The complexity of climate model drifts

Transcription

1 The complexiy of climae model drifs Davide Zanchein Angelo Rubino Maeregu Arisido Carlo Gaean Universiy of Venice, Dep. of Environmeal Sc., Informaics and Saisics A conribuion o PREFACE-WP10: (Saisical mehods o assess and improve forecas of Tropical Alanic variabiliy) PREFACE-PIRATA-CLIVAR TAV Conference, Paris, UPMC

2 Wang e al Annual-mean SST bias averaged in 22 climae models. The SST bias is calculaed by he SST difference beween he model SST and exended reconsruced SST. Sanchez-Gomez e al Climae model drifs: Spaghei plo of he baroropic sreamfuncion averaged over he wesern SPG region for decadal hindcass (DEC, red) and hisorical simulaions (HIST, gray) as a funcion of leadime; ensemble means (hick red and black lines).

3 Drifs occur a differen ime scales for differen variables, can obscure he iniialcondiion forecas informaion and is usually removed a poseriori by an empirical, usually linear, adjusmen (IPCC-AR5, 2013) DCPP guidelines for daa and bias correcion for decadal climae predicions : 10 forecass, j=1,,n iniial imes; =1,,m forecas range observaion-based daa Under full-field iniializaion he model drif is and he bias-correced forecas is:

4 MOTIVATION We need o beer characerize spaial-emporal feaures of model errors and he uncerainies involved in heir esimaion and o opimally merge informaion from observed and simulaed daa in space and ime (i s he goal of PREFACE-WP10).

5 A STATE-SPACE APPROACH Dynamical linear models (DLMs) use unobservable sae variables which allow direc modelling of he processes generaing he observed variabiliy. y x F x G x v 1, w, v ~ N( 0, V ) w ~ N( 0, W ) p(y x, θ) OBSERVATION UNCERTAINTY p(x x -1, θ) PROCESS UNCERTAINTY 1,..., n y : observaion vecor a ime{ p} x :( hidden) sae vecor a ime{ m} G F : sysem operaor : observaion operaor V : observaion error covariance mxm pxm W : sysem error covariance : saic parameers vecor P(x, y) BAYESIAN ANALYSIS P(y x, ) P(x ) P( ) The DLM formulaion can be seen as a special case of a general hierarchical saisical model wih hree levels: daa y, process x, parameers θ = {G,F,V,W} (e.g., Cressie and Winkler, 2011). The classical Kalman filer formulas and Mone Carlo Markov Chain (MCMC) provide efficien and well founded compuaional ools o deermine all he relevan saisical disribuions.

6 STRUCTURAL DECOMPOSITION OF THE ERROR The process of ineres incorporaes sysemaic conribuions o he decadal climae predicion errors: sysemaic mean error δ() wih sochasic rend τ() annual and semi-annual seasonal biases, namely β 12 () and β 6 () D() = d() + β 12 () + β 6 () d() = d(-1) + (-1) + e d () e d ~ N(0, s 2 d) () = (-1) + e () e ~ N(0, s 2 ) The process model above can be easily exended o include he effec of exernal facors, by including addiional explanaory variables. For one covariae X(), he model becomes D*() = D() + γ()x() γ() = γ(-1) + e g () e g ~ N(0, s 2 g)

7 SST bias, K A FIRST APPLICATION of he DLM Tropical and Souh Alanic monhly sea-surface emperaures from he MiKlip full-field GECCO r1 decadal hindcass wih MPI-ESM-LR*. From: Jungclaus e al., 2013 Bayesian analysis applied on error covariances V and W (a oal of 3 parameers), use lognormal priors [logn(0,1)] For spaial analysis, individual grid poins are processed individually, parallelizaion speeds up calculaion. The MCMC (10000x) is based on he slicesampler algorihm. Use he dlmsmo rouine from he dlm oolbox by Markko Laine * Couresy of Wolfgang Mueller, MPIM

8 RESULTS REGIONAL ERROR: SST in he Angola Benguela fron Dj: empirical hindcas error δ: drif/bias τ: sochasic rend componen β: seasonal bias componen (annual and semiannual)

9 RESULTS REGIONAL ERROR: SST in he Angola Benguela fron Dj: empirical hindcas error δ: drif/bias τ: sochasic rend componen β: seasonal bias componen (annual and semiannual)

10 RESULTS REGIONAL ERROR: SST in he Angola Benguela fron Dj: empirical hindcas error δ: drif/bias τ: sochasic rend componen β: seasonal bias componen (annual and semiannual)

11 RESULTS A LOOK AT RESIDUALS (DRIFT-CORRECTED ERRORS), SST ABF Temporal evoluion of poseriori means of monhly-mean residuals in SSTs for he Angola-Benguela fron region. C

12 RESULTS A LOOK AT RESIDUALS (DRIFT-CORRECTED ERRORS), SST ABF Temporal evoluion of poseriori means of monhly-mean residuals in SSTs for he Angola-Benguela fron region. C

13 RESULTS REGIONAL ERROR: SST in he Angola Benguela fron Effec of covariaes Dj: empirical hindcas error δ: drif/bias τ: sochasic rend componen β: seasonal bias componen (annual and semiannual)

14 RESULTS REGIONAL ERROR: SST in he Angola Benguela fron Effec of covariaes Dj: empirical hindcas error δ: drif/bias τ: sochasic rend componen β: seasonal bias componen (annual and semiannual)

15 RESULTS REGIONAL ERROR: SST in he Angola Benguela fron Effec of covariaes Dj: empirical hindcas error δ: drif/bias τ: sochasic rend componen β: seasonal bias componen (annual and semiannual)

16 RESULTS REGIONAL ERROR: SST in he Angola Benguela fron Effec of covariaes Dj: empirical hindcas error δ: drif/bias τ: sochasic rend componen β: seasonal bias componen (annual and semiannual)

17 RESULTS REGIONAL ERROR: SST in he Angola Benguela fron Effec of covariaes Dj: empirical hindcas error δ: drif/bias τ: sochasic rend componen β: seasonal bias componen (annual and semiannual)

18 RESULTS PROPAGATION OF SEASONAL SST ERRORS, THE ROLE OF SALINITY ERRORS Longiudinal secion a 44 S C C/psu

19 CONCLUSIONS (WIP) AND OUTLOOK We propose a srucural decomposiion of sysemaic decadal climae predicion errors (drif/climaological bias and seasonal biases), which is implemened via a sae-space model buil wihin a Bayesian hierarchical framework. Resuls help characerizing he grea complexiy behind drif/climaological bias and seasonal biases. Do we undersand he differen physical sources, propagaion mechanisms and implicaions of such model error componens? There is an inimae connecion beween (esimaed) drif developmen and inerdecadal climae evoluion. Furhermore, he hindcas error in a cerain locaion can be subsanially shaped by he effec of sysemaic errors over remoe regions (e.g., PDO). Do he found uncerainies in drif componens call for improved drif esimaion and adjusmen echniques?

20 CONCLUSIONS (WIP) AND OUTLOOK We propose a srucural decomposiion of sysemaic decadal climae predicion errors (drif/climaological bias and seasonal biases), which is implemened via a sae-space model buil wihin a Bayesian hierarchical framework. Resuls help characerizing he grea complexiy behind drif/climaological bias and seasonal biases. Do we undersand he differen physical sources, propagaion mechanisms and implicaions of such model error componens? There is an inimae connecion beween (esimaed) drif developmen and inerdecadal climae evoluion. Furhermore, he hindcas error in a cerain locaion can be subsanially shaped by he effec of sysemaic errors over remoe regions (e.g., PDO). Do he found uncerainies in drif componens call for improved drif esimaion and adjusmen echniques? THANK YOU FOR YOUR ATTENTION

21 RESULTS PROPAGATION OF SEASONAL SST ERRORS Pulse error signals generaed around 50 W apparenly ravel easward o abou 25 W, wih a speed of approximaely 4 cm/s C m 3 /s

22 RESULTS - MARGINAL POSTERIOR DISTRIBUTIONS OF SST ERRORS grid-poin analysis Shading: poseriori median (drif componen); large (small) dos mark grid poins where he 0-value lies wihin he 40h- 60h (5h-95h) percenile range of he poseriori disribuion

23 RESULTS IMPACTS OF NUMBER OF OBSERVATIONS ON DRIFT ESTIMATION

24 A STATE-SPACE APPROACH We use DLM o deermine: uncerainy of unknown saes and heir evoluion condiional o observaions and model parameers: p ( x 1 1: y : n n, ) by means of Kalman based simulaion smooher Uncerainy of unknown saes and parameers and heir evoluion condiional o all available observaions (Bayesian approach): p( x1 : n y1 : n) p( x1 : n, y1 : n) d by means of Mone Carlo Markov Chain (MCMC). This is possible hanks o he Markov propery inheren in he definiion of our model: he sae a ime is saisically condiionally independen on he whole hisory as i only depends on he sae a -1.

25 BAYESIAN ANALYSIS 1. KF forward recursion Assuming he iniial disribuions a ime =0 are known, he Kalman filer forward recursion can be used o calculae he disribuion of he sae vecor x, given observaions up o ime : p(x y, θ). This is done by calculaing, as prior, he mean and covariance marix of one-sep-ahead prediced saes: p(x x -1, y -1, θ) = N(x, C ) xˆ Cˆ C G y, G F C x 1 1 Cˆ 1 G T F T prior mean for W V x prior covariance for covariance for predicing x y Then he poserior sae and is covariance are calculaed using he Kalman gain marix: K v x C Cˆ y xˆ Cˆ F T F K C xˆ 1 y, v K F Cˆ Kalman gain predicion residual poserior mean for poserior covariance for x x Equaions are ieraed for =1,,N

26 BAYESIAN ANALYSIS 2. Kalman smooher backward recursion KF provides disribuions of x given observaions up o ime. We wan o accoun for all observaions, so: p(x y 1:n, ) (all gaussian). The Kalman smooher backward recursion provide socalled smoohed saes for =N,N-1,,1. Seing r N+1 and N N+1 equal o zero: L r N G F F G C C v K F F L L r N auxiliary L auxiliary variable auxiliary variable variable ~ x xˆ Cˆ r smoohed sae mean ~ C Cˆ Cˆ N Cˆ smoohed sae covariance T T 1 y, 1 y, T T We need full join poseriori disribuion of all saes given all observaions (see 1 and 2) and parameers: p(x 1:N y 1:N, θ). This disribuion does no have a closed form soluion, bu we can draw realizaions for i using he so-called simulaion smooher algorihm. In pracice, he algorihm proceeds as follows: - Sample from sae space equaions o ge x I 1:N and y I 1:N ( sands for ilde, smoohed values) - Use Kalman smooher wih he new observaion y I 1:N o ge smoohed saes x Is 1:N - Add he sae residual o he original smoohed saes o obain x* 1:N = x I 1:N-x Is 1:N+x s 1:N

27 BAYESIAN ANALYSIS 4. Uncerainy on parameers We do no wan θ o be fixed, insead we wan o esimae i using Bayesian saisics. We need he marginal likelihood funcion p(y 1:n θ) wih he uncerainy of saes accouned for (which means inegraed ou). For each θ, such likelihood is provided as a byproduc of he Kalman filer. Due o he Markov propery of he sae space equaions, we can calculae he marginal likelihood as: N p( y1 : N ) p( y1 ) p( y y1 : 1, ) 2 Which for a Gaussian linear model is proporional o: N 1 T 1 exp ( y F xˆ ) Cy, ( y F xˆ ) log( Cy, MCMC ) A MCMC is performed o calculae he marginal poserior disribuion p(θ y 1:N ), using he likelihood defined in sep 4 and wih proper priors. 6. Seps 5 and 1-3 are combined o draw samples from he disribuion p(x 1:N, θ y 1:N )

28 BAYESIAN ANALYSIS We can apply he Bayesian inference on error covariances W and V. We mus specify priors (all Gaussian) and likelihoods for all such unknown parameers. Pracically, o reduce compuaional requiremens, we define priors/likelihoods for he sandard deviaions of he following parameers: one prior for V (acually fixed and no esimaed in presen analysis of MiKlip hindcass) four priors for W (one for DF, one for B, one common for SF1 and SF2, one common for BSF1 and BSF2). An adapive Meropolis algorihm is ieraively used o sample from he full poserior disribuion of he unknown parameers. Kalman filer and Kalman smooher are hen used o ieraively sample he sysem saes along he MCMC (i.e., we derive associaed marginal disribuions for each of he sae componens)