downscalingprojectionsofindianmonsoonrainfallusinga non...

Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. 137: 347–359, January 2011 B

Downscaling projections of Indian monsoon rainfall using anon-homogeneous hidden Markov model

Arthur M. Greenea*, Andrew W. Robertsona, Padhraic Smythb and Scott Trigliab

a International Research Institute for Climate and Society, Columbia University, New York, USAb University of California, Irvine, USA

*Correspondence to: A. M. Greene, IRICS, Lamont Campus, Columbia University, Palisades,New York 10964, USA. E-mail: [email protected]

Downscaled rainfall projections for the Indian summer monsoon are generated usinga non-homogeneous hidden Markov model (NHMM) and information from both adense observational dataset and an ensemble of general circulation models (GCMs).The projections are conditioned on two types of GCM information, correspondingapproximately to dynamic and thermodynamic components of precipitation change.These have opposing effects, with a weakening circulation compensating not quitehalf of the regional precipitation increase that might otherwise be expected. GCMinformation is taken at the largest spatial scales consistent with regional physics andmodelling constraints, while the NHMM produces a disaggregation consistent withthe observed fine-scale spatiotemporal variability. Projections are generated usingmultimodel mean predictors, with intermodel dispersion providing a measure ofthe uncertainty due to GCM differences. The downscaled simulations exhibit smallincreases in the number of dry days, in the average length of dry spells, in mean dailyintensity and in the exceedance frequency of nearly all daily rainfall percentiles.Copyright c© 2011 Royal Meteorological Society

Key Words: precipitation; climate projections; NHMM

Received 24 May 2010; Revised 12 January 2011; Accepted 14 January 2011; Published online in Wiley OnlineLibrary

Citation: Greene AM, Robertson AW, Smyth P, Triglia S. 2011. Downscaling projections of Indianmonsoon rainfall using a non-homogeneous hidden Markov model. Q. J. R. Meteorol. Soc. 137: 347–359.DOI:10.1002/qj.788

1. Introduction

Precipitation fields in general circulation models (GCMs)do not capture detail at the fine spatial scales of interestin many climate risk management applications. GCMparametrizations also tend to produce biased rainfalldistributions (Dai, 2006). These factors preclude thedirect imputation of localized, temporally disaggregatedprecipitation changes from GCM simulations. On theother hand, GCMs are the most comprehensive toolsyet devised for the quantitative characterization of climatechange.

Observational data do represent the fine spatial detaillacking in GCM simulations, and are less likely to be biased

with respect to spatial patterns or rainfall distributions.However, they are records of past climate, fundamentallymute about the future.

Here these complementary data types are combined in theframework of a non-homogeneous hidden Markov model(NHMM). The NHMM decomposes the observed spatio-temporal rainfall variability over a network of locations viaa small, discrete set of ‘hidden’ states (so termed becausethey are not directly observed). Each state comprises a setof rainfall probabilities and wet-day distribution functionsfor all locations in the network; the states proceed on a dailytime step following a first-order Markov process. Thesecharacteristics enable the NHMM to represent both spatialcovariance over the network and persistence of large-scale

Copyright c© 2011 Royal Meteorological Society

348 A. M. Greene et al.

weather patterns, which can be associated with the states.It is thus well-suited for the representation of daily rainfallin climate regimes that can be characterized in terms ofvariably persistent large-scale weather patterns, as can theIndian monsoon.

In the ‘non-homogeneous’ construction utilized here,the state transition matrix is conditioned on an exogenouspredictor, giving it an implicit time dependence. This allowsfor realistic simulation of the seasonal cycle and, moreimportantly, provides a pathway for the introduction ofGCM-based climate change information.

To generate projections of future rainfall, two predictivefields from an ensemble of GCMs are utilized: the regionalmean rainfall change and a dynamical index of monsooncirculation strength. The former serves as a constrainton the NHMM-simulated rainfall change, while the latteris used to condition transition probabilities among thehidden states. These large-scale predictors are analogous tothe GCM-derived boundary conditions used to constrainregional dynamical models. The NHMM is used to generatestochastic daily rainfall sequences for both the twentiethand twentyfirst centuries, from which statistics of interestare extracted. Information from the GCMs is utilized in theform of multimodel means; individual models then serveto delineate the range of uncertainty owing to differencesamong GCMs.

Markov and hidden Markov models of varying structurehave been utilized previously for rainfall modelling (e.g.Richardson and Wright, 1984; Hughes and Guttorp, 1994a,b;Bates et al., 1998; Hughes et al., 1999; Robertson et al.,2004; Mehrotra and Sharma, 2005; Robertson et al., 2006,2009; Ailliot et al., 2009; Gelati et al., 2010). A particularnovelty in the present study is the implied decomposition ofmonsoon rainfall change into dynamic and thermodynamiccomponents, as manifest in the exogenous variable chosento drive the NHMM and the concomitant scaling of dailyintensity distributions. This decomposition is of particularrelevance in the context of long-term climate change (e.g.Held and Soden, 2006).

Section 2 places the NHMM in the context of otherstatistical downscaling methods. Section 3 provides amodelling overview, while section 4 describes both theobservational and GCM data utilized. Section 5 covers keytheoretical underpinnings of the method as well as detailsof its present application. The NHMM simulations areconsidered in section 6; this is followed in section 7 bya discussion of rainfall changes on both fine space- andtime-scales. Sections 8 and 9 present a validation by wayof method comparison and a discussion of uncertainties,respectively. A summary and conclusions are provided insection 10.

2. NHMM in context

As a downscaling method, the NHMM could be classifiedas a hybrid between weather-typing and regression-basedmodels (Wilby et al., 2002). Weather states are inferredfrom patterns of observed daily precipitation and modelledas first-order Markov dependent in time, while a large-scalecirculation index is utilized as predictor in a multinomiallogistic regression, the dependent variables being theprobabilities in the Markovian transition matrix.

Beyond this general affinity, the NHMM bears someresemblance to the WGEN model described by Richardson

and Wright (1984) as adapted by Wilks (1992) for thepurposes of downscaling climate change scenarios andeventually (Wilks, 1998) to model spatially correlated rainfallover a network of sites. These models include two states (wetand dry), with daily rainfall fit by either a gamma distributionor, in Wilks (1998), a two-component exponential mixture.Such models require specification of a large number oflocation-specific parameters. The NHMM, by contrast, caninclude an arbitrary number of states and implicitly accountsfor spatial covariance, in the form of the rainfall patternsassociated with the states. As in Wilks (1998), daily rainfalldistributions are modelled as a two-component exponentialmixture.

The NHMM’s hidden states, although nominally abstrac-tions, were shown in an earlier study (Greene et al., 2008)to be sensibly associated with large-scale anomalies in themonsoonal circulation. The persistence of these anomaliesin time is captured by the Markovian dependence of theday-to-day state transitions.

3. Modelling overview

The NHMM is first trained on observational data for theperiod 1951–2004, along with a seasonally varying indexof monsoon circulation strength, taken from the GCMensemble in the form of a multimodel mean. In this step,both the parameters of the hidden states and the relationshipbetween the circulation index and the transition probabilitiesgoverning the day-to-day progression of the states areinferred. A bias correction is introduced in order to bringthe simulated seasonal cycle into better agreement with theobserved. The NHMM is then used to generate stochasticsequences of daily rainfall for the periods 1951–2000 and2070–2099, using in each case the corresponding circulationindex. Differences between these two sets of simulationsprovide a measure of rainfall change owing to changes in themonsoon circulation, and may be thought of as dynamicallyinduced.

In going to the twentyfirst century the circulationweakens. Given the relationships inferred by the NHMMin the learning step, this brings about a reduction inregional mean rainfall. Such a reduction, however, isinconsistent with the multimodel response, in which rainfallincreases. The discordance is attributed to thermodynamiceffects, and resolved by scaling the NHMM daily wet-dayamount distributions so that the simulated regional meanrainfall change comes into agreement with the multimodelmean. Attention then focuses on the statistical features ofclimatically induced rainfall change, disaggregated in bothtime and space. It is this downscaling that constitutes theproper domain of the NHMM

4. Data

4.1. Observations

A daily rainfall dataset for 1951–2004 developed bythe India Meteorological Department (Rajeevan et al.,2006) constitutes the observational record. This is a 1◦gridded product derived from quality-controlled stationrecords and is complete. Only the June–September (JJAS)values, representing the southwest (summer) monsoon, areutilized. The data are geographically truncated to 15◦–29◦N,68◦–88◦E, which coincides approximately with the main

Copyright c© 2011 Royal Meteorological Society Q. J. R. Meteorol. Soc. 137: 347–359 (2011)

Downscaling Projections of Indian Monsoon Rainfall 349

(a) (b) (c)

Figure 1. Climatological JJAS rainfall for the thinned IMD data: (a) mean daily rainfall, (b) probability of rain, and (c) mean daily intensity.

‘monsoon zone’ (Sikka and Gadgil, 1980). This region hasbeen identified in a cluster analysis (V. Moron, personalcommunication) as having a homogeneous seasonal cyclefor precipitation. This facilitates the analysis, since thepresent NHMM structure is not designed to accommodateseasonal-cycle mixtures. To allow for multiple simulations,given the computer resources at hand, the data were thinnedto a resolution of 3◦, resulting in a dataset having 24gridpoints. On a single-processor 2.8 GHz workstation,fitting the NHMM to 54 years of JJAS daily data on thisgrid requires ≈20 min. Note that the effective resolution ofthe thinned data remains 1◦.

Climatological rainfall and its decomposition in termsof probability of rain, P(R) and mean daily intensity, I(amount on wet days) is shown in Figure 1, where (a) and(b) exhibit a clear gradient, extending from arid Rajasthanand the Thar desert southeast toward the wet coast of theBay of Bengal. The gradient in I, shown in Figure 1(c), is lessclear, owing in part to high intensity values in the centralregion, but this variable also reflects relative dryness in thenorthwest.

The behaviour of the most westerly gridpoint just northof 20◦N seems somewhat anomalous in the context of thisgradient, and signals a moisture source to the west interactingwith the escarpment of the Deccan plateau, as opposed tothe convective systems that propagate northwestward overthe continent from the Bay of Bengal. This is consistent withhigh rainfall at coastal locations further to the south that arenot part of the study dataset.

4.2. Information from GCMs

Precipitation and zonal wind fields from an ensemble of 23GCMs from the Coupled Model Intercomparison Projectversion 3 dataset (CMIP3; Meehl et al., 2007) are utilized.Simulations from the CMIP3 models underlie much of theresearch considered in the Fourth Assessment Report ofthe Intergovernmental Panel on Climate Change (IPCC,2007). The models employed (Table I) comprise all thosefor which these two fields were available, for both the20C3M (twentieth-century) and A1B (twentyfirst-century)stabilization experiments. The data utilized have monthlytime resolution. In addition, the 500 mb vertical velocity fieldfrom a subset of these models is employed for a validationexercise.

Information regarding the circulation is introduced inthe form of the ‘Westerly Shear Index 1’ (WSI1), as definedin Wang and Fan (1999). This is the vertical shear of thezonal wind (u850 − u200) averaged over the box 5◦–20◦N,

Table I. GCMs in the CMIP3 archive (Meehl et al., 2007)for which both precipitation and zonal wind fields were

available for both 20C3M and A1B simulations.

Group Model

BCCR (Norway) BCCR-BCM2.0CCCMA (Canada) CGCM-3.1CCCMA CGCM-3.1(T63)CCSR/NIES/JAMSTEC (Japan) MIROC3.2(hires)CCSR/NIES/JAMSTEC MIROC3.2(medres)CNRM (France) CM3CSIRO (Australia) CSIRO-Mk3.0CSIRO CSIRO-Mk3.5GFDL (USA) GFDL-CM2.0GFDL GFDL-CM2.1GISS (USA) GISS-AOMGISS GISS-EHGISS GISS-ERHadley Centre (UK) HadCM3Hadley Centre HadGEM1IAP (China) FGOALS-g1.0INGV (Italy) INGV-ECHAM4INM (Russia) INM-CM3.0IPSL (France) IPSL-SXGMPI (Germany) ECHAM5/MPI-OMMRI (Japan) MRI-CGCM2.3.2NCAR (USA) CCSM3NCAR PCM1

40◦–80◦E. Figure 2(a) shows that this box encloses alocal maximum in the climatological JJAS shear field, asrepresented in the NCEP–NCAR reanalysis (Kalnay et al.,1996).

Wang and Fan, using monthly mean values for JJAS overthe 1974–1997 period, found that WSI1 was both stablein its physical configuration and significantly correlated(r = 0.61) with outgoing long-wave radiation, indicativeof monsoon rainfall, over the Indian subcontinent. Of theclimate models utilized (Table I), a majority also show alocal maximum for the shear field in the WSI1 box. Forthe exceptions, the box invariably sits astride a ridge, themaximum then occurring to the east. The zonal elongationof shear contours is also a robust feature of the GCMs. Thispattern concordance extends across centuries, and likelyresults from the broad land–ocean contrasts and extendedhigh-amplitude topographic relief that characterize this



Eq

30N

30E 60E 90E 120E

(a)-30

-30-20

-10

10

1020

30

Jan

Feb Mar Apr

May Ju

n Jul

Aug Sep OctNov Dec

-20

-10

0

10

20

30

40

She

ar, m

s−1

(b)

Obs

20C3M

A1B

Figure 2. Vertical shear of the zonal wind (u850 − u200, m s−1): (a) climatological JJAS field from the NCEP–NCAR reanalysis for 1951–2000, showingthe box defining WSI1, and (b) the climatological annual cycle, showing the 1951–2000 reanalysis (Obs), the multimodel means for 1951–2000 (20C3M)and the A1B scenario for 2070–2099 (A1B).

region. On the large scale there is thus relatively robustmodel agreement, stable across centuries, with respect tomonsoon circulation structure.

The WSI1 index tracks the annual cycle of monsoonwinds, as shown in Figure 2(b). During the southwest(summer) monsoon, extending from June to September,values are strongly positive; outside this seasonal windowthe winds reverse and WSI1 changes sign. The figureshows the NCEP–NCAR reanalysis climatology as wellas multimodel means for the twentieth and twentyfirstcenturies. The GCM simulations match the reanalysis fairlyclosely in shape, although the latter has a slightly greateramplitude. Figure 2(b) also shows that the amplitude ofthe seasonal cycle decreases in going from the twentiethto twentyfirst centuries. This is also a robust feature ofthe multimodel dataset, as illustrated in Figure 3. Thisfigure shows the distribution across GCMs of the differencebetween twentyfirst and twentieth century WSI1 values, bymonth.

The largest decreases occur in July and August. This isthe core of the wet season, when rainfall is greatest. Shiftsfor June and September would have a reduced influence onoverall rainfall change, but are still negative for most models.Like the spatial pattern of the shear field, the seasonal cycleshape is rather stable across centuries, weakening slightlybut showing no phase shift. The NHMM assumes a degreeof stationarity in the large-scale flow structure; both thespatial and temporal stability of the WSI1 cycle support thisassumption.

WSI1 is not unique among climate variables inhaving a well-defined seasonal cycle. What is importantfor the exogenous predictor, rather, is that it linkmonsoon circulation strength and rainfall over the Indiansubcontinent in a physically sensible manner. Its change,in going from the twentieth to twentyfirst centuries, thenbecomes an indicator of dynamically mediated shifts inmonsoon rainfall. It is this linkage that imbues its use in theNHMM with meaning.

5. Methodology

5.1. Theoretical basis

The NHMM is similar to the homogeneous model describedin Greene et al. (2008), in that it decomposes the dailyprecipitation field over a network of locations into a smallset of discrete ‘states’, that proceed in time according to a

Figure 3. Intermodel spread in the difference between WSI1 values, goingfrom the twentieth to the twentyfirst century, for each month.

first-order Markov process. In both of these models, rainfallat each of the locations is conditionally independent: giventhe state, it does not depend on rainfall at other locations (orat other times). Spatial and temporal dependence are thenrepresented implicitly, the former through the existence ofmultiple states, the latter via the Markov property of thestate transition process. For each location, conditional onstate, the rainfall ‘emitted’ by the model on a given day isdrawn from a three-component mixture, a delta functionat zero (representing no rain) and two exponentials. Two-component exponential mixtures have been found effectivein the modelling of daily rainfall distributions (Woolhiserand Roldan, 1982). The homogeneous model comprisesthese exponential parameters and mixing weights as well asthe matrix of probabilities governing day-to-day transitionsamong the states.

The present model differs from that of Greene et al.(2008) by the inclusion of an exogenous predictor. Lackingsuch a predictor, the homogeneous model is not suitablefor prognosis, as there is no covariate that might assumediffering values during different climate epochs. Here theWSI1 index is taken as the covariate; its value is related tothe state-to-state transition probabilities according to thepolytomous logistic regression (Robertson et al., 2004):

P(St = j|St−1 = i, Xt = x) = exp(σij + ρjx)K∑

k=1

exp(σik + ρkx)

. (1)



Here, t is time, discretized in terms of days, and P theprobability of an i → j transition in going from time t − 1to t, given value x of predictor X at time t. The σij and ρj

are parameters to be fit and K is the number of modelledstates, here taken as four in correspondence with the modelof Greene et al. (2008).

Expression (1) refers to transition probabilities P amongthe states but makes no reference to the state characteristicsthemselves, which include rainfall probabilities and intensitydistributions for each of the stations. However, through theagency of the transition matrix, X may influence stateoccupation statistics, and thus, inter alia, total seasonalrainfall, shape of the seasonal cycle or the fraction of rainfallderiving from each of the states. The underlying assumptionis that future rainfall will arise from the same assemblageof large-scale weather patterns as presently exists, but thattheir relative occurrence frequencies, timing or persistencecharacteristics may shift as climate changes.

This stationarity of weather states, with the forcedresponse taking the form of shifts in occupation statistics,is consistent with a nonlinear-dynamical view of climate, inwhich the states represent basins of attraction toward whichtrajectories in the climate phase space are drawn (Lorenz,1963; Palmer, 1999). There is also a connection with theidea of fluctuation dissipation, i.e. that the response of anonlinear system to small perturbations can be understoodby studying fluctuations of the unperturbed system (Leith,1975).

A final element in the modelling scheme involvesthe scaling of NHMM intensity distributions so as tobring simulated regional-mean precipitation change intoagreement with the GCM ensemble mean. This is performedoutside the NHMM fitting process, and reflects a relianceon the GCMs for large-scale climate information. From amore process-oriented perspective, this procedure, whichconditions rainfall amounts on wet days, may be thoughtof as representing variation due to changes in atmosphericmoisture, the ultimate source of wet-day amounts, given adynamical predisposition to moisture flux convergence.

Use of multimodel mean fields follows from work showingthat such fields as a rule provide more consistently faithfulrepresentations of the observed climate than do individualmodels (Gleckler et al., 2008), and provides a natural wayof estimating uncertainties arising from differences in GCMformulation.

5.2. Model fitting

WSI1 values (Figure 2(b), ‘20C3m’ and ‘A1B’) were linearlyinterpolated to daily resolution for ingestion by the NHMM.Since daily, or even interannual, variations of modelWSI1 cannot be expected to correspond with observedrainfall changes, these cycles were utilized in the form ofclimatologies, repeated for each year for both training andsimulation. One may think of the historical sequences asequally likely draws, given the WSI1 climatology, in theconditioning of a statistical forecast model. This is similar tothe model output statistics (MOS) idea, in which localizedweather forecast statistics are conditioned on more coarselyresolved model output.

The model is fitted using the observed rainfall andinterpolated twentieth-century WSI1, producing both thedecomposition of the rainfall by states and the parametersof the logistic regression (1). As in Greene et al. (2008),

maximum-likelihood estimates of the model parametersare obtained using the expectation-maximization algorithm(Dempster et al., 1977).

In section 5.3, it will be seen that the spatial patternsof rainfall probability and mean daily amount associatedwith each of the hidden states are quite similar to thoseinferred using the homogeneous model of Greene et al.(2008), in which precipitation from a smaller and somewhatmore localized network of stations was fitted. This suggeststhat (a) the smaller network is in fact extensive enough tocapture most of the spatiotemporal variability present overthe 24 locations of the present study, and (b) inclusionof the exogenous variable has little influence on the statedecomposition. As we will see, it is rather in the simulationstep that the effect of WSI1, realized through its conditioningof the state transition matrix, becomes manifest.

5.3. State descriptions

The four states from the fitted model are summarized in thefour columns of Figure 4, with rainfall probabilities in the toprow, mean intensities below (cf. Greene et al., 2008, Figure 4).Because rainfall sequences include many dry days, HMMdecomposition typically identifies a state having both lowoccurrence probabilities and intensities; here this is state 4(Figures 4(d, h)). State 3 exhibits the uniformly highestprobability of rain (Figure 4(c)), along with relatively highintensities in both the west and adjacent to the Bay of Bengal(Figure 4(g)). This reflects enhanced convection over most ofthe subcontinent. In state 1, convection occurs preferentiallytoward the Himalayan foothills, while state 2, with enhancedprobabilities toward the southeast but moderate intensities,may represent an early stage in the propagation across thesubcontinent of intraseasonal oscillations (Goswami andMohan, 2001). State 2 differs more than the others in theearlier analysis, but in that study the southeastern sector,where the probability of rain is seen to be concentrated,was not sampled. Since the states described here are quitesimilar to those inferred in Greene et al. (2008), the readeris referred to that work for more detailed discussion.

5.4. Effect of exogenous predictor

The learned relationship between WSI1 and transitionprobabilities is summarized in Figure 5(a), which showsthe four probabilities associated with the equilibriumstate distribution as a function of WSI1. The vector ofstate probabilities p is advanced in time by the lineartransformation

T′pt−1 = pt , (2)

(Norris, 1997), where T′ is the transpose of the transitionmatrix. Identification of a state probability vector that isinvariant under this transformation corresponds to theeigenvalue problem

T′p = λp , (3)

where we seek the eigenvector having λ = 1. The existenceand uniqueness of such a vector is implied by the stochasticcharacter of T (its rows each sum to unity) and the ergodicproperty of the Markov chain modelled by the NHMM (itis aperiodic, with finite return times).



(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 4. (a)–(d) probability of rain and (e)–(h) mean daily intensity, for states 1–4 in the corresponding columns.

(a) (b)

Figure 5. (a) Equilibrium state probabilities for the four states, as a function of WSI1, and (b) seasonal climatology of the Viterbi sequence, shown withmatching colours.

To generate Figure 5(a), the matrix T was computed on agrid of x-values spanning the seasonal range of WSI1, using(1). For each fixed T thus obtained, the eigendecompositionsuggested by (3) was performed and the eigenvector havingunit eigenvalue identified. The figure shows, for each x,the four components of this vector. The distributions soidentified represent points of stable equilibrium (Cox andMiller, 1965).

5.5. Viterbi sequence

Complementary to the equilibrium state probabilitydistributions shown in Figure 5(a) is the sequence of most-likely states. Computed using the Viterbi algorithm (Forney,1973), this is the most likely day-by-day trajectory throughstate space, given the data (IMD observations and twentieth-century WSI1) and the fitted model. A daily climatology ofthe Viterbi sequence, in terms of relative state occupationfrequencies, is shown in Figure 5(b). (Note that in thisfigure relative state frequencies correspond to the heightof the bands representing the respective states, not to theirabsolute positions along the ordinate.)

For low WSI1 values, Figure 5(a) indicates that state 4, thedry state, is most likely. This is consistent with Figure 5(b),where early in the season state 4 occupies most of thevertical probability space. Figure 5(b) also indicates thatthis state occurs infrequently during the July–August coreof the rainy season, corresponding to its relatively lowequilibrium probabilities at high values of WSI1. The coreof the rainy season, on the other hand, is dominated bystates 1 and 3, the equilibrium probabilities for which risecorrespondingly with increasing WSI1. Conditioning of thetransition probabilities by WSI1 thus acts as an effectivecontrol on the subseasonal evolution of precipitation.

6. Simulations

6.1. Twentieth century

6.1.1. Seasonal cycle and bias correction

Using the learned NHMM parameters and twentieth-century shear index, an ensemble of ten 54-year simulationswas generated. Biases were noted in these simulations in



regional mean rainfall (overestimated by ∼ 2%) and in theshape of the seasonal cycle. As we shall see, these biases arenot unrelated.

The climatological seasonal cycle of spatially averagedobserved rainfall is shown in Figure 6(a), along withthe initial simulation. It can be seen that the latteroverestimates rainfall in the early and late parts of the season,underestimating during July and the first part of August.Since the state frequency distribution differs considerablybetween the central and peripheral parts of the season(Figure 5(b)), such a mismatch could have significant effectson the spatial pattern of simulated rainfall change; dailyrainfall statistics would also likely be biased. A correctionwas therefore undertaken.

From the modelling, it is inferred that circulation-drivenrainfall changes account for only a portion of the totalchange. The remainder is assigned to thermodynamic effects,essentially the moistening of the lower atmosphere owingto climatic warming. This idea is applied in the climatechange sense by scaling intensity distributions so as to bringNHMM-simulated rainfall change into agreement with theGCM-simulated change. A similar compositing of effects isinvoked here for bias correction.

Lower-tropospheric moisture exhibits a seasonal cyclequite similar in shape to the shear cycle (Figure 2(b)), butthis significant component of intraseasonal variation hasso far been ignored. The seasonal variation shown in the‘SIM’ trace of Figure 6(a) is due only to shifts in relativestate occupation frequencies. These represent changes in thestatistics of large-scale flow anomalies, but do not capturethe contribution to the seasonal cycle of intensity changesdue to atmospheric moistening. It is perhaps not surprisingthat the simulated seasonal cycle is less ‘peaked’ than theobserved, to which the seasonality of lower troposphericmoisture undoubtedly contributes as well.

The correction was implemented by first computing,for each day in the seasonal cycle climatology, the factorrequired to compensate for the simulation excess or deficit,as shown in Figure 6(a). A third-order polynomial was fittedto the resulting ‘compensation curve’ (Figure 7) and thefitted values then used to rescale the rainfall within eachsimulated season.

The climatology of the bias-corrected twentieth-centurysimulation is shown in Figure 6(b), where it can be seenthat a considerably better fit to observations has nowbeen obtained. Even for September, where high-frequencyfeatures render the correction somewhat less effective thanduring the remainder of the season, the RMS differencebetween series is reduced by 48%. (For the season as awhole the reduction is 52%.) Since the shear cycle doesnot undergo any significant change in shape as we gothe twentyfirst century, it seems reasonable to assumethat seasonal-cycle bias will also persist. The correction,computed as above, is thus applied uniformly, across bothtwentieth- and twentyfirst-century simulations.

As a test, an 8-state model was fitted to the data, thesupposition being that with more degrees of freedom theNHMM might better be able to ‘tune’ the seasonal cycle,by fitting a more precisely calibrated sequence of evolvingstates (thus exponential parameters). This did not result inan improved representation, suggesting that a good fit toseasonal-cycle shape would not likely be obtained by drivingthe model with WSI1 alone. This result also tends to validatethe procedure utilized.

(a)

(b)

Figure 6. (a) Seasonal cycle of location-averaged precipitation (mm d−1)in the observations (IMD) and initial twentieth-century simulations. (b) isas (a), with simulations corrected for seasonal-cycle shape bias. This figureis available in colour online at wileyonlinelibrary.com/journal/qj

Figure 7. Daily correction factor required to remove the seasonal-cycle biasin the twentieth-century simulations, and the third-order polynomial fit(dashed) used to scale the simulations. This figure is available in colouronline at wileyonlinelibrary.com/journal/qj

The correction process effectively eliminates the negativemean bias initially noted in the twentieth-century sim-ulations, reducing it to <0.01%. The mean bias is thusvery likely attributable to the mismatch in seasonal cycleshape. Small mean biases are typically noted in HMM cli-matologies (Robertson et al., 2009); this result suggests onepossible cause. Correcting the twentieth and twentyfirst-century simulations produces almost identical shifts in themean, so inferred climatic changes should not be affectedby the bias correction process, which is applied uniformlyacross centuries.



Figure 8. Daily rainfall distribution for the centrally located point 23.5◦N,78.5◦E, for observations (IMD) the ECHAM5 model (MPI) and thetwentieth-century NHMM simulations (Sim).

6.1.2. Daily rainfall distributions

Daily rainfall distributions are an area of pervasive GCMbias, with typical parametrizations producing too muchdrizzle and too few heavy rainfall events. Figure 8 comparesa typical GCM distribution with that simulated by theNHMM, for the central location 23.5◦N, 78.5◦E. (The closestGCM gridpoint is utilized.) GCM daily amounts are sharplytruncated around 70 mm, while the NHMM captures themuch longer tail of the observed distribution. There is asmall degree of mismatch at levels above about 300 mm,suggesting that close attention to distribution extremesmight be a fruitful path for extending this work (althoughthis discrepancy could also be attributable to samplingvariability). Vrac and Naveau (2007) have implemented anNHMM utilizing extreme-value mixtures that is possiblyof relevance, but this potential refinement of distributionalform lies beyond the scope of the present study.

6.1.3. Spatial pattern

Figure 9 shows spatial patterns of the GCM ensemble-mean climatological JJAS rainfall, the observations and thebias-corrected twentieth-century simulations. The first ofthese (Figure 9(a)) captures the general northwest–southeastprecipitation gradient seen in the observations (Figure 9(b)),but misses various small-scale features, including the sharpmaximum on the western coast, as well as variations alongsouthwest–northeast diagonals. The multimodel mean isgridded at 2◦ × 2.5◦, fine enough to discern these details ifthey were present (although the resolution of some of theGCMs is coarser). The NHMM simulation, constrained as itis by the observed patterns, captures these details faithfully.

6.2. Projections

A second ensemble of simulations was generated using theA1B shear index. Because the seasonal shear cycle weakens ingoing to the twentyfirst century (Figure 2(b)), and owing tothe WSI1 dependence of state distributions (Figure 5(a)), thebalance in these simulations is tilted away from the wetterstates and toward the dry, resulting in an overall reduction inseasonal precipitation. Figures 10(a, b) show the mean JJASrainfall and the decrease owing to this effect, respectively.

Averaged over locations, seasonal rainfall is reduced via thismechanism by 4.3%.

Figure 10(d) indicates that the multimodel meanrainfall changes in the opposite sense, with seasonalvalues everywhere increasing. Thus, while monsoon dryingmay be consistent with dynamical GCM behaviour, itis not consistent with the total GCM response. Asdiscussed previously, the discrepancy is here attributed tothermodynamic effects, and resolved by scaling the NHMMdaily intensity distributions so that the simulated regionalmean fractional precipitation change (or equivalently,precipitation sensitivity) comes into agreement with themultimodel mean. This scaling (not to be confused withthe bias correction described in section 6.1.1) is applieddirectly to the means of the exponential distributions in theNHMM, from which wet-day amounts are drawn duringthe simulation step. All 192 distributions (two at eachlocation and for each state) are scaled identically; givensufficient predictive information it might be possible toimplement a more granular scaling scheme, but this wasjudged impractical for the present study.

A third set of simulations was then generated, using boththe A1B shear and the scaled rainfall distributions. Theresulting daily sequences reflect the dynamically inducedshift in state occupation statistics but are now consistentwith the GCM-inferred total precipitation change; they alsoincorporate the corrected seasonal-cycle pattern.

The multimodel mean fractional JJAS rainfall change(i.e. the average of the models’ fractional changes) for thestudy area (15◦–30◦N, 65◦–90◦E) amounts to an increaseof ∼ 5.7%. (The averaging was performed in this way inorder to reduce potential effect of mean rainfall biaseswithin the ensemble.) The required increase for the rainfallintensity distribution means was then found to be 10.4%,close to the simple difference between total and circulation-induced changes. The overall picture is thus one in whichonly about half of the thermodynamically driven potentialincrease in regional mean precipitation is realized, owing toa compensating weakening of the large-scale circulation.

It is perhaps of interest to note that the increase inmonsoon rainfall, as represented in the IPCC ensemble (andin the A1B scenario), occurs principally during 2030–2060.Before this period, there is little apparent change in levelfrom that of the second half of the twentieth century, whileafterward there are some significant decadal swings butagain little change in overall level. Conditional on shiftsin dynamics (as in the above analysis), adjustments to thescaling factor utilized would be expected to follow suchfluctuations. We do not speculate here about the causesor likely realism of rainfall variations in the multimodelensemble, but the presence of decadal-scale fluctuationsprovides, ex post facto, some justification for the use of 30-year averaging periods in computing the ‘time-slice’ statisticsdescribed herein.

The change in seasonal mean rainfall in the third ensembleof simulations is shown in Figure 10(c), and can be comparedwith the multimodel mean change in Figure 10(d). Thelarge-scale patterns are similar, with a broad minimumextending from northwest to southeast, and higher valuestoward the southwest and northeast, although, as expected,there are differences at finer scales. Given that the patternsof Figures 10(c) and (d) are the products of quite differentanalytical processes–the simple multimodel mean versus thedisaggregation into states, logistic regression of (1) and the



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Figure 9. Average daily rainfall from (a) multimodel mean climatology, (b) observations and (c) twentieth-century simulations.

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Figure 10. (a) The mean simulated JJAS twentieth-century precipitation field, (b) percent changes for 2070–2099, using only WSI1, (c) percent changesusing WSI1 and scaled daily intensity distributions, and (d) the multimodel mean percent precipitation change. In (b) and (c), the filled markers showvalues < 0, and unfilled markers > 0.

constraints imposed by observed patterns of variability–theagreement is noteworthy.

7. Fine-scale rainfall changes

7.1. Spatial patterns

Figure 10(b) indicates that the largest fractional rainfallreductions occur toward the northwest. These magnitudesare large in part because the twentieth-century valuesare small, but also because of the shift in relative statefrequencies: states 1, 2 and 4 are all drier in this region, inthe sense of both rainfall probabilities and daily intensities,than state 3. A shift away from state 3 (such as would beexpected from a weakening of WSI1) would thus tend to drythe northwest more than it would subregions where states 1,2, and 4 (which substitute for state 3 to varying degrees asWSI1 weakens) agree less well.

The largest fractional rainfall increase occurs at 17.5◦N,75.5◦E. Figure 10(b) shows that circulation changes alonebring about an increase in seasonal rainfall at this location.This occurs because of the low probability and intensityvalues in state 3, the intensity actually being less than that ofthe dry state. State 2, meanwhile, exhibits considerablyhigher rainfall frequency and intensity at this point.A dynamically mediated increase here, unlike at mostother locations, is thus consistent with a reduction instate 3 occurrence frequency. This increase increments thatmediated by the intensity scaling.

These observations confirm the basic functioning of theNHMM, modulating relative state occupation frequenciesaccording to the dependencies shown in Figure 5(a) whileincreasing intensity-distribution means uniformly across

stations within states. Figure 11(a) shows changes in statefrequencies, and indicates that the core monsoon wetstates 1 and 3 become less frequent, and the dry state 4more frequent, in going from the twentieth to twentyfirstcenturies.

Figure 11(b) shows daily twentieth- and twentyfirst-century rainfall distributions pooled across stations. Foralmost all bins, frequencies are greater in the twentyfirstcentury, the notable exception being the lowest, whichincludes dry days. Note that the log scale of Figure 11(b)causes the substantial difference in counts in the lowest binto appear quite small, when in fact it is sufficient to balanceall the differences of opposite sign in the higher bins. Theuniformity of response is no doubt related to the uniformapplication of distribution scaling.

7.2. Temporal aspects of precipitation change

As a downscaled example, we again consider the centrallocation 23.5◦N, 78.5◦E. This point receives most of itsrainfall from state 3, with secondary contributions fromstates 1 and 2 (in that order) and very little from state 4.The rainfall increase at this location is 2.6%, about halfthe regional mean. This increase occurs with a concurrentreduction in the frequency of rain, the number of dry days inthe 122-day JJAS season increasing by 7%, from an averageof 43.7 to 46.8. Mean daily intensity, meanwhile, increases by6.8%, from 14.7 to 15.7 mm. Applied to the slightly-reducednumber of wet days, this is sufficient to bring about thecomputed net increase.

The length of dry spells, averaged over stations, increasesby about the same percentage that rainfall decreases,4.5 ± 2.5% (1σ ). This amounts to only a small fraction



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Figure 11. Change in (a) relative state occupation frequencies and (b) precipitation (mm d−1), pooled over stations.

of a day, since the mean length of twentieth-century dryspells is ∼ 2.5 day. The percentage increase is smaller if oneconsiders only breaks longer than seven days, with a meanincrease of 1.1 ± 1.7%, the noisier statistics owing to thereduced number of samples. The mean dry-spell length, forspells longer than seven days, is ∼ 11 day.

Possible changes in the date of monsoon onset wereinvestigated using the agronomic definition of Moron et al.(2009). As employed here, onset is defined locally, as thefirst wet day (> 1 mm) of the first five-day wet spell in theseason for which accumulated rainfall is at least as greatas the climatological five-day wet spell, provided that thisperiod is not followed, during the succeeding 30 days, by aspell longer than 15 days in which accumulated rainfall isless than 5 mm. (The latter condition is imposed in orderto avoid ‘false positives’, since a significant dry spell wouldnegate the effects of the earlier rains.) Onset is thus sensitivenot only to accumulated rainfall but to the temporal detailsof how this rainfall is received.

With this definition it was found that monsoon onset isdelayed in going to the twentyfirst century, but only by anaverage of one day, with a standard deviation over locationsof 1.1 day. The interannual standard deviation in the classicalonset time, over Kerala state, is about 6 day, (Pai andRajeevan, 2007), so a shift of 1 day would in fact be difficultto discern, and by itself would be expected to have littleagronomic impact. This minimal change is in some contrastto the shift that might be expected from changes in WSI1alone (Figure 2(b)). Based on the 15 June value from the dailyinterpolation, the change in WSI1 suggests an onset delayof 4 day, as well as a 3 day advance in monsoon withdrawal,leading to a net reduction in wet-season length of about7 day. Evidently the increase in intensities is sufficient tocompensate for a delay in ‘dynamical’ monsoon onset.The agronomic calculation also suggests an insignificantdelay in monsoon withdrawal of 0.1 ± 0.8 da, which wouldhave the effect of compensating very slightly the estimatedreduction in wet-season length. It should be borne in mindthat other factors, including temperature and atmosphericcarbon dioxide concentrations, may also have appreciableeffects on crop yields (e.g. Challinor et al., 2009).

The above result may be contrasted with that of Nayloret al. (2007) for Indonesian rice agriculture. Those authorsfound significant increases, for year 2050, in the probabilityof onset delay exceeding 30 day, particularly in EasternJava and Bali. Methodology, onset definition, region and,

not incidentally, monsoon system differ from those of thepresent study, but the considerably greater climate sensitivityimplied is notable.

A shift toward higher intensities may be expectedto increase the frequency of heavy rainfall events. Thistendency is apparent in the projected distributions shown inFigure 11(b), but for the A1B scenario the changes are notdramatic. The 90th percentile for daily intensity during thetwentieth century ranges from 19 to 51 mm across locations.The intensity increase at this percentile is 9.5 ± 2%, fairlyuniform across stations. Holding amounts constant, theshift in percentiles is only about–1.5, meaning that eventscorresponding to the twentieth-century threshold occuronly slightly more often during 2070–2099. It should bekept in mind that the present analysis is not optimized inparticular for inference regarding extreme events, for whichother distributional forms may be more appropriate (e.g.Coles, 2001).

8. Dynamics and thermodynamics

Although the experimental design does not contemplatea direct estimation of the thermodynamic contribution tomonsoon rainfall changes, the use of a circulation indexas covariate implies at least an approximate partitioning ofeffects along dynamic/thermodynamic lines. An attempt ismade here to validate this partitioning by comparing thecomponents of rainfall change estimated in this way withresults obtained via an independent method.

Bony et al. (2004) performed such a decomposition forcloud radiative forcing, by first inferring an empiricalrelationship between the forcing and the probabilitydistribution of 500 mb vertical velocity, ω500. Shifts incloud radiative forcing due solely to changes in thisdistribution were then identified as dynamically induced,while independent changes were assigned thermodynamicorigin. (A small component was found to result fromcovariation.) Since rainfall is strongly coupled to thevertical velocity, such an analysis is also appropriate here.Accordingly, probability distributions were computed forω500 and rainfall using pooled monthly JJAS values forthe region and time periods of the preceding analysis. Thecomputations were carried out using a sub-ensemble of 18models from the 23-member group for which the requisitevertical velocity fields were available. The multimodel meanfractional rainfall change, now weighted by the probability



distribution function of ω500, was 5.3%, quite similar to themultimodel mean as previously computed. This indicatesthat the inferred quantitative association between ω500 andprecipitation is a reasonable one.

The dynamically induced regional mean rainfall changecomputed in this way and averaged over GCMs is −2.3%,the thermodynamic component 7.2%. There is an additionalsmall contribution of 0.4% arising from covariation betweenchanges in the probability distribution function of ω500

and precipitation. (In the foregoing analysis, this would beassigned to the non-dynamical component.) The dynamicalshift is only about half of that inferred with the NHMM, butin both analyses the thermodynamically mediated increaseis compensated by dynamical effects to a similar degree: by32% in the ω500 decomposition, 41% in the NHMM analysis.Thus the overall picture does not differ greatly between themethods. The quantitative discrepancy may be attributableto use of daily data in the NHMM analysis, as opposed tomonthly values for the ω500 decomposition.

9. Uncertainty

The tendency of model representations to improve withincreasing spatial scale is a recognized characteristic ofGCMs (IPCC, 2007), and can be ascribed, at least partly, tothe finite resolution of model grids. The analysis presentedabove utilizes GCM information on the largest spatial scalesconsistent with the statistical and physical constraints ofthe problem: if a significantly larger region were chosen,the degree of applicability to the study area would comeinto question, since there are significant local variations inrainfall change within the larger Indian Ocean domain.

Figure 12 illustrates intermodel spread for the dynamiccomponent and for the total JJAS precipitation change forthe study region (left two boxes), as well as the spread intotal JJAS precipitation change for two larger regions. Toobtain the first of these distributions, the NHMM was fittedto the IMD data using each GCM’s twentieth-century shearcycle. Simulations were then generated for the twentiethand twentyfirst centuries, as was done previously using themultimodel mean values. The third and fourth boxes showdistributions for total JJAS (land-only) precipitation changefor the IPCC South Asia region (‘SAS’) and for the globaltropics (‘TRO’). Means for the four distributions are –4.0%,5.7%, 9.5% and 3.5%, respectively. For the study area,uncertainty is dominated by the spread in total precipitationchange, for which the standard deviation is about 14%. Thecorresponding value for region SAS is 4.2% and for globaltropical land 1.5%.

The relatively large intermodel spread in rainfall changeover India is intrinsic to the CMIP3 ensemble, and for thepresent analysis represents irreducible uncertainty. This isconsistent with deployment of the NHMM as a downscalingtool, rather than as a method of superseding the GCMswith respect to large-scale climate projection. There is lessuncertainty associated with the rainfall change in regionSAS, so one might consider applying the NHMM to thislarger region, but the daily observational data at hand islimited to India, precluding such an option for the present.The necessity of a homogeneous shear cycle also limits theextent the modelled domain. The area of region SAS is aboutdouble that of the study region.

∆pr,

%

Figure 12. Distributions of area-mean rainfall change for the shear-inducedcomponent (Dyn), for the study area 15◦–30◦N, 65◦–90◦E (Total), for theIPCC South Asia region 5◦–30◦N, 65◦–100◦E (SAS) and for the globalTropics 30◦S–30◦N (TRO). Values are percent changes for JJAS rainfallover land.

10. Summary and conclusions

A non-homogeneous hidden Markov model was utilized toinvestigate potential changes in Indian monsoon summer(JJAS) rainfall, comparing the 2070–2099 period in the IPCCA1B scenario with the second half of the twentieth century.A dynamical index of monsoon circulation was utilized asthe exogenous predictor; rainfall observations derive froma 1◦ gridded precipitation product produced by the IndiaMeteorological Department.

The dynamical index, in the form of a 23-model ensemblemean, conditions the NHMM transition matrix, and byextension the timing and relative occupation frequencies ofthe NHMM’s hidden states. A weakening of the circulationin going to the twentyfirst century, as translated by thismechanism, induces rainfall reductions at most locations,with an area-averaged JJAS decrease of ∼ 4.3%. At the sametime, multimodel mean precipitation increases by ∼ 5.7%.A ‘non-dynamical’ increase of ∼ 10.4%, obtained by scalingthe daily rainfall intensity distributions, is required to bringthe area-averaged NHMM and multimodel mean changesinto agreement. This factor is assigned to thermodynamicprocesses. An independent method of decomposingprecipitation change into dynamic and thermodynamiccomponents yields a similar proportionality. The scalingprocedure is performed outside the NHMM fitting process,owing to the quite different time-scales on which the relevantprocesses operate and the fact that conditions correspondingto the late twentyfirst-century basic state fall outside therange of twentieth-century variability.

Shifts in subseasonal rainfall statistics arise from the statedecomposition effected by the NHMM, as conditioned onthe dynamical predictor, and from the concomitant scalingof intensity distributions. These shifts include small increasesin the number of dry days, in the average length of dry spells,in mean daily intensity, and in the exceedance frequency ofnearly all daily rainfall percentiles. No significant delay inmonsoon onset was indicated, owing to a compensation of‘dynamical’ delay by increased daily intensities.

These inferences are drawn from stochastic daily rainfallsequences generated by the NHMM. Such sequences haverealistic daily distributions and, given the appropriateobservational data, may be generated for arbitrarily preciselocations. As such, they are expected to be useful in a



variety of applications, such as the driving of streamflowor agricultural models, and should be helpful in testing,thus potentially increasing, the resilience of proposedadaptations.

Owing to significant subregional variations in precipita-tion change, GCM information must be extracted at a scaleno larger than that of the study area. Although there existsa reasonably strong model consensus regarding the sign ofprojected change for this region, agreement is not universaland there is considerable intermodel spread. In consideringthe results presented, this uncertainty should be borne inmind.

The modelled shifts tend to be smooth, giving noindication of possible abrupt changes in rainfall statistics.This smoothness should not be taken as a guarantee thatsuch changes are outside the realm of possibility, but ratherarises as a consequence of model assumptions regardingstructural stationarity of the large-scale flow, including theanomalous departures associated with the NHMM’s hiddenstates. The smooth dependence of the equilibrium statevector on WSI1 (Figure 5(a)) is implicit in this assumption.The uniformity with which the thermodynamic scaling isapplied may also play a role in the generally ‘incremental’nature of the modelled response. None of this is to saythat such a response is unrealistic, however; the stationarityassumption may well be a reasonable one, at least for modestdepartures from the twentieth-century basic state. But it iswell to keep in mind the role of model structure andassumptions, in contemplating the results of this study.

The NHMM fits a two-exponential mixture to the dailyintensity distributions, permitting a degree of independencein fitting distribution means and tails. Scaling is applieduniformly, across locations, states and both exponentials.While this is consistent with the use of GCM informationonly at large spatial scales, a more complex statistical model(accompanied by suitable conditioning information) mightpermit some spatial disaggregation of distribution scaling,or differentiation in its application to the two exponentialparameters.

The feasibility of the modelling enterprise describeddepends to a significant degree on the existence ofspecific regional dynamics, which we have shown to berobust among the GCMs of the multimodel ensemble.We believe that the methodology is in fact more broadlyapplicable, but extension to other settings would necessarilydepend on regional details, including the identification of asuitable predictor field (or fields). Ongoing work includesexploration of this possibility.

A key assumption is that the multimodel ensemble isthe optimal source of quantitative information regardingexpected climatic changes. The NHMM is not utilized for thisaspect of the projections, but rather, for the disaggregation ofdaily rainfall variability over a network of precise locations,for which the GCMs provide only ‘generalized’, i.e. large-scale, information. These complementary tools are thusfused, in a process that exploits the most appropriateinformation from each. While the merging of such datasources per se is not a unique idea, we believe thatthe deployment presented herein, in the regional settingof the Indian monsoon and respecting basic theoreticalexpectations through the use of a dynamical exogenousvariable and the implied thermodynamic scaling, representsa novel application.

Acknowledgements

Many colleagues from the International Research Institutefor Climate and Society, Lamont–Doherty Earth Obser-vatory of Columbia University and the NASA GoddardInstitute of Space Studies have provided the authors withhelpful insights and comments. The NHMM computationswere performed using the MVNHMM toolbox∗, written andmaintained by Sergey Kirshner, Purdue University. We aregrateful to M. Rajeevan of the India Meteorological Depart-ment for providing the rainfall data, and acknowledge themodelling groups, the Program for Climate Model Diagnosisand Intercomparison (PCMDI) and the WCRP’s WorkingGroup on Coupled Modelling (WGCM) for their roles inmaking available the WCRP CMIP3 multi-model dataset.Support of this dataset is provided by the Office of Science,US Department of Energy. This research was supported byUS Department of Energy grants DE-FG02-02ER63413 andDE-FG02-07ER64429.

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