Understanding the joint behavior of temperature and

Theor Appl Climatol DOI 10.1007/s00704-016-1774-1

ORIGINAL PAPER

Understanding the joint behavior of temperature and precipitation for climate change impact studies Arun Rana 1 & Hamid Moradkhani 1 & Yueyue Qin 1

Received: 20 August 2015 / Accepted: 13 March 2016 # Springer-Verlag Wien 2016

Abstract The multiple downscaled scenario products allow us to assess the uncertainty of the variations of precipitation and temperature in the current and future periods. Probabilistic assessments of both climatic variables help better understand the interdependence of the two and thus, in turn, help in assessing the future with confidence. In the present study, we use ensemble of statistically downscaled precipitation and temperature from various models. The dataset used is multi-model ensemble of 10 global climate models (GCMs) downscaled product from CMIP5 daily dataset using the Bias Correction and Spatial Downscaling (BCSD) technique, generated at Portland State University. The multi-model ensemble of both precipitation and temperature is evaluated for dry and wet periods for 10 sub-basins across Columbia River Basin (CRB). Thereafter, copula is applied to establish the joint distribution of two variables on multi-model ensemble data. The joint distribution is then used to estimate the change in trends of said variables in future, along with estimation of the probabilities of the given change. The joint distribution trends vary, but certainly positive, for dry and wet periods in sub-basins of CRB. Dry season, generally, is indicating a higher positive change in precipitation than temperature (as compared to historical) across sub-basins with wet season inferring otherwise. Probabilities of changes in future, as estimated from the joint Electronic supplementary material The online version of this article (doi:10.1007/s00704-016-1774-1) contains supplementary material, which is available to authorized users. * Hamid Moradkhani [email protected]

1

Remote Sensing and Water Resources Laboratory, Department of Civil and Environmental Engineering, Portland State University, 1930 S.W. 4th Avenue, Suite 200, Portland, OR 97201, USA

distribution, indicate varied degrees and forms during dry season whereas the wet season is rather constant across all the sub-basins.

1 Introduction Many global services are closely related/dependent on climate of the region e.g., agriculture, water, ecosystem, fisheries, and energy. Climate change resulting from increasing emissions of greenhouse gases will threaten these by increase in temperature, changes in the amount, and intensity of precipitation, among others, and thus affecting the variability of climatic variables (IPCC 2007, 2013). Drought will threaten the water supplies for the crop irrigation, which will directly reduce the production of crops, whereas floods would wipe out the yields and thus causing direct damage. This is a threat to global food security where we have forever-increasing population. Climate change is also likely to shrink the water supply and simultaneously extend the demand of the said resource. Water quality will be an issue to tackle with areas experiencing excessive precipitation. Heavy precipitation will increase amount of runoff into water bodies and will also lead to increased sediments, pollutants, etc. Changes in climate have significant consequences on the ecosystem, including changes of habitat and alteration of species. The same will also affect energy production. Changing climate has been altering the hydrology and exerting global scale impacts on our environment with significant implications for water resources (Barnett et al. 2008; Milly et al. 2008). A detailed analysis of the cause and effects of climate change is presented by Karl and Melillo (2009), especially in USA. Temperature and precipitation are between two main climatic variables. Changes in these are highly indicative of climate change, and it is difficult to simulate rainfall and

Rana A. et al.

temperature simultaneously due to the interdependence (correlation) between them. Researchers have focused their resources to study the (future) changes in these climatic variables across the world. Piao et al. (2010) studied the projected changes and trends of temperature and precipitation to assess the impacts of climate change on water resource and agriculture in China. Recently, Feng et al. (2014) studied the shifts in climate regimes from 20 GCMs participating in phase 5 of the Coupled Model Intercomparison Project (CMIP5). Najafi and Moradkhani (2015) conducted a multi-model ensemble analysis of extreme events within a Bayesian framework over the Pacific Northwest of USA. Wuebbles et al. (2014) studied the climate extremes in the contiguous USA using CMIP5 data. Future climate impact studies based in the USA have also been conducted by many researchers, including Brunsell et al. (2010); Halmstad et al. (2012); Najafi and Moradkhani (2014), and Pierce et al. (2013). Specifically, in the Pacific Northwest, research has been focused on climate change effects using various variables from climatic variables e.g., Rupp et al. (2013a, b) and Klos et al. (2014). Abatzoglou et al. (2014) evaluated seasonal climate variability and change in PNW. Demirel and Moradkhani (2015) evaluated the seasonality and timing of precipitation in PNW using CMIP5 dataset. Recently, Rana and Moradkhani (2015) evaluated the spatial, temporal, and frequency-based changes of precipitation and temperature in PNW for future climate scenarios. All the studies have used climatic variables from models or observations but have not emphasized the interrelationship of these variables in evaluating the effect of climate change. Currently, most of the studies consider temperature and precipitation independently while assessing the climate change impact. The attention should be directed towards the combined effects of temperature and precipitation change and thus associated impacts. Limited number of studies have looked into this aspect, for example, temporal relationships between the two variables was studied by Rajeevan et al. (1998) (monthly basis) and Huang et al. (2009) (annual basis). Collins and Knight (2007) studied the applicability of ensemble for climate change projections. Tebaldi and Sansó (2009) established the joint projections of future temperature and precipitation trends and changes by applying a Bayesian hierarchical model to the general circulation models. The statistical methods for probabilistic projections have been applied to establish the joint behavior of two or more correlated variables. Hydrological and climatological variables are usually correlated to each other and thus the statistical methods could be used to explain interdependence. Over the past decade, copula functions have widely been used to develop the joint distribution of variables for various applications, including rainfall analysis (Zhang and Singh 2007a, b; Salvadori and Michele 2006; Kao and Govindaraju 2010; Serinaldi 2008; Laux et al. 2009, 2011), flood analysis

(Favre et al. 2004; Zhang and Singh 2007a, b; Salvadori and Michele 2010), drought analyses (Shiau et al. 2007; Dupuis 2007; Kao and Govindaraju 2010; Wong et al. 2009; Madadgar and Moradkhani 2013, 2014a). Recently, Madadgar and Moradkhani (2014b) improved multimodeling by introducing copula in the Bayesian model averaging (BMA) for hydrological forecasts. For temperature and rainfall interdependence, Schölzel and Friederichs (2008) used a simple statistical model based on the copula approach. They found that the cold periods were accompanied by small precipitation amounts. Recently, Cong and Brady (2012) applied copula to develop the joint distribution between temperature and precipitation over the historical data and studied the interdependence of these variables on the future period. In the present study, we have applied copula-based approach to evaluate future climatic conditions using multimodel scenario ensemble data. We have established joint distribution between precipitation and temperature climate variables in the historical period (1970–1999) using copula. The method is applied over 10 sub-basins over the Columbia River Basin (CRB) using multi-model ensemble of 10 downscaled global climate models (GCMs) from CMIP5 daily data. The multi-model ensemble resulting in probability density function (PDF) of both precipitation and temperature is evaluated for the drier period (i.e., April–September) and the wetter period (i.e., October–March) in historical data. The joint PDFs are then used to evaluate the trends over sub-basins in future multi-model ensemble scenario period (2070–2099) along with estimation of the probabilities of the given change. This paper is organized in following manner: section 1 BIntroduction^ is followed by section 2 describing the study area, data used, and methodology (multi-model ensemble and joint distribution) applied for analysis. Section 3 ("Results") presents multi-model ensemble product and for application of copula in studying the joint change of temperature and precipitation. The section also deals with discussion of obtained results. Section 4 provides concluding remarks.

2 Study area, data used, and methodology 2.1 Study area This study aims to analyze the multi-model ensemble climate projections over the CRB (Fig. 1), in the northwestern region of the USA. The Columbia River is the largest river in the Pacific Northwest region of North America. It drains an area of about 567,000 km2 consisting of portions of seven states in the western USA (Washington, Oregon, Idaho, Montana, Wyoming, Nevada, and Utah), contributing to almost 85 % of the basin and part of British Columbia, in western Canada (15 % of the basin). The 10 sub-basins delineated are namely the Dalles (dalles), TW Sullivan (sul), Priest Rapids (pr), Ice

Joint behavior of temperature and precipitation for climate change impact

Fig. 1 Study area, 10 sub-basins delineated from Columbia River Basin (CRB) in the Pacific Northwest USA. A location of CRB in USA, B elevation (DEM) map of CRB with red color representing the highest elevation followed by yellow and green, C 10 sub-basins with specific stream flow location points used for delineation of watershed from DEM, D the Dalles and TW Sullivan sub-basins in CRB, E Priest Rapids and Ice

Harbor sub-basins (within the Dalles) in CRB, F Oxbox sub-basin (within Ice Harbor) in CRB, G Chief Joseph sub-basin (within Priest Rapids) in CRB, H Waneta sub-basin (within Chief Joseph) in CRB, I Corra Linn and Revelstoke sub-basin (within Waneta) in CRB, J Mica sub-basin (within Revelstoke) in CRB

Harbor (ib), Oxbox (oxbow), Chief Joseph (cj), Waneta (wan), Corra Linn (cor), Revelstoke (rev), and Mica (mica), summarized in Table 1. For simplicity, we use the names in brackets for the rest of the paper.

multi-model ensemble techniques was done on historical data (1970–1999) and RCP 4.5 future data (2070–2099). The analysis of changes, trends, and probabilities for joint distribution is performed on multi-model ensemble future data period 2070–2099 as compared to multi-model ensemble historical data. Daily records of precipitation (P) and near-surface temperature (T) in the study region (Fig. 1) were collected for 10 GCMs (Table 2) of the CMIP5 data (Taylor et al. 2012) and divided among seasons. Readers are referred to Taylor et al. (2012) and its references for more details about the scenarios and CMIP5 data. Details of the models, institutions, GCM resolution, and scenario used are provided in Table 2, along with total other information used in the present study. The analysis is performed for the two periods of April– September and October–March across all 10 sub-basins in the CRB.

2.2 Data used In the present study, daily (statistically downscaled) climate data for 10 GCMs from CMIP5 dataset over CRB was used (Ahmadalipour et al. 2015). The authors used uni-variate and multivariate statistical approaches to evaluate 20 GCM pools and selected 10 GCMs that better represent the precipitation and temperature properties in CRB. They were then downscaled using the bias correction and spatial downscaling (BCSD) method (Wood et al. 2002) at Portland State University (Rana and Moradkhani 2015). The analysis of

Rana A. et al. Table 1 Description of different sub-basins delineated and used in this study (refer to Fig. 1 for details on geographical locations of each)

Sub-basin number

Sub-basin name

Representative name

Area (1000 km2)

Average annual P (mm)

Average annual TMax (°C)

Average annual TMin (°C)

1

Chief Joseph

cj

193.6

1024

9

−3

2 3

Corra Linn The Dalles

cor dalles

121.7 614.3

1188 684

7.4 12.2

−4.3 −1.7

4

Ice Harbor

ib

278.5

578

13.4

−1.7

5 6

Mica Oxbow

mica oxbow

8.4 187

1164 493

6 13.7

−5.3 −2

7 8

Priest Rapids Revelstoke

pr rev

245.2 13.4

915 1189

9.8 6

−2.3 −5.1

9

TW Sullivan

sul

24.6

1661

13.9

2.1

10

Waneta

wan

151.9

1132

7.9

−4

2.3 Methodology 2.3.1 Multi-model ensemble (bootstrap sampling) Before deducing marginal distributions of climatic variables (temperature and precipitation), we ought to select the most suitable theoretical distributions. In case of climate change scenarios, it is advisable to do so on ensemble of simulations to be able to characterize the uncertainty in climate models in a probabilistic way. Advantages of using multi-model ensemble instead of single scenario/projection have been pointed out in various studies previously (Krishnamurti et al. 2000; Raftery et al. 2005; Tebaldi and Knutti 2007; Duan and Phillips 2010; Najafi and Moradkhani 2015). Multi-model ensemble provides an alternative to improve projections, in terms of reduction of spread from large sets of climate projections, requires less computation capacity, and improves accuracy in model predictions.

Various studies on multi-model ensemble have been conducted with different techniques (e.g., Giorgi and Mearns 2002; Knutti et al. 2010; Feng et al. 2014). Even though several advanced methods have been proposed to combine multimodel ensemble (e.g., Duan and Phillips 2010; Sillmann et al. 2013), research has indicated that ensemble mean outperforms all or most of the individual models (e.g., Tebaldi and Knutti 2007; Knutti et al. 2010; Madadgar and Moradkhani 2013, 2014a, b). The primary objective of this research is not centered towards the multi-model method used; the authors have chosen the combination of ensemble mean with bootstrap sampling for generating multi-model ensemble. Here we will be applying the ensemble mean using bootstrap sampling to pre-process our multi-model GCM simulations in historical period (1970–1999). The same approach would be applied to future data (2070–2099) for multi-model ensemble, on satisfactory application in historical data, to analyze the trends of changes in said climatic variables as compared to historical ones.

Table 2 Models used in this study and their characteristics. All the models used were statistically downscaled using BCSD and MACA techniques for entire CRB for RCPs 8.5 and 4.5 in future period (2010–2099) Sub-basin number

Model

Center

Atmosphere resolution (Lon × Lat)

Vertical levels in atm

1 2 3 4 5 6 7 8 9 10

bcc-csm1–1 CanESM2 CCSM4 GFDL-ESM2G GFDL-ESM2M inmcm4 IPSL-CM5A-LR IPSL-CM5A-MR IPSL-CM5B-LR MIROC5

Beijing Climate Center, China Meteorological Administration Canadian Centre for Climate Modeling and Analysis National Center of Atmospheric Research, USA NOAA Geophysical Fluid Dynamics Laboratory, USA NOAA Geophysical Fluid Dynamics Laboratory, USA Institute for Numerical Mathematics, Russia Institut Pierre Simon Laplace, France Institut Pierre Simon Laplace, France Institut Pierre Simon Laplace, France Atmosphere and Ocean Research Institute (The University of Tokyo), National Institute for Environmental Studies, and Japan Agency for Marine-Earth Science and Technology

2.8 2.8 1.25 2.5 2.5 2.0 3.75 2.5 3.75 1.4

26 35 26 48 48 21 39 39 39 40

× × × × × × × × × ×

2.8 2.8 0.94 2.0 2.0 1.5 1.8 1.25 1.8 1.4


The bootstrap sampling infers the estimates or statistics, such as mean, variance, and error about population from sample datasets. It is simply random sampling from population with replacement based on the desired statistics by users (the expected value in this case and thus reduction of variance). For bootstrap sampling, the sample dataset represents Bpopulation^ (10 GCM projections), and re-sample dataset (multi-model ensemble) is regarded as sampling from population, which enables us to measure the quality of inference (mean) from resample data to population. Furthermore, bootstrap sampling can provide the typical distribution and uncertainty inherent in the estimation of statistics. Suppose we have a sample of n points with x1 , x2 , ⋯ , xn. For this data, we are interested in the statistical estimation: s ¼ S ðx1 ; x2 ; ⋅⋅⋅; xn Þ

ð1Þ

where s is the statistical estimation from sample; S is the statistical estimation function of the samples provided. In the present study, we have used bootstrap sampling (Ebtehaj et al. 2010) to re-sample our dataset (daily historical precipitation and temperature) 200 times and thus creating a new multi-model ensemble time series for both variables. Each of this is obtained by randomly sampling n points from original sample (x1, x2, ⋯ , xn) with replacement. In turn, each re-sample dataset will be constructed as x*1 , x*2 , ⋯ , x*n It is quite likely that some of the original sample elements are selected more than once but others are omitted, which is the intention behind bootstrap sampling to resample as many times as possible in order to take all the elements into consideration. Mathematically, we will be able to compute its statistical estimation as follows: s*i ¼ S x*i1 :x*i2 ; ⋅⋅⋅; x*in ð2Þ where i = 1 , ⋯ N, and N represents the total number of resamples and i represents each re-sample; x*i1, x*i2, ⋯ , x*in represent randomly sampled n points from original sample set. In addition to a set of statistical estimation siconstructed, a distribution of statistical estimations s will be obtained and in turn be used to approximate the distribution of the actual statistical estimation (s). The statistical estimation in consideration for the study is representation of mean of the population for each time step. Thus, the re-sampled dataset is representative of the mean of 10 GCM simulations at that time step (day), and the procedure is repeated for each day for entire period both in historical and future data. 2.3.2 Copula Various methods can be deployed to establish the joint projections/distribution, copula being one such statistical

method. Its advantages include the ability to join several variables with different correlation and dependence structures. It develops the joint distribution by estimating the marginal distributions of variables and their correspondence separately, which are not restricted to any parametric distribution (e.g., Gaussian distribution). Copula is a statistical procedure to join multivariate distribution functions of few variables by their marginal distribution functions (Nelson 2007). The method has been applied widely in hydrological and climatological studies (e.g., Salvadori and Michele 2010; Kao and Govindaraju 2010; Madadgar and Moradkhani 2013, 2014a, b; Behrangi et al. 2015). The physical relationship (interdependence/correlation) between the temperature and precipitation will determine the inherent relationship between these two climatological variables. The interdependence of these two climatic variables using copula have not been extensively investigated (Schölzel and Friederichs 2008; Cong and Brady 2012). In the present study, we have investigated the same using five parametric copula families (discussed below) to establish correlation. We have established the joint distribution of temperature and precipitation to study the changes in trends and probabilities of future multi-model ensemble scenario in respect to historical multi-model ensemble scenario, for all 10 sub-basins. Thus, the changes across subbasin are studied with joint distribution and probabilities instead of considering them independently. Mathematically, assume a pair of random variables X and Y, with cumulative distribution functions (CDFs) F(x) = P(X ≤ x) and G(y) = P(Y ≤ y), respectively, and a joint distribution of H(x, y) = P(X ≤ x, Y ≤ y). Each pair of (x, y) is associated with three numbers: F(x), G(y), and H(x, y) (each of these numbers falls in the range of [0,1]). To be specific, each pair (x, y) of real numbers associates with a point ,ðð F ðxÞ; GðyÞÞ and, in turn, this ordered pair corresponds to a number H(x, y) in [0,1] (Nelson 2007). The copula function to describe this correspondence, which is assigning the value in [0, 1] equal to the value of the joint distribution function H(x, y), can be written as follows: H ðx; yÞ ¼ H F −1 ðuÞ ; G−1 ðvÞ ¼ C ðu; vÞ

ð3Þ

where u and v are equal to CDFs of X and Y respectively: u = F(x), v = G(y), ranging from 0 to 1 (here u and v are strictly increasing functions); C is the copula function; F-1 and G-1 are the inverse CDFs of X and Y, respectively. Sklar’s theorem states that any multivariate joint distribution can be written in terms of cumulative distribution function of each variable and copula is used to describe the dependence or any level of correlation between variables (Sklar 1959; Rüschendorf 2009). Copula function in question satisfies the following properties:

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(1) The joint probability density function: ∂cðu; vÞ cðu; vÞ ¼ ∂u∂v

(3) Gumbel copula:

ð4Þ

∂cðu; vÞ ∂u ∂v f ðx; yÞ ¼ * * ¼ cðu; vÞ* f x ðxÞ*f y ðyÞ ∂u∂v ∂x ∂y

h i 1θ C ðu; vÞ ¼ exp − ð−lnuÞθ þ ð−lnvÞθ

ð13Þ

where θ measures the dependency between u and v. ð5Þ

(4) Clayton copula: −1 C ðu; vÞ ¼ u−θ þ v−θ −1 θ

(2) The boundary condition:

0 ≤ C ðu; vÞ≤ 1

ð6Þ

C ðu ¼ 0; vÞ ¼ C ðu; v ¼ 0Þ ¼ 0

ð7Þ

C ðu ¼ 1; v ¼ 1Þ ¼ 1

ð8Þ

C ðu; v ¼ 1Þ ¼ u

ð9Þ

C ðu ¼ 1; vÞ ¼ v

ð10Þ

Here we have shown just two random variables, and the joint distribution of multiple variables has the same properties. There are many parametric copula families available, which usually have parameters that control the strength of dependence. Below are five copula functions that are considered in the present study:

ð14Þ

where θ measures the dependency between u and v. (5) Frank copula:

C ðu; vÞ ¼

−θu −θv e −1 e −1 −1 ln 1 þ e−θ −1 θ

ð15Þ

where θ measures the dependency between u and v. The joint distribution (of multi-model ensemble data) after copula is then used to get probabilistic projections/scenarios for future and historical data which, in turn, is used to study changes, trends, and probabilities of those variables in future climate change scenario.

(1) Gaussian copula:

3 Results and discussion Z C ðu; vÞ ¼

∅−1 ðuÞ Z ∅−1 ðvÞ −∞

exp

−∞

1

ð11Þ

1

2πð1−ρ2 Þ 2

−x2 þ y2−2ρxy dxdy⋅⋅⋅⋅⋅⋅x; y∈R 2ð1−p2 Þ

where ρ is the linear correlation coefficient and ∅ is the standard normal cumulative distribution function. (2) T copula: Z C ðu; vÞ ¼

Z t−1 t −1 v ðuÞ v ð vÞ −∞

−∞

1

ð12Þ

1

2πð1−ρ2Þ 2

x2 þ y2−2ρxy exp 1 þ vð1−ρ2 Þ

Þ −ðvþ2 2

dxdy⋅⋅⋅⋅⋅⋅x; y∈R

where ρ is a linear correlation coefficient, tv is cumulative distribution function of t distribution with v degree of freedom.

3.1 Verification of bootstrap sampling and multi-model ensemble The estimates of mean statistics and variance would be tested in order to judge the performance of multi-model ensemble scenario against 10 individual GCMs. For both historical and future data, the performance is tested in two different seasons, namely, the wetter season (October to March) and the drier season (April to September), for both precipitation and temperature variables. Thus, the seasonal average for the 30 years period in consideration for multi-model ensemble was tested against the individual GCMs averages. Mean absolute error (MAE) is calculated for the multimodel ensemble mean (with bootstrapping) of 10 individual GCMs and observation data from the same historical period. Mathematically, MAE can be summarized as below: XT XN jSimti −Obst j t¼1 i¼1 M AE ¼ ð16Þ N *T where T represents each year of the historical period. N is the total number of elements included in the ensemble for each


year. For original ensemble data, N is equal to 10 and for the bootstrap sampling ensemble data, N is equal to 200. ti is the value of each ensemble member from each year, and Obst is the observation value of each year. MAE represents the measure of proximity (closeness) of simulations to observations, with matching profiles being 0. Figure 2 represents MAE values for precipitation and temperature in all sub-basins across CRB. It can be observed that the bootstrap-sampled multi-model ensemble outperforms the simple mean ensemble of 10 GCMs against the observations in both seasons and climatic variables and all sub-basins. The percentages of MAE improvement averaged over 10 subbasins of CRB from bootstrap-sampled ensemble compared to mean-original ensemble datasets are 15 % for precipitation in dry season and 13 % in wet season, whereas for temperature it is 19 and 18 % for dry and wet seasons, respectively. Therefore, it could be concluded that bootstrap-sampled ensemble datasets have an advantage of combining multimodels’ strengths and result in decreasing the bias between simulations and observations than using simple mean over the models. Further, frequency-based histogram of bootstrap-sampled multi-model ensemble is compared to 10 GCMs in all 10 subbasins of CRB, to compare mean and variance statistics. Figures 3 and 4 represent the frequency-based histograms of

precipitation for bootstrap-sampled multi-model ensemble against original GCMs ensemble in historical period (1970– 1999) for dry and wet season, respectively; Figs. 5 and 6 represent the same for temperature. It can be noted from Figs. 3, 4, 5, and 6 that the histograms of bootstrap-sampled ensemble are narrow in spread compared with original GCMs ensemble in all 10 sub-basins in both seasons and variables under consideration. The narrow-spread histograms of bootstrap sampled multi-model ensemble reduce the uncertainty/ variance due to its shorter tails as compared to those of the histograms in original ensemble data, which in turn provides more accurate assumption of mean and variance of the data from 10 GCMs in consideration. Similar observations were made in accordance to bootstrap-sampled multi-model ensemble over original 10 GCMs in all the sub-basins and season. Readers are referred to the supplementary material for figures. 3.2 Joint relationship between temperature and precipitation (copula application) Since bootstrap-sampled multi-model ensemble (herein called as multi-model ensemble) is found to be a sufficient representation of the mean and variation of 10 GCMs in both historical and future period, the same application of copula would be

Fig. 2 Mean absolute value (MAE) for bootstrap-sampled multi-model ensemble and original GCMs mean-ensemble datasets in the historical period against observation data for precipitation (a dry season and b wet season) and temperature (c dry season and d wet season)

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Fig. 3 Frequency-based histograms of daily precipitation average datasets in dry season for original GCMs and bootstrap-sampled multimodel ensemble over the historical period (1970–1999) in 10 sub-basins

of CRB. The pink color represents the original GCMs dataset, and the blue color represents the bootstrap sampled multi-model ensemble dataset

performed. The joint distribution/projections of both temperature and precipitation variables are developed for both historical and future periods and both seasons. The application of copula starts with fitting theoretical distributions to temperature and precipitation multi-model ensemble data. Seven distributions, namely, gamma, generalized extreme value (GEV), lognormal, Gaussian, Weibull, Gumbel, and exponential, are fitted to temperature and precipitation data in both dry and wet seasons, both for historical (1970–1999) and future periods (2070–2099), respectively. This is performed separately for each of the sub-basins. The parameters of marginal distributions are estimated by the maximum likelihood estimation method. The KolmogorovSmirnov (K-S) test is used to find the appropriate theoretical distributions, for a particular variable in all season and subbasin, in accordance to acceptable p value (at significance level α = 0.05). If the p value is less than 0.05, the ensemble dataset belongs to this reference theoretical distribution and is thus disregarded (refer to supplementary table). Further, the Akaike Information Criterion (AIC) values are used to choose the best theoretical distribution from the approved theoretical distributions. AIC value should be minimum for best fit (Akaike 1974; Madadgar and Moradkhani 2013). Readers are referred to supplementary table for details on selection of acceptable theoretical distributions AIC values and ultimately

the distribution chosen. This selection is made separately for each sub-basin, season, and climate variable. Thereafter, the CDFs of temperature and precipitation are linked with copula. As discussed in the methodology section, we have employed five copula functions for the same, namely, Gaussian and T copulas from elliptical copulas and Gumbel, Clayton, and Frank from Archimedean copulas. The goodness-of-fit (GOF) test is employed to assess the copula function that describes the relationship between temperature and precipitation. GOF is based on the statistics relating the distance between the empirical and theoretical copulas. Eventually, the parametric bootstrap procedure was employed to estimate the Cramér-von Mises statistics (Sn) and the related p value in order to choose the best copula function (Genest and Rémillard 2008; Madadgar and Moradkhani 2014a). If the variable’s dataset fits to the parametric copula, representing null hypothesis, (H 0 : C n ∈C θn ), with p value greater than the significance level α = 0.05, then null hypothesis is accepted; otherwise, rejected. Therefore, among a group of copula, the one with the greatest p value and smallest Sn is preferred. Table 3 represents the statistics for choosing best copula to represent the relationship between temperature and precipitation. Copula selection is done on annual data for all sub-basins in both historical and future periods. Once copula selection is performed, the same copula is used to evaluate


Fig. 4 Frequency-based histograms of daily precipitation average datasets in wet season for original GCMs and bootstrap-sampled multimodel ensemble over the historical period (1970–1999) in 10 sub-basins

of CRB. The pink color represents the original GCMs dataset, and the blue color represents the bootstrap-sampled multi-model ensemble dataset

the changes on seasonal basis in future multi-model ensemble data and observed multi-model ensemble data. It can be observed from Table 3 that in historical data, T copula is selected in 4 sub-basins followed by Frank (3), Gaussian (2), and Clayton (1), whereas, in future data, T copula is describing the relationship between temperature and precipitation in 7 sub-basins; Frank, Gaussian, and Clayton are chosen in 1 sub-basin each. Gumbel copula is not found appropriate to describe the temperature and precipitation relationships in any of the data periods. There is no correlation between the size and location of the sub-basin and the chosen copula (the relationship between temperature and precipitation is not related to the size and location of the sub-basin in CRB).

3.3.1 Dry season

3.3 Future trends and probabilities The following section would deal with trends in multi-model ensemble future data as compared to multi-model ensemble historical data from combined distribution of temperature and precipitation. We would also be evaluating the future probabilities of precipitation and temperature based on their joint distribution. This would be done separately for dry and wet seasons and for all the sub-basins.

Trends and changes in joint distribution of temperature and precipitation Figure 7 represents the changes and trends as pointed by joint distribution of temperature and precipitation for all the 10 sub-basins in CRB in dry season for both historical and future periods based on their joint CDFs. The copula-based relationship (CDFs) of temperature and precipitation is depicted for 10 sub-basins. For each of the sub-basin, e.g., the Chief Joseph sub-basin), the solid lines represent the future period (2070–2099) and the dash lines represent the historical period (1970–1999) with color of the line indicating the probability associated with each color (refer to figure color bar). In all the sub-basins, it was noted from Fig. 7 that the trends are positive in future scenario for both the climatic variables. There is variation among the sub-basins for the degree of that change from historical values. It can also be observed that every sub-basin has its own change characteristics i.e., change in precipitation as compared to temperature for particular sub-basin. Since temperature is believed to be better predicted than precipitation and also some sub-basins have similar temperature ranges (based on geographical location, Fig. 1), we would categorize them (sub-basin) into five classes, according to temperature ranges for further

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Fig. 5 Frequency-based histograms of daily temperature average datasets in dry season for original GCMs and bootstrap-sampled multimodel ensemble over the historical period (1970–1999) in 10 sub-basins


evaluation of trends and changes. Similar categorization would be followed in wet season as well.

reason behind low precipitation and higher temperature ranges. This also corresponds to the observations in the area. In the future period, the joint CDFs of temperature and precipitation depict positive and upward trends in both variables (slope is much flat). However, the ranges of precipitation in these two sub-basins only have changed slightly from 0.5– 2.5 mm (1 mm increase from historical). It could be conclusively said that these two sub-basins will potentially experience drought in the future scenario, with sharp rise in temperatures. The TW Sulliwan sub-basin has relatively high temperature, similar to the above two sub-basins but also has quite high precipitation (range of 1.5–3.5 mm), for the historical period with similar slope inclination towards temperature. In the future period, in addition to moving upward (positive trend), the joint CDF hints towards increase in ranges of precipitation (1–4 mm) and flat slope for both variables. Temperature and precipitation changes are more synchronized in future scenario with the former affecting the latter with more likelihood. TW Sullivan sub-basin receives the highest precipitation in the CRB along with relatively high temperature due to its southwestern location and proximity to the ocean. The sub-basin will potentially be wetter and warmer in future scenario.

Category 1 (sub-basins with highest temperature ranges) The Ice Harbor, Oxbow, and TW Sulliwan are the sub-basins with highest seasonal average temperatures among all the subbasins in dry season. For historical period (1970–1999), temperature ranges among these three basins vary around 11 to 14 °C; however, for future period (2070–2099), temperatures vary between 14 to 18 °C. The lowest seasonal average temperature rise is 3 °C with higher seasonal average temperature rise being 4 °C in the dry seasons of future period compared to historical one. For precipitation, Ice Harbor and Oxbow share similar characteristics, both of these sub-basins have relatively high temperature but less amount of precipitation. Seasonal average precipitation for these two sub-basins in dry seasons ranges from 1 to 2 mm in the historical period. Joint CDFs of temperature and precipitation in the historical period have sharp slope against temperature axis, indicating sharp and higher changes in temperature with small/negligible changes in precipitation. Geographical location of these sub-basins in CRB, southwest of CRB, where Cascade mountain range creates a barrier from receiving much precipitation is the primary


Fig. 6 Frequency-based histograms of daily temperature average datasets in wet season for original GCMs and bootstrap-sampled multimodel ensemble over the historical period (1970–1999) in 10 sub-basins


Category 2 (sub-basins with second high temperature range) The Dalles sub-basin has the second highest seasonal average temperature in dry season with historical temperature range around 10.5–13.5 °C, whereas temperatures vary from 13.5–16.5 °C in the future. The lowest and highest seasonal average temperature rise is 3 °C in the dry seasons of future period compared to the historical period. As per precipitation, Dalles share similar characteristics as two above sub-basins (Ice Harbor and Oxbow). Dalles has relatively high temperature but less precipitation with seasonal average of 1–2 mm. Joint CDFs of temperature and precipitation, in the historical period, exhibit similar behavior as the abovementioned sub-basins (sharp slope to temperature) indicating that the change range of temperature is wider than that of precipitation. In the future period, the joint CDFs of temperature and precipitation is predicting positive and upward trend in comparison to historical one with slight change in ranges of seasonal average precipitation (1.5–2.5 mm, ∼0.5 mm increase).

average temperature ranges from 9 to 11.5 °C, and in the future period, seasonal average temperatures vary from 11 to 14 °C. Thus, the lowest seasonal average temperature in the dry season rise 2 °C and the highest seasonal average temperature rise is around 2.5 °C. For Priest Rapids basin, in the historical period, temperature ranges from 10 to 12.5 °C, whereas it varies from 12 to 14.5 °C in future scenario indicating 2 °C rise in both the lowest and highest average temperature values. For precipitation, both sub-basins exhibit the same ranges of 1.5–3 mm in the historical dataset, which depicts a sharp rise of range to 1–4.5 mm in future scenario. Joint CDFs of temperature and precipitation are similar to above-noted features in all sub-basins (sharp slope inclined towards temperature) in historical data. Whereas the joint CDFs, for future period, is more inclined towards precipitation i.e., more precipitation change as compared to temperature changes. This indicates that the two sub-basins would be warmer and wetter as compared to the historical scenario.

Category 3 (sub-basins with medium temperature ranges) The Chief Joseph and Priest Rapids sub-basins have medium seasonal average temperatures in dry seasons among all the sub-basins. For Chief Joseph, in the historical period, seasonal

Category 4 (sub-basins with second lowest temperature ranges) The Corra Linn and Waneta have the second lowest seasonal average temperatures in dry season. For Corra Linn basin, in the historical period, seasonal average

Rana A. et al. Table 3 Best copula selection in historical and future periods over all 10 sub basins of CRB (preferred copula is with greatest p value and smallest Sn value)

Sub-basin

Copula

Statistics

Gaussian

T

Chief Joseph

Historical period

p value Sn

0.975 0.019

0.965 0.018

Future period

p value

0.064

0.114

NaN

NaN

0.025

Historical period

Sn p value

0.051 0.748

0.041 0.797

NaN NaN

NaN NaN

0.044 0.797

Sn

0.032

0.028

NaN

NaN

0.028

Future period

p value Sn

0.035 0.044

0.163 0.041

NaN NaN

NaN NaN

0.025 0.044

Historical period

p value Sn

0.272 0.044

0.411 0.038

NaN NaN

NaN NaN

0.312 0.039

Future period

p value

0.046

0.050

0.111

0.047

0.050

Sn

0.062

0.049

0.136

0.054

0.052

p value Sn p value

0.629 0.035 0.252

0.579 0.035 0.391

NaN NaN NaN

NaN NaN NaN

0.450 0.035 0.153

Corra Linn

The Dalles

Ice Harbor

Historical period Future period

Mica


Oxbow


Priest Rapids


Revelstoke


TW Sullivan


Waneta


Clayton

Gumbel

Frank

0.965 0.015

0.896 0.023

0.995 0.020

Sn

0.038

0.032

NaN

NaN

0.034

p value Sn p value Sn

0.262 0.037 0.066 0.071

0.738 0.025 0.065 0.061

0.777 0.022 0.025 0.082

0.173 0.041 0.005 0.057

0.411 0.034 0.068 0.061

p value Sn p value

0.738 0.031 0.153

0.559 0.034 0.144

NaN NaN NaN

NaN NaN NaN

0.658 0.033 0.045

Sn p value Sn

0.043 0.906 0.024

0.042 0.837 0.024

NaN 0.837 0.020

NaN 0.807 0.026

0.043 0.946 0.024

p value Sn p value Sn p value

0.045 0.089 0.153 0.041 0.658

0.074 0.073 0.688 0.027 0.698

NaN NaN 0.520 0.028 NaN

NaN NaN 0.153 0.043 NaN

0.025 0.081 0.322 0.038 0.579

Sn

0.028 0.252 0.048 0.074 0.051 0.748 0.026 0.292 0.034

0.025 0.292 0.045 0.203 0.036 0.876 0.023 0.718 0.024

NaN NaN NaN NaN NaN 0.827 0.019 NaN NaN

NaN NaN NaN NaN NaN 0.876 0.026 NaN NaN

0.025 0.252 0.045 0.094 0.038 0.896 0.026 0.550 0.026

p value Sn p value Sn p value Sn p value Sn

Values in italics are the selected copula for respective period for joint distribution of temperature and precipitation

temperatures ranges from 7.5–10.5 °C and 9.5–12.5 °C for future period. Both the lowest and highest average temperatures in the area is indicating towards 2 °C rise. For Waneta basin, in the historical period, temperature ranges from 8 to 10.5 °C and 10–12.5 °C for future data. The rise in lowest and highest temperature exhibits similar range as Corra Linn (2 °C).

For precipitation, both the sub-basins provide similar variations as compared to temperature in historical period (slope inclined towards temperature). Ranges for Corra Linn vary from 2 to 4 mm and that for Waneta is 2–3 mm, in historical data. Joint CDFs, depicting changes and trends, hint towards positive and upward trends but with varying degrees for both the sub-basins. The changes in precipitation are more


Fig. 7 Comparison of joint cumulative distribution functions (CDFs) of temperature and precipitation, using multi-model ensemble, for dry season over the 10 sub-basins of CRB. The solid lines represent the future period (2070–2099), and the dash lines represent the historical

period (1970–1999). Color of the line indicates the probability of values for temperature and precipitation. The probabilities associated with each color line refer to figure color bar

profound for Corra Linn sub-basin ranging from 2 to 6 mm (∼2 mm increase) with the joint distribution slope inclined highly towards precipitation, i.e., changes in precipitation would be higher with every degree change in temperature. Whereas in case of Waneta, the changes are positive (ranges from 1.5–4.5 mm, ∼1 mm increase) but the slope is not as inclined towards precipitation as was in Corra Linn. The CDFs for Waneta suggests equal changes in both climatic variables. Thus, it can be conclusively said that both the subbasins would be warmer and wetter with Corra Linn, projecting higher precipitation amounts.

and that of Revelstoke sub-basin is 2–4 mm, with similar slope distribution in joint CDFs for temperature and precipitation. For the future period, both of Mica and Revelstoke subbasins indicate positive and upward change, as was the case in all sub-basins. The slope of joint CDFs is becoming flatter and inclined towards precipitation (more so the case in Mica than in Revelstoke). Although precipitation change range in both the sub-basin is 2–5 mm, Mica is predicted to show more increase in precipitation with every degree rise in temperature in future scenario and thus wetter and warmer than historical ones. Overall, we can deduce that categories close together i.e., similar spatial distribution (geographic space) and seasonal average temperature, in dry season, will have positive change in precipitation and temperature which is relatively decreasing from north to south of CRB. Sub-basins in category 1 and category 2 with high seasonal average temperature in dry season, which are located in southern CRB, exhibit higher changes in temperatures than precipitation and vice versa for other categories north of the basin.

Category 5 (sub-basins with lowest temperature ranges) Mica and Revelstoke sub-basins have the lowest seasonal average temperatures in dry seasons. For Mica basin, in the historical period, temperature ranges from 6.5–9 and 9– 11 °C in the future period. The lowest average temperatures rise is around 2.5 °C, and the highest temperature rise is 3 °C. For Revelstoke sub-basin, in the historical period, temperature ranges from 7 to 9 °C and 9–11 °C in the future period with 2 °C rise in both the lowest and highest average temperature. Mica and Revelstoke are smallest sub-basins with lowest temperatures, located in the far north of CRB. In the historical period, precipitation ranges of Mica sub-basin is 1.5–4 mm

Probabilities of joint distribution of temperature and precipitation in future scenario Figure 8 represents the probabilities/likelihood of two variables with joint distribution

Rana A. et al.

Fig. 8 Likelihood of precipitation ranges given a particular temperature in future data (2070–2099) during dry season in each of the 10 sub-basins in CRB. Conditional probability density function of precipitation against certain temperature is plotted and scaled for probability (0 to 1), for

visualization purpose (red color–most likelihood and blue color–least likelihood). The conditional probability density associated with each color line refers to this figure color bar

for all the 10 sub-basins in CRB in dry season for future period based on their joint CDFs. For each of the sub-basin, e.g., first sub-plot top left (Chief Joseph sub-basin), the likelihood of precipitation ranges given a particular temperature in future data (2070–2099) during dry season is plotted and scaled for probability (0 to 1). Red color represents most likelihood and blue color least likelihood of amount of precipitation given a particular temperature in future scenario. Thus, it can be said that the areas with dark blue colors indicate the tails of the precipitation PDF, at given temperature, which shows the least likely occurring precipitation given certain temperature and vice versa for red color. Therefore, if given temperatures are expected, precipitation magnitudes associated with the red areas (closely surrounding the mode of the condition PDFs of precipitation given temperature) are most likely to happen. From Fig. 8 it can be noted that 7 out of the 10 subbasins (namely Chief Joseph, Ice Harbor, Oxbow, Priest Rapids, Revelstoke, TW Sullivan, and Waneta) depict a negative relationship between the most likely occurrence of precipitation with temperature i.e., the amount of precipitation in dry season is decreasing with increasing precipitation for most likelihood scenarios. The geographic locations of these 7 sub-basins do not show any obvious pattern. It also indicates that the temperature is more

dominant variable in the joint probabilities and is in inverse relation to precipitation in dry season. During this season, we can expect higher temperatures and drier conditions in future scenario (as already pointed out that temperatures are increasing in future scenario and thus precipitation would relatively be lower increase). It should be pointed out that there is increase in amount and intensity of both variables in future scenario but their interdependence is varied in different sub-basins. For 2 sub-basins i.e., Dalles and Mica, the most likely occurrence of precipitation at a given temperature is constant indicating constant precipitation increase in these subbasins irrespective of increase in temperature. Thus, the expected increase in precipitation would be independent of increase in temperature. In case of Corra Linn, the precipitation does not show any specific patterns. For higher temperatures (more than 9 °C), the precipitation amounts are relatively constant whereas the probabilities of precipitation are very varied for lower temperatures. The precipitation amount (likelihood of getting that–red color) is depicting both increase and decrease with temperatures from 7 to 9 °C, with wider ranges towards the lower temperatures. The inverse relationship across most of the subbasin could be related to water holding capacity of clouds with particular temperatures and vice versa.


3.3.2 Wet season Trends and changes in joint distribution of temperature and precipitation Figure 9 represents the same as Fig. 7 for wet season. Similar nomenclature is used for the plots as was in dry season. Similar observations as Fig. 7 can be made in Fig. 9 in reference to trends and changes in future scenario i.e., positive and upwards changes and trends in both variables. There is variation among the sub-basins for degree of that change from historical values. Similarly, for comparison within sub-basin, same categorization of temperature is studied in wet season as well (not necessarily same sub-basins would fall in same category).

the ranges (upwards increase) are higher. The slope is more inclined towards precipitation and thus indicating higher changes of precipitation amounts than temperature. The subbasin depicts the likely possibility of higher precipitation along with rising temperature events in future period. Increased intensity of precipitation would lead to the high possibility of flood in the area.

Category 1 (sub-basins with highest temperature ranges) The TW Sullivan depicts the highest seasonal average temperatures among all the sub-basins in wet seasons, 3–5 °C for the historical period (1970–1999) and 5.5–9 °C for future period. There is a rise of 2.5 and 4 °C in lowest and highest temperatures, respectively, for the future scenario. In addition, TW Sullivan sub-basin has maximum amount of average seasonal precipitation among all of the 10 sub-basins in the wet seasons of historical period ranging from 5 to 12 mm, whereas it increases to 5–15 mm in the future scenario. The slope of joint CDF is not changing in the historical and future data but

Category 2 (sub-basins with second highest temperature ranges) The Dalles, Ice Harbor, and Oxbow sub-basins have second high seasonal average temperatures in wet seasons ranging from −2 to 0.5 °C in the historical period (1970– 1999); however, for the future period (2070–2099), temperatures vary around from 0 to 4 °C for Ice Harbor and Oxbow sub-basins and from −0.5 to 3 °C for Dalles sub-basin. For Ice Harbor and Oxbow, the rise in lowest and highest seasonal average temperatures is 2 and 3.5 °C respectively, whereas for Dalles it is 1.5 and 2.5 °C respectively. All these 3 sub-basins share similar characteristics i.e., they have relatively high temperature but lower amounts of precipitation. Seasonal average precipitation for Dalles and Ice Harbor sub-basins ranges from 2 to 3 mm and for Oxbow it is 1-2 mm, with around 0.5 mm increase in future for all three. The joint CDFs of temperature and precipitation for all of these three, both in historical and future periods, have sharp slope to temperature axis indicating

Fig. 9 Comparison of joint CDFs of temperature and precipitation, using multi-model ensemble, for wet season over the 10 sub-basins of CRB. The solid lines represent the future period (2070–2099), and the dash

lines represent the historical period (1970–1999). Color of the line indicates the probability of values for temperature and precipitation. The probabilities associated with each color line refer to figure color bar

Rana A. et al.

that the changes in range of temperature is higher than that of precipitation. In the future period, the joint CDFs of temperature and precipitation depict a positive change but still higher changes in temperature than in precipitation. The following scenario hints towards possible droughts in the future with higher temperature rise and relatively low or no precipitation increase. Category 3 (sub-basins with medium temperature ranges) The Chief Joseph and Priest Rapids have medium seasonal average temperatures, with Chief Joseph historical temperatures ranging −5 to −2 °C, and −3 to 1 °C for future, whereas Priest Rapids historical temperature ranges from −4 to −1.5 °C, and −2 to 1.5 °C for the future. The changes/rise in lowest and highest average temperature for both the sub-basin are 2 and 3 °C respectively. For precipitation, Chief Joseph has 2.5–5 mm range in historical period (1–2 mm increase in future) and Priest Rapids has 2.5–4.5 mm range in same period (∼1 mm increase in future). The joint change trends (CDFs) of temperature and precipitation are quite similar to the subbasins discussed in category 2 of wet season, i.e., for both subbasins, the changes in temperature are more pronounced than in precipitation. For future period, the CDFs indicate a higher rise in temperature ranges with slight increase in precipitation ranges. It hints towards a similar fate of these sub-basins in future as was for category 2 but with lower temperatures than those three sub-basins in category 2. Thus, in the future, the temperature rise dominates the precipitation rise. Category 4 (sub-basins with second lowest temperature ranges) The Corra Linn and Waneta have second lowest seasonal average temperatures in wet seasons. For Corra Linn, in the historical period, seasonal average temperatures ranges from −6 to −4 °C, whereas for the future period, the ranges are −4 to 0 °C. The lowest seasonal average temperature rise is 2 °C, and the highest seasonal average temperature rise is 4 °C. For Waneta, the historical temperature ranges are from −6 to −3.5 °C, and future temperatures vary from −4 to 0 °C depicting 2 and 3.5 °C rise in lowest and highest temperatures. Historical precipitation ranges for Corra Linn are 2.5–6 and 2.5–5 mm for Waneta which increases to 4–7.5 and 3.5– 6.5 mm in future for the two sub-basins, respectively. The joint change trends of temperature and precipitation in both the historical and future periods are similar in both of these two sub-basins. The CDFs depict similar behavior as categories 2 and 3 for wet season i.e., higher increase in temperature ranges than in precipitation values. Thus, a wetter and warmer scenario should be expected in the future. Category 5 (sub-basins with lowest temperature) The Mica and Revelstoke sub-basins have the lowest seasonal average temperatures in wet seasons, located in the far north of CRB. For both Mica and Revelstoke, in the historical period,

temperature ranges from −8 to −5 °C and −5.5 to −2 °C in the future period. The lowest average temperatures rise is 2.5 °C, and the highest temperature rise being 3 °C in the wet season. In historical period, precipitation ranges of both Mica and Revelstoke are 2.5–5 mm which increase to 3– 7.5 mm in future scenario. Both sub-basins depict similar properties of joint changes of precipitation and temperature in the historical and future periods, indicating higher precipitation accompanied with increasing temperature events in the future as compared to historical. The positive changes in precipitation are accompanied with similar degree of positive changes in temperature, as depicted from joint CDFs in future scenario. Probabilities of joint distribution of temperature and precipitation in future scenario Figure 10 represents the probabilities/likelihood of two variables with joint distribution for all the 10 sub-basins in CRB in wet season for future period based on their joint CDFs. The figure should be read in similar context to Fig. 8. In wet season, the most likely occurrence of precipitation ranges at a given temperature range is relatively constant for all the 10 sub-basins in CRB, but the ranges of precipitation are different for sub-basins. For Mica, at higher temperatures, the range is precipitation is relatively higher than any of the sub-basins; otherwise, a constant range is maintained (within a particular sub-basin) irrespective of temperature changes. Thus, it can be conclusively said that the range of precipitation (most likelihood) and independent of changes in temperature values. In other words, the most likely precipitation is not related to temperature changes. This can be attributed to the fact that although the temperatures are rising, average temperatures are still sub-freezing and thus the holding capacity of clouds will not change. There are insignificant variations with some of the sub-basins where precipitation ranges are increasing with temperature but comparatively very low changes e.g., Priest Rapids, Revelstoke, and Waneta. This would have a severe consequence on spring flooding and harvesting season in the area.

4 Conclusions We have applied copula-based coupling of temperature and precipitation across 10 sub-basins in CRB for future scenario (2070–2099). Trends and changes in these climate variables are compared with copula-based distribution from historical, and probabilities of one variable in relation to other are also studied for future changes using CDFs. All the analyses were performed on multi-model ensemble of 10 GCMs (downscaled with BCSD) across CRB (divided into sub-basins) using ensemble mean with bootstrap sampling for generating multi-model ensemble. The method was helpful in producing mean statistics and variability from 10 GCMs in terms of


Fig. 10 Likelihood of precipitation ranges given a particular temperature in future data (2070–2099) during wet season in each of the 10 sub-basins in CRB. Conditional probability density function of precipitation against certain temperature is plotted and scaled for probability (0 to 1), for

visualization purpose (red color–most likelihood and blue color–least likelihood). The conditional probability density associated with each color line refers to this figure color bar

MAE and frequency-based analysis, both in dry and wet seasons across the sub-basins. We further explored the relationship between temperature and precipitation in each sub-basin using copula functions, which enabled us to study/model any level of dependence or independence regardless of each variable’s suitable theoretical distribution and complex relationships. The main summary from joint CDFs of precipitation and temperature is as follows:

predicting inverse relation i.e., with increasing temperature, the ranges of precipitation received would decrease. 6. Whereas in wet season, precipitation ranges seemed to be independent of temperature changes, and the range of precipitation is not affected by how much the temperature rises.

1. The temperature and precipitation are depicting positive and upward trends in all sub-basins and all time scales considered. 2. The geographical location of sub-basins plays important roles in the degree of changes observed in both parameters. 3. Southeast of the basin is usually projecting warmer and drier conditions, whereas southwest projections hint towards wetter and warmer trends. 4. Trends of positive (wetter) and warmer conditions are significant in north of the CRB with central parts predicting higher temperatures with little changes in precipitation. 5. Interdependence of temperature and precipitation is varied during dry season with most of the sub-basins

The most important application of the above study is to analyze the impact of climate on the CRB with combined relationships of precipitation and temperature and overcoming the usual impasse that is mostly worked on by providing only separate projections of temperature and precipitation. For most impact studies/applications, the joint behavior of said climatic variables is especially used, e.g., drought analysis, agriculture production, irrigation, ecosystem services, fisheries, water quality and availability, flood analysis, urban planning, and storm-water management. The results of this study provide additional and valuable information about how precipitation and temperature are jointly affected under climate change scenario, which in turn will provide sufficient management and strategies to meet the challenges in future for decision makers. It would be helpful improvising decisionmaking abilities under different levels of risks with high level of certainty/confidence. In continuation to this study, we intend to focus on identification and quantification of climate

Rana A. et al.

changes impacts on certain key and strategic areas that are important for CRB e.g., drought, flood, agriculture, and stream flow, using joint relationship between various variables involved. Acknowledgments The authors would like to acknowledge the partial financial support provided by the Institute for Sustainable Solutions at Portland State University. The authors would also like to acknowledge the World Climate Research Programme’s Working Group on Coupled Modelling, which is responsible for CMIP, and we thank the climate modeling groups for producing and making available their model outputs. For CMIP, the US Department of Energy’s Program for Climate Model Diagnosis and Intercomparison provides coordinating support and leads development of software infrastructure in partnership with the Global Organization for Earth System Science Portals.

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