Gaussian Copula Method for Bias Correction of Daily Precipitation Generated by a Dynamical Model

Moosup Kim APEC Climate Center, Busan, South Korea

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Yoo-Bin Yhang APEC Climate Center, Busan, South Korea

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Chang-Mook Lim APEC Climate Center, Busan, South Korea

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Abstract

The daily precipitation data generated by dynamical models, including regional climate models, generally suffer from biases in distribution and spatial dependence. These are serious flaws if the data are intended to be applied to hydrometeorological studies. This paper proposes a scheme for correcting the biases in both aspects simultaneously. The proposed scheme consists of two steps: an aggregation step and a disaggregation step. The first one aims to obtain a smoothed precipitation pattern that must be retained in correcting the bias, and the second aims to make up for the deficient spatial variation of the smoothed pattern. In both steps, the Gaussian copula plays important roles since it not only provides a feasible way to correct the spatial correlation of model simulations but also can be extended for large-dimension cases by imposing a covariance function on its correlation structure. The proposed scheme is applied to the daily precipitation data generated by a regional climate model. We can verify that the biases are satisfactorily corrected by examining several statistics of the corrected data.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Moosup Kim, moosupkim@gmail.com

Abstract

The daily precipitation data generated by dynamical models, including regional climate models, generally suffer from biases in distribution and spatial dependence. These are serious flaws if the data are intended to be applied to hydrometeorological studies. This paper proposes a scheme for correcting the biases in both aspects simultaneously. The proposed scheme consists of two steps: an aggregation step and a disaggregation step. The first one aims to obtain a smoothed precipitation pattern that must be retained in correcting the bias, and the second aims to make up for the deficient spatial variation of the smoothed pattern. In both steps, the Gaussian copula plays important roles since it not only provides a feasible way to correct the spatial correlation of model simulations but also can be extended for large-dimension cases by imposing a covariance function on its correlation structure. The proposed scheme is applied to the daily precipitation data generated by a regional climate model. We can verify that the biases are satisfactorily corrected by examining several statistics of the corrected data.

© 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Moosup Kim, moosupkim@gmail.com

1. Introduction

Dynamical downscaling using regional climate models (RCMs) is a prominent method for obtaining local or regional precipitation data with a fine time resolution (e.g., daily) from a GCM forecast. In contrast to statistical methods, even without any highly reliable observation data collected over a long period, the dynamical model can simulate regional weather including precipitation with physical consistency. Moreover, the dynamical method does not require a stationary relation between large-scale circulations and the local-scale climate of a target basin that is a basic assumption of many statistical methods. Thus, it has been widely adopted by researchers for hydrometeorological studies of ungauged basins and climate change impacts [cf. Anderson et al. (2007), Boé et al. (2007), Schmidli et al. (2007), and Kure et al. (2013)].

However, the daily precipitation data generated by dynamical models including RCMs generally show a statistical bias, which is a serious flaw that undermines their application. Thus, a suitable correction process must follow. This study aims to correct the bias in the marginal distribution and spatial correlation of the daily precipitation data generated by an RCM for the climate period of 1988–2010. In fact, the distributional bias can be readily corrected by using the quantile mapping (QM) technique, which is very widely used for this purpose (cf. Maraun (2013) and Rajczak et al. (2016)). However, owing to its monotonicity, this technique cannot correct the bias in spatial correlation. For the general limitation of this technique, we refer the readers to Maraun et al. (2017). Recently, the copula model has received attention as a method that overcomes this drawback of the QM technique [cf. Laux et al. (2011), Mao et al. (2015), and, Vrac and Friederichs (2015)]. Among the diverse parametric copula models, this paper concentrates on Gaussian copula due to its own properties (see properties P1 and P2 and the following paragraph in section 4a) and proposes a correction scheme.

The proposed scheme mainly consists of two steps: an aggregation step and a disaggregation step. The scheme needs to retain essential precipitation information produced by the dynamical model in its correction process. The aggregation step aims to obtain such information. This study assumes that the information is represented by a smoothed spatial variation of daily precipitation. In particular, for the data used in this paper, the information is summarized into a bivariate pattern index, which is supported by a principal component analysis (PCA). However, at the same time, the analysis shows that the index of the dynamical model has a bias in correlation. The proposed scheme corrects the bias by using a Gaussian copula–based method. The Gaussian copula assumes that the spatial dependence of daily precipitation is determined by the correlation matrix of the underlying Gaussian variables. Thus, it enables us to obtain independent components by rotating the Gaussian variables suitably and to correct the bias in spatial correlation by adjusting their variances independently [cf. Bárdossy and Pegram (2012) and Cannon (2016)]. However, for daily precipitation, the underlying Gaussian variables are partially observed due to frequent zero amounts. This gives rise to technical issues in estimating the Gaussian copula correlation matrix and in obtaining/adjusting independent components. This paper deals with these issues in a mathematically sound manner.

The smoothed pattern produced by the aggregation step naturally has a smaller spatial variation than that of the observation. A disaggregation step makes up for the deficient spatial variation of the previous step so that the resulting spatial variation has the observational spatial dependence structure. For this, we use the stochastic analog technique adopted by Hwang and Graham (2013), which requires a proper spatial dependence model with a large dimension. Since the Gaussian copula can be extended with a mild complexity for large-dimension cases by imposing a covariance function on its correlation structure, it is employed to derive such a model.

The rest of the paper is organized as follows. Section 2 describes the observational and model data used in this paper with the related configuration. In section 3, we investigate the biases in the model simulation. In particular, an EOF analysis is presented together for understanding the structure of bias in spatial correlation. In section 4, we propose the main bias correction scheme with the related methodologies and show its correction result for the data described in section 2. The conclusions of this paper along with some discussion are presented in section 5. Finally, the appendix provides some derivations and a Gaussian copula estimation method.

2. Data

a. Observational data

As reference data for the bias correction, we employ Automatic Synoptic Observation System (ASOS) data (available at https://data.kma.go.kr). The data period used here is from 1988 to 2013, totaling a period of 26 years. This study focuses on the daily precipitation on a basin in South Korea during the boreal summer, June–August; thus, the sample sizes are 780 = 30 × 26 for June and 806 = 31 × 26 for both July and August. The main stream of the basin is the Nakdong River, which is one of the longest rivers on the Korean Peninsula. We use data from 14 stations around the Nakdong River. Figure 1 shows the locations of the stations, which are spread widely in the meridional direction. Note that the period covers the rainy season of the Korean Peninsula and, therefore, is crucial for water resource management.

Fig. 1.
Fig. 1.

Map of the target basin. The Nakdong River has one of the largest basins on the Korean Peninsula. There are 14 stations nearby. The numbers on the map identify the stations. The seven red- and blue-colored circles indicate the stations in the northern and southern parts of the basin, respectively. In section 4d, they are denoted by and , respectively.

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-18-0089.1

b. Model data

The model data were obtained from simulations of the Global/Regional Integrated Model System [GRIMS; see Hong et al. (2013)]. The regional part of the GRIMS, namely the Regional Model Program (RMP), was used in this study. Model simulation was performed during summer (June–August) from 1988 to 2010 at 30-km resolution. All experiments were forced by the National Centers for Environmental Prediction (NCEP) Climate Forecast System (CFS) (Saha et al. 2014). The physics packages used in this study have already been evaluated in East Asia [cf. Lee et al. (2014) and Yhang et al. (2017)].

3. Statistical bias in model precipitation

In this section, we investigate the statistical bias in model precipitation. In the following, is the grid system of finitely many grid points, including the station locations; is the observation stations , where is the number of stations; is the observational period; is the period of model simulation, and, s, o, and t denote a generic spatial point in , the station location, and the daily time point, respectively. For the precipitation, is the daily cumulative precipitation of model simulation at and at time t and is the real daily cumulative precipitation at at time t. For the cumulative distribution function (cdf) G, denotes the quantile function of G, . For set A, denotes the number of elements in A.

a. Bias in distribution

Let be the cdf of . It is implicitly assumed that the real precipitation is strictly stationary with respect to time, and, thus, its cdf does not depend on t. However, it is unknown and is usually estimated by its empirical cdf based on for . We also consider the empirical cdf based on denoted by , . As is expected to be similar to , we draw quantile-to-quantile (Q–Q) plots to check the distributional similarity. Figure 2 shows the Q–Q plot for four randomly selected stations in July. The distributions are fairly different; that is, the model precipitation has a serious distributional bias. Specifically, rainfall events more frequently occur in the model simulation than in the observations, whereas model extreme rainfalls are weak in terms of the amount of precipitation. For the other months, similar patterns of bias are observed, but are not presented in this paper. To correct the distributional bias, we adopt a QM technique where the related transfer function is
e3.1
Note that, with this method, the distributional bias can be effectively corrected.
Fig. 2.
Fig. 2.

The Q–Q plots of the raw model simulation at randomly selected stations in the basin. Serious distributional biases can be observed.

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-18-0089.1

b. Bias in spatial correlation

In this section, we concentrate on the spatial correlation for wet days. In this paper, a wet day represents a day when the amount of rainfall at some stations is greater than 0. Let
e3.2
be the QM-corrected values. Then, the collections of wet days are formally defined as
eq1

To check the similarity between the spatial correlations of the observations and the model simulation, we compare the correlation coefficients of and for with . Figure 3 presents the plots of the correlation coefficient values of all pairs in July. The spatial correlation of the model simulation is weaker than that of the observations.

Fig. 3.
Fig. 3.

Plots of the Pearson and Spearman correlation coefficients for all station pairs in July. The colors indicate the distances between the two stations. The spatial correlation of the model simulation is weaker than that of the observations.

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-18-0089.1

To investigate the bias in the spatial correlation, we carry out a PCA. Let and ; we then obtain the variances of the principal components and their empirical orthogonal function (EOF) modes from the eigenvalue decompositions of the sample covariance matrices of and . Figure 4 indicates that the first two EOF modes of the observations and the model simulation are almost identical for every month. The first and second modes are clearly interpreted as basin-wide and dipole (north–south contrast) modes, respectively. However, the variance portions of the first principal components are significantly different, as seen in Fig. 5; for example, the variance portion in the observations is around 60%, whereas that in the model simulation is 40% in July. The basin-wide mode of the model simulation has a lower variability than that of the observations, which leads to a weak positive spatial correlation in the model simulation.

Fig. 4.
Fig. 4.

Profiles of the meridional direction of the first two modes of the observations and the model simulation. The results are fairly similar. The first and second modes are interpreted as the basin-wide and the north–south contrast dipole, respectively.

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-18-0089.1

Fig. 5.
Fig. 5.

Variance portions of the principal components. Those of the first principal components are significantly different.

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-18-0089.1

4. Bias correction scheme

To correct the bias of the model precipitation mentioned previously, we propose a new method that aims to generate synthetic precipitation amounts retaining an essential spatial pattern of model precipitation and exhibiting the observational structure of spatial correlation.

Conceptually, the essential spatial pattern is a representative spatial variation of daily precipitation and may be thought as the core information of the model. We intend to introduce a pattern index summarizing such spatial patterns. The index may be defined as a pair of principal components of the basin-wide and dipole modes as they explain at least 60% of the total spatial variation in both the observations and the model simulation; furthermore, the mode shapes are reproduced almost perfectly by the dynamical model. Figure 6 shows how much of the spatial variation of the observation is explained by the patterns reconstructed by the first two principal components (i.e., the sum of the corresponding EOF loading vectors multiplied by the principal components plus the sample mean vector of the daily precipitation). Furthermore, one may consider a correction scheme in terms of the principal components, as the bias in the spatial correlation seems to be caused by the variance portion difference of the basin-wide mode principal component, as indicated in section 3b. However, the problem of considering and developing such a correction scheme is intractable owing to dependency between the principal components. Figure 7 shows that the range of the principal component of the dipole mode depends on the principal component of the basin-wide mode, and, thus, it is infeasible to adjust the variability of each principal component independently. To circumvent the difficulty, we consider a Gaussian copula-based scheme with a different pattern index definition.

Fig. 6.
Fig. 6.

Observations at randomly chosen days (black colored and denoted by 1) and the spatial pattern reconstructed by the first two principal components, i.e., the sum of the corresponding EOF loading vectors multiplied by the principal components plus the sample mean vector of the daily precipitation (red colored and denoted by 2) in the meridional direction. Observational spatial variations are represented well by the reconstructed pattern.

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-18-0089.1

Fig. 7.
Fig. 7.

Dependence between the first two principal components. The range of the principal component of the dipole mode depends on that of the basin-wide mode.

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-18-0089.1

The main scheme conceptually consists of two steps:

  1. an aggregation step—this step smooths out a noisy portion in the spatial variation of the model precipitation to obtain the value of a bivariate pattern index; however, since the resulting index reveals a bias in the correlation, it is corrected by a Gaussian copula–based method; and

  2. a disaggregation step—this next step performs a spatial disaggregation of the pattern index onto a grid system, which generates a synthetic precipitation with the spatial pattern indicated by the corrected index. The generation is conducted by the simulation on the basis of a spatial dependence model derived by a Gaussian copula–based method.

Hereinafter, this section presents the details of the above steps together with the related methodologies, and thus, this section consists of many subsections: the first two subsections present the general concept of Gaussian copula and its adaptation to daily precipitation modeling, respectively. In section 4c, we consider a multivariate bias correction scheme based on the adapted Gaussian copula model. Section 4d deals with the details of the aggregation step. In sections 4e and 4f, a spatial dependence model is derived on the basis of the adapted Gaussian copula model, and the details of the disaggregation step and the main scheme are provided. The last subsection provides the verification results of the proposed scheme.

a. Gaussian copula

This section begins with the general concept of a copula. Unfortunately, the distribution of multisite daily precipitation is far from the multivariate normal, which provides a popular methodology for multivariate modeling. Thus, we need to take a general approach for modeling multisite precipitation. A copula is a multivariate distribution whose marginals are uniformly distributed on the unit interval [0, 1]. It is well known that every multivariate distribution can be represented in terms of its marginal distributions and a copula describing the dependence among the marginals (cf. Sklar 1959). Especially, it implies that the dependence modeling can be performed separately from the marginal estimation. Specifically, let be a generic random vector and be the cdf of , , (). Then, there exists a random vector , each component of which is uniformly distributed on [0, 1], such that X is distributed as .

However, since copulas do not constitute a parametric model class, we need to take a suitable subclass for a feasible estimation. Among the many popular subclasses, we consider the Gaussian copula that assumes ; that is, the dependence structure is determined by a positive-definite covariance matrix (here, denotes the cdf of the standard normal distribution). This model has some remarkable features:

  • P1—rotating the underlying Gaussian random vector suitably produces a random vector of independent components and

  • P2—for a large positive integer p, the model can be readily extended with a mild complexity by imposing a covariance function on .

P1 is a key property for a Gaussian copula to be adopted for the first step of the main scheme; meanwhile, P2 is useful for deriving the spatial dependence model used in the second step. Although many parametric copula models have been proposed, none of them has the above features to the best of our knowledge.

b. Adapted Gaussian copula model for daily precipitation

Hereinafter, let denote a vector of daily precipitation amounts on a wet day rather than generic ones. Its components are always nonnegative. Moreover, by the definition of wet day in section 3b,
  • for each , occurs with a strictly positive probability;

  • furthermore, X is observed only if for some .

Since a naïve Gaussian copula model cannot capture the above features, we adapt it as follows: let follow the conditional distribution of a Gaussian copula vector given for some , and
e4.1
where are called threshold probabilities in this paper and is a continuous cdf with . Then, we have the following:
The derivation of A1–A5 and the parameter estimation are dealt with in the appendix.

c. Multivariate bias correction

In this section, we present a multivariate bias correction scheme based on the adapted Gaussian copula model. In section 4d, we introduce a bivariate pattern index representing a spatial pattern of the daily precipitation; however, the index of the model simulation has a bias in correlation as seen there. The scheme presented in this section will be applied for correcting the bias.

Let be a random vector representing a model simulation of X. Thus, Y shares some properties with X: each component of Y is nonnegative, and the cdf of , , is continuous at but jumps at (). Furthermore, it is assumed that Y also follows an adapted Gaussian copula model whose positive-definite covariance matrix and threshold probabilities are denoted by and , respectively. For simplicity, let the parameters and cdfs be known. In practice, they are substituted with their estimates.

We have and , where and are the diagonal matrices of their eigenvalues and and are orthonormal matrices consisting of their eigenvectors. Note that the orthonormal matrices are utilized for rotating the underlying Gaussian random vector, and the eigenvalues are the marginal variances of the rotated vectors. Furthermore, the loading vectors and eigenvalues are assumed to be arranged in their matrices in the following manner: let and .
  1. The eigenvalues in are arranged in descending order.

  2. There exists a permutation of , and a sequence , , such that and for each j.

  3. Then, is rearranged into , and the eigenvalues in are rearranged so that holds.

This arrangement is based on the fact that is large as and indicate similar modes. See the remarks in section 4c(1) for its details. The problem is that Y has a fairly different dependence structure from that of X. Below, we first present the correction scheme, where denotes the resulting corrected value of Y:
  1. Suppose that for each . Then, letting
    e4.2
where , we take
e4.3
e4.4
where , .
  • (ii) Suppose that for some j but not all. For convenience, let and the other components be nonzero, and and , (). Similarly as in the previous step, . Generate
    e4.5
    eq2
where , , and are the matrices of , , and , respectively, such that
eq3
Then, we obtain from (4.3) and (4.4) with replaced by . Note that the resulting value depends on , which is randomly generated in (4.5). Thus, to obtain the final corrected value, we take the conditional expectation of with respect to given for each .
  • (iii) If , then take .

The first step in the above scheme is to obtain the underlying Gaussian variables. In case i, they can be easily obtained by using (4.2), (cf. A5 in section 4b). P1 in section 4a plays an important role in the next step [(4.3)]: independent components are obtained from rotating by , and, then, their variances are adjusted to those of X by applying to the rotated vector. Note that the adjustment is carried out in a component-wise fashion unlike in the case of the principal components. The step is completed by rotating the adjusted independent components by . Finally, the corrected value is obtained from the resulting Gaussian vector through model specification (4.4) [cf. (4.1)]. In case ii above, some of the underlying Gaussian variables are censored; that is, implies that the corresponding Gaussian variable does not exceed , but its exact value is not observed. In this case, we simulate the censored elements of . Note that (4.5) is the conditional distribution of the censored elements given the completely observed ones, and thus, it is adopted for the simulation. Case iii reveals that the day is dry in the model simulation. For this case, we treat the day as dry in real-world settings.
Note that the above scheme depends on , , (), , , , and . We define the transfer function multivariate bias correction (MBC) of the scheme by
e4.6

Remark 1

Note that the eigenvalue decomposition is not unique in a rigorous sense. For example, even though the eigenvectors in are rearranged in a different order or have their signs altered, still holds provided that the eigenvalues of are also rearranged in the same order. However, the rearranging and sign change crucially influence the above scheme and its interpretation. The arrangement order and the sign determine how to match the loading vectors in to those of Y; in terms of letting and , the variational magnitude of loading vector is adjusted and then delivered to for by (4.3). Furthermore, the loading vectors are associated with the variational modes. Therefore, the arrangement order and the sign are crucial for the main scheme to retain the spatial pattern of the model simulation.

d. Aggregation step

Now, we present the details of the aggregation step. As seen in section 3b, the basin-wide and dipole modes dominate the spatial variation in both the observations and the model simulation. Therefore, we divide into two parts, that is, and , which represent the northern and southern parts in the basin, respectively (see Fig. 1). Then, we let
e4.7
be the local averages of the QM-corrected model simulation and the observations. We define the pairs of local averages as the pattern index for the model simulation and the observations, respectively. Note that the sum and difference of the local averages correspond to the basin-wide and dipole mode variations with high correlation, respectively. Figure 8 shows that the local averages represent the observational spatial variations as much as the principal components do. Therefore, by this pattern index, we take the precipitation information from the model simulation as much as that by the principal components.
Fig. 8.
Fig. 8.

Observations (black colored and denoted by 1) and the spatial pattern reconstructed by the local averages (red colored and denoted by 2) in the meridional direction. Spatial variations of the observations are also represented by the local averages as much as the principal components are.

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-18-0089.1

However, the correlation of is significantly different from that of for every month, as seen in Table 1. To correct the correlation of the QM-corrected local averages, we apply the correction method presented in section 4c. Below, it is helpful for reading to match and to X and Y, respectively, within the context of section 4c. We carry out the following steps for each month.
  1. First, we estimate the parameters of the adapted Gaussian copula model, say, , , and of the observational local averages, by maximizing
    eq4
    where is the likelihood function given in (A.1), and
    eq5
  2. Similarly, we estimate the parameters , , and of the model local averages by maximizing
    eq6
  3. For , we take
    e4.8
    where the transfer function MBC is given in (4.6) and and () are the empirical cdfs of and , respectively.
The following are the estimates of the Gaussian copula covariance matrices and their eigenvalue decompositions for July:
eq7
eq8
Note that in step 3, MBC is applied with , , and
eq9
within the context of section 4c (cf. the arrangement 1–3 in section 4c). Therefore, the scheme rotates the underlying Gaussian random vectors of the local averages so that the resulting components correspond to the variational magnitude in the basin-wide and dipole modes, whose corresponding loading vectors are and , respectively. Moreover, each underlying loading vector of the observational local averages meets that of the same mode of the QM-corrected ones in the sense of the remarks in section 4c(1). Thus, we can interpret that the correction scheme increases the variance portion of the basin-wide mode by and decreases that of the dipole mode by . Similar interpretations are established for the other months, but are not presented here for brevity. Figure 9 shows the moves of the QM-corrected pattern index values by MBC. Most of them move to be more spatially coherent than before without a big difference. Table 1 shows the correction result in terms of the Pearson and Spearman correlation coefficients; meanwhile, Figs. 10 and 11 present the Q–Q plots for checking the similarity of the marginal distributions and the scatterplots for dependence, respectively. As seen in the results, the correlation between the QM-corrected local averages is satisfactorily corrected by the correction scheme.
Table 1.

Correlation coefficients of the pattern indices.

Table 1.
Fig. 9.
Fig. 9.

Plot of the QM-corrected pattern index values (black dot) and their corrected values by MBC (red dot). We randomly chose 20 points with more than 10 mm of the basin average. Most of the points move to be more spatially coherent than before.

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-18-0089.1

Fig. 10.
Fig. 10.

The Q–Q plots of the corrected local averages by MBC for each month.

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-18-0089.1

Fig. 11.
Fig. 11.

Scatterplots of the pattern indices. The dashed lines indicate several quantiles along an angle from 0° to 90° and aid in recognizing the bivariate distribution shape. The shape became closer to that of the observations by using the proposed method.

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-18-0089.1

e. Spatial dependence model and simulation

For the second step of the main scheme, a spatial dependence model for real daily precipitation is required. The adapted Gaussian copula model is also employed owing to its feature P2: let be the amount of rainfall on a wet day at site , and let , , be its conditional cdf. For conciseness, the cdfs are assumed to be known; however, in practice, they are replaced with the empirical counterparts. We assume that there exists a class of uniformly distributed random variables and satisfying
e4.9
The spatial dependence structure of is described by the covariance structure of , that is, together with . Note that the covariance structure itself is of high complexity: the number of parameters is ) and for most of , observations are not available. For a feasible estimation and simulation, an anisotropic covariance function is imposed on the covariance structure:
e4.10
where
eq10
eq11
This covariance function has been popularly used for kriging (cf. Stein 1999). With this function, as the distance between and increases, the correlation decreases. The decreasing rate depends not on the location but on the direction, which is determined by .
Let and be a matrix where the -th element is (). The covariance function estimation is carried out by maximizing
eq12
where is given in (A.1). For the observational data used in this study, Table 2 presents the estimation results for June–August.
Table 2.

Estimation results of the spatial dependence model.

Table 2.
Now, we consider the simulation of real precipitation on grid system by using the spatial dependence model. To estimate quantiles at ungauged grid points, we employ
eq13
We present a simulation algorithm of .

Algorithm 1

A new is generated by carrying out the following steps.

  1. Take a realization of the Gaussian field given in (4.9) and (4.10).

  2. Let
    eq14

    for .

  3. If for some , it is then completed. Otherwise, go back to 1.

f. Disaggregation step

The second step of the main scheme is a disaggregation of the pattern index, which is conducted by applying the stochastic analog technique proposed by Hwang and Graham (2013) but is based on the spatial dependence model presented in (4.9) and (4.10). We carry out disaggregation by using the algorithm presented below. Note that the role of step 3 therein is to keep a given pattern index within the tolerance.

1) Algorithm 2

Let be a given pattern index with . Fix the tolerance .

  1. Generate a synthetic precipitation through algorithm 1.

  2. Calculate its local averages for .

  3. Take as a disaggregation of if . Otherwise, go back to 1.

2) Remark 2

Hwang and Graham (2013) proposed a similar disaggregation scheme that is called the bias correction and stochastic analog (BCSA). Therein, the spatial correlation is estimated by the Pearson correlation coefficients among the pseudo–z scores of the gridded observations. On the other hand, our spatial dependence model uses raw observations, but, by virtue of the covariance function, we can conduct the disaggregation onto a grid system even without gridded observations.

Finally, we propose the main scheme.

3) Main scheme

We begin by calculating

  • ,forand , and

  • ,forand .

  1. Take
    eq15

    See (4.8) for the details.

  2. If , then we treat day as a dry day, and, thus, we take for every ; otherwise, we obtain by using algorithm 2 with therein.

  3. Let be a corrected value of .

We obtain diverse correction results by carrying out step 2 in the above scheme repeatedly as the disaggregation result of algorithm 2 is not unique. Figure 12 presents several different correction results for the raw model precipitation on a wet day with the tolerance ε in algorithm 2 set to 0.1. Note that all of the results retain the corrected pattern index of the model precipitation within the tolerance.

Fig. 12.
Fig. 12.

(top left) Raw model precipitation, (bottom left) its QM-corrected version, and some examples of its disaggregation result produced by the proposed method in the remaining panels. The red- and blue-colored circles indicate the locations of the stations in the northern and southern parts, respectively. The local average vector of the QM-corrected version is (15.28, 9.62), which is corrected to (17.36, 11.19) by the proposed method. All the disaggregation results retain the corrected local averages within a tolerance of 0.1.

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-18-0089.1

g. Results and verification

The proposed main scheme is applied to the model precipitation data, where the tolerance ε in algorithm 2 is set to 0.1. Then, we verify whether the biases are corrected appropriately. As mentioned previously, the disaggregation result is not unique but random. Thus, we produce 1000 correction datasets for representing the randomness. In the following, we present several figures revealing the verification, where the gray-colored bands indicate the confidence ranges of 95% coverage probability that are obtained from the datasets.

First, we check whether the distributional bias is corrected. Figures 13 and 14 show the Q–Q plots of the correction result for randomly selected stations in the northern and southern parts, respectively. As can be seen in these figures, the distributional bias is corrected successfully. Furthermore, Figs. 1517 compare some prominent characteristics of the precipitation data reproduced by the raw model simulation and its correction result with those of the observation. In the figures, we deal with precipitation intensities, 50% and 95% quantiles, and probabilities of zero amounts at the sites on a wet day. On the basis of the results of the comparison, we confirm that the proposed method definitely improves the statistical characteristics of the raw data. Note that the distinct feature of the proposed method is to correct the bias in the spatial dependence as well as that in the distribution. Figures 18 and 19 compare their capabilities in reproducing the observational spatial dependence. The Spearman correlation coefficient and log odds ratio are employed as the measure of spatial dependence in Figs. 18 and 19, respectively. The Spearman correlation coefficient indicates the correlation between the precipitation amounts at two sites, whereas the log odds ratio measures the coherence between rainfall event occurrences, specifically, the logarithm of the ratio between the number of days when two sites are either rainy or dry at the same time and the number of days in the opposite case. Both analyses show that the proposed method reproduces a fairly similar spatial dependence to that of the observations.

Fig. 13.
Fig. 13.

The Q–Q plots of the corrected model precipitation by the proposed method for four randomly selected northern stations in the basin. The distributional bias is acceptably corrected.

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-18-0089.1

Fig. 14.
Fig. 14.

The Q–Q plots of the corrected model precipitation by the proposed method for four randomly selected southern stations in the basin. The distributional bias is acceptably corrected.

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-18-0089.1

Fig. 15.
Fig. 15.

Several statistical characteristics of the observations, model simulation, and its correction result for June. All the characteristics are manifestly improved by the proposed method.

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-18-0089.1

Fig. 16.
Fig. 16.

Several statistical characteristics of the observations, model simulation, and its correction result for July. All the characteristics are manifestly improved by the proposed method.

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-18-0089.1

Fig. 17.
Fig. 17.

Several statistical characteristics of the observations, model simulation, and its correction result for August. All the characteristics are manifestly improved by the proposed method.

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-18-0089.1

Fig. 18.
Fig. 18.

Spearman correlation coefficients of all pairs for the corrected model data by the QM technique and the proposed method. The colors indicate the distances between the two stations. The biased coefficients are suitably corrected by the proposed method.

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-18-0089.1

Fig. 19.
Fig. 19.

Log odds ratios of all pairs for the corrected model data by the QM technique and the proposed method. The colors indicate the distances between the two stations. The proposed method suitably corrects the results.

Citation: Journal of Applied Meteorology and Climatology 58, 2; 10.1175/JAMC-D-18-0089.1

5. Conclusions and discussion

a. Conclusions

In this paper, we proposed a new scheme to correct biases in spatial correlation as well as in the marginal distributions of the simulated precipitation. Simultaneously, the scheme retains a representative spatial variation of the raw model precipitation summarized into the pattern index. The proposed scheme is based on the Gaussian copula, which plays a prominent role in the correction process: it provides a feasible way to adjust the biased correlation structure. Moreover, combined with a parametric covariance function, it derives a spatial dependence model that is used for the disaggregation of the pattern index onto a grid system. We verify that the proposed scheme exhibits good performance as several biased features of the model simulation are corrected satisfactorily. Hence, we conclude that the proposed scheme is an appropriate solution for correcting the biases of the model simulation without distortion of the spatial pattern.

b. Discussion

The proposed scheme begins by taking a pair of local averages of the daily precipitation as a pattern index, technically for a suitable dimensional reduction. If the scheme is applied to another wide basin, then the local averages have to be taken in a different fashion to represent appropriately the precipitation spatial pattern on the basin. Although an EOF analysis is not necessary, it may provide a nice clue for taking the local averages, as shown in this paper. In general, we wish to divide into several parts (i.e., , ), such that letting and , ,

  1. p is small,

  2. is small, and

  3. the distribution of is similar to that of .

Conditions i and ii require a simple pattern index that can explain a large portion of the total precipitation variation. Moreover, by condition iii, the pattern index is intended to be simulated by the dynamical model at hand owing to the correction burden. In a future study, we will explore such methods.

Similar correction schemes have been proposed: Bárdossy and Pegram (2012) proposed a scheme based on a matrix recorrelation technique, Cannon (2016) utilized Choleskey decomposition instead of eigenvalue decomposition and Cannon (2018) applied an N-pdft scheme, which is based on a random rotation method combined with univariate quantile mapping, to climate change scenario data. Among them, that of Bárdossy and Pegram (2012) is most related to our scheme. Here, we compare both methods: in their scheme, (4.3) is replaced with . Let , , and, , and then, the jth element of , (), is with (i.e., a weighted sum of ). Note that and that is close to 1 as and indicate similar modes. In the case of the proposed scheme, for each j, for k such that is locally maximized at , and for . In summary, the difference between the two methods is the choice of , which determines the connection between and : in Bárdossy and Pegram (2012), each is connected to all the loading vectors in with different weights proportional to the similarity, whereas in our scheme it is connected to just one loading vector with the most similarity. A comparison study is left as a future project.

Acknowledgments

This research was supported by the APEC Climate Center.

APPENDIX

Derivation of the Properties and Estimation of the Adapted Gaussian Copula Model

a. Derivation of A1–A5

Here, we derive A1–A5 presented in section 4b. In this appendix, all the conditions and notations therein are used.

A1 is trivially verified by definition in (4.1). Moreover, for ,
eq16
eq17
eq18
eq19
Since for , A2 holds. A3 and A4 are verified as follows:
eq20
eq21
where , and, thus, for ,
eq22
As if and only if ,
eq23
which implies A5.

b. Estimation of the adapted Gaussian copula model

We consider estimating the dependence structure of X under the adapted Gaussian copula model (i.e., and ). Let be an independent sample of X, and , . Since the underlying Gaussian variables are incompletely observed, the ordinary maximum likelihood principle is not feasible. Instead, we consider a maximum composite likelihood method for the estimation (cf. Varin and Vidoni 2005). Let
eq24
which is the estimate of , and take satisfying . For the data used in this study, we chose and . Note that the role of is to reduce the influence of extremely small or large rainfall amounts on the estimation. For with , set
eq25
eq26
eq27
eq28
where , , , (), , ,
eq29
and
eq30
eq31
Meanwhile, set
eq32
for . Let be the parametric model of , where θ is the parameter vector. Then, the estimates of θ and are obtained by maximizing
eea_1_1
where is the th component of and
eq33

REFERENCES

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  • Schmidli, J., C. Goodess, C. Frei, M. Haylock, Y. Hundecha, J. Ribalaygua, and T. Schmith, 2007: Statistical and dynamical downscaling of precipitation: An evaluation and comparison of scenarios for the European Alps. J. Geophys. Res., 112, D04105, https://doi.org/10.1029/2005JD007026.

    • Search Google Scholar
    • Export Citation
  • Sklar, M., 1959: Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Stat. Univ. Paris, 8, 229231.

  • Stein, M. L., 1999: Statistical Interpolation of Spatial Data: Some Theory for Kriging. Springer Series in Statistics, Springer-Verlag, 249 pp.

    • Crossref
    • Export Citation
  • Varin, C., and P. Vidoni, 2005: A note on composite likelihood inference and model selection. Biometrika, 92, 519528, https://doi.org/10.1093/biomet/92.3.519.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Vrac, M., and P. Friederichs, 2015: Multivariate—intervariable, spatial, and temporal—bias correction. J. Climate, 28, 218237, https://doi.org/10.1175/JCLI-D-14-00059.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Yhang, Y.-B., S.-J. Sohn, and I.-W. Jung, 2017: Application of dynamical and statistical downscaling to East Asian summer precipitation for finely resolved datasets. Adv. Meteor., 2017, 2956373, https://doi.org/10.1155/2017/2956373.

    • Crossref
    • Search Google Scholar
    • Export Citation
Save
  • Anderson, M., Z. Chen, M. Kavvas, and J. Yoon, 2007: Reconstructed historical atmospheric data by dynamical downscaling. J. Hydrol. Eng., 12, 156162, https://doi.org/10.1061/(ASCE)1084-0699(2007)12:2(156).

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bárdossy, A., and G. Pegram, 2012: Multiscale spatial recorrelation of RCM precipitation to produce unbiased climate change scenarios over large areas and small. Water Resour. Res., 48, https://doi.org/10.1029/2011WR011524.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Boé, J., L. Terray, F. Habets, and E. Martin, 2007: Statistical and dynamical downscaling of the Seine basin climate for hydro-meteorological studies. Int. J. Climatol., 27, 16431655, https://doi.org/10.1002/joc.1602.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cannon, A. J., 2016: Multivariate bias correction of climate model output: Matching marginal distributions and intervariable dependence structure. J. Climate, 29, 70457064, https://doi.org/10.1175/JCLI-D-15-0679.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cannon, A. J., 2018: Multivariate quantile mapping bias correction: An N-dimensional probability density function transform for climate model simulations of multiple variables. Climate Dyn., 50, 3149, https://doi.org/10.1007/s00382-017-3580-6.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hong, S.-Y., and Coauthors, 2013: The Global/Regional Integrated Model system (GRIMs). Asia-Pac. J. Atmos. Sci., 49, 219243, https://doi.org/10.1007/s13143-013-0023-0.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hwang, S., and W. D. Graham, 2013: Development and comparative evaluation of a stochastic analog method to downscale daily GCM precipitation. Hydrol. Earth Syst. Sci., 17, 44814502, https://doi.org/10.5194/hess-17-4481-2013.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kure, S., S. Jang, N. Ohara, M. Kavvas, and Z. Chen, 2013: WEHY-HCM for modeling interactive atmospheric-hydrologic processes at watershed scale. II: Model application to ungauged and sparsely gauged watersheds. J. Hydrol. Eng., 18, 12721281, https://doi.org/10.1061/(ASCE)HE.1943-5584.0000701.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Laux, P., S. Vogl, W. Qiu, H. Knoche, and H. Kunstmann, 2011: Copula-based statistical refinement of precipitation in RCM simulations over complex terrain. Hydrol. Earth Syst. Sci., 15, 24012419, https://doi.org/10.5194/hess-15-2401-2011.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lee, J.-W., S.-Y. Hong, E.-C. Chang, M.-S. Suh, and H.-S. Kang, 2014: Assessment of future climate change over East Asia due to the RCP scenarios downscaled by GRIMs-RMP. Climate Dyn., 42, 733747, https://doi.org/10.1007/s00382-013-1841-6.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mao, G., S. Vogl, P. Laux, S. Wagner, and H. Kunstmann, 2015: Stochastic bias correction of dynamically downscaled precipitation fields for Germany through copula-based integration of gridded observation data. Hydrol. Earth Syst. Sci., 19, 17871806, https://doi.org/10.5194/hess-19-1787-2015.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Maraun, D., 2013: Bias correction, quantile mapping, and downscaling: Revisiting the inflation issue. J. Climate, 26, 21372143, https://doi.org/10.1175/JCLI-D-12-00821.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Maraun, D., and Coauthors, 2017: Towards process-informed bias correction of climate change simulations. Nat. Climate Change, 7, 764773, https://doi.org/10.1038/nclimate3418.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rajczak, J., S. Kotlarski, and C. Schär, 2016: Does quantile mapping of simulated precipitation correct for biases in transition probabilities and spell lengths? J. Climate, 29, 16051615, https://doi.org/10.1175/JCLI-D-15-0162.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Saha, S., and Coauthors, 2014: The NCEP Climate Forecast System version 2. J. Climate, 27, 21852208, https://doi.org/10.1175/JCLI-D-12-00823.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Schmidli, J., C. Goodess, C. Frei, M. Haylock, Y. Hundecha, J. Ribalaygua, and T. Schmith, 2007: Statistical and dynamical downscaling of precipitation: An evaluation and comparison of scenarios for the European Alps. J. Geophys. Res., 112, D04105, https://doi.org/10.1029/2005JD007026.

    • Search Google Scholar
    • Export Citation
  • Sklar, M., 1959: Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Stat. Univ. Paris, 8, 229231.

  • Stein, M. L., 1999: Statistical Interpolation of Spatial Data: Some Theory for Kriging. Springer Series in Statistics, Springer-Verlag, 249 pp.

    • Crossref
    • Export Citation
  • Varin, C., and P. Vidoni, 2005: A note on composite likelihood inference and model selection. Biometrika, 92, 519528, https://doi.org/10.1093/biomet/92.3.519.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Vrac, M., and P. Friederichs, 2015: Multivariate—intervariable, spatial, and temporal—bias correction. J. Climate, 28, 218237, https://doi.org/10.1175/JCLI-D-14-00059.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Yhang, Y.-B., S.-J. Sohn, and I.-W. Jung, 2017: Application of dynamical and statistical downscaling to East Asian summer precipitation for finely resolved datasets. Adv. Meteor., 2017, 2956373, https://doi.org/10.1155/2017/2956373.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    Map of the target basin. The Nakdong River has one of the largest basins on the Korean Peninsula. There are 14 stations nearby. The numbers on the map identify the stations. The seven red- and blue-colored circles indicate the stations in the northern and southern parts of the basin, respectively. In section 4d, they are denoted by and , respectively.

  • Fig. 2.

    The Q–Q plots of the raw model simulation at randomly selected stations in the basin. Serious distributional biases can be observed.

  • Fig. 3.

    Plots of the Pearson and Spearman correlation coefficients for all station pairs in July. The colors indicate the distances between the two stations. The spatial correlation of the model simulation is weaker than that of the observations.

  • Fig. 4.

    Profiles of the meridional direction of the first two modes of the observations and the model simulation. The results are fairly similar. The first and second modes are interpreted as the basin-wide and the north–south contrast dipole, respectively.

  • Fig. 5.

    Variance portions of the principal components. Those of the first principal components are significantly different.

  • Fig. 6.

    Observations at randomly chosen days (black colored and denoted by 1) and the spatial pattern reconstructed by the first two principal components, i.e., the sum of the corresponding EOF loading vectors multiplied by the principal components plus the sample mean vector of the daily precipitation (red colored and denoted by 2) in the meridional direction. Observational spatial variations are represented well by the reconstructed pattern.

  • Fig. 7.

    Dependence between the first two principal components. The range of the principal component of the dipole mode depends on that of the basin-wide mode.

  • Fig. 8.

    Observations (black colored and denoted by 1) and the spatial pattern reconstructed by the local averages (red colored and denoted by 2) in the meridional direction. Spatial variations of the observations are also represented by the local averages as much as the principal components are.

  • Fig. 9.

    Plot of the QM-corrected pattern index values (black dot) and their corrected values by MBC (red dot). We randomly chose 20 points with more than 10 mm of the basin average. Most of the points move to be more spatially coherent than before.

  • Fig. 10.

    The Q–Q plots of the corrected local averages by MBC for each month.

  • Fig. 11.

    Scatterplots of the pattern indices. The dashed lines indicate several quantiles along an angle from 0° to 90° and aid in recognizing the bivariate distribution shape. The shape became closer to that of the observations by using the proposed method.

  • Fig. 12.

    (top left) Raw model precipitation, (bottom left) its QM-corrected version, and some examples of its disaggregation result produced by the proposed method in the remaining panels. The red- and blue-colored circles indicate the locations of the stations in the northern and southern parts, respectively. The local average vector of the QM-corrected version is (15.28, 9.62), which is corrected to (17.36, 11.19) by the proposed method. All the disaggregation results retain the corrected local averages within a tolerance of 0.1.

  • Fig. 13.

    The Q–Q plots of the corrected model precipitation by the proposed method for four randomly selected northern stations in the basin. The distributional bias is acceptably corrected.

  • Fig. 14.

    The Q–Q plots of the corrected model precipitation by the proposed method for four randomly selected southern stations in the basin. The distributional bias is acceptably corrected.

  • Fig. 15.

    Several statistical characteristics of the observations, model simulation, and its correction result for June. All the characteristics are manifestly improved by the proposed method.

  • Fig. 16.

    Several statistical characteristics of the observations, model simulation, and its correction result for July. All the characteristics are manifestly improved by the proposed method.

  • Fig. 17.

    Several statistical characteristics of the observations, model simulation, and its correction result for August. All the characteristics are manifestly improved by the proposed method.

  • Fig. 18.

    Spearman correlation coefficients of all pairs for the corrected model data by the QM technique and the proposed method. The colors indicate the distances between the two stations. The biased coefficients are suitably corrected by the proposed method.

  • Fig. 19.

    Log odds ratios of all pairs for the corrected model data by the QM technique and the proposed method. The colors indicate the distances between the two stations. The proposed method suitably corrects the results.

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