Ensemble Kalman Filtering for Glacier Modeling

2025-04-22 0 0 4.41MB 21 页 10玖币

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Emily Corcoran , Logan Knudsen, Talea Mayo, Hannah

Park-Kaufmann, Alexander Robel

Abstract

Working with a two-stage ice sheet model, we explore how statistical data assim-

ilation methods can be used to improve predictions of glacier melt and relatedly,

sea level rise. We ﬁnd that the EnKF improves model runs initialized using incor-

rect initial conditions or parameters, providing us with better models of future

glacier melt. We explore the necessary number of observations needed to produce

an accurate model run. Further, we determine that the deviations from the truth

in output that stem from having few data points in the pre-satellite era can be

corrected with modern observation data. Finally, using data derived from our

improved model we calculate sea level rise and model storm surges to understand

the aﬀect caused by sea level rise.

Keywords: Data Assimilation, Glacier Modeling, Kalman Filter, Dynamical Systems

1 Introduction

Research has shown that climate change will likely impact storm surge and make storm

surges more severe ([1]). Storm surge occurs when high winds and low pressure from

a tropical storm force ocean water into coastal regions. Inundation caused by storm

surge has the potential to cause a great deal of damage in coastal regions when a storm

makes landfall. Models have been developed in order to help predict storm surge during

storm events, and researchers have begun to include future sea-level rise as a factor in

these models. When studying the impact of recent tropical storms, [1] found that for

all 14 storms simulated in the Gulf of Mexico and Atlantic Ocean, storm inundation

increased by an average of 25%, likely due to the impacts of climate change. This

illustrates the importance of modeling sea level rise and therefore improving glacier

models by incorporating data assimilation. Doing so increases our understanding of

how glacier melt will contribute to sea level rise, and in turn aﬀect storm surge.

Sea level rise caused by climate change plays a signiﬁcant part in this impact. To

better model sea-level rise, we turn to glacier modeling, speciﬁcally marine-terminating

glaciers. These have a natural ﬂow towards the ocean, which contributes to sea level

rise ([2]). By the year 2300, the Antarctic ice sheet is projected to cause up to 3 meters

arXiv:2210.02647v3 [math.DS] 20 May 2023

of sea level rise globally ([3]). Due to the severe impacts of glacial melting, modeling

changes in ice sheets is an important task. There are challenges to modeling sea level

rise, as ice sheet instability leads to signiﬁcant sea-level rise uncertainty ([2]).

Data assimilation is a method to move models closer to reality using real-world

observations by readjusting the model state at speciﬁed times ([4]). Data assimilation

was initially developed for use in weather models in the late 1990s in order to improve

weather forecast accuracy on short-range weather models ([5]). These models would

typically have a run time of 1-6 hours, and during that process real-time observations

would be pulled for this purpose in order to readjust the current model run. Since

its introduction, data assimilation has been incorporated into other geoscience models

and radars, to name a few uses.

Data assimilation can also play a factor in decision making when it comes to

collecting ﬁeld data. In the ﬁeld of glaciology, data is largely collected via satellite or

in-person ﬁeld data collection. While both methods collect valuable data for modeling

glaciers, both are expensive and time-consuming. Using data assimilation can help to

inform the glacier modelers and glaciologists who collect data about how to collect

data in an eﬃcient way. This can help researchers to more eﬃciently utilize funding

and avoid unnecessarily expensive data collection that does not signiﬁcantly improve

glacier models.

2 Methods

2.1 Glacier Model

We have chosen a simpliﬁed two-stage ice sheet model for our explo-

ration (Figure 1). This model describes the changes in ice mass of marine-

terminating glaciers, which may be impacted over time by climate change ([6]).

Fig. 1 Diagram of a marine-terminating glacier from [6].

A glacier can be repre-

sented with a simpliﬁed box

model that has a length L, pre-

cipitation P, and height and

ﬂux at the grounding line, hg

and Qg, respectively. The two-

stage model used in this paper

incorporates a nested box into

the system to more accurately

represent a glacier. This new

box has a thickness, H, and an

interior ﬂux, Q. The change in

length and height of the glacier

can be described with these diﬀerential equations:

dt =P−Qg

L−H

hgL(Q−Qg) (1)

dt =1

(Q−Qg).(2)

The following equations are used to calculate the variables used in this system:

hg=−λb(L) (3)

Q=γH2n+1

Ln(4)

Qg= Ωhβ

g,(5)

where γand Ω are assumed to be constants for our purposes, and λ=ρw

ρi

, where ρw

is seawater density and ρiis glacial ice density.

Parameters and Initial Conditions

Parameter Value

smbo0.3

smb10.15

smbf0.0

Ho2.18

Lo4.44

bx-0.001

sillmin 415

sillmax 425

sillslope 0.01

Fig. 2

Despite the simplicity of this model, it

oﬀers a suﬃcient approximation of glacier

melt using Qand Qg. As a result, ﬁnd-

ings we learn from this simpliﬁed model are

worth evaluating in a more complex model.

Further, for most of this study we will

assume the initial conditions and parame-

ters are what we see in Figure 2. We let

smbo,smb1, and smbfdescribe the surface

mass balance (P in Equation 1) at three

points over time. Hoand Loare initial con-

ditions for height and length at year 0.

The parameter bxis the slope of the Earth

underneath the glacier. Finally, sillmin and

sillmax describe the start point of the glacial

sill (region of reverse slope) with sillslope

representing the slope of the sill. For a more in-depth explanation and justiﬁcation of

the model, see [6].

The program used to model the glacier behavior and assimilate the data requires

choosing a set of initial conditions (Figure 3). Once the initial conditions are input

to the model, a Runge-Kutta 4th order method is used to advance the model in time,

and the model output can be used along with a data assimilation method. Then the

analyzed data from the assimilation is fed back into the model, where time is advanced

again.

Model

Initial Conditions

Initialize DA Function

Analysis

Forecast at time t

Fig. 3 Code Structure Diagram

Next, we give an overview of the data assimilation techniques used in this work.

2.2 Kalman Filter

The Kalman Filter is a data assimilation technique that uses the model, observations,

and corresponding error covariance matrices in order to adjust model output to be

closer to reality. The goal of the Kalman Filter is to compute an optimal estimate from

a combination of the results of a previous forecast and observations. The key to this

is the Kalman Gain, Kt, which decides the balance of how much this analysis relies

on the model and the observations.

2.2.1 Ensemble Kalman Filter

The Ensemble Kalman Filter (EnKF) is a nonlinear version of the Kalman ﬁlter ﬁt

for large problems. The model state is represented by an ensemble of states, and the

covariance matrix is replaced by the sample covariance. An analysis is performed for

each member of the ensemble.

Consider the following state estimation system:

xf,(i)

t=Mx(i)

t−1+w(i)

t(6)

Pf=1

N−1

i=0

[x(i)

t−xa

i][x(i)

t−xa

i]0(7)

Let Mrepresent our model and xa,(0)

t,xa,(1)

t,...,xa,(N)

t,be the ensemble members

at time t. We initialize this ensemble by choosing values normally distributed around

an estimated value. Here, xf,(i)

tis the state system forecasted from the prior probability

distribution (PD) at time t, and Pfis the error covariance of that state.

Let ytbe an observation at time t; the noise on our observations is assumed to be

multivariate and normally distributed around 0. We reﬂect this using the measurement

covariance matrix, Rt, i.e. w(i)

t∼ Nn(0,Rt); let the model noise be multivariate

normally distributed around 0 using the model system covariance matrix, Qt, or v(i)

t∼

Nmt(0,Qt); Htis the observation operator and Ct=˜

Stwhere ˜

Stis the sample

covariance of the current ensemble. The three equations that describe the analysis

step are as follows:

xa,(i)

t=xf,(i)

t+Ka

t(yt−H(i)

txf,(i)

t) (8)

f=1

N−1

i=0

[xf,(i)

t−xa

i][xf,(i)

t−xa

i]0(9)

t=CtH0

t(HtCtH0

t+Rt)−1(10)

Here (3) is the state from data posterior PD, where (4) is the covariance matrix for

the new prediction, and (5) is the Kalman gain. The notation for the sub-/superscripts

are: f= forecast, a= analysis, t= time, and (i) indexes the ensembles.

It is worth noting that often the convention Ct=Qtis used, and that is used for

this research. Further, we calculate the covariance matrix of the analysis with each

assimilation, P=1

N−1PN

i=1[x(i)

t−xa

i][x(i)

t−xa

i]0.Algorithm 1 outlines the algorithm

we use for a model in the time range of t={0,1, . . . , T }.

Can data assimilation also provide more accurate and timely forecasts of glacier

Algorithm 1 Ensemble Kalman Filter

Generate xa,(0)

0,xa,(1)

0,...,xa,(N)

for t= 0,1, . . . , T do

if t in Tobs then

Calculate Ct

Calculate Rt

Calculate Ka

t=CtH0

t(HtCtH0

t+Rt)−1

for i= 0,1, . . . , N do

xf,(i)

t=Mx(i)

t−1

y(i)

t=yt+v(i)

xa,(i)

t=xf,(i)

t+Ka

t(y(i)

t−Htxf,(i)

end for

Calculate xa

t=1

NPN

i=0 xa,(i)

Calculate Pa

f=1

N−1PN

i=0[xf,(i)

t−xa

i][xf,(i)

t−xa

i]0

else

for i= 0,1, . . . , N do

x(i)

t=Mx(i)

t−1

end for

Calculate xa

t=1

NPN

i=0 xa,(i)

Calculate Pf=1

N−1PN

i=0[x(i)

t−xa

i][x(i)

t−xa

i]0

end if

end for

melt? What is the computational cost, and can our experiments with data assimilation

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EnsembleKalmanFilteringforGlacierModelingEmilyCorcoran,LoganKnudsen,TaleaMayo,HannahPark-Kaufmann,AlexanderRobelAbstractWorkingwithatwo-stageicesheetmodel,weexplorehowstatisticaldataassim-ilationmethodscanbeusedtoimprovepredictionsofglaciermeltandrelatedly,sealevelrise.WendthattheEnKFimprovesmodelr...

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Ensemble Kalman Filtering for Glacier Modeling

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