Data Assimilation: Mathematical Concepts and Instructive Examples

140 Pages·2016·1.696 MB·English

by  Rodolfo Guzzi (auth.)

SPRINGER BRIEFS IN EARTH SCIENCES
Rodolfo Guzzi
Data Assimilation:
Mathematical
Concepts and
Instructive
Examples
123
SpringerBriefs in Earth Sciences
More information about this series at http://www.springer.com/series/8897
Rodolfo Guzzi
Data Assimilation:
Mathematical Concepts
and Instructive Examples
123
RodolfoGuzzi
System Biology Group
University La Sapienza
Rome
Italy
ISSN 2191-5369 ISSN 2191-5377 (electronic)
SpringerBriefs inEarth Sciences
ISBN978-3-319-22409-1 ISBN978-3-319-22410-7 (eBook)
DOI 10.1007/978-3-319-22410-7
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Preface
Data Assimilation is a set of mathematical techniques allowing us to use all the
information available within a time frame. This includes observational data, any a
priori informationwe may haveanda deterministic orstochastic model describing
our system, and encapsulating our theoretical understanding. The mathematical
basis is the estimationtheory or theory ofthe inverse problem that isan organized
set of mathematical techniques for obtaining useful information about the physical
world on the basis of observation.
In a conventional problem one would use a set of known prior parameters to
predictthestate ofthephysical system.Thisapproach isusually calleda “forward
problem.”Inthe“inverseproblem”oneattemptstouseavailableobservationofthe
state of the system to estimate poorly known parameters of the state itself. In both
these cases Data Assimilation can be treated as a Bayesian system. The Bayesian
theoremorthelawofinverseprobabilityallowsustocombineaprioriinformation
about theparameterswith theinformation contained intoobservations toguide the
statistical inference process.
The reason why the Data Assimilation is so effective is that it seeks to produce
an analysis that fits a set of observations taken over a time frame (not just the
observations made at one instant in time). This is subject to the strong constraint
that the evolution of the analyzed quantities is governed by a deterministic model
describing the given observation.
Due to its flexibility Data Assimilation has been applied to several fields, from
thenumericalweatherprediction,whereitwasfirstdeveloped,toplanetaryclimate
analysis up to the evolution of biological cells.
One begins with a forecast model, often called background. In order to make
usefulpredictionsthebackgroundmustbeupdatedfrequentlywithnoisyandsparse
measurements. This procedure updates the background in light of the new obser-
vation to produce an analysis, which, under suitable assumptions is the maximum
estimate of the model state vector. Later, the model is restarted from analysis and
produces a new background forecast.
v
vi Preface
Data Assimilation and model forecasts can be combined into an observing
system simulation experiment in order to quantify the effect of changes in the
observation accuracy, type, location, and frequency on the accuracy of the
numerical forecast.
The method is prone to severe limitations, because eventually the forecasted
parameters diverge from the true values. The optimal procedures would combine
the most exhaustive theoretical and observational knowledge taking into account
the errors of observation and those due to the model.
Theaimofthisbookistogiveanin-depthandcompleteoverviewintheframe
oftheSpringerBriefshortbookconcept,toconstructDataAssimilationindifferent
and emerging fields from environment to biology, as outlined in the Applications
chapter.
The book serves both teachers and college students and other interested parties
providingthealgorithmsandformulastomanagetheDataAssimilationwherevera
dynamic system is present.
In writing this book I have followed the history of the data assimilation evo-
lution.Thefirstchaptergivesawideoverviewofthedataassimilationstepsstarting
from Gauss’ first methods to the most recent as those developed under the Monte
Carlomethods.Thesecondchaptertreatstherepresentationofthephysicalsystem
as an ontological basis of the problem. The third chapter deals with the classical
Kalman filter, while the fourth chapter deals with the advanced methods based on
recursive Bayesian Estimation. A special chapter, the fifth, deals with the possible
applications,fromthefirstLorenzmodel,passingthroughthebiologyandmedicine
uptoplanetaryassimilation,mainlyonMars.Theearthquakeapplication,wherethe
modelishighlynonlinearandtheerrorisnon-Gaussian,isalsoreportedinorderto
offer another point of view to the reader.
The book endeavors to give a concise contribution to understanding the data
assimilation and related methodologies. The mathematical concepts and related
algorithms are fully presented, especially for those facing this theme for the first
time.
Finally, I want to express my special thanks to Judith De Paul and George C.
KieffertowhomIamindebtedfortheirlinguistictipsandtheeffectivecorrections,
and Luca Montabone for his suggestions on the chapter of applications, mainly on
Mars data assimilation.
Contents
1 Introduction Through Historical Perspective . . . . . . . . . . . . . . . . . 1
1.1 From Gauss to Kolmogorov. . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Approaching the Meteorological System . . . . . . . . . . . . . . . . . . 6
1.3 Numerical Weather Prediction Models. . . . . . . . . . . . . . . . . . . . 10
1.4 What, Where, When. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Representation of the Physical System. . . . . . . . . . . . . . . . . . . . . . 19
2.1 The Observational System and Errors . . . . . . . . . . . . . . . . . . . . 19
2.1.1 The Estimation Problem . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.2 The Linear Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.3 Optimal Estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.4 Minimization Methods of Cost Functions . . . . . . . . . . . . 28
2.1.5 Some Properties of Estimation. . . . . . . . . . . . . . . . . . . . 30
2.1.6 Estimation of the quality of analysis. . . . . . . . . . . . . . . . 31
2.2 Variational Approach: 3-D VAR and 4-D VAR . . . . . . . . . . . . . 32
2.3 Assimilation as an Inverse Problem . . . . . . . . . . . . . . . . . . . . . 34
2.3.1 An Illustrative Example. . . . . . . . . . . . . . . . . . . . . . . . . 35
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Sequential Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 An Effective Introduction of a Kalman Filter. . . . . . . . . . . . . . . 39
3.1.1 Linear System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.2 Building up the Kalman Filter . . . . . . . . . . . . . . . . . . . . 42
3.2 More Kalman Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.1 The Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . . 46
3.2.2 Sigma Point Kalman Filter (SPKF). . . . . . . . . . . . . . . . . 49
3.2.3 Unscented Kalman Filter (UKF). . . . . . . . . . . . . . . . . . . 56
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
vii
viii Contents
4 Advanced Data Assimilation Methods . . . . . . . . . . . . . . . . . . . . . . 61
4.1 Recursive Bayesian Estimation. . . . . . . . . . . . . . . . . . . . . . . . . 61
4.1.1 The Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.2 The Forecast Step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1.3 The Analysis Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.1.4 Prediction by Stochastic Filtering. . . . . . . . . . . . . . . . . . 70
4.2 Ensemble Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.1 The Stochastic Ensemble Kalman Filter Menu. . . . . . . . . 74
4.2.2 The Deterministic Ensemble Kalman Filter . . . . . . . . . . . 76
4.2.3 The Analysis Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.4 The Deterministic Ensemble Kalman Filter Menu. . . . . . . 79
4.3 Issues Due to Small Ensembles . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3.1 Inbreeding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3.2 Filter Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3.3 Spurious Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.4 Methods to Reduce Problems of Undersampling. . . . . . . . . . . . . 83
4.4.1 Spatial Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.4.2 Covariance Inflation. . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4.3 Covariance Localization . . . . . . . . . . . . . . . . . . . . . . . . 84
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.1 Lorenz Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.1.1 Solution of Lorenz 63 Model. . . . . . . . . . . . . . . . . . . . . 92
5.1.2 Lorenz Model and Data Assimilation . . . . . . . . . . . . . . . 93
5.2 Biology and Medicine. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.2.1 Tumor Growth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.2.2 Growth Tumor Data Assimilation with LETKF . . . . . . . . 99
5.2.3 LETFK Receipt Computation. . . . . . . . . . . . . . . . . . . . . 104
5.3 Mars Data Assimilation: The General Circulation Model. . . . . . . 105
5.3.1 Mars Data Assimilation: Methods and Solutions. . . . . . . . 112
5.4 Earthquake Forecast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.4.1 Renewal Process as Forecast Model . . . . . . . . . . . . . . . . 115
5.4.2 Sequential Importance Sampling and Beyond. . . . . . . . . . 116
5.4.3 The Receipt of SIR. . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Chapter 1
Introduction Through Historical Perspective
Abstract The first approach to the data assimilation methods started with the
applicationsoftheNewtonianequationstothemotionoftheplanetsmadebyLaplace
andGauss.However,onlywiththeintroductionofaprobabilisticview,thedynamical
equationswereabletomakearealforecast.Thefirstnumericalweatherprediction
(NWP)models,basedondeterminism,soonbecameprobabilistic,mainlywiththe
contributeofthegroupleadedbyVonNeumannatPrinceton’sInstituteforAdvanced
Study.Thischaptercontainsthehistoryoftheevolutionofthedataassimilationmeth-
ods,fromtheearlydeterminismalgorithmstothemostrecentprobabilisticmethods
astheensembleforecasting.ThefundamentalworkbyEdwardLorenzandbythe
groupofLosAlamos,whichdevelopedtheMonteCarlomethod,arealsoreported.
1.1 FromGausstoKolmogorov
Forecastsfrommodelsprovideusefulinformationtohelppredictthefuturestateofa
system.However,inevitablypredictionswilldivergefromrealityastimeprogresses,
that is an inescapable property of dynamical systems. Data assimilation (DA) is a
formalapproachusedtohelpcorrectforecastsbyintroducinginformationobserved
fromtheenvironment.
ThefirstDynamicDataAssimilation,whichweintendtodescribeinthisbook,
whichisthatrequirestheexistenceofdataandequationsordynamicmodels,was
made to calculate the comet’s orbit, the moving bodies per excellence. This is a
classicalproblemoftwobodieswhereoneseekstodeterminethepositionandthe
componentsofthebodyataparticulartime,(PrincipiabookIIIpropXLI).
Thefirstmethodoffindingtheorbitofacometmovinginaparabolawastriedby
Newton[1],usingthreeobservations.He,however,wrote,“Thisbeingaproblemof
verygreatdifficulty,Itriedmanymethodsofresolvingit”.
WhentheobservationsaretakenfromthesurfaceofEarth,theapparentposition,
in space, of the comet is non-given and therefore the velocity components are not
determined. This fact implies that the observer has to make more observations, at
different times. During the interval of time, Earth and comet have moved, so the
©TheAuthor(s)2016 1
R.Guzzi,DataAssimilation:MathematicalConceptsandInstructiveExamples,
SpringerBriefsinEarthSciences,DOI10.1007/978-3-319-22410-7_1