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Kalman Filtering in Climate Models - Applied Mathematics · 2020-04-13 · Kalman Filtering in Climate Models Kalman Filtering Energy Balance Application Kalman Filter for Earth’s

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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 ak ∼ 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−1Pak−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 Kkthat satisfies

    xak = xfk + Kk(yk − Hkx fk )

    (yk − Hkx fk ) 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 + KkRkKTk

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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 P

    ak .

    Tr [Pak ] = Tr [Pfk ]− 2Tr [KkHkP fk ] + Tr [Kk(HkP fkHTk + Rk)KTk ]

    ∂Tr [Pak ]

    ∂Kk= 0 =⇒ Kk = P fkHTk (HkP fkHTk + Rk)−1

    Concise form for covariance is

    Pak = (I − KkHk)P fk

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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−1Pak−1A

    Tk−1 + Qk A Priori Covariance

    Kk = PfkH

    Tk (HkP

    fkH

    Tk + Rk)

    −1 Kalman Gain

    xak = xfk + Kk(yk − Hkx fk ) Posterior Estimate

    Pak = (I − KkHk)P fk 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−2855 )

    1 + e(T−285

    5 )

    )

    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− �σT 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

    � 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. Pa0 =

    (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.

    Climate and StatisticsIntroductionData Assimilation

    Kalman FilteringIntuition and ProofEnergy Balance ApplicationEnsemble Kalman Filter

    Conclusion