seasonal climate prediction using linear weighted multi model system w. t. yun apcn

Multi Model Ensemble Prediction

Seasonal climate prediction using linear weighted

multi model system

W. T. Yun

APCN/

Korea Meteorological Administration


Contents

Introduction

What is Multi Model Ensemble?

Construction of Multi Model Ensemble System

- Gauss-Jordan Elimination

- Singular Value Decomposition (SVD)

- Synthetic multi model ensemble

- Generating of Synthetic Dataset

Multi Model Ensemble Seasonal Forecast

Skill of Multi Model Forecast

Application


Regional climate change and climate variability have various impacts on the socio-economic activities. The impacts increase as the socio-economic activities become complex and active.

One of important and challenging task in areas of meteorology is climate seasonal prediction.

The advance climate seasonal prediction of droughts, monsoon etc. is now scientifically feasible.

This can be enormously beneficial in national planning, e.g. in areas of water resources management, disaster management, and agricultural planning and food production.

Introduction


What is multi model ensemble?

Multi Model Ensemble

An Ensemble comprising different models

weighted Multi Model Ensemble

Weighted Combination of Multi Models


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Biased Ensemble Mean

Bias Corrected Ensemble Mean

weighted Combination of Multi Models

Multi-Model Ensemble Anomaly Forecast?


ECMWF

GFDL

MPI

AMIP Model Forecasts (Dec. 1988) Superensemble Forecast

Obs

Sup

Sup-Obs

Why Multi-Model Ensemble Forecast?


The climate system can be regarded as a dynamic nonlinear systemPrediction

Linear statistical methods

Nonlinear statistical methods, Artificial neural network methods

Construction of Multi-Model Ensemble


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Input layerxk

Hidden layerhj

Output layeryi

x1

x2

x3

xk

Prediction

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Error

A feed-forward neural network with one hidden layer, where the jth neuron in this hidden layer is assigned the value hj.

A linear combination of the neurons in the layer just before the output layer.

The cost function is minimized by means of gradient descent.

Whole vector of weight are updated according to the back propagation learning rule.

The learning can be more efficient by including a momentum term, which refers to previous updating.

Local minimum in the network can be avoided by introducing noise to the gradient descent updating rule, which in the case considered here is following Manhattan updating rule.

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Neural Network Model with Back-Propagation


RMSE of Global Precipitation for 12Months (Jan.-Dec. 1988) ANN Forecasts (using AMIP data)

1.1

1.15

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1 2 3 4 5 6 7 8 9 10 11 12

R M

S

Bias Corrected

Climatology

ANN

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Skill Score of Non-Linear Multi-Model Ensemble Forecast


Construction of weighted linear Multi-

Model Ensemble Prediction System


Observed Analysis

Training Phase Forecast Phase

MME Forecast

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t=0

Multi-Model Ensemble Prediction System


t=0

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B

C

D

E

t=t-1…

MME with Pointwise Regression

(1)

t=0

A

B

C

D

E

t=t-1…(2) MME with Pattern

Regression

t=0

A

B

C

D

E

t=t-1…(3) MME with Spatio-

Temporal Regression

Weighted Multi-Model Ensemble Techniques


Superensemble Based on

Gauss-Jordan Elimination


Superensemble Forecast :

Where, Fi is the ith model forecast , is the mean of the ithforecast over the training period, is the observed mean over the training period, are regression coefficients obtained by a minimization procedure during the training period, and N is the number of forecast models involved.

For obtaining the weights, the covariance matrix is built with the seasonal cycle-removed anomaly ( )

,

where, t and i, j denote time and ith - ,jth – forecast model, respectively. After calculation of the covariance matrix C, we can construct the weighting component for each grid point of each model.

T.N. Krishnamurti et al., 1999, Science

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Construction of Superensemble


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Superensemble Based on SVD


Where, Fi is the ith model forecast, is the mean of the ith forecast over the training period, is the observed mean over the training period, ai are regression coefficients obtained by a minimization procedure during the training period, and N is the number of forecast models involved. For obtaining the weights, the covariance matrix is built with the seasonal cycle-removed anomaly (F’).

Where, t and i, j denote time and ith- ,jth– forecast model, respectively. After construction of the covariance matrix C, weights are computed for each grid point of each model.

Best Linear Unbiased Estimation (BLUE)

This will be the solution-vector of smallest length |x|2 in the least-square sense. x which minimizes r ≡||C·x - b||. SVD realizes a completely orthogonal decomposition for any matrix.

W.T.Yun, et al., 2003, J. Climate

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MME System based on SVD


This will be the solution-vector of smallest length in the least-square sense.

x which minimizes

In the case of an underdetermined system, m<n, fewer equations than unknowns, SVD produces a solution whose values are smallest in the least-square sense.

In the case of an overdetermined system, m>n, more equations than unknowns, SVD produces a solution that is the best approximation in the least-square

sense.

The SVD technique removes the singularity problem.

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SVD realizes a completely orthogonal decomposition for any matrix A


Error Covariance Matrices (AMIP) (Precipitation: Gauss-Jordan and SVD)

Total Variance

UnexplainedVariance (%)

ExplainedVariance


%)100or (

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variancetotal

varianceexplained2 r Relative unexpl. Variance = 1 - r2

Variables

Total

Variance

r2 (%)Gauss-Jordan

SVDw(1) w(1-2) w(1-3) w(1-4) w(1-5) w(1-6)

Precipitation 1.2496 85.0723 92.2724 90.4899 89.1822 87.8679 86.5430 85.2433

T850 2.3328 90.3815 97.4425 95.8743 94.4600 93.2157 91.9906 90.5030

u200 19.3385 87.1593 93.6309 92.3347 91.1363 89.8606 88.6386 87.3436

u850 4.2623 90.1600 96.2329 95.0812 93.7804 92.7039 91.5405 90.3010

v200 10.7958 92.5053 98.0942 96.9052 95.9238 94.8459 93.7451 92.6036

v850 2.3297 92.7304 98.6048 97.2439 96.1230 95.0179 93.9193 92.8188

Relative explained variance r2 (%) of regression models using Gauss-Jordan elimination and SVD with zeroing the small singular values. All values are averaged.

Explained Variance of Regression Models


Training ForecastConventional Superensemble SVD

SVD Mean RMSE

Conventional Superensemble

Simple Ensemble

RMSE of MME based on SVD

(Global, Precipitation)


The condition number of a matrix is defined as the ratio of the largest (in magnitude) of the wj’s to the smallest of the wj’s. A matrix is singular if its condition number is infinite, and it is ill-conditioned if its condition number is too large.

The solution vector x obtained by zeroing the small wj’s and then using the equation (1) is better than SVD solution where the small wj’s are left nonzero. It may seem paradoxical that this can be so, since zeroing a singular value corresponds to throwing away one linear combination of the set of equations that we are trying to solve. The resolution of the paradox is that we are throwing away precisely a combination of equations that is corrupted by roundoff error. If we let the small wj’s nonzero, it usually makes the residual larger. We don’t know exactly what threshold to zero the small wj’s is acceptable.

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Singular Values n

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Singular Values & Variance in the estimate of xj

(Precipitation, for one grid point)


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w (1), (1-2), (1-3), (1-4), (1-5), (1-6) G Clim w (1), (1-2), (1-3), (1-4), (1-5), (1-6) G Clim

w (1), (1-2), (1-3), (1-4), (1-5), (1-6) G Clim w (1), (1-2), (1-3), (1-4), (1-5), (1-6) G Clim

Global mean precipitation RMSE Global mean T850 RMSE

Global mean v200 RMSEGlobal mean u200 RMSE

Global Mean RMSE

(with Zeroing the Small Singular Values)


Cancellation of bias among different models

Not directly influenced by the model’s systematic errors

Maximization of explained variance

Removes singularity in matrix

Best Linear Unbiased Estimator (BLUE)

Zeroing the small singular values wj

High Prediction Skill of Multi-Model Ensemble


Synthetic Multi Model Ensemble


The MME prediction skill during the forecast phase could be degraded if the training

was executed with either a poorer analysis or poorer forecasts.

This means that the prediction skills are improved when higher quality training data sets

are deployed for the evaluation of the multi model bias statistics.



E(2) – Minimization

Actual Data SetSynthetic Ensemble Prediction

Synthetic Data Set

Superensemble Prediction

Schematic chart for the synthetic superensemble prediction system. The synthetic data are generated from the FSU coupled multi-model outputs by minimizing the residual error variance E(2).

W.T. Yun, 2004, Tellus accepted



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2 - Minimization


Actual Data Set (N) Synthetic Data Set (N) Prediction

E(2) -Minimization

Schematic chart of the multi model synthetic MME prediction. The synthetic data set is generated from the actual data set.

N - Actual Data Set

N - Synthetic Data Set

Estimating Consistent PatternWhat is matching spatial

pattern in forecast data, Fi(x,T), which evolves according to PC time series O(t) of observation

data, O(x,t)?

Observed Analysis

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Generating Synthetic Data Set


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Observed Analysis

Training Phase Forecast Phase

SyntheticMME Forecast

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The synthetic data set generated is separated into training and forecast phases. During training phase, optimal weights are computed which are used for producing synthetic MME forecast.

Synthetic MME Prediction


Atmospheric Global Spectral Model (T63L14)+Hamburg Ocean Model HOPE

Starting from 31 December 1986 (to Dec. 2002), every 15 days three months forecasts were made with the four different versions of the coupled model. The multimodels are constructed using two cumulus parameterization schemes (modified Kuo’s scheme following Krishnamurti and Bedi, 1988; and Arakawa-Schubert type scheme following Grell, 1993) and two radiation parameterization schemes (an emissivity-abosrbtivity based radiative transfer algorithm following Chang 1979 and a band model for radiative transfer following Lacis and Hansen 1974) in the atmospheric model only. KOR – Kuo type convection with Chang radiation computationsKNR – Kuo type convection with Lacis and Hansen radiation computation AOR – Arakawa Schubert type convection with Chang radiation computationsANR – Arakawa Schubert type convection with Lacis and Hansen radiation computation

FSU Unified Model Data Set


DEMETER (Development of a European Multi-Model Ensemble System for Seasonal to Inter-Annual Prediction) system comprises 7 global coupled ocean-atmosphere models.

CERFACS (European Centre for Research and Advanced Training in Scientific Computation, France), ECMWF (European Centre for Medium-Range Weather Forecasts, International Organization), INGV (Istituto Nazionale de Geofisica e Vulcanologia, Italy), LODYC (Laboratoire d’Océanographie Dynamique et de Climatologie, France), Météo-France (Centre National de Recherches Météorologiques, Météo-France, France), Met Office (The Met Office, UK), MPI (Max-Planck Institut für Meteorologie, Germany)

The DEMETER hindcasts have been started from 1st February, 1st May, 1st August, and 1st November initial conditions. Each hindcast has been integrated for 6 months and comprises an ensemble of 9 members.

The multi-model synthetic ensemble/superensemble is formed by merging the 15 yr (1987-2001) ensemble hindcasts of the seven models, thus comprising 7x9 ensemble members.

DEMETER Model Data Set


Quality of Data Set

Actual data setSynthetic data set


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ACC & RMS of the DEMETER Multi Model & Synthetic Data Set (Average over 2-4 months Global Precipitation Forecast, JJA)(ECMWF, UKMO, Meteo France, MPI, LODYC, INGV, CERFACS)

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RMS of Synthetic Data Set

ACC of Actual Data Set

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ACC of Synthetic Data Set

87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 Mean

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87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 Mean


ACC & RMS for FSU Unified Model Data Set & Synthetic Data Set (Average over 1-3 months Global Surface Temperature Forecast, JJA; ANR, AOR, KNR, KOR)

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RMS of Synthetic Data Set

ACC of Actual Data Set

ACC of Synthetic Data Set

87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 Mean

87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 Mean

87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 Mean

A : Arakawa Schubert cumulus parameterization K : FSU- modified Kuo cumulus parameterization algorithm. NR : Band model radiation code (New radiation scheme) OR : Emissivity absorbtivity radiation code (old radiation scheme)

87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 Mean


The Global Distribution of Weights for the DEMETER Multi Model & Synthetic Data Set (Average over 2-4 months Global JJA 2001 v-Wind at 850hPa Forecast)

(ECMWF, UKMO, Meteo France, MPI, LODYC, INGV, CERFACS)

Weights of Actual Data Set Weights of Synthetic Data Set

ECMWF

INGV

CERFACS

LODYC

Meteo France MPI

UKMO ECMWF

INGV

CERFACS

LODYC

Meteo France MPI

UKMO


The Synthetic Seasonal Forecasts


FSU Unified Model Synthetic Ensemble/Superensemble Prediction (Precipitation, 30S-30N, JJA 2001)

Obs.

EM

SEM

SSF


DEMETER Multi Model Synthetic Ensemble/Superensemble Prediction (Precipitation, 5N-40N 150W-50W, JJA 2001)

Obs.

SSF

EM

SEM


DEMETER Multi Model Synthetic Ensemble/Superensemble Prediction (Surface Temperature, 5N-40N 150W-50W, JJA 2001)

Obs.

SSF

EM

SEM


DEMETER Multi Model Synthetic Ensemble/Superensemble Prediction (Wind Speed at 850hPa, India 10SN-35N 50E-110E, JJA 2001)

Obs.

SSF

EM

SEM


The Skill Score of Synthetic Forecasts


The Skill Metrics of Forecasts in a Deterministic Sense

The AC is a measure of how well the phase of the forecast anomalies corresponds to the observed anomalies. The overbar denotes mean, and the summation can be either in space or in time, depending on whether spatial or temporal anomaly correlation is computed and G is the number of either grid points or time points.

The RMSE is a measure of the average magnitude of the forecast error.

Despite the fact AC is a good measure of phase error and doesn’t take bias into account, it is possible for a forecast with large errors to still have a good correlation coefficients. So, it is necessary to evaluate the average magnitude of the forecast errors.

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Skill Score Metrics


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SEM SF SSF

JJA-TR

1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

DJF-TRMember ModelEM

SEM SF SSF

The summer (JJA) and Winter (DJF) precipitation anomaly correlation skill scores for tropical domain (30S-30N). The bars in diagram indicate skill scores of the 4 FSU member models, bias corrected ensemble mean (EM), synthetic ensemble mean (SEM), superensemble (SF), and synthetic superensemble (SSF) from left to right.

Cross Validated ACC for FSU unified Model & synthetic MME


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Cross-validated RMS & ACC for FSU Unified Model & Synthetic Superensemble (30-30N JJA, Average over 1-3 months Precipitation Forecast, ANR, AOR, KNR, KOR)

A : Arakawa Schubert cumulus parameterization K : FSU- modified Kuo cumulus parameterization algorithm. NR : Band model radiation code (New radiation scheme) OR : Emissivity absorbtivity radiation code (old radiation scheme)

ANR, AOR, KNR, KOR FSU EM SEM SSF

1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 Mean

1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 Mean


Cross-validated RMS & ACC of the DEMETER Multi Model & Synthetic Superensemble (30°S-30°NJJA, Average over 2-4 months Surface Temperature Forecast)

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1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 Mean

1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 Mean

ECMWF, UKMO, Meteo France, MPI, LODYC, INGV, CERFACS DEMETER EM SEM SSF


Cross-validated RMS & ACC of the DEMETER Multi Model & Synthetic Superensemble (30°S-30°N JJA, Average over 2-4 months Precipitation Forecast)

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1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 Mean

1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 Mean

ECMWF, UKMO, Meteo France, MPI, LODYC, INGV, CERFACS DEMETER EM SEM SSF


MAM JJA SON DJF

PODy

EM 0.55 0.55 0.55 0.56

SEM 0.58 0.58 0.58 0.58

SF 0.55 0.55 0.55 0.55

SSF 0.58 0.58 0.58 0.60

PODn

EM 0.60 0.61 0.60 0.61

SEM 0.61 0.60 0.61 0.61

SF 0.60 0.60 0.60 0.61

SSF 0.61 0.59 0.60 0.61

ETS

EM 0.11 0.11 0.10 0.11

SEM 0.13 0.13 0.13 0.12

SF 0.10 0.11 0.10 0.10

SSF 0.13 0.12 0.13 0.13

TSS

EM 0.16 0.16 0.15 0.16

SEM 0.19 0.18 0.19 0.19

SF 0.14 0.15 0.15 0.16

SSF 0.19 0.17 0.19 0.20

Overall average statistics of seasonal precipitation categorical forecast. Statistics are given for March-April-May (MAM), June-July-August (JJA), September-October-November (SON), and December-January-February (DJF). EM, SEM, SF, and SSF indicate unbiased ensemble mean, synthetic ensemble mean, superensemble based on SVD, and synthetic superensemble forecast, respectively.

Statistics of seasonal precipitation categorical forecasts


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GL-DJF TR-DJF NH-DJFGL-JJA TR-JJA NH-JJA

GL-SON TR-SON NH-SONGL-MAM TR-MAM NH-MAM

Member ModelEM

SEM SF SSF

16 years (1987-2002) averaged (Fischer Z-Transform) AC precipitation skill scores of all seasons (MAM, JJA, SON, DJF) for global, tropical (30S-30N), and north hemispheric (0-60N) domains. The bars in the diagram indicate the 4 member models, unbiased ensemble mean (EM), synthetic ensemble mean (SEM), superensemble based on SVD (SF), synthetic superensemble (SSF) of FSU model.

Averaged ACC for All Season (FSU unified Model & synthetic MME)


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RMS-TR-DJF

16 years (1987-2002) averaged RMS precipitation skill scores of all seasons (MAM, JJA, SON, DJF) for tropical (30 S-30N) domains. The bars in the diagram indicate the 4 member models, unbiased ensemble mean (EM), climatology (CLIM), synthetic ensemble mean (SEM), superensemble based on SVD (SF), synthetic superensemble (SSF) of FSU model.

Averaged ACC for Tropic (FSU unified Model & synthetic MME)


Superensemble Precipitation Forecast for JFM 1988, 9 Year Training(AMIP: MPI, CSI, ECMWF, GFDL, NMC, UKMO, ECMWF Reanalysis)

OBS

SUP

DIF

JANUARY FEBURARY MARCH


Superensemble Precipitation Forecast for AMJ 1988, 9 Year Training(AMIP: MPI, CSI, ECMWF, GFDL, NMC, UKMO, ECMWF Reanalysis)

OBS

SUP

DIF

APRIL MAY JUNE


Superensemble Precipitation Forecast for JAS 1988, 9 Year Training(AMIP: MPI, CSI, ECMWF, GFDL, NMC, UKMO, ECMWF Reanalysis)

OBS

SUP

DIF

JULY AUGUST SEPTEMBER


Superensemble Precipitation Forecast for OND 1988, 9 Year Training(AMIP: MPI, CSI, ECMWF, GFDL, NMC, UKMO, ECMWF Reanalysis)

OBS

SUP

DIF

OCTOBER NOVEMBER DECEMBERAMIP Model Forecast for December 1988

ECMWF

GFDL

MPI


Application of Multi Model Ensemble Technique


Cor: 0.72

Cor: 0.41

Forecasting Floods from the Superensemble

One of the areas of strength of the superensemble is in its ability to predict

heavy rains better than any existing models.

Mozambique Floods, Feb. 2000


Skill of Numerical Weather Prediction


Real Time Hurricane Forecasts (Floyd of 1999)


Input of Multi Model Dataset

Make AnomaliesConstructCovariance

MatrixSolve

CovarianceMatrixCompute

WeightsReconstruction Forecast-FieldsMME

Forecast

Construction of MME Program


Summary

• The synthetic multi model algorithm for climate predictions shows better skill scores than the individual member models, and more importantly, better than the unbiased ensemble of member models.

• The synthetic algorithm can be applied to weather and climate forecasting (Short-, Medium-, Long-Range forecasts, and Hurricane prediction).

• The synthetic algorithm can be incorporated into a state-of-the-art dynamic model. Given a number of physical parameterizations of a given process in a dynamical model, the statistical multi model approach can be applied towards the calculation of an optimal unified parameterization scheme.

seasonal climate prediction using linear weighted multi model system w. t. yun apcn

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