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©2014 Royal Statistical Society 0035–9254/14/63000
Appl. Statist. (2014)
A Bayesian hierarchical downscaling model for
south-west Western Australia rainfall
Yong Song,
Commonwealth Scientific and Industrial Research Organisation, Melbourne,
Australia
Yun Li and Bryson Bates
Commonwealth Scientific and Industrial Research Organisation, Perth,
Australia
and Christopher K. Wikle
University of Missouri, Columbia, USA
[Received September 2012. Revised October 2013]
Summary. Downscaled rainfall projections from climate models are essential for many
meteorological and hydrological applications. The technique presented utilizes an approach
that efficiently parameterizes spatiotemporal dynamic models in terms of the close association
between mean sea level pressure patterns and rainfall during winter over south-west Western
Australia by means of Bayesian hierarchical modelling. This approach allows us to understand
characteristics of the spatiotemporal variability of the mean sea level pressure patterns and
the associated rainfall patterns. An application is presented to show the effectiveness of the
technique to reconstruct present day rainfall and to predict future rainfall.
Keywords: Bayesian hierarchical modelling; Climate change simulation; Downscaled rainfall
projection; Downscaling; Empirical orthogonal functions
1. Introduction
There is evidence that the global pattern of annual streamflow trends is partly due to the effects of
anthropogenic climate change resulting from a poleward expansion of the subtropical dry zone
(Lu et al., 2007). However, uncertainties in projected rainfall changes for later this century plague
estimates of the effects on future run-off and water storage (Milly et al., 2008). In particular,
several difficulties are associated with interpreting changes in variables simulated at a resolution
of 100–200 km in terms of changes to be expected at smaller catchment scales. This affects
the ability to simulate, for example, expected changes in surface run-off and streamflows into
water catchments. Resource managers, who are required to make long-term decisions regarding
future infrastructure demands (e.g. the need for new reservoirs, pipelines and drainage), have
an increasing need for reliable estimates in these changes to surface run-off and streamflow.
For instance, south-west Western Australia (SWWA) has already been severely affected by a
downturn in rainfall and even more serious reductions in run-off (Bates et al., 2008), and there
Address for correspondence: Yong Song, Land and Water, Commonwealth Scientific and Industrial Research
Organisation, PO Box 56, Graham Road, Highett, Melbourne, VIC 3188, Australia.
E-mail: Yong.Song@csiro.au
2Y. Song, Y. Li, B. Bates and C. K. Wikle
is growing concern that this pattern of reduced rainfall may be extending to the Australian
eastern states.
SWWA (south-west of the line joining 30◦S, 115◦E, and 35◦S, 120◦E) receives the bulk of its
annual rainfall during the winter months (i.e. May–October). Furthermore, rainfall during the
austral winter months, from June to August, is primarily controlled by large-scale atmospheric
circulation patterns, as opposed to convectively dominated systems that govern the northern
regions and during the summer (Smith et al., 2008). The winter weather patterns include incur-
sions of cold air outbreaks from higher latitudes that are associated with depressions originating
in either the Indian Ocean or Southern Ocean. In contrast, the persistence of anticyclones is
associated with relatively dry conditions. Variability of rainfall is strongly linked to the relative
strength and location of these weather patterns that subsequently affect wind strength and direc-
tion (Sturman and Tapper, 1996). Several studies (e.g. Smith et al. (2000)) have documented the
strong inverse relationship between the mean sea level pressure (MSLP) and SWWA regional
winter rainfall on monthly and seasonal time scales. Allan and Haylock (1993) suggested that
southern Australian winter rainfall is modulated by a long-term MSLP pattern, with a pro-
nounced trend evident in recent decades. This large-scale pressure pattern is referred to as the
southern annular mode (SAM) and influences rainfall across southern parts of the Australian
continent in some seasons (Meneghini et al., 2006; Hendon et al., 2007). Li et al. (2005) modelled
winter extreme rainfall over SWWA and its associated changes with the SAM. They showed
that the upward trend of the SAM is consistent with the decrease in extreme winter rainfall
observed over SWWA that has been apparent since the mid-1960s.
Climate models sometimes have difficulty in realistically simulating rainfall at regional, and
smaller, scales (Hewitson and Crane, 2006) and this affects the confidence that can be placed in
such models simulating future change scenarios. The difficulty arises for various reasons, includ-
ing the fact that different models use different parameterization schemes for various physical
processes. Therefore, the results not only reflect the effects of forced changes but also reflect
internal variability resulting from the chaotic nature of the system (R¨
ais¨
anen, 2007). In particu-
lar, ensemble rainfall projections for Australia are characterized by relatively large uncertainties
(Whetton et al., 2005). For example, results from increased carbon dioxide experiments (Cai and
Watterson, 2002; Hope, 2006) show that SWWA becomes drier as mid-latitude MSLP increases.
Although these results are useful, it also needs to be recognized that the degree of drying that is
suggested by these global climate models (GCMs) is accompanied by considerable uncertainty,
particularly as the simulated rainfall levels for present-day conditions tend to underestimate
those that have been observed. Consider the seven GCMs that were analysed by Hope (2006).
The percentage errors that are associated with estimates for present-day (1961–2000) June and
July mean rainfall for SWWA range from −80% to 0%. These same models yield percentage
changes for later this century (2081–2100) that range from −3% to −25%, with the average
reduction being −13%.
Various approaches to overcoming the uncertainties accompanying future climate change
projections are being developed, including the assessment of the performance of individual
models as a guide to the reliability of their predicted changes (Maxino et al., 2008; Perkins
et al., 2009; Suppiah et al., 2007; Whetton et al., 2007; Perkins and Pitman, 2009; Smith and
Chandler, 2010). Downscaling is another method that can potentially assist in the assessment of
climate models. A simple test for a model is that it provides an accurate estimate for present-day
regional rainfall and it should also simulate the observed relationship between regional rainfall
and other key variables, such as the MSLP. If these criteria can be satisfied, then it is arguable that
any simulated changes in rainfall are more likely to be reliable than could otherwise be estimated.
Downscaling not only provides an indication of any such relationships; it can also potentially
Downscaling Model for South-west Western Australia Rainfall 3
provide alternative estimates for rainfall changes if the simulated changes in the key variables
are believed to be more reliable than the rainfall estimates themselves (e.g. Benestad (2001)).
There are two main approaches to downscaling: dynamical (Christensen et al., 1998;
Murphy, 1999; Schmidli et al., 2006) and statistical (Zorita and von Storch, 1997, 1999; von
Storch et al., 2000). The former approach involves nesting regional models of relatively high
horizontal resolution with a coarser resolution GCM. The latter approach is based on historical
(empirical) studies of the relationship between the large-scale climate anomalies and local climate
fluctuations. There are numerous ways to develop statistical downscaling models (Zorita and
von Storch, 1997; Fowler et al., 2007), but it is important to note that a statistical downscaling
approach to climate change implicitly assumes that any derived historical relationships also
hold for the future (Wilby, 1997).
Several studies have involved statistical downscaling as a tool in seasonal rainfall predictions
for Australia (Hughes et al., 1999; Charles et al., 2004). Hughes et al. (1999) developed a non-
homogeneous hidden Markov model for SWWA using three atmospheric predictors: the mean
large-scale value for the MSLP, the north–south MSLP gradient and the east–west gradient
in geopotential height at 850 hPa, RH850. They found that the model could provide credi-
ble reproductions of precipitation and its spatial association as well as dry and wet spell length
statistics for a range of site locations. In a subsequent study, Charles et al. (2004) also recognized
that the dewpoint temperature depression at 850 hPa is an important factor and demonstrated
some predictability of rainfall occurrence probabilities. However, there was less success with
predicting actual rainfall levels. Another technique, which was referred to as analogue-based
downscaling, was used in an attribution study of the rainfall decline over SWWA (Timbal
et al., 2006) and in interpreting climate change results for the whole of Australia (Timbal and
McAvaney, 2001; Timbal, 2004; Timbal and Jones, 2008). Hope et al. (2006) used a self-
organizing map to identify the typical synoptic patterns that govern daily rainfall events over
SWWA and showed how a change in the frequency of rainfall bearing events could explain the
observed decline in winter rainfall since the mid-1970s. Li and Smith (2009) applied a principal
component regression (PCR) technique to downscale MSLP patterns to rainfall over southern
Australia during winter.
In recent years, Bayesian methods have attracted the attention of climate modelling
researchers by providing a flexible modelling alternative in statistical downscaling approaches.
Mendes et al. (2006) modelled the spatiotemporal relationships between large-scale sea level
pressure fields and local daily precipitation in a simple Bayesian hierarchical modelling (BHM)
framework. Coelho et al. (2006) used a Bayesian forecast assimilation model to calibrate the
rainfall prediction that was generated by three climate models and found that the process
improved the rainfall prediction skill. Berrocal et al. (2010) proposed a Bayesian statistical model
that downscaled numerical model predictions from the level of grid cells to that of observational
points within climate monitoring networks. Fasbender and Ouarda (2010) used a geographi-
cal regression model for downscaling minimum and maximum temperatures of coarse scale
atmosphere–ocean coupled general circulation model output to finer scale application.
The research that is presented in this paper is focused on the area of Bayesian modelling and
approaches the problem from a different perspective. We utilize an approach that efficiently
parameterizes spatiotemporal dynamic models in terms of the association between the MSLP
patterns and rainfall during winter over southern Australia. This approach, implemented within
a BHM framework, allows the modelling of the complex spatiotemporal dynamical processes
within regular and irregular geometric domains. The BHM approach allows us to account
directly for multiple sources of uncertainties. Also, because of its reduced rank formulation, the
Bayesian spatiotemporal dynamical model can be implemented in high dimensions.
4Y. Song, Y. Li, B. Bates and C. K. Wikle
This paper is organized as follows. In Section 2, we describe the data that are under study. In
Section 3, we present the principal component analysis of the MSLP fields and determine the
dominant principal components. In Section 4, we present our Bayesian hierarchical downscaling
model (BHDM). The description of the model implementation and the analysis of the results
is given in Section 5. Section 6 is devoted to the description of the model application to climate
change simulation. Conclusion and extensions of the current model are presented in Section 7.
2. Data
The MSLP data were extracted from the National Centers for Environmental Prediction (NCEP)
–National Center for Atmospheric Research global reanalysis archived data set with a horizontal
resolution of 2:5◦×2:5◦covering the region 60◦S–0◦S, 90◦E–180◦E. Additional information
is available on line from http://www.cdc.noaa.gov/index.html (Kalnay et al., 1996).
The rainfall data that are analysed here were provided by the National Climate Centre of the
Bureau of Meteorology and consist of 53 observation locations in SWWA. In particular, the
target region was selected as the region south-west of the line joining 30◦S, 115◦E and 35◦S,
120◦E (Fig. 1). Average winter (June–August) rainfall totals for the period 1948–2004 were gen-
erated.
The model output that was used to demonstrate the application of the downscaling method-
ology was generated as part of the Commonwealth Scientific and Industrial Research Organ-
isation’s (CSIRO’s) participation in the ‘World climate research programme coupled model
inter-comparison project’, phase 3. The specific model that was used was the CSIRO mark 3.5
climate model (Gordon et al., 2002; Hirst, 2007), and the climate change experiment was based
on the A2 emissions scenario. Rotstayn et al. (2009) assessed the mark 3.5 climate model in terms
of its ability to simulate El Ni˜
no–southern oscillation teleconnection patterns over Australia and
found that it performs as well as, if not better than, other international models.
3. Bayesian hierarchical downscaling model for rainfall
Li and Smith (2009) proposed a PCR model for downscaling large-scale MSLP patterns to
rainfall over regions across southern Australia. In this section, we give a brief introduction to
this PCR approach and its strengths and limitations. We then introduce a BHDM downscaling
approach to account for these limitations.
In general, a PCR model for downscaling large-scale MSLP patterns to local rainfall can be
given by (Li and Smith, 2009)
Yt=ΨZt+ςt,.1/
where Yt=.y1,t,:::,ym,t/is the vector of observed rainfall values at locations 1, :::,mfor time t,
and Zt=.z1,t,:::,zk,t/is the vector of length kthat results from projecting the MSLP anomalies
over the wider Australian region onto the first kempirical orthogonal functions (EOFs) (i.e.
Xt=ΦZt, where Φis the n×ktruncated EOF matrix and Xt=.x1,t,:::,xn,t/is the vector of the
gridded MSLP observations at time t). Here Ψis a k×mparameter matrix and ςtis the vector of
residuals. The jth element of Ytand the associated regression equation with summation notation
illustrates the simplicity of the model:
yj,t=
k
i=1
ψi,jzi,t+ςj,t,.2/
where ψj=.ψ1,j,:::,ψk,j/can be estimated on the basis of the least squares fit to the T-year
training data {.Xt,Yt/:t=1, :::,T}.
Downscaling Model for South-west Western Australia Rainfall 5
Fig. 1. Locations of 53 observation stations over SWWA
Clearly, the spirit of this method can be described as one that derives the EOFs from the
anomalies of large-scale MSLP fields in the training period, and then projects these anomalies
in the prediction period onto these EOFs. On the basis of the estimated parameter vector ψj, the
MSLP anomalies are downscaled onto the changes in local rainfall. The PCR downscalingmodel
shows robust predictive ability of local rainfall, particularly at regional scales (Li and Smith,
2009). Although the PCR model has been proven to be an effective downscaling model, the
model has some weaknesses. Firstly, it tends to ignore the spatial correlation between different
climate recording stations or regions. Secondly, since the parameter vector ψjis estimated by
using a least squares fit to the fixed T-year training data {.Xt,Yt/:t=1,:::,T}, it tends to
ignore the uncertainties from some underlying processes. Thirdly, the fixed number of principal
components that are used in the prediction equation (equation (2)) may induce small-scale
uncertainty in the MSLP data decomposition and, thus, downscaling uncertainty. Additionally,
the linear-regression-based PCR model may not realistically account for forecast uncertainties.
To address the limitations of the PCR downscaling model, a flexible framework is needed
to accommodate complex relationships between large-scale atmospheric circulation variables
6Y. Song, Y. Li, B. Bates and C. K. Wikle
and local climate information, as well as incorporating various sources of uncertainty. To
accommodate these aspects, we adopt a BHM approach which provides an effective framework
to account for uncertainty in the data, the processes and the parameters. Indeed, it is known
that the BHM may provide a coherent and statistically meaningful assessment of uncertainty in
spatiotemporal settings (Cressie and Wikle, 2011). More importantly, with a BHM framework,
the link between large-scale atmospheric circulation variables and local climate information can
be effectively modelled through conditional independence on dynamical processes of interest.
Consequently, such approaches can explicitly account for the uncertainty due to downscaling
from climate fields with different spatial scales. In our study, we assume that the relationship
between the large-scale MSLP circulation and local rainfall are independent conditional on
a first-order Markov process. The details of the framework, including the data model, process
model and parameter model, are described below.
3.1. Data model
The first stage of the BHM hierarchy is the data model. Let Xtdenote the n×1 vector (n=925
grid points covering the region 60◦S–0◦S, 90◦E–180◦E in our study) of the gridded MSLP
observations at time t. The data model for MSLP can then be given by
Xt=Φxat+"t,"t
IID
∼N.0, σ2
"In/,.3/
where Φxis an n×kmatrix of orthonormal basis functions and at=.a1,t,:::,ak,t/isak×1
vector of time varying random coefficients. Here, "tis assumed to represent measurement error
and small-scale random variation. Consequently, we assume that the error process in the MSLP
data model is uncorrelated across space and time. This is reasonable since MSLP typically
shows fairly large-scale variation at mesoscales and synoptic scales, and is thus likely to be well
represented by a fairly low dimensional hidden dynamic process, if the associated basis functions
represent large-scale modes of variability, such as with EOFs. Following Li and Smith (2009),
we chose to use EOFs as our basis functions in this study.
We note that atis considered as a random dynamical process. We emphasize that the modelled
ats need not coincide with the original Zts (equation (1)) determined by the EOF decomposition.
The modelling of the dynamical relationships will be implemented on the hidden process at,
which has a much lower dimension than the observed MSLP pattern (i.e. kn). This rank
reduction is critical for spatiotemporal statistical models in high dimensions.
Let Ytdenote the m×1 vector (m=53) of rainfall observations at time t, corresponding to
mspatial locations. Then Ytcan be modelled in the data stage as
Yt=Λat+ηt,ηt
IID
∼N{0, σ2
ηR.θ/},.4/
where Λis an m×kdownscaling projection matrix that provides a convenient framework to
account for downscaling from large-scale MSLP patterns to regional winter rainfall in the target
region; ηtis the downscaling error vector. The correlation matrix R.θ/represents the residual
spatial correlation based on the Euclidean distance dbetween locations where θis the spatial
range parameter in a stationary and isotropic exponential covariogram model:
R.θ/=exp.−θd/: .5/
More complicated spatial models could also be considered (e.g. Cressie and Wikle (2011)). How-
ever, it is reasonable to assume that the non-stationary spatial components of the precipitation
field are largely accounted for by the hidden dynamic process at, suggesting the plausibility of
the simple stationary spatial model.
Downscaling Model for South-west Western Australia Rainfall 7
The hierarchical data model provides a natural model framework to combine data sets (e.g.
Wikle (2010)). Conditioned on the dynamical process at,Xtand Ytare assumed to be indepen-
dent. In this context, the combined data model can be written as
[Yt,Xt|at,Λ,σ2
",σ2
η,θ]=[Yt|at,Λ,σ2
η,θ][Xt|at,σ2
"]:.6/
The conditional independence of Xtand Ytdoes not mean that these data sets areunconditionally
independent. Indeed, the dependence exists between the two data sets owing to the dynamical
process at. The fact that both the MSLP and precipitation data are dependent on the same latent
dynamical process is a novel contribution of our approach to the downscaling literature. This
allows atto learn from both components.
3.2. Process model
In the second stage of the model hierarchy, we specify the model for the dynamical process
at. In previous studies, several researchers have provided evidence that MSLP anomalies can
be modelled in a Markov model framework (e.g. Bates et al. (1999) and Donald et al. (2006)).
In our BHM framework, we model the latent process as a first-order Markov process (vector
auto-regression of order 1) with Gaussian noise:
at=Hat−1+ςt,ςt
IID
∼N.0, Σς/,.7/
where Hisak×kpropagator matrix, and ςtis a Gaussian noise process that is independent
in time. Here Σςis the covariance matrix, where Σ−1
ςis assumed to have a Wishart distribu-
tion.
The propagator matrix His parameterized generally to capture the complex dynamical inter-
actions across space–time that are realistic for the evolution of MSLP and precipitation fields
(Cressie and Wikle, 2011). Following Percival et al. (2001), a first-order auto-regressive (AR(1))
process can be fitted to the winter-averaged sea level pressure time series for the Aleutian low
(the North Pacific index). Thus, it is reasonable to model the process atlinearly as a first-
order Markov process with Gaussian spatial noise. We could fit a more sophisticated process
model, such as a general quadratic non-linear model, or base the parameterizations on classes of
deterministic dynamical models (e.g. partial differential equations, integrodifference equations,
matrix models and agent-based models) (Wikle and Hooten, 2010). However, most of the focus
for the application is on downscaling the MSLP anomalies onto the local rainfall pattern. From
this point of view, the first-order Markov process model is a reasonable choice for building a
parsimonious model and can be motivated as linearization of the underlying dynamics.
One could consider basis functions such that expansion coefficients are indexed in physical
space (e.g. discrete kernel convolutions or predictive process bases; Cressie and Wikle (2011)).
However, in the context of atmospheric data such as of interest here, there are likely to be
multiscale interactions and it is common to use a spectral basis representation (e.g. orthogonal
polynomials, normal modes or empirical orthogonal functions) to accommodate the dynamical
interactions across scales (Cressie and Wikle, 2011). In this case, the relatively low dimensional
nature of the coefficients allows us to estimate an unstructured propagator matrix Hwhich adds
flexibility.
3.3. Parameter model
To complete the BHM framework, we need to specify the distributions of parameters from the
previous stages. For simplicity, we consider basic conjugate parameter models. The following
8Y. Song, Y. Li, B. Bates and C. K. Wikle
prior distributions (with hyperparameters chosen such that the priors are relatively vague) are
considered for the unknown parameters:
h|·≡vec.H/|·∼N.μh,Σh/,
λ|·≡vec.Λ/|·∼N.μλ,Σλ/,
σ2
η∼inverse-gamma.qη,rη/,
σ2
"∼inverse-gamma.q",r"/,
Σ−1
ς∼Wishart{.vςsς/−1,vς},
a0∼N.μ0,Σ0/,
θ∼gamma.qθ,rθ/: .8/
Note that a prior distribution for process atis specified at t=0. Hence, we can write the joint
posterior distributions of a0,a1,:::,aT,Λ,H,σ2
η,θ,σ2
"and Σ−1
ς, given the observed rainfall and
MSLP, as
[a0,a1,:::,aT,Λ,H,σ2
η,θ,σ2
",Σς|Y1,:::,YT,X1,:::,XT]
∝T
t=1
[Yt|Λ,at,σ2
η,θ] T
t=1
[Xt|at,σ2
"] T
t=1
[at|at−1,H,Σς][a0][Λ][H][σ2
η][θ][σ2
"][Σς]:.9/
Samples from the posterior for conjugate model parameters can be calculated by using a Gibbs
sampler, and the non-conjugate parameter θcan be sampled via Metropolis–Hasting updates
within the Gibbs sampler (Robert and Casella, 2004). The quality of the model performance is
not overly sensitive to the choice of hyperparameters. The Markov chain Monte Carlo simulation
was run for 2000 iterations with a ‘burn-in’ period of 500 iterations. Simple diagnostic checks
were performed; convergence is verified through both visual inspection and a diagnostic check
(Gelman and Rubin, 1992).
3.4. Selecting the number of empirical orthogonal functions
Atmospheric data sets typically have a large number of spatial locations. EOF decomposition of
such data sets provides an efficient way to represent the dominant patterns in their time varying
nature as well as to explain the maximum variation in the space–time system. In addition, the
computational efficiency can be improved while retaining the most important information of
the variability in the system (e.g. Berliner et al. (2000)). One of the crucial issues in applying
this dimension reduction technique is to choose the optimum number of EOFs to be retained,
which is a problem of model selection. Underestimation of the number of EOFs can lead to
poor estimation, along with the loss of important information. In contrast, overestimation can
bring in extraneous information and possibly overfit the data. This can also then reduce the
computational efficiency and result in poor predictive performance.
Analysis of the MSLP data yields eight leading modes from the EOFs, which explain 95%
of the total variance. Each mode explains a certain pattern of the MSLP anomalies. All these
modes are strongly linked to both the June–August southern oscillation index and the SAM
index (Li and Smith, 2009).
Our focus is on the skill of the proposed BHDM in forecasting future rainfall. Among many
model selection techniques, cross-validation is an effective method for model selection based
Downscaling Model for South-west Western Australia Rainfall 9
on the assessment of predictive skill. The simplest form of cross-validation is leave-one-out
cross-validation (LOOCV). However, since LOOCV needs to use each single observation from
the original sample for validation, it can be computationally inefficient. In addition, Efron
(1983) showed that LOOCV, although giving an unbiased estimate of error, tends to inflate the
experimental variance. Shao (1993) observed that LOOCV is inconsistent and tends to select
an unnecessarily large model. Also, the purpose of this study is to downscale the large-scale
MSLP patterns to rainfall over southern Australia during winter. If one station has been left
out, we cannot assess the downscaling predictive skill in the specific station. Picard and Cook
(1984) proposed Monte Carlo cross-validation (MCCV), which randomly splits the data set
into training and validation data. Then, for each spilt, the observations in the training data are
used for fitting the model, whereas the observations in the validation data are used for assessing
the predictive skill. Shao (1993) showed that MCCV is asymptotically consistent and has a
larger probability of better performance than LOOCV. Xu et al. (2004) performed real case
comparisons and found that MCCV can avoid selecting an unnecessarily large model and can
also decrease the risk of overfitting. The downside of MCCV is that the performance is subject to
the selection of the validation subsets. Since the validation subsets are randomly selected, there
is a possibility that some observations may not appear in all of the validation subsets, whereas
some observations may appear more than once. Therefore, to reduce variability, a large number
of splits is expected. However, such a procedure is difficult to implement for computationally
expensive models.
As a more computationally efficient compromise, a rule-of-thumb approach is proposed here,
which follows the fundamental principles of MCCV. The model is calibrated by using the rainfall
and MSLP data from 1951–1990. From these data, we create 14 subsets with 10 observations in
each subset. For the first four subsets, the data are grouped into 10-year periods (i.e. 1951–1960,
1961–1970, 1971–1980 and 1981–1990) and data are selected from each period to ensure that
a representative set of data is used for training and validation. The remaining 10 subsets are
constructed by randomly selecting 10 observations from the period 1951–1990. It should be
noted that subsets might overlap (i.e. the same data may be present in two or more subsets).
According to the test, a single subset is retained as the validation data for testing the model,
and the remaining observations are used as training data. The cross-validation process is then
repeated 14 times, with each of the 14 subsets used exactly once as the validation data. The
cross-validation method selects the model by minimizing the mean-squared prediction error.
Specifically, we split the data into a validation set svand a calibration set scwhere svcontains nv
observations and sccontains ncobservations. The selection criterion can then be summarized
as
CV.k/ =1
14nv
14
i=1
Ysv.i/ −ˆ
Yk,sv.i/2,.10/
where kis the number of EOFs that we retain, and Ysv.i/ is the subvector of observations
corresponding to the ith validation set. Note that ˆ
Yk,sv.i/ is the predicted vector of Ysv.i/ by
using the BHDM with kEOFs. In this setting, the size of the validation set is nv=10.
The results are presented in Table 1 and we see that the model with the first six EOFs gives
the lowest cross-validation score. Thus, from a prediction point of view, this suggests that it is
reasonable to fit the model with the first six EOFs.
4. Downscaling analysis and results
Estimation of posterior distributions of model parameters provides important information on
10 Y. Song, Y. Li, B. Bates and C. K. Wikle
Tab le 1 . Values of CV(k)
for the prediction set
Number of CV(k)
EOFs
3 44.98
4 41.49
5 41.23
6 39.98
7 40.57
8 40.51
the underlying dynamics. Considerable uncertainty can be observed in the posterior distribution
of the at-process (of the first six leading latent process components defined by equation (7)) in
the training period from 1948 to 1990. The uncertainty is associated with the dynamical system.
Recall that ataccounts for the variation in the relationship between the MSLP anomalies and the
amount of rainfall over the target area in each time step. It is worth noting, again, that the process
atis analogous to the subset of principal component time series Ztin the PCR downscaling
model (1) (Li and Smith, 2009). However, unlike the PCR approach, atis not fixed during
the training time period and is also influenced by the precipitation data. The distributional
information of atis critical for accounting for the uncertainties due to downscaling. This also
illustrates the strength of the BHDM approach over the traditional PCR model.
Figs 2(a) and 2(b) show the posterior mean of the propagator matrix Hand the downscaling
projection matrix Λ. The posterior mean of the propagator matrix indicates the association
between atlatent process components across time, and the posterior mean of the downscaling
projection matrix illustrates the dynamical link between this latent process and local rainfall over
the target area. Particularly, the change in dynamics of large-scale MSLP patterns is projected
and linked to the winter extreme rainfall over SWWA through the at-process.
A plot representing the posterior mean of the rainfall data model error matrix is presented in
Fig. 2(c). Inspection of this error covariance matrix reveals that spatial dependence in the errors
is present.
To investigate the forecasting skill, the BHDM was calibrated by using the observed rainfall
and the NCEP MSLP data over the training period 1948–1990. The predictive model used
the NCEP MSLP data over the verification period with the estimated posterior distribution of
the downscaling projection matrix Λfrom the training period. The downscaled rainfall values
after 1990 therefore indicate the true forecasting skill of the BHDM approach. The uncertainty
in terms of 95% credible intervals associated with predictions after 1990 are also indicated in
Fig. 3 by the shaded envelope. Table 2 summarizes the skill by showing the corresponding
associated correlation coefficients, the root-mean-square error RMSE and the ratio of RMSE
to the climatology of June–August rainfall.
As shown in Fig. 3 and Table 2, BHDM performance in the training period is maintained
during the verification period. Indeed, forecasting skill in the verification period is higher than it
is in the training period. In the verification period, note that all the observations fall in the 95%
credible interval. On closer inspection, we see that some extreme wet years have been predicted
very well, implying that our model may provide good guidance in the prediction of extreme
rainfall events. However, downscaling rainfall extremes is beyond the scope of this study, and
we shall explore it elsewhere. The correlation coefficient for the five selected observation stations
and the average rainfall for all 53 observation stations is higher than 0.7, which implies that a
Downscaling Model for South-west Western Australia Rainfall 11
(a)
(c)
(b)
Fig. 2. (a) Posterior mean of propagator matrix H, (b) posterior mean of the downscaling projection matrix
Λand (c) image corresponding to the posterior mean of the error covariance matrix from the rainfall data
model
strong link exists between the MSLP anomalies and the amount of winter rainfall over SWWA.
It is important to note that the associated RMSEs give a reasonable representation of the true
forecast error.
To detect where this BHDM technique is most useful, we show in Fig. 4 the performance
of BHDM forecasting skill by plotting the correlation map between observed and predicted
rainfall in the verification period 1991–2004 over 53 stations across SWWA. As shown in Fig. 4,
the forecasting skill is highest in the central part of the region. The skill is moderate along the
west coast and in the south-western part of the region, whereas the skill tends to be lowest in
the south-eastern part of the region. This could be related to the interpolation scheme and the
sparse distribution of observation stations in that area. Also, van Ommen and Morgan (2010)
have suggested that the local rainfall on the south coast of SWWA is associated with on-shore
northward flow.
Fig. 4(b) illustrates the distribution of RMSE over the target region, with the west coast
region exhibiting relatively high RMSE. However, owing to the large difference in magnitude
12 Y. Song, Y. Li, B. Bates and C. K. Wikle
(a) (b)
(c) (d)
(e) (f)
Fig. 3. Predicted (— —) versus observed June–August rainfall levels ( ) for each of (a) Perth, (b) Gingin, (c) Culicup Estate, (d) Hawthornden and
(e) Hillcroft, and (f) the total average: the results for 1948–1990 correspond to the training period, whereas the results for 1991–2004 correspond to the
verification period ( , 95% credible interval for the verified prediction)
Downscaling Model for South-west Western Australia Rainfall 13
Tab le 2 . Performance of the BHDM at reproducing observation station scale rainfall
over the training period (1948–1990) and the verification period (1991–2004)†
Station name Results for training Results for verification
period (1948–1990) period (1991–2005)
Correlation RMSE ρ(%) Correlation RMSE ρ(%)
Perth 0.682 59.3 16 0.814 32.7 12
Gingin 0.685 57.6 16 0.868 40.4 10
Culicup 0.716 47.2 23 0.822 30.6 16
Hawthornden 0.756 67.1 20 0.885 41.6 15
Hillcroft 0.673 42.8 22 0.846 26.7 11
Average 0.685 54.7 22 0.867 44.1 19
†The correlation is between the predictions and observations, and ρ% is the ratio of RMSE
to the climatology of June–August rainfall.
Tab le 3 . Regional June–August mean rainfall for present-day (1971–2000) and the future (2071–2100) over
SWWA: comparison between observed, GCM-simulated, the PCR downscaling model and BHDM results†
Source Results for the present (1971–2000) Results for the future (2071–2100)
Amount Difference with Difference Amount Difference with Difference
(mm) observed (mm) (%) (mm) results (%)
(1971–2000) (mm)
Observed 300
GCM 121 −179 −60 100 −21 −17
PCR 205 (154, 261) −95 −32 196 (142, 254) −9(−12,−7) −5
BHDM 249 (177, 327) −51 −17 225 (153, 302) −24 (−31,−21) −11
†The 95% confidence or credibility levels are shown in parentheses.
in the mean rainfall across the observation stations, the ratio of RMSE to the climatological
mean of June–August rainfall represents more information of forecast uncertainty. As Fig. 4(c)
shows, the ratio of RMSE to the mean of June–August rainfall varies locally, ranging from 0.1
to 0.3. The spatial distribution of this ratio is close to the spatial distribution of the correlation
coefficient. Thus, the spatial distribution of forecast uncertainty is consistent with the spatial
distribution of forecast skill.
5. Application to climate change simulation
We apply the downscaling model to the MSLP data derived from GCM simulations for both
the present-day and future climate. The CSIRO mark 3.5 climate model was used in an exper-
iment in which greenhouse gases were prescribed to increase according to the A2 emissions
scenario (Solomon et al., 2007). Mean winter (June–August) MSLP fields were calculated for
both present-day conditions (1971–2000) and for later this century (2071–2100). Fig. 5 com-
pares the observed (NCEP) long-term average June–August MSLP with those simulated by the
GCM for present-day and future conditions. The pattern correlation between the observed and
14 Y. Song, Y. Li, B. Bates and C. K. Wikle
(a)
(c)
(b)
Fig. 4. (a) Correlation between predicted and observed June–August rainfall at 53 observation stations
across SWWA in the verification period 1991–2004, (b) the same as (a), but for RMSE, and (c) the same as
(a), but for the ratio of RMSE to the mean rainfall over the verification period 1991–2004
Downscaling Model for South-west Western Australia Rainfall 15
(a)(b)
(c) (d)
Fig. 5. Long-term average winter MSLP (hPa) (a) from the NCEP data for 1970–2000, (b) simulated by
the CSIRO mark 3.5 climate model for the present day (1970–2000), (c) simulated by the CSIRO mark 3.5
climate model for the future (2071–2100) and (d) difference in MSLP between the future and the present
the GCM-simulated long-term average June–August MSLP was 0.969 for the present day and
0.967 for the future. The MSLP values simulated for the future are higher than present-day
values over most of the continent, with the largest increases found just to the south. Decreases
are a feature of the differences towards the very high latitudes.
The observed, GCM-simulated and downscaled rainfall totals are summarized in Table 3.
For the averaged rainfall over SWWA, the mean downscaled values from the BHDM and the
PCR downscaling model are shown with their respective 95% confidence or credibility intervals.
These intervals reflect the estimated uncertainty that is associated with downscaling through
the BHDM and PCR. As shown in Table 3, the downscaled values from both the BHDM and
PCR represent a significant underestimate of observed rainfall. However, note that the raw
GCM values underestimate with a higher percentage error of −60%. This is comparable with
16 Y. Song, Y. Li, B. Bates and C. K. Wikle
the average underestimate (−56%) from the seven GCMs that were analysed by Hope (2006).
Thus, the downscaled values are higher than the GCM values but the percentage errors are
much less: −17% for the BHDM and −32% for the PCR downscaling model. Additionally,
these results suggest that the BHDM is superior to PCR in downscaling MSLP patterns to
SWWA rainfall.
The GCM results for later this century all indicate decreases in rainfall compared with the
present-day values. These results are consistent with the assumption of long-term winter rainfall
trends in regions of SWWA (e.g. Bates et al. (2008)). The GCM-simulated decrease of −17% is
comparable with the average (−18%) from the seven GCMs that were analysed by Hope (2006).
The downscaled values from both the BHDM and PCR also indicate a decrease: −11% for
the BHDM and −5% for PCR. Moreover, the mean values for present and future downscaled
rainfall are significantly different from 0, on the basis of the 95% credible intervals (Table 3). By
taking into account the uncertainty in the downscaled rainfall, it is evident that winter rainfall
over SWWA displays a substantial decrease. This decrease is consistent with the increases in
simulated MSLP in Fig. 5.
Fig. 6 shows a comparison of the spatial distribution between observed and downscaled
rainfall values over SWWA from the BHDM in the present (1971–2000) and the future (2070–
2100) periods. In general, the overall patterns from the downscaled values in the present period
are consistent with the observed rainfall values (Fig. 6(b) compared with Fig. 6(a)). Although
there are underestimates at stations in the south-east and north-west, and overestimates for
stations across the middle and south-west SWWA, the ratio of the difference between the GCM
present and observed values is relatively small (from about −0:2 to 0.2) (Fig. 6(d)), which is
around 10–20 mm in magnitude. Therefore, the downscaling rainfall values from future MSLP
from the BHDM (Fig. 6(c)) can be deemed a good approach to project future rainfall values in
SWWA.
The decrease in magnitude for rainfall over SWWA in the future is depicted by the percentage
difference between the downscaled rainfall values over present-day climate and the downscaled
rainfall values over the future climate (Fig. 6(e)). As shown in Fig. 6(e), the decrease in rainfall
can be observed for the entire SWWA, with a decrease of approximately −10% in south-east
SWWA. It is worth noting that Fig. 6(e) also shows the spatial pattern of future rainfall trends
since rainfall in this region has been declining since the late 1960s. Allan and Haylock (1993)
suggested that the spatial MSLP pattern is dominated by a sequence that begins with low or
high anomalies over southern Australia and ends with the opposite configuration of high or low
respectively. This slowly evolving feature may be indicative of the passage of a low frequency
wave that could provide a modulation of the long wave pattern in this sector. The decrease in
SWWA June–August rainfall appears to be the result of the development of positive MSLP
anomalies over southern Australia under the influence of this low frequency signal. This is
consistent with the largest increases of the simulated, future MSLP values, which are found just
over southern Australia (Fig. 5(d)).
In summary, these results indicate that it is beneficial to apply the BHDM for statistical
climate downscaling. The performance of the BHDM in climate downscaling of precipitation
is significantly improved compared with the PCR downscaling model and GCM estimation.
Specifically, whereas the GCM estimate for present-day winter rainfall totals represents relatively
large underestimates (−60%), the BHDM provides much more realistic estimates (−17%) which
is better than the PCR downscaling estimate (−32%). The improvement in present-day rainfall
estimation demonstrates that the BHDM may provide more accurate future climate projection
than the PCR downscaling model and GCM estimation in SWWA. As shown in Table 3, the
PCR estimate for future changes to rainfall suggests a relatively small decrease of −9 mm com-
Downscaling Model for South-west Western Australia Rainfall 17
(a)(b)
(d) (e)
(c)
Fig. 6. (a) Observed mean rainfall between 1971 and 2000 across SWWA; (b) downscaled rainfall values for the present-day climate (1971–2000), (c)
downscaled rainfall values for the future climate (2070–2100), (d) percentage difference between the downscaled rainfall values for the present-day climate
and the observed rainfall values and (e) percentage difference between the downscaled rainfall values for the present-day climate and the downscaled
rainfall values for the future climate
18 Y. Song, Y. Li, B. Bates and C. K. Wikle
pared with the GCM value of −21 mm, which can be translated to a percentage change of −5%
compared with the GCM value of −17%. The BHDM estimate for future changes is −24 mm,
corresponding to a decrease of −11%.
6. Discussion
The purpose of this paper was to demonstrate how the BHDM can be applied to downscale
large-scale MSLP circulations to regional rainfall over SWWA. It is found that the BHDM may
provide a way for better rainfall future projection on the basis that the downscaling values of the
BHDM are more close to the observed values in present-day (1971–2000) climate. Compared
with the PCR downscaling model, some aspects of the improvement are related to the skill of
rainfall prediction and climate change projections.
The downscaling method that is presented in this paper represents only changes in rainfall
linked to changes in the atmospheric circulation. This is because we used only one predictor
field, the MSLP, in the downscaling model. Changes in future rainfall may occur due to changes
in other factors that were not considered in the downscaling model, e.g. changes in atmospheric
humidity or temperature. To some extent, changes in humidity may be accounted for by changes
in the MSLP patterns in so far as they can affect the direction of the prevailing winds. They
will not, however, be able to account for large-scale changes in humidity that are associated
with global warming, but this is an effect that is difficult to incorporate by using present-day
observations (Charles et al., 1999).
Also, there is considerable scope for extensions of the BHDM in this study. One obvious
extension would be to fit a more sophisticated process model, such as a general quadratic non-
linear model or using a priori scientific knowledge to motivate the parameterization (Wikle and
Hooten, 2010), to describe the evolutionary processes of MSLP anomalies better.
Finally, in addition to the assumptions of linearity and stationarity, downscaling results are
constrained by the quality of the predictors (e.g. the MSLP) simulated by the GCMs. In this
case, a greater consensus about the expected changes to the MSLP patterns may lead to an
improved consensus of the expected changes to rainfall by downscaling techniques. Multimodel
combination of GCMs and effective model bias correction techniques can provide better cali-
bration than individual models. Further work will investigate the application of these techniques
to an ensemble of GCM results.
Acknowledgements
The authors thank the Associate Editor and two reviewers for their constructive comments,
which led to a great improvement of the manuscript. We thank Q. J. Wang, David Robertson
and Anthony Woolley for valuable discussions and suggestions. We acknowledge the CSIRO
internal review by Lawrence Murray. YS was supported by the Water Information Research and
Development Alliance between the Australian Bureau of Meteorology and the CSIRO Water
for a Healthy Country Flagship. CWK was supported by the National Science Foundation,
under grant DMS-1049093. YL and BB were supported by the CSIRO Climate Adaptation
Flagship affiliation and the Indian Ocean Climate Initiative.
Appendix A
The full conditional distributions can be given as follows:
(a) the full conditional distribution of a0
Downscaling Model for South-west Western Australia Rainfall 19
[a0|·]∝[a1|a0,H,Σς][a0],
a0|·∼N.A−1
0b0,A−1
0/,.11/
where
A0=.HΣ−1
ςH+Σ−1
0/,
b0=.HΣ−1
ςa1+Σ−1
0μ0/;
(b) the full conditional distribution of at
[at|·]∝[Yt|at,Λ,σ2
η,θ][Xt|at,σ2
"][at+1|at,H,Σς][at|at−1,H,Σς],
at|·∼N.A−1
tbt,A−1
t/,.12/
where
At=.Φ
xΦx=σ2
"+Σ−1
ς+ΛR.θ/−1Λ=σ2
η+HΣ−1
ςH/,
bt=.Φ
xXt=σ2
"+Σ−1
ςHat−1+HΣ−1
ςat+1+ΛR.θ/−1Yt=σ2
η/;
(c) the full conditional distribution of aT
[aT|·]∝[YT|aT,Λ,σ2
η,θ][XT|aT,σ2
"][aT|aT−1,H,Σς],
aT|·∼N.A−1
TbT,A−1
T/,.13/
where
AT=.Φ
xΦx=σ2
"+Σ−1
ς+ΛR.θ/−1Λ=σ2
η+HΣ−1
ςH/,
bT=.Φ
xXT=σ2
"+Σ−1
ςHaT−1+ΛR.θ/−1YT=σ2
η/;
(d) the full conditional distribution of H
[H|·]∝
T
t=1
[at|at−1,H,Σς][H],
h|·≡vec.H/|·∼N.V−1b,V−1/,.14/
where
V=.A
t−1⊗Ik/˜
Σ−1
ς.A
t−1⊗Ik/+Σ−1
h,
b=.A
t−1⊗Ik/˜
Σ−1
ςvec.At/+Σ−1
hμh,
At=.a1,:::,aT/,
At−1=.a0,:::,aT−1/,
˜
Σς=IT⊗Σς,
where Ikis k×kidentity matrix;
(e) the full conditional distribution of Λ
[Λ|·]∝
T
t=1
[Yt|at−1,Λ,R][Λ],
λ|·≡vec.Λ/|·∼N.V−1b,V−1/,.15/
where
V=.A
t−1⊗Im/˜
R.θ/−1.A
t−1⊗Im/=σ2
η+Σ−1
λ,
b=.A
t−1⊗Im/˜
R.θ/−1vec.Yt/=σ2
η+Σ−1
λμλ,
At=.a1,:::,aT/,
At−1=.a0,:::,aT−1/,
Yt=.Y1,:::,YT/,
˜
R.θ/=IT⊗R.θ/,
where Imis an m×midentity matrix and ITis an T×Tidentity matrix;
20 Y. Song, Y. Li, B. Bates and C. K. Wikle
(f) the full conditional distribution of σ2
η
[σ2
η|·]∝
T
t=1
[Yt|at,Λ,σ2
η,θ][σ2
η],
σ2
η|·∼inverse-gamma
qη+mT
2,1
rη
+1
2
T
t=1
.Yt−Λat/R.θ/−1.Yt−Λat/−1;.16/
(g) the full conditional distribution of θ
[θ|·]∝
T
t=1
[Yt|at−1,Λ,σ2
η,θ][θ],
θ|· ∼
T
t=1
N.Yt|at−1,Λ,σ2
η,θ/gamma.θ|qθ,rθ/,.17/
(h) the full conditional distribution of σ2
"
[σ2
"|·]∝
T
t=1
[Xt|at,σ2
"][σ2
"],
σ2
"|·∼inverse-gamma
q"+nT
2,1
r"
+1
2
T
t=1
.Xt−φxat/.Xt−φxat/−1;.18/
(i) the full conditional distribution of Σς
[Σς|·]∝
T
t=1
[at|H,at−1,Σς][Σς],
Σ−1
ς|·∼WishartT
t=1
.at−Hat−1/.at−Hat−1/+rςsς−1
,rς+T:.19/
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