Joint bias correction of temperature andprecipitation in climate model simulationsChao Li1, Eva Sinha1,2, Daniel E. Horton2,3, Noah S. Diffenbaugh2,3, and Anna M. Michalak1

1Department of Global Ecology, Carnegie Institution for Science, Stanford, California, USA, 2Department of EnvironmentalEarth System Science, Stanford University, Stanford, California, USA, 3Woods Institute for the Environment, StanfordUniversity, Stanford, California, USA

Abstract Bias correction of meteorological variables from climate model simulations is a routine strategy forcircumventing known limitations of state-of-the-art general circulation models. Although the assessment ofclimate change impacts often depends on the joint variability of multiple variables, commonly used biascorrection methodologies treat each variable independently and do not consider the relationship amongvariables. Independent bias correction can therefore produce non-physical corrections and may fail to captureimportant multivariate relationships. Here, we introduce a joint bias correctionmethodology (JBC) and apply it toprecipitation (P) and temperature (T) fields from the fifth phase of the Climate Model Intercomparison Project(CMIP5) model ensemble. This approach is based on a general bivariate distribution of P-T and can be seen as amultivariate extension of the commonly used univariate quantile mapping method. It proceeds by correctingeither P or T first and then correcting the other variable conditional upon the first one, both following the conceptof the univariate quantile mapping. JBC is shown to not only reduce biases in the mean and variance of P andT similarly to univariate quantile mapping, but also to correct model-simulated biases in P-T correlation fields.JBC, usingmethods such as the one presented here, thus represents an important step in impacts-based researchas it explicitly accounts for inter-variable relationships as part of the bias correction procedure, thereby improvingnot only the individual distributions of P and T, but critically, their joint distribution.

1. Introduction

Estimates of the potential impacts of anthropogenic climate change rely on climate model projections.Although global climate modeling has evolved rapidly in recent years, output from climate models is stillprone to biases [Ehret et al., 2012; Mehran et al., 2014; Mueller and Seneviratne, 2014]. To circumvent thisobstacle, bias correction is routinely applied to climate model output [Ashfaq et al., 2010; Harding et al., 2011;Warszawski et al., 2014; Horton et al., 2014].

Commonly used bias correction methods [e.g., Ines and Hansen, 2006; Christensen et al., 2008; Piani et al.,2010a, 2010b; Li et al., 2010; Haerter et al., 2011; Johnson and Sharma, 2012; Grillakis et al., 2013; Hempel et al.,2013] correct variables independently, with variable interactions typically not considered. As such,independent variable bias correction methods do not correct errors in multivariate relationships [Dosio andParuolo, 2011] and may result in physically unrealistic corrections [Thrasher et al., 2012]. A well-formulatedjoint correction methodology that considers the co-variability of meteorological fields is thus needed forapplications where the joint variability among several climate variables is important.

One such area of application is the assessment of the impacts of climate change, such as the effects of analtered hydrological cycle on flooding risk, crop production, and public health. In such applications, projectedimpacts often depend on multiple variables and their interactions. In studies of climate change impacts onhuman health, for example, impacts are oftenmediated via a combination of different meteorological factors.For instance, a combination of high temperatures and high humidity has been found to increase humandiscomfort, morbidity, and mortality [Fisher et al., 2012; Sherwood and Huber, 2010], while the co-occurrenceof precipitation-free days and light winds has been linked to pollutant accumulation, air quality degradation,and an increase incidence of respiratory ailments [Horton et al., 2014].

The joint distribution of projected temperature (T) and precipitation (P) in particular is of critical importanceinmany such impact studies. For example, in the process of runoff generation, T determines the proportion ofP that occurs as rain or snow, and partly controls the subsequent snow melt; P affects soil moisture, which,

LI ET AL. ©2014. American Geophysical Union. All Rights Reserved. 1

PUBLICATIONSJournal of Geophysical Research: Atmospheres

RESEARCH ARTICLE10.1002/2014JD022514

Key Points:• A joint precipitation and temperaturebias correction algorithm is presented

• The algorithm is applied to CMIP5simulations and evaluated againstCRU data

• The algorithm improves the jointdistribution of P and T,including correlations

Supporting Information:• Readme• Text S1, Table S1, and Figures S1–S4

Correspondence to:C. Li,lichao@stanford.edu

Citation:Li, C., E. Sinha, D. E. Horton, N. S.Diffenbaugh, and A. M. Michalak (2014),Joint bias correction of temperature andprecipitation in climate model simula-tions, J. Geophys. Res. Atmos., 119,doi:10.1002/2014JD022514.

Received 3 SEP 2014Accepted 6 NOV 2014Accepted article online 10 NOV 2014

together with T, impacts evapotranspiration; all of the above processes further contribute to runoffgeneration. Therefore, climate model simulated biases in the correlations among meteorological fields willaffect simulations of processes such as, snowmelt, evapotranspiration, and runoff generation [Buishand andBrandsma, 2001; Maurer et al., 2010; Mueller and Seneviratne, 2014; Immerzeel et al., 2014]. Furthermore,because regional hydrology is very sensitive to P and T, even small biases could change the hydrologicalequilibrium by, for example, switching a water-limited regime to an energy-limited one or vice versa[Bárdossy and Pegram, 2011]. Another example is the impact of climate change on crop yields, for whichconcurrent changes in P and T are of greater interest than independent projections of each variable[Tebaldi and Lobell, 2008], with P and T jointly explaining 29% to 65% of the variation in global average yieldsof major crops (e.g., barley, wheat, maize, and soybean) [Lobell and Field, 2007]. Based on these and otherexamples, it is clear that a comprehensive assessment of climate change impacts should take into accountthe uncertainties in all contributing variables as well as in their interactions [Fisher and Knutti, 2012].

Although the need for joint bias correction has been suggested previously [Hagemann et al., 2011; Ehret et al.,2012; Johnson and Sharma, 2012; White and Tourmi, 2013], to date only a few studies have probed suchapproaches. Thrasher et al. [2012] corrected biases in daily minimum and maximum temperatures by firstcorrecting maximum (or minimum) temperature and diurnal temperature range, and then deriving theremaining variables by subtracting (or adding) the corrected diurnal range. This approach, however, is onlysuitable for variables with explicit and deterministic relationships. Piani and Haerter [2012] performed joint P-Tcorrection by correcting T first, then grouping P-T pairs into a number, typically less than 5, of temperature bins,and finally correcting P within each temperature bin, but because no rigorous statistical P-T relationship wasestablished, the benefits of this approach were limited (particularly when assessing P-T correlation coefficients).

To develop an effective and easy-to-implement method for joint bias correction (JBC) that corrects the jointdistribution of P and T, here we define a general bivariate probability distribution of P-T, which is used as the basisfor the proposed JBC method. Although the method is designed for P and T, the concept is directly applicableto other variables (with appropriate modifications to the functional form of their joint distribution) and couldpotentially also be extended tomultiple variables. We focus on P and T due to their wide usage as climate inputsin impact models [Piani and Haerter, 2012], and due to the existence of high quality observational standardsfor methodological calibration and evaluation. We evaluate the developed method by correcting P and Tsimulations from climate models participating in the fifth phase of the Climate Model Intercomparison Project(CMIP5), and compare it with the commonly used independent quantile mapping bias correction method.

This paper is organized as follows. Formulating the research objective in this section 1, a short discussion of thedata set to be used and a detailed description of the developed JBC methodology are presented in section 2.Methodological evaluation and comparison are presented in section 3. This paper is concluded in section 4.

2. Data and Methodology2.1. Data

The reference observational P and T data used in this work are the 0.5° × 0.5° gridded monthly observationsfrom the Climate Research Unit Version 3.1 (CRU) for 1951 to 2005 [Harris et al., 2013]. Simulated monthly Pand T for the same time period from 38 GCMs (Table S1) participating in the CMIP5 “historical” experiment[Taylor et al., 2012] are used as the basis for the bias correction experiments. For models withmultiple ensemblemembers, only the first one was used.

Both CRU observations and GCM simulations were regridded to a 1° × 1° resolution, using bilinearinterpolation. Then, they were partitioned into a “historical” calibration set (1951–1980) and a “future”validation set (1981–2005). The “historical” set was for the estimation of the correction parameters and the“future” set for methodological validation.

2.2. Joint Distribution of P and T

We treat P and T at the scales of interest to impact models (from daily to annual) as random variables, anddenote them as X and Y, respectively, with X ≥ 0 and Y ∈ R. The joint probability distribution function of X and Ycan be expressed as

Φ x; yð Þ ¼ P X ≤ x; Y ≤ yð Þ ¼ 1� pwð ÞHY yð Þ þ pwC F xð Þ;G yð Þð Þ (1)

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where pw=P(X> 0), HY(y) = P(Y ≤ y|X= 0),C(�) is the Gaussian copula function that enables us to formulate thedistribution of X and Y given X> 0 as a function of F(x) and G(y) and captures the dependence structurebetween P and T of wet days/months, F(x) = P(X ≤ x|X> 0) and G(y) = P(Y ≤ y|X> 0). The Gaussian copula wasemployed because it can represent both positive and negative correlations while also being quite simple toimplement. The joint distribution in equation (1) characterizes the joint distributional behavior of P and T.To model the marginal distributions of P and T, we adopt gamma and Gaussian distributions, respectively.Parameters of the joint distribution can be estimated in a manner similar to Li et al. [2013]. The formulation ofJBC relies on the marginal and conditional distributions of P and T, presented in the next subsection.

2.3. Derived Marginal and Conditional Distributions

For simplicity, we assume that temperature on wet and dry days/months is identically distributed.Mathematically, this means that for any y ∈ R, HY(y) =G(y). Under these conditions, it can be shown that themarginal distribution function of T is

ΦY yð Þ ¼ P Y ≤ yð Þ ¼ G yð Þ (2)

and the conditional distribution function of P given T is

ΦXjY ¼ y xð Þ ¼ P X ≤ x Y ¼ yÞ ¼ 1� pw þ pwC2 F xð Þ;G yð Þð Þjð (3)

where C2(�) is the partial derivative of C(�) with respect to its second argument. The corresponding p-quantilefunctions are

yY pð Þ ¼ G�1 pð Þ (4)

xXjY¼y pð Þ ¼ F �1 C �121 p� 1þ pwð Þ=pw ;G yð Þð Þ� �

(5)

where G� 1(�), F� 1(�), and C�121 �ð Þ are the inverse of G(�), F(�), and C2(�) with respect to its first argument,

respectively. Likewise, the marginal distribution function of P is

ΦX xð Þ ¼ P X ≤ xð Þ ¼ 1� pwð Þ þ pwF xð Þ (6)

and the conditional distribution function of T given P is

ΦYjX¼x;x¼0 yð Þ ¼ P Y ≤ y X ¼ x; x ¼ 0Þ ¼ G yð Þjð (7a)

ΦYjX¼x;x > 0 yð Þ ¼ P Y ≤ y X ¼ x; x > 0Þ ¼ C1 F xð Þ;G yð Þð Þjð (7b)

Correspondingly, the p-quantile functions are

xX pð Þ ¼ F�1 p� 1þ pwð Þ=pwð Þ (8)

yYjX¼x;x¼0 pð Þ ¼ G �1 pð Þ (9a)

yYjX¼x;x > 0 pð Þ ¼ G �1 C �112 F xð Þ; pð Þ� �

(9b)

Derivations of these marginal and conditional distributions are provided in the Supporting Online Material.

2.4. JBC Procedure

Once the joint P-T distributions have been quantified for the “historical” observations and simulations, thegoal becomes to correct the “future” simulations. JBC can proceed either by correcting T first and then Por vice versa, as summarized below and illustrated in Figure 1 using JBCp as an example. Here we usethe super- or subscripts hs, ho, fs, and fc to denote historical simulations, historical observations, futuresimulations, and bias-corrected future simulations, respectively. For example, Gho(�) denotes the distributionfunction of observed T in the “historical” period; xfs represents simulated P in the “future” period.

Correcting T first and then P (termed JBCt) proceeds as follows:

1. Evaluate p1 =Ghs(yfs) (equation (2)).

2. Set yfc ¼ G�1ho p1ð Þ (equation (4)).

3. Evaluate p2 ¼ 1� phsw þ phsw Chs2 Fhs xf sð Þ;Ghs yf sð Þð Þ (equation (3)).

4. Evaluate p3 ¼ p2 � 1þ phow� �

=phow and determine whether p3 thus obtained is positive; if so, set

xfc ¼ F�1ho Cho�1

12 p3;Gho yfcð Þð Þ� �

(equation (5)); otherwise, set xfc = 0.

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Correcting P first and then T (termed JBCp) proceeds as follows:

1. Evaluate p1 ¼ 1� phsw þ phsw Fhs xf sð Þ (equation (6)).

2. Evaluate p2 ¼ p1 � 1þ phow� �

=phow and determine whether p2 thus obtained is positive; if so, set

xfc ¼ F�1ho p2ð Þ (equation (8)); otherwise, set xfc = 0.

3. Determine whether xfs is positive; if so, evaluate p3 ¼ Chs1 Fhs xf sð Þ;Ghs yf sð Þð Þ (equation (7b)); if xfs is zero,

evaluate p3 =Ghs(yfs) (equation (7a)).

4. Determine whether xfc is positive; if so, set yfc ¼ G� 1ho Cho�1

12 Fho xfcð Þ; p3ð Þ� �

(equation (9b)); if xfc is 0,set yfc ¼ G�1

ho p3ð Þ (equation (9a)).

Note that JBC can be viewed as a bivariate extension of the univariate quantile mapping method withgamma and Gaussian distributions for P and T (henceforth denoted by IBC), respectively; see Ines and Hansen[2006] and Piani et al. [2010a]. If both observed and simulated P and T were independent, JBC reduces to IBC.Quantile mapping approaches are generally preferred to approaches based on linear transformation,especially when the objective is to bias correct higher moments of a variable beyond its mean and/orvariance [Berg et al., 2012; Teutschbein and Seibert, 2012; Bennett et al., 2013; Chen et al., 2013].

Note also that JBC implicitly assumes that the simulated precipitation frequency is no less than the oneobserved. This is reasonable because climate models are known to produce more drizzle and fewer dryperiods relative to observations. Under this assumption, like most quantile mapping bias correctionmethods,

Figure 1. Schematic illustration for joint bias correction (JBC) (using JBCp as an example): Suppose that all observed andsimulated P are strictly positive. Before applying JBC, one should first estimate the joint P-T distributions from “historical”simulations and observations, as illustrated by the contour plots in Figures 1c and 1d, respectively. From the joint distri-butions, marginal distributions of simulated and observed P can be derived, as illustrated by the plots in Figures 1a and 1b,respectively. Now given a pair of “future” P-T simulations, i.e., xfs and yfs, Step 1 is to evaluate the marginal distribution ofsimulated P at the “future” P simulation xfs, resulting in a quantile value. Step 2 is to evaluate the inverse of the marginaldistribution of observed P at the quantile value thus obtained. The resulting value xfc is the bias-corrected P. Step 3 is toevaluate the conditional distribution of simulated Tgiven P equal to xfs, derived from the joint distribution of simulated P-Tand shown by the plot in Figure 1e, at the “future” T simulation yfs, yielding a quantile value. Step 4 is to evaluate the reverseof the conditional distribution of observed T given P equal to xfc, derived from the joint distribution of observed P-T andshown by the plot in Figure 1f, at the quantile value thus obtained. The resulting value yfc is the bias-corrected T.

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JBC will correct simulated wet days/months with P less than a certain threshold (depending on observationsand simulations of P in the “historical” period) to be dry, and will not introduce additional wet periods.

3. Results and Discussions3.1. Observed P-T Correlation Fields

Observed P-T Pearson correlation coefficient fields vary regionally and seasonally. In boreal summer(Jun–Jul–Aug), negative correlations prevail over the global land, with strong “hot spots” observed in the GreatPlains of North America, the Pacific Northwest, the Sahel, equatorial Africa, and India (Figure 2a). In boreal winter

Figure 2. P-T Pearson correlation coefficient fields for boreal (left column) summer and (right column) winter from(a, b) Climate Research Unit Version 3.1 (CRU) observational data and (c, d) HadCM3 native, (e, f ) IBC-corrected, and(g, h) JBCt-corrected simulations over the validation period 1981–2005. Highlight boxes outline regions with the mostnoticeable P-T correlation improvement after JBCt correction. White regions indicate no observational data.

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(Dec–Jan–Feb), the relationship isreversed in high latitudes (Figure 2b).This is especially true for the NorthernHemisphere, with exceptions in theRockyMountain region and the southerntip of Greenland. P-Tcorrelation fields aresimilar to each other during the twotransition seasons, but distinct fromsummer and winter, with morepronounced and expansive positivecorrelations occurring in the middle-to-high latitudes and East and SoutheastAsia (Figure S1).

3.2. P-T Correlation in NativeModel Simulations

We use HadCM3 as a prototypical modelfrom the CMIP5 archive to illustratedifferences between observed(Figures 2a–2b) and simulated(Figures 2c–2d) P-T correlationcoefficients. HadCM3 approximatelycaptures large-scale correlation patterns,but systematic model-simulated biasesare apparent. Magnitudes of simulatedcorrelations are generally larger thanobserved, particularly for positivecorrelations. In summer, an extraneousanti-correlation (positive correlation)“hot spot” appears near the western(eastern) edge of the Sahel, and the anti-correlation over India is overly intense(box L1 in Figure 2c). In winter, simulatedpositive correlations over high latitudesare substantially larger than observed(box R2 in Figure 2d).

In addition to biases in magnitude, thesign of the correlation is reversed in someregions. In summer, in the NorthernHemisphere high latitudes (box L2 inFigure 2c) and Southeastern China,negative observed P-T correlations arereplaced by positive correlations inthe simulations. In winter, positive

correlations are evident in the simulated fields in Greenland and most parts of North America (box R1 inFigure 2d) and China, whereas the observed correlations are weak or negative. In summer, more complex mixedbiases in both the sign and the magnitude can be found in South America (box L3, Figure 2c).

Biases also exist in individual P and T statistics (e.g., mean and standard deviation), in other seasons and inother climate models, as shown by the larger root-mean-square errors (RMSE) in different P and T statistics ofthe native simulations for different seasons (Figure 3). Overall, the results demonstrate that climate modelsimulated P-T fields are afflicted with biases not only in the marginal distributions of P and T but also intheir correlations. This result confirms the need for a bias correction procedure that can correct the jointdistribution of P-T as a whole.

Figure 3. Fifth phase of the ClimateModel Intercomparison Project (CMIP5)multi-model ensemble root-mean-square errors in (a) P-T correlations,(b) mean of P, (c) P standard deviation, (d) mean of T, and (e) T standarddeviation in native, JBCt-corrected, and JBCp-corrected simulations over thevalidation period 1981–2005. For a given climate model before and aftercorrection, RMSE is calculated for each statistics across all grid cells,assuming an equal weight for each cell. Thus, it serves as a cumulativemeasure of the bias over the land. The spread of the boxplot reflects theinter-model variability. The white dot represents the climatemodel with themedian RMSE. The height of the box represents the 25th and 75th percen-tiles. The whiskers extend to 1.5 times the interquartile range from the box.

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3.3. P-T Correlation Fields After IBC

We use the “historical” set (1951–1980) to calibrate bias correction models (IBC and JBC) and then evaluatethe calibrated models over the “future” set (1981–2005). Both calibration and validation are carried outseparately for each season, each grid cell, and each CMIP5 model.

The IBC-corrected correlation fields are nearly identical to the simulated fields across all seasons for theprototypical example using HadCM3 (Figures 2e–2f and S1e–S1f), demonstrating the limitation of IBC inrepresenting the dynamical linkage between P and T. This is consistent with Dosio and Paruolo [2011], whofound that monotonically increasing quantile mapping does not alter the rank order correlation betweenvariables. We note here that it does not alter the Pearson correlation coefficient either, if the variables(here monthly P and T) have a dependence structure that is close to Gaussian. As expected, IBC does reducebiases in the statistics of the individual P and T fields (Figures 3b–3e).

3.4. P-T Correlation Fields After JBC

In contrast to IBC, JBCt substantially reduces biases in the simulated P-Tcorrelation fields. Most noticeably, theswath of spuriously high correlations over northern high latitudes is suppressed, muted, or reversed forsummer and winter (boxes L2 and R2 in Figure 2); the misrepresented summer anti-correlation “hotspot” inthe Saheland India (box L1 in Figure 2) and the correlation field in South America (box L3 in Figure 2) arecorrected; and the region of winter anti-correlations ranging from Central America to the central Great Plainsof North America and to the Pacific Northwest is restored (box R1 in Figure 2). This leads to a much closerresemblance of the corrected P-T correlation fields to those observed (Figures 2g–2h vs. Figures 2a–2b;Figures S1g–S1h vs. Figures S1a–S1b), as also can be seen from Figure S2. Similar improvements are observedas well when applying JBCp (Figure S3).

JBC yields comparable improvements for the other models in the CMIP5 archive (Figure 3), confirming itsrobust and reliable performance when applied across different climate models. Relative to the nativesimulations, JBC-corrected simulations have smaller RMSE, not only for the P-T correlations, but also for otherindividual statistics, regardless of the order of the correction (i.e., JBCt vs. JBCp). These results also confirmthat the proposed approach overcomes a noted problem with an existing joint P-T correction method[Piani and Haerter, 2012], namely that this earlier approach did not yield appreciable improvements in P-Tcorrelation coefficients.

Results in Figure 3 also reveal that JBC’s skill depends, to some extent, on the order of correction. JBCt isslightly superior to JBCp in correcting the correlation biases, and JBCt and JBCp are better at reproducing theindividual statistics of T and P, respectively, although in general the differences are quite small. Theasymmetric performance of JBCt and JBCp is most likely caused by the differing performance of climatemodels in simulating temporally coherent P and T fields. Rather than selecting JBCt or JBCp, we recommendimplementing both to obtain a more complete picture of the uncertainty of impact projections.

3.5. Evaluation of JBC Relative to IBC

The most important advantage of JBC over IBC lies in its ability to correct biases in P-T correlations. For theindividual P and T statistics, Figure 3 reveals that the performance of JBC is identical to that of IBC for the firstvariable to be corrected, and marginally compromised for the second variable. This cost is rather smallcompared to the benefits gained in the improved fidelity of the P-T correlations, especially in situationswhen the interaction of P-T is of interest, as is the case for many impact models [Tebaldi and Lobell, 2008;Bárdossy and Pegram, 2011; Warszawski et al., 2014].

It is worth pointing out two caveats of JBC. First, JBC cannot completely correct the errors in the sign of the P-Tcorrelation coefficient, and in some regions may introduce new sign errors. This occurs, however, only in asmall number of regions, and primarily ones where the observed and JBC-corrected correlations are both toosmall to be statistically significant (Figure 4), implying that the potential impacts of such sign errors arelimited. Increasing the length of the calibration period is expected to alleviate this problem.

Second, although JBC reduces the correlation biases for most of the global land, it may increase the bias in asmall fraction of grid cells. A similar feature was also found for other individual statistics and seems to be acommon characteristic shared by most of the current bias correction methods. There are several hypothesesthat potentially explain increased biases. There could be (1) imperfect generalization of the P-T dependence

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structure, (2) violation of the stationarity assumption, (3) lack of sufficiently long data for calibration andvalidation, and (4) uncertainty inherent in the CRU observational data as well as in the simulations. To pin downthe contributions of each of these is beyond the scope of the current work. Instead, we note that specific gridcells where adverse effects are evident vary across the CMIP5 models (Figure S4), implying that using amulti-model ensemble approach would alleviate the impact of such localized errors.

4. Concluding Remarks

We present a method for correcting the joint distribution of P and T. Extensive evaluation of the methodshows that it reduces biases not only in the mean and variance of P and T, but also in their correlations.This feature makes the method distinctive from existing univariate bias correction methods and particularlyattractive for climate impact studies.

Although we demonstrate the JBC methodology using monthly data, its discrete-continuous mixed structurefacilitates application to analyses that require annual- to daily scale temporal resolutions. Impact metrics andprocess-based models often require daily scale meteorological data to capture phenomena relevant tohuman system [Diffenbaugh et al., 2012; Scherer and Diffenbaugh, 2014]. Extreme, high-impacts events, for

a b c d

e f g h

i j k l

Figure 4. Regions with errors in the sign of the P-T correlation coefficient for different seasons (columns 1–4) from (a–d) HadCM3 native, (e–h) IBC-corrected, and(i–l) JBCt-corrected simulations over the validation period 1981–2005. Colored regions are those where the sign of the P-T correlations is incorrect and are categorizedinto two types: blue represents regions where both the observed and simulated P-T correlations are statistically insignificant at the 95% level, and the lack ofconsistency in the sign is therefore of lesser concern; red represents regions where the observed and/or the simulated P-T correlations are statistically significant,and the error in sign is therefore of greater concern. After JBCt correction, sign errors occur in much more limited regions relative to the native and IBC-correctedsimulations, and moreover most of the errors occur in regions where the observed and JBCt-corrected correlations are too small to be statistically significant(blue colored). This is not the case for native or IBC-corrected simulations, where large regions with statistically significant incorrect correlations can be observed.

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example, often result from the synergistic effects of multiple climate variables [Field et al., 2012]. Due to thelack of high quality gridded daily observations, however, applying JBC or any other bias correction methodsat daily scale is still limited in practice. Several approaches are feasible to circumvent this obstacle, such asdisaggregation of corrected monthly scale data to daily scale data [e.g., Ashfaq et al., 2010], among others.In such applications, observational uncertainties must of course be considered, especially those introducedby the lack of consistent high-resolution gridded daily observational data.

We note that joint bias correction of P-T does not capture relationships among other variables, or between Por T and those other variables, and JBC cannot therefore be guaranteed to produce corrections that arephysically consistent across all simulated variables. Future research is required to extend the bivariatecorrection into amultivariate framework. One simple solution might be to select a pilot variable first and thento apply JBC to other variables by conditioning upon the pilot variable. Also of note is that JBC can beused directly to bias-correct T or P data sets that exhibit autocorrelation, but it is not designed to correct anybiases in the autocorrelation itself. In situations where biases in the autocorrelation itself are of particularinterest, sequentially applying JBC and the autocorrelation-bias method in Johnson and Sharma [2012] mightbe a possible option. Exploring such an option deserves further research.

Overall, the presented method incorporates P-T correlations into the bias correction procedure and cantherefore be used to inform impact studies driven by climate model projections by improving therepresentation of not only the marginal distributions of P and T, but critically their joint distribution.

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AcknowledgmentsWe thank Vijay P. Singh at Texas A&MUniversity, Claudio Piani at theAmerican University of Paris and UweEhret at Karlsruhe Institute ofTechnology for constructive discussionsduring the development of the jointbias correction approach. We acknowl-edge the World Climate ResearchProgramme’s Working Group onCoupled Modelling, which is responsi-ble for CMIP, and we thank the climatemodeling groups (listed in Table S1 ofthis paper) for producing and makingavailable their model output. For CMIP,the U.S. Department of Energy’sProgram for Climate Model Diagnosisand Intercomparison provides coordi-nating support and led the develop-ment of the software infrastructure, inpartnership with the GlobalOrganization for Earth System SciencePortals. CMIP5 climate model simula-tions used in this paper can bedownloaded at http://pcmdi9.llnl.gov/esgf-web-fe/ and the CUR 3.21 obser-vational data at http://www.cru.uea.ac.uk/. This material is based upon worksupported by the National ScienceFoundation (NSF) under grant 1313897.

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