Kalman (1930-2016),is commonly used as a concise, easily implemented sequentialfiltering algorithm for linear systems with gaussian noise.
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Kalman Filtering in Climate Models
Kalman Filtering
Intuition and Proof
Forecast and Analysis
Each iteration of the Kalman Filter is composed of
1 A forecast step
Uses the transition model to find the a priori distributionat time k given the posterior distribution at time k − 1.
2 An analysis step
Uses the observation model to find the posteriordistribution at time k given the a priori distribution attime k .
Note that we assume the process is Bayesian.
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Kalman Filtering in Climate Models
Kalman Filtering
Intuition and Proof
Visualization
Laura Silvinski, Brown University
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Kalman Filtering in Climate Models
Kalman Filtering
Intuition and Proof
Notation
Our new notation uses gaussian random variables.
A Priori and Posterior Distributions at Time k
X fk ∼ N(x fk ,P
fk ) X a
k ∼ N(xak ,Pak )
xk is the mean estimate, and Pk is the covariance matrix.
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Kalman Filtering in Climate Models
Kalman Filtering
Intuition and Proof
Transition and Observation Models
Transition Model:
xk = Ak−1xk−1 + ξk
Ak−1 ∈ Rn×n is the linear transition matrix at time k − 1.
ξk ∈ Rn is a RV for gaussian noise, ξk ∼ N(0,Qk).
Observation Model:
yk = Hkxk + ζk
Hk ∈ Rm×n is the linear operator for the observation at timek .
ζk ∈ Rm is a RV for gaussian noise, ζk ∼ N(0,Rk).
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Kalman Filtering in Climate Models
Kalman Filtering
Intuition and Proof
Forecast Step
The forecast step follows directly from linear algebra.
The a priori estimate follows directly from the transitionmodel
x fk = Ak−1xak−1
Calculating the covariance from the expression for thetransition model gives
P fk = Ak−1P
ak−1A
Tk−1 + Qk
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Kalman Filtering in Climate Models
Kalman Filtering
Intuition and Proof
A Leap of FaithFor tha Analysis Step, let us assume that there is some matrix Kk
that satisfiesxak = x fk + Kk(yk − Hkx
fk )
(yk − Hkxfk ) is called the innovation.
Kk ∈ Rn×m is known as the Kalman gain matrix.
To find the posterior covariance, we can brute force using theexpectation
Pak = E[(xk − xak )(xk − xak )T ]
= (I − KkHk)P fk (I − KkHk)T + KkRkK
Tk
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Kalman Filtering in Climate Models
Kalman Filtering
Intuition and Proof
Deriving the Kalman Gain Matrix
The trace of the covariance matrix is the sum of mean squareerror, so we can find the optimal Kk by minimizing the trace of Pa
k .
Tr [Pak ] = Tr [P f
k ]− 2Tr [KkHkPfk ] + Tr [Kk(HkP
fkH
Tk + Rk)KT
k ]
∂Tr [Pak ]
∂Kk= 0 =⇒ Kk = P f
kHTk (HkP
fkH
Tk + Rk)−1
Concise form for covariance is
Pak = (I − KkHk)P f
k
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Kalman Filtering in Climate Models
Kalman Filtering
Intuition and Proof
Kalman Filter Algorithm
We finally arrive at our algorithm.
Kalman Filtering
Inputs: xa0 ,Pa0 , {Ai−1}, {Hi}, {Qi}, {Ri}, i ∈ {1, 2, . . .}
For k = 1, 2, . . .
x fk = Ak−1xpk−1 A Priori Estimate
P fk = Ak−1P
ak−1A
Tk−1 + Qk A Priori Covariance
Kk = P fkH
Tk (HkP
fkH
Tk + Rk)−1 Kalman Gain
xak = x fk + Kk(yk − Hkxfk ) Posterior Estimate
Pak = (I − KkHk)P f
k Posterior Covariance
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Kalman Filtering in Climate Models
Kalman Filtering
Energy Balance Application
Using the Kalman Filter
Example: Energy Balance
Let us simulate Kalman Filtering for Earth’s Energy Balance asa process for temperature, with observation variables in threedimensions to represent three independent observations.
The transition model will be a discretization of
∂T
∂t= Ein − Eout
The observation model will use the matrix
H =(1 1 1
)T
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Kalman Filtering in Climate Models
Kalman Filtering
Energy Balance Application
AlbedoAlbedo, denoted as α, is a coefficient for the measure of solarradiation reflected from Earth. For our purposes, we representalbedo as a function of temperature using a sigmoid function.
α(T ) = 0.7− 0.4
(e(T−285
5 )
1 + e(T−2855 )
)
α(T ) ≈ 0.3 when T > 305K , indicative of Earth’s surfacebeing mainly water.
α(T ) ≈ 0.7 when T < 265K , indicative of Earth’s surfacebeing mainly ice.
Note that this is a positive feedback loop.
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Kalman Filtering in Climate Models
Kalman Filtering
Energy Balance Application
Energy Budget ODE
∂T
∂t=
1
C
((1− α(T ))
S
4− εσT 4
)
C is the specific heat capacity of the Earth
S is the solar constant, the average spatial energy density ofsunlight received at Earth’s atmosphere
ε is the ”greenhouse factor,” the permittivity of theatmosphere
σ is the Stefan-Boltzmann constant
T is temperature in Kelvin
t is time in seconds
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Kalman Filtering in Climate Models
Kalman Filtering
Energy Balance Application
Kalman Filter for Earth’s Energy Budget
Parameters set up so that ∂T∂t
= 0 at an semi-stable pointat T = 264K .
Our demonstration uses xa0 = 265 with full confidence,i.e. Pa
0 =(0)
We discretize the ODE using Runge-Kutta-4 with timesteps on the order of 1013 seconds.
Transition covariance: Qi =(0.6)∀i
Observation covariance matrix: Ri = 1.5(I3), ∀i , where Iis the identity matrix.
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Kalman Filtering in Climate Models
Kalman Filtering
Energy Balance Application
Results
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Kalman Filtering in Climate Models
Kalman Filtering
Energy Balance Application
Results
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Kalman Filtering in Climate Models
Kalman Filtering
Ensemble Kalman Filter
More Complex Cases
What if the dynamical system is multi-dimensional and/orlacks a closed form solution to linearize?
Providing an accurate linear transition matrix becomesdifficult; ODE solver approach earlier only works in onedimension.
Can approximate linear transition using the Jacobian, butthat requires closed form solution.
Known as Extended Kalman Filter
Idea: Monte Carlo Approach
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Kalman Filtering in Climate Models
Kalman Filtering
Ensemble Kalman Filter
Ensemble Kalman Filter (EnKF)
For the forecast step:
1 Sample from previous posterior distribution.2 Update mean estimate by finding the sample mean of
the transitions.3 Update covariance by finding the sample covariance of
the transitions.
Analysis step remains the same.
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Kalman Filtering in Climate Models
Kalman Filtering
Ensemble Kalman Filter
Lorenz 96
Arbitrary dynamical system governed by
∂xi∂t
= (xi+1 + xi−2)xi−1 − xi + F
Can set to any dimension. Used as a standard example indynamical system analysis.
Indices in xi are interpreted as modular with thedimension for boundary conditions.
F is the forcing term.
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Kalman Filtering in Climate Models
Kalman Filtering
Ensemble Kalman Filter
Lorenz 96 EnKF Parameters
Process dimension and observation dimension of 40.
Initial value is xa0 =(0.1 0.1 0.1 . . .
)T. Initial
covariance is the zero matrix.
Transition covariance matrices are Qi = 0.6(I40) ∀i .Observation covariance matrices are Ri = 1.5(I40) ∀i .We set F to 8, known to have chaotic behavior.
Number of ensemble members is 100.
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Kalman Filtering in Climate Models
Kalman Filtering
Ensemble Kalman Filter
Results
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Kalman Filtering in Climate Models
Conclusion
Ending Remarks
Numerical Weather Prediction and Climate Modeling isactually one of the fastest advancing fields.
Scientists and other specialists in these areas literallyhave to predict the future.Data assimilation, first incorporated in the early 1990s,has been integral to geophysical modeling, by usingadvances in computer science and fluid dynamics forbetter modeling.