The Intergovernmental Panel on Climate Change (IPCC, 2007) has concluded that global mean atmospheric temperature will likely increase between 1.8 and 4.0 °C by the end of this century, and will consequently lead to an intensified global hydrological cycle. Many global and regional climate models (GCMs and RCMs) have consistently projected considerable increases in (1) the frequency and magnitudes of extreme events including both droughts and heavy downpours during the century, (2) average daily rainfall intensity, especially for extreme rainfall events, and (3) the spatial and temporal variability of precipitation (IPCC, 2007). The frequency and magnitude of extreme precipitation events was projected to increase during the last century for the contiguous US. The projected increases are in line with the observed historical trends that precipitation has increased by some 10% since 1910 with the increase being primarily in the form of heavy and extreme daily storms (Karl and Knight, 1998). Groisman et al. (2001) analysed the trends in the share of total annual precipitation occurring in heavy (>95th percentile), very heavy (>99th), and extreme (>99.9th) daily precipitation events in the contiguous US between 1910 and 1969, and 1970 and 1999, and reported that the linear trends between the two periods increased by 4.6, 7.2, and 14.1% per decade, respectively, which are in contrast to the 1.2% increase per decade for total annual precipitation. The potential increase towards more intense rainfall events in future is of great concern for assessing the potential impacts on surface hydrology, soil erosion, crop production, and environmental protection because catastrophic environmental destruction often results from infrequent heavy storms.
For the site-specific impacts of climate change on natural resources of particular fields or farms, the major obstacle is the spatiotemporal mismatch between GCM/RCM projections and the fine- resolution data requirements of hydrological and agricultural systems models (Hansen and Indeje, 2004). Arguably, GCMs generally provide skillful data at the monthly scale for grid boxes of hundreds of kilometers, while agricultural system models require daily weather data and operate at a field or farm scale. Both dynamical and statistical (empirical) approaches are used to bridge the spatiotemporal gap. Dynamical downscaling is used to achieve a higher spatiotemporal resolution by nesting a RCM within GCM output fields. RCMs provide daily values at a spatial resolution of tens of kilometers. Though the spatiotemporal resolution of RCMs is greatly improved over GCMs, the direct use of RCM output in impact models should be avoided, as suggested in the IPCC guidance for using RCM output (Mearns et al., 2003). This is firstly because daily precipitation statistics at a station or point are different from those averaged over a 20-km grid box (Guo and Senior, 2006; Boberg et al., 2010). Secondly, it is well known that RCMs have considerable systematic errors in predicting daily precipitation, as RCMs consistently overpredict the number of wet days and light daily precipitation events, yet underestimate rainfall amounts of heavy storms (Guo and Senior, 2006; Semenov, 2007; Rosenberg et al., 2010; Herrera et al., 2010; Maraun et al., 2010; Themeßl et al., 2010; van Roosmalen et al., 2010;). Rivington et al. (2008) evaluated the Hadley Centre's RCM with observed data from 15 meteorological stations in the UK for the 1960–1990 period, and concluded that the generated daily data were unsuitable for detailed site-specific impact studies in their current form. The reason, as stated by Lenderink et al. (2010), is that ‘Convection is not resolved in the current generation of RCMs, and models use parameterisations to represent convection. It is known that these parameterisations have shortcomings …’. Since the systematic bias in reproducing daily precipitation distribution as well as distributions of consecutive dry/wet days will profoundly alter simulated hydrology and plant growth, many researchers have used several model output statistics (MOS) method to correct the RCM output bias for use in impact assessment (Guo and Senior, 2006; Rosenberg et al., 2010; Schoof et al., 2009; Themeßl et al., 2010; van Roosmalen et al., 2010). In essence, bias correction of GCM/RCM projections is an integral part of the MOS methods. Themeßl et al. (2010) evaluated seven different MOS methods for correcting errors in raw RCM output and for providing suitable daily climate data at particular locations for impact studies. They reported that quantile mapping of daily precipitation between the native RCM grids and target locations performed the best in bridging the gap between RCM output and end- user needs. Those MOS methods are able to correct RCM precipitation intensity bias (overestimation of light events and underestimation of heavy precipitation) and to adjust precipitation frequency (number of wet days), but are unable to improve temporal structure (representation of the lengths of dry and wet spells) (Maraun et al., 2010). The proper representation of precipitation sequence is important for correctly simulating plant growth and surface hydrology.
A diverse range of statistical spatial downscaling techniques were developed, including transfer function approaches, weather typing schemes, and weather generators (Wilby et al., 1998b; von Storch et al., 2000). These techniques, in general, fall in two categories, i.e. the MOS and Perfect Prognosis (PP) approaches. The transfer function is the most widely used approach, which involves deriving statistical relationships between predictands of observed local climatic variables and predictors of large-scale observed variables (PP), or GCM output fields (MOS), using regression-type methods such as multivariate linear or nonlinear regressions (e.g. Kidson and Thompson 1998; Kilsby et al.1998; Wilby et al.1998a), principal component analysis (e.g. Karl et al., 1990; Murphy, 1999), canonical correlation analysis (e.g. von Storch et al., 1993; Busuioc et al., 1999), Kriging, and artificial neural networks (e.g. Crane and Hewitson, 1998; Wilby et al., 1998b; Trigo and Palutikof, 2001). Most conventionally used predictors from GCM output for precipitation include vorticity, airflow indices, wind strength and direction, mean sea-level pressure, geopotential heights at 500 and 700 hPa, and relative humidity (e.g. Wilby et al., 1998b; Sailor and Li, 1999; Solman and Nuñez, 1999; Wilby and Wigley, 2000; Trigo and Palutikof, 2001). In addition, Kilsby et al. (1998) included altitude, distance from coast, and geographical location of the target station as predictors in their regression equations.
Widmann et al. (2003) found that statistical downscaling of precipitation directly using GCM precipitation as a singular predictor performed considerably better than conventional methods of using other predictors as mentioned above. Wood et al. (2004) used GCM precipitation as a sole predictor for bias correction using quantile plots between GCM output and measured precipitation at the same (regional) spatial scale. This method was used to develop ‘bias-corrected and spatially downscaled climate projections derived from CMIP3 data’, which is available for the contiguous US at resolutions as fine as a 12-km grid (http://gdo-dcp.ucllnl.org/downscaled_cmip3_projections/). Comparedwith dynamical downscaling, the advantages of statistical downscaling are its easiness to implement at any spatial scale and ability to calibrate to local conditions (Solman and Nuñez, 1999). One constraint for the approach is the availability of suitable observational data at local levels. As observation networks expand and density of weather stations increase throughout the world, this limitation is expected to diminish in the near future. However, the major concern of the statistical approach is whether statistical relationships derived based on the present climate are fully applicable to future climate, especially when non-stationary change occurs (e.g. Busuioc et al., 1999; Charles et al., 1999; Solman and Nuñez, 1999; von Storch et al., 2000; Wilby and Wigley, 2000; Wilby et al., 2004; Vrac et al., 2007). Many statistical downscaling methods implicitly assume the stationarity in their statistical relationships or transfer functions, and thus they are not applicable to non-stationary climate change if the change has altered the relationships. In general, the stationarity of the statistical relationships is one of the most essential requirements in empirical statistical downscaling. To address this potential problem, better methods or alternatives need to be developed and verified.
Some RCM-projected daily precipitation data for certain greenhouse gas emission scenarios are available in a number of regions. For example, the ENSEMBLES program provides RCM-projected daily precipitation data at a spatial scale of 25 km across Europe for climate research purposes (http://ensembles-eu.metoffice.com/data.html). The CMIP5 datasets are also available on a daily basis. However, those daily data cannot be directly used for site-specific impact studies without bias correction and further downscaling to a station. Alternatively, GCM/RCM monthly projections are widely available online and, arguably, more skillful than daily values, especially for GCM projections. Those monthly values at spatial scales of > 20 km can be readily downscaled to daily series for particular locations for use in process-based models such as plant growth models. For example, 45 km grid monthly outputs from a Canadian RCM are available to download for Canada (http://www.cccma.bc.ec.gc.ca/), whilst daily outputs are available mainly for climate research such as CMIP5. Monthly climate data, statistically downscaled from GCM to resolutions as fine as 12 km, are available for the contiguous US (http://gdo-dcp.ucllnl.org/downscaled_cmip3_projections/).
Stochastic weather generators, meanwhile, are the most widely used tools for temporally disaggregating monthly climate projections to daily time series, which is required by many process-based agricultural system models for impact assessment. Input parameters for daily precipitation distributions primarily include the mean and variance of wet days and conditional probabilities of precipitation occurrence. These parameters are often parameterized for each calendar month to reflect seasonality, and can be readily manipulated to simulate arbitrary changes in mean and variance quantities for sensitivity analysis, or be deliberately modified to mimic changes in mean and variance as predicted by GCMs/RCMs for impact assessment. Parameter values for a particular climate change scenario are often developed by perturbing parameter values of the present day climate under the guidance of GCM-projected relative changes (e.g. Wilks, 1992; Semenov and Porter, 1995; Katz, 1996; Semenov and Barrow, 1997; Mavromatis and Jones, 1998; Zhang, 2005; Semenov, 2007), or are directly estimated using statistical relationships developed between model parameters and GCM output fields (Kilsby et al., 1998; Wilby et al., 1998a).
The objective of this work is to verify (1) that a temporal disaggregation method for disaggregating monthly precipitation to daily series at a station scale is applicable to non-stationary climate conditions using historical station data that experienced significant changes in precipitation in Oklahoma during the past 100 years, and (2) whether the method is capable of generating correct precipitation intensity and frequency as well as wet/dry spell sequence in comparison with the observed daily records. The use of historical precipitation for verification is based on the premise that measured data are the best available ‘ground truths’ for testing and validating a method.
2. Statistical temporal disaggregation
Proper spatial and temporal treatments of climate change during downscaling are essential to capture the potential shift towards more intense storms in future climate change, and therefore, are critical to yielding reliable impact assessment at a particular location. Zhang (2005) developed a statistical downscaling method that explicitly treats spatial and temporal climate variations in two separate steps. First, GCM- or RCM-projected monthly precipitation is spatially downscaled from a grid box to a target station using quantile mapping (i.e. QQ-plot). Secondly, the spatially downscaled monthly precipitation is disaggregated to daily precipitation series at the target station using the CLImate GENerator (CLIGEN) (Nicks and Gander, 1994). Compared with many conventional statistical downscaling approaches, this method imparts more detailed treatment of spatiotemporal climate variations on generated daily series including variance of daily precipitation amounts as well as distributions of consecutive wet/dry days (Zhang, 2007), which are crucial for properly simulating soil hydrology and crop production. Owing to the systematic bias associated with RCMs-projected daily precipitation amounts, the use of MOS methods to correct the bias and to provide suitable climate scenario data for climate change impact research is inevitable (Themeßl et al., 2010). The MOS methods can correct daily precipitation intensities and to adjust precipitation frequency, but they cannot improve the temporal structure or precipitation sequence (Maraun et al., 2010). However, the presented method is intended to amend the drawbacks mentioned above.
2.1. Stochastic climate generator (CLIGEN)
CLIGEN generates daily precipitation occurrence,amounts, duration, peak storm intensity, time to peak, and daily values of maximum, minimum, and dew- point temperature, solar radiation, and wind speed and direction individually for each calendar month. Only precipitation amounts and occurrence are presented here. A first- order, two-state Markov chain algorithm is used to generate precipitation occurrence for a day given the previous day being wet or dry. If a random number that is drawn from a uniform distribution for each day is less than the conditional precipitation probability for the given previous day's status, a precipitation event is predicted. For a predicted wet day, a skewed normal distribution (also referred to as Pearson type III distribution), which is parameterized for each calendar month (Table I), is used to generate daily precipitation amounts for the calendar month in question (Nicks and Lane, 1989)
where x is the standard normal deviate, R is the daily precipitation amount, and µ, σ, and g are the mean, standard deviation, and skew coefficient of the daily amounts of wet days, respectively, for the month. The Box-Muller method is used to generate the standard normal deviate x, which is then used in Equation (1) to compute the daily amount R.
Table I. Disaggregated and interpolated CLIGEN precipitation input parameter values for the validation period (1985–2007) for Lahoma, Oklahoma
Rd is the mean daily precipitation of wet days.
σd is the standard deviation.
Skewness is the skewness coefficient that was directly taken from the validation period without any adjustment.
Equation (1) was tested for about 200 stations scattered throughout the US (Nicks and Lane, 1989). Various versions of CLIGEN have been evaluated in the past 20 years. Johnson et al. (1996) evaluated version 4.x for six locations dispersed across the contiguous US and concluded that annual and monthly precipitation statistics including means, standard deviations, and extremes were adequately replicated by the model. But daily amounts, particularly extreme amounts in any given year, were not entirely satisfactorily generated. They further reported that the first-order Markov models used in the model adequately replicated sequences of wet and dry days. Headrick and Wilson (1997), who evaluated CLIGEN (v4.x) for five Minnesota locations, reported that CLIGEN replicated daily precipitation amounts reasonably well. Since the year 2000, a number of major changes have been made to CLIGEN v5.x (Yu, 2000; Flanagan et al., 2001), including fixing code bugs, correcting errors in databases, refining algorithms, and implementing random number quality control. Meyer et al. (2008) imposed confidence interval tests on uniform random number streams and generated standard normal deviates to select only those numbers that conform to the expected distributions for subsequent use. Zhang and Garbrecht (2003) evaluated v5.107 for four Oklahoma stations in the study region and reported that mean absolute relative errors for simulating daily, monthly, and yearly precipitation were 4.7, 1.7, and 1.5% for the means and 3.7, 6.7, and 15% for the standard deviations, respectively. Mean absolute relative errors for the all-time maxima of daily, monthly, and yearly precipitation were 17.7, 8.9, and 6.5%, respectively. Chen et al. (2009) evaluated v5.22564 for the Loess Plateau of China, and found that mean absolute relative errors for simulating daily, monthly, annual, and annual maximum daily precipitation depth across all 12 stations were 3.5, 1.7, 1.7, and 5.0% for the means, and 5.0, 4.5, 13.0, and 13.6% for the standard deviations, respectively. CLIGEN reproduced distributions of monthly and annual precipitation amounts very well (P > 0.3), but in comparison distributions of daily precipitation amounts were less well reproduced. Zhang and Garbrecht (2003) and Chen et al. (2009) found that frequencies of both wet and dry periods were relatively well replicated by CLIGEN (v5.x), indicating that the first-order, two-state Markov chain algorithm is adequate for generating daily precipitation occurrence.
2.2. Temporal disaggregation using CLIGEN
Zhang (2005) developed a two-step downscalingapproach in which GCM monthly precipitation projections are first spatially downscaled from a grid box to a target station using a transfer function approach, and the downscaled monthly values are then disaggregated to daily series using CLIGEN (v5.3). Only the temporal disaggregation step is presented here in more detail. Readers are referred to Zhang (2005) for more information of the procedures. For temporal disaggregation, adjusted daily precipitation parameters of CLIGEN include mean (Rd) and variance (σd2) of daily precipitation amounts (excluding zeros) and conditional transition probabilities of a wet day following a dry day (Pw/d) and a wet day following a wet day (Pw/w) (Table I). For transition probability adjustment, linear relationships between transition probabilities and mean monthly precipitation (Rm) were developed for each calendar month using historical station records. For each calendar month (say January), approximately, the 30 wettest and 30 driest months selected from the calibration period were used to compute Pw/w, Pw/d, and Rm for each group. Linear functions between dependent Pw/w and independent Rm as well as between Pw/d and Rm were derived algebraically using the two endpoints (one pair for the wet group and another for the dry group). The straight line between the two endpoints was then used to calculate new Pw/w or Pw/d using Rm of the validation period or spatially downscaled Rm in the case of climate change. In addition, a linear regression between Pw/w (or Pw/d) and Rm was carried out using four data points (i.e. the two endpoints plus the two points for the 1910–1939 and 1940–1969 slices) for each month to see any improvement over the direct connection method discussed above. For convenience, the adjusted new conditional transition probabilities are equivalently expressed in term of an unconditional probability of daily precipitation occurrence (π) and a dependence parameter (r) defined as the lag-1 autocorrelation of daily precipitation series thus:
The adjusted new mean daily precipitation per wet day (Rd) is estimated as
where Nd is the number of days in the month and Ndπ is the expected number of wet days in the month. Since the variance of monthly precipitation (σm2) is known for the validation period at the target station or can be estimated using spatially downscaled monthly values in the case of future climate change, the adjusted new daily precipitation variance (σd2) is estimated using Equation (11b) of Wilks (1999) as
All adjusted daily precipitation parameter values for each calendar month (Table I) were used in CLIGEN (v5.3) to generate any length of daily precipitation time series using Equation (1). For this work, daily series of 100 years were generated and used in the analysis.
3. Dataset, analysis procedure, and statistics
Five stations (Hooker, Weatherford, Lahoma, Chandler, and Idabel) in Oklahoma were selected to cover a wide precipitation range (Table II). There were approximately 100 years of measured daily precipitation records at each station. The measured mean annual precipitation for the entire records varied from about 470 mm at Hooker to 1207 mm at Idabel. The measured daily precipitation records at each station were split into a calibration period and a validation period. An effort was made to maximize the difference in mean precipitation between the two periods while attempting to keep the validation period in the vicinity of 30 years. The mean annual precipitation for the validation period numerically decreased by 5.6%, compared with the calibration period at Hooker (P = 0.12 with a one-tailed t-test). In contrast, the mean annual precipitation of each validation period increased significantly over the corresponding calibration period by 13–15% at the other four stations (P = 0.05).
Table II. Station location, elevation, calibration and validation periods, and average annual precipitation for each period
Precipitation (mm yr−1)
Precipitation (mm yr−1)
The driest 30 months (dry end) and the wettest 30 months (wet end) selected from the calibration periods were used to derive linear relationships between conditional transition probabilities and mean monthly precipitation for each calendar month. The linear relationship was compared graphically with the transition probabilities and mean monthly precipitation computed for each of the three consecutive 30-year slices of the measured data (i.e. 1910–1939, 1940–1969, 1970–1999) as well as those for the validation period for each calendar month and location. The straight lines connecting the endpoints along with monthly statistics (e.g. mean and variance of monthly precipitation) of the validation periods were used in Equations (2–5) to estimate daily precipitation statistics for each calendar month for the validation periods (Table I), which were input to CLIGEN to generate daily time series for the periods. Measured and generated daily and monthly precipitation were statistically compared in the forms of mean, standard deviation, coefficients of skewness and kurtosis, percentiles, and extreme values. Relative error (RE), computed as the difference between generated and measured values divided by measured values, was presented. Measured and generated numbers of consecutive dry days and wet days were compared for all stations.
An F-test and one-tailed t-test were used to test the equality of standard deviations and means, respectively, for generated and measured monthly and annual precipitation amounts for each station. On the basis of the central limit theorem, monthly and annual precipitation amounts tend to be normally distributed, though daily precipitation amounts are not. Thus, application of these tests to monthly and annual precipitation is deemed acceptable. In addition, a nonparametric, two-sample Kolmogorov-Smirnov (K-S) test, which is applicable to any distribution, was used to test the null hypothesis that two samples are from the same distribution (SAS Institute, 1990). The test statistic is the K-S distance D, which is the maximum absolute deviation between two empirical distribution functions of the two samples.
In addition, model efficiency (ME), which measures how well a model reproduces measured data as defined by Nash and Sutcliffe (1970), was calculated as
where Yobs = measured value, Ypred = predicted value, and Ymean = measured mean. ME can range from − ∞ to 1. If ME = 1, the model produces the exact prediction for each data point. A zero value of ME implies that a single measured mean is as good an overall predictor as the model. A negative value of ME indicates that the measured mean is a better predictor than the model.
4. Results and discussion
4.1. Linearity between precipitation occurrence probability and mean monthly precipitation
Conditional transition probabilities Pw/w and Pw/d and mean monthly precipitation amounts, calculated with measured station data for the month of interest for the 30 driest and wettest months, the three 30-year slices (1910–1929, 1930–1959, and 1960–1999), and the validation periods, were plotted for each calendar month in Figure 1 for the driest Hooker site, and in Figure 2 for the wettest Idabel site for illustration purpose. Also shown were the straight lines connecting the driest and the wettest values, which were used to interpolate Pw/w and Pw/d of the validation periods using the mean monthly precipitation of the periods. Conditional transition probabilities Pw/w and Pw/d generally exhibit a linear increasing trend with mean monthly precipitation for all months and sites. An increase in Pw/d means an increase in the occurrence of wet-day-following-dry-day events and, therefore, a decrease in the number of consecutive dry days (meaning less drought stress for plant growth). In contrast, an increase in Pw/w with mean monthly precipitation denotes an increase in the occurrence of wet-day-following-wet-day events and, thereby, an increase in the number of consecutive wet days (critical for flooding forecast). An increase in number of consecutive wet days and a decrease in number of consecutive dry days are consistent with a general increase in monthly precipitation. In addition, we carried out a linear regression using four data points (excluding the 1960–1999 slice and the validation period) for each month and location. The results showed that 47 out of 60 (12 months × 5 sites) were significant at P = 0.05 for Pw/d while 19 in 60 were significant for Pw/w, and the slopes of the connection lines (between the two endpoints) were within the 95% confident intervals of all regression slopes.
The linear interpolation with the connection line between the two endpoints worked well for Pw/d for all months and sites, with the four data points of the three 30-year slices and the validation period (labelled with ‘v’) being fairly close to the connection lines (Figures 1 and 2). In comparison, the data points for Pw/w were much more scattered because there were fewer wet-following-wet events than wet-following-dry events. A smaller number of wet-following-wet events resulted in more variable and less reliable estimates of Pw/w for the 30-year duration, indicating that a longer period should be used to obtain more accurate estimates of Pw/w, even though the 30-year period has been widely used to develop standard climatology. This was especially true for the driest Hooker site, which exhibited the most scattering of Pw/w among the five sites (figures not shown). In contrast, the best results of Pw/w were obtained for the wettest Idabel site due to an increased occurrence of wet-following-wet events on the site. This result indicates that increasing the sample size or duration holds the key to improving the accuracy of Pw/w estimation, implying that the use of a larger sample size or greater number of months for each end point would enhance the overall quality of the linear interpolation method. It should be pointed out that the use of a longer duration would be unlikely to violate the stationary assumption of CLIGEN, because each end group contains sorted dry or wet months. For cross-site comparisons, linear correlations between monthly mean precipitation amounts and probabilities of precipitation occurrence (Pw/d, Pw/w, or π), calculated using the observed data, were carried out for the 12-month pooled data (n = 6 data points × 12 months) for each site (Table III). Generally, the correlation between Pw/d and mean monthly precipitation was much stronger than that between Pw/w and mean monthly precipitation for all sites. Unsurprisingly, the unconditional probability (π), which relates to both Pw/w and Pw/d in Equation (2) and directly determines the number of wet days, showed the strongest correlation with mean monthly precipitation for each site. The correlation coefficients for the calibration period of 1910–1970 were 0.870, 0.623, and 0.895 for Pw/d, Pw/w, and π, respectively, for all months and sites, and were 0.831, 0.570, and 0.847 for the validation period, showing little change in the correlation coefficients, though mean annual precipitation changed more remarkably between the two periods. Overall, the relatively large linear correlation coefficients (P < 0.001) for all five sites indicate that the linear relationship is applicable to a wide range of precipitation conditions. Furthermore, the linear relationship that connects the driest and wettest hypothetical climate conditions is acceptable for interpolation to any possible climate condition within the two extremes. This is particularly useful if observed records are relatively short, say, less than 60 years. For longer record durations, linear regression may be used to improve the accuracy of the Pw/d and Pw/w interpolation.
Table III. Pearson linear correlation coefficients between mean monthly precipitation amounts and probabilities of precipitation occurrence, calculated with observed data, for all 12 months for the 5 stations
For general comparison, the unconditional probability of precipitation occurrence (π) was further calculated with Equation (2) using all the 72 pairs of Pw/w and Pw/d for each station. The resultant π values were plotted with the corresponding mean monthly precipitation for each station in Figure 3. The overall results showed that the ‘measured’ π was positively correlated with, and strongly dependant upon the measured mean monthly precipitation for all sites (Table III). However, a quadratic rather than linear function appeared to be the best fit for all sites due to seasonality, indicating that the linear interpolation is applicable only to each calendar month, or a season at most, but not an entire year. Similar results were obtained when Pw/w or Pw/d instead of π were plotted. Though the plots were somewhat more scattered, especially with Pw/w, the nonlinearity was evident for all sites, indicating that Pw/w and Pw/d interpolation are better done on a monthly basis in a linear fashion rather than by a best quadratic fit for all months. The strong correlations between transition probabilities and mean monthly precipitation amounts indicate that a non-stationary change in mean precipitation would come with considerable changes in transition probability (i.e. precipitation frequency). Such non-stationarity can be overcome by employing linear interpolations of transition probabilities in conjunction with Equations (2–5) and a weather generator.
The conditional transition probabilities Pw/w and Pw/d linearly interpolated using the connection lines for the validation periods for each month as well as the unconditional probability π computed with Equation (2) for each Pw/w and Pw/d pair were plotted with those directly calculated using the measured daily precipitation records of the periods (Figure 4). The interpolated and measured precipitation probabilities agreed well, with ME ranging from 0.722 to 0.887. We also calculated ME using the linear regression interpolation method instead of the connection line method as was done in Figure 4 (data not shown). The ME was decreased for the Chandler station but slightly improved for the other four stations, indicating the potential for using the linear regression method, especially when long observation records are available. The major source of errors for either interpolation method are from Pw/w estimation on all sites, suggesting that (1) the 30 driest months used to develop the linear interpolation functions might not be long enough, and (2) the validation periods (<30 years on four sites) might also not be long enough to produce reliable Pw/w estimates for comparison. These results suggest that use of longer durations may improve Pw/w estimation and, therefore, the quality of the disaggregated data.
4.2. Downscaled and measured distributions of dry and wet spells
Ability to generate realistic distributions of consecutive dry day lengths and wet day lengths under climate change is important for simulating climatic impacts on water resources and crop production. Daily wet-dry sequence is stochastically generated for a month by comparing a uniformly distributed random number stream with Pw/d or Pw/w of the month, depending on the previous day state (dry or wet). If a random number is less than or equal to Pw/d or Pw/w for a given day, then a wet day is predicted. Otherwise, a dry day is produced. Thus, the quality of Pw/d and Pw/w estimation is the key for generating realistic distributions of wet spells and dry spells for any climate condition in question. The cumulative frequencies of dry and wet spells, generated using Pw/d and Pw/w interpolated for the validation periods using the connection method, were compared with those directly calculated from the measured data of the same period (Figure 5, note the logarithmic scale on the X-axis). The K-S statistics D, which is the maximum absolute deviation or vertical displacement between the two empirical distributions, is also given in Figure 5. Since the sample sizes are fairly large (n > 4000 for generated and > 1000 for measured) on all sites, the critical Dα at P = 0.01 is about 0.053 (Dα is about 0.048 at P = 0.05) for all sites. The conclusion that the two samples are from different distributions is arrived if D ≥ Dα. On the basis of the tests, about half the datasets, as denoted by the asterisk in Figure 5, are significantly different at P = 0.01. The significant difference may be partially attributed to the large sample size, as the K-S test becomes extremely stringent and powerful as the sample size becomes large (Zhang and Garbrecht, 2003). However, the degree of agreement between the disaggregated (interpolated) and measured wet-dry sequences (Figure 5) was comparable to the ability of CLIGEN to reproduce measured wet-dry sequences as reported by Zhang and Garbrecht (2003) for Oklahoma and by Chen et al. (2009) for the Loess Plateau of China.
4.3. Disaggregated and measured daily precipitation amounts
The Pw/w and Pw/d interpolated with the connection method and the subsequently calculated mean (Rd) and standard deviation (σd) of wet day precipitation were used in Equation (1) to generate 100 years of daily precipitation sequences for the validation periods with CLIGEN. The skewness coefficient in Equation (1) was not adjusted because there was no apparent correlation between monthly precipitation and skewness coefficients in all the observed data at the five stations. To evaluate the goodness of the other adjusted or estimated parameters, values calculated for the validation period were used. Note, if a two-parameter gamma distribution of daily precipitation amounts is used in a weather generator such as WGEN, the skewness parameter is not needed, and the two parameters (i.e. shape and scale) can be directly calculated from the adjusted mean and variance of daily precipitation. Statistics of the disaggregated daily precipitation amounts were compared with those calculated with the measured daily data of the validation periods (Table IV). The daily precipitation means of the disaggregated series were consistently greater than those of the measured data for all five sites. The largest relative error (RE) was 18.1% at Hooker, and the smallest was 6.4% at Chandler, with an average of 11.9% across all sites. This consistent overestimation of the mean precipitation was mainly caused by the consistent underestimation of the mean number of wet days for all sites (last row in Table IV). The mean number of wet days per year was underestimated from 3 to 8 days for all sites. The greater daily precipitation amounts were to compensate for the fewer rain days to achieve the target precipitation total as governed by Equation (4). As a result of overestimation of the precipitation means, standard deviations were also overestimated for four out of five sites, with RE ranging from 0.6% at Lahoma to 16.7% at Hooker. The standard deviation was underestimated only at Chandler (−4.3%), partially because the generated and measured precipitation means were the closest among all five sites. The mean absolute RE for standard deviation was about 7.1% for all sites. The skewness coefficients between generated and measured data were fairly close, except for the driest Hooker site (−15.6%). The overall mean of the absolute RE for all sites was about 4.8%. The Kurtosis coefficients of generated and measured data agreed reasonably well on four out of five sites, except, again, at Hooker. The mean absolute RE for all sites was 12.6%. The cumulative distributions of disaggregated and observed daily precipitation were relatively close, as indicated by the percentiles. However, the generated percentiles tended to be slightly greater than the observed percentiles for all sites except for the 99th percentile at Chandler and Idabel. The K-S tests rejected the null hypothesis that the two samples were from the same population for all sites at very significant levels (P < 0.0001). These test results may have been strongly influenced by the extremely large sample size of the pooled data of all 12 months (n > 5000 for generated data and > 1500 for observed data). When the sample size is very large (as it is here), the K-S test becomes extremely stringent and powerful. This is partially illustrated by the test results for each calendar month on each site (i.e. for month-site combinations, about 92% reduction in the sample size). Out of the 60 combinations, only 17 tests rejected the null hypothesis at P < 0.0001. Approximately 52% of the K-S tests rejected the null hypothesis at P = 0.01, which was very close to the 54% rejection rate when CLIGEN-reproduced daily values were tested against observed daily data on four stations across the same study region (Zhang and Garbrecht, 2003). In their study, all the CLIGEN input parameters including Pw/w, Pw/d, mean and standard deviation of daily precipitation amounts and skewness coefficient, which were directly calculated from observed daily data, were used to generate 100 years of daily series, and the generated and measured daily data were then compared to evaluate the reproducibility of the CLIGEN model. The degree of agreement between disaggregated and observed daily values as obtained here was generally comparable to the agreement between reproduced and observed daily values as shown in Zhang and Garbrecht (2003), though the former was slightly lower than the latter as expected. Surprisingly, the all-time maximum daily precipitation amounts of the disaggregated data were very close to those of observed data, indicating CLIGEN's ability (or suitability of Equation (1)) for generating extreme precipitation events for the study region. It should be mentioned that there was little difference in interpolated Pw/w and Pw/d between the connection and the linear regression methods, thus the daily statistics using the linear regression interpolation should be comparable to those reported here.
Table IV. Statistics of observed and CLIGEN-disaggregated daily precipitation amounts of wet days with precipitation > 1 mm for the validation period for 5 stations. The Kolmogorov-Smirnov tests for pooled data of 12 months for each of the 5 sites are significant at P < 0.0001
4.4. Disaggregated and measured annual maximum daily precipitation
Statistics of annual maximum daily precipitation, which is the maximum daily amount of each year, are given in Table V. The RE for the mean values ranged from − 9.9% at Chandler to 11.5% at Hooker, with the mean absolute RE across all 5 sites being 6.2%. The RE for the standard deviation ranged from − 13.2% at Idabel to 10.7% at Weatherford, with the mean absolute RE being 10.6%. Zhang and Garbrecht (2003) reported that the mean absolute RE was about 5.1% for the mean and 9.0% for the standard deviation when CLIGEN-reproduced (v5.107) and observed daily sequences were compared across four stations dispersed in the same study region. The similarity in these mean absolute RE between disaggregated and reproduced values indicate that the disaggregated method allows CLIGEN to generate annual maximum daily precipitation amounts at or near its maximal reproduction skills. The distribution of the annual series of extreme values was much less skewed and peaked than that of the daily precipitation series. The disaggregated percentiles were slightly larger than the corresponding observed percentiles at Hooker, Weatherford, and Lahoma; smaller at Chandler; and mixed at Idabel. However, the K-S tests showed that differences between the two distributions were insignificant at all sites (Table V), meaning that the two samples were likely from the same population. This result indicates that the distribution of annual maximum daily precipitation amounts was better generated by CLIGEN than was the distribution of daily precipitation amounts.
Table V. Statistics of observed and CLIGEN-disaggregated annual maximum daily precipitation amounts of validation periods for five stations
4.5. Generated and measured monthly precipitation amounts
The 100 years of disaggregated daily precipitationamounts were summed up to monthly values and compared to observed monthly data of the validation periods. The generated monthly means were almost identical to the observed means, with a mean absolute RE across all five sites of 0.8% (Table VI). The t-test showed that the differences in means were insignificant for all sites (P > 0.40). The RE for standard deviation ranged from − 5.0% at Hooker to 1.6% at Chandler, with a mean absolute RE for all sites of 1.6%. The F-test showed no significant differences in standard deviation for all sites, with a mean P value of 0.66 across all five sites. The skewness and kurtosis coefficients, relative to those of observed monthly data, were reasonably well preserved by the disaggregated method except for the kurtosis at Chandler. All percentiles of the generated monthly series were rather close to those of the observed series for all five sites. The K-S test statistics in Table VI showed that the two samples were likely from the same population on all five sites. Similar K-S P values were obtained when each of the 60 month-site combinations was tested separately (data not shown). The closeness between the generated and observed monthly precipitation data demonstrates that this temporal disaggregation method is capable of preserving statistics of target monthly precipitation amounts. Specifically, linear interpolation of Pw/w and Pw/d in conjunction with Equations (2–5) is able to preserve statistics of monthly precipitation series at a target station. This verification work assumes ‘perfect’ spatial downscaling of monthly precipitation from GCM or RCM grid scales to target locations. This assumption is largely met if the quantile mapping (QQ-plot) method is used for spatial downscaling and bias correction as was used by Zhang (2005). The method matches the distribution of GCM-projected monthly precipitation (hindcast) with that of observed monthly precipitation at a target station by fitting a transfer function. The fitted function is then applied to future projection for bias correction and spatial downscaling, assuming that the bias remains the same in future projection as in hindcast for the grid box of interest. This bias correction does not alter any climate change trend or perturb non-stationary changes. The quantile mapping method has been successfully used to downscale GCM/RCM projections to a station scale in US, China, and Canada (Li et al., 2010; Zhang et al., 2011).
Table VI. Statistics of observed and CLIGEN-generated monthly precipitation amounts for the validation period for five stations
The presented disaggregation method is particularly useful to provide daily climate data for impact assessment for regions where only GCM projections are available. As pointed out by Hewitson (2003), the skillful resolution of a GCM is typically some spatial and temporal aggregation of the native GCM resolution. That is, the spatial skill resolution is typically below the native grid box of the GCM, and the temporal skill resolution is the monthly scale at best, suggesting daily values are of little direct use. Even for areas where RCM daily projections are available, this method can still be a useful tool to generate quality daily precipitation data. Compared with GCMs, spatial and temporal skillful resolutions of RCMs are greatly improved. However, the direct use of the RCM daily output for impact studies is not recommended because it contains systematic errors (Mearns et al., 2003). Several MOS methods are used to satisfactorily correct daily precipitation intensity and to adjust precipitation frequency (number of wet days in a year), but they are unable to improve the representation of the lengths of dry and wet spells (Maraun et al., 2010). However, this work shows that this disaggregation method is capable of amending the drawbacks mentioned above.
The spatiotemporal downscaling method of Zhang (2005) has been successfully used to evaluate the site-specific impacts of climate change on surface hydrology, soil erosion, and crop production (Zhang, 2007; Li et al., 2010; Zhang et al., 2011). GCM monthly projections at grid scales are first spatially downscaled to a target station using transfer functions developed from quantile mapping, and the downscaled monthly values at the station are then further disaggregated to daily series using stochastic weather generators. This study seeks to verify the validity of the temporal disaggregation step for generating daily weather series for non-stationary changes. Approximately 100 years of observed data at each station have been split into calibration and validation periods in such a way so as to validate the method's applicability for non-stationarity in mean precipitation. The validation results show that the method is valid for up to 15% increases in mean precipitation (i.e. the maximum increase in the validation period over the calibration period among the five sites). However, in principle, the method can be applied to much larger changes, provided the potential changes are within the range between the two hypothetically dry and wet extremes. Specifically, the average increase in monthly mean precipitation of the wet end over that of 1910–1999 ranged from 61 to 84% for the five sites, whilst the decrease at the dry end relative to the 1990–1999 means ranged from − 74 to − 60%. Given the wide ranges between the dry and wet ends, the method is suitable to disaggregate any climate change scenarios projected for the century, in which the projected changes are within ± 30% globally as averaged over multiple GCMs (IPCC, 2007).
Strong linear correlation between transition probabilities of precipitation occurrence (Pw/w and Pw/d) and mean monthly precipitation was evident for all five stations dispersed in Oklahoma, with mean annual precipitation ranging from 450 to 1330 mm. Transition probabilities generally increased in a linear fashion with an increase in mean monthly precipitation for each calendar month. The straight line connecting the two endpoints is adequate for transition probability interpolation, especially if observed records are relatively short (e.g. < 60 years). The linear regression method may be used for longer records to potentially improve the interpolation accuracy. The linearity was much stronger for Pw/d than for Pw/w. The more variable estimates of Pw/w, compared with Pw/d, were largely because fewer wet-following-wet events than wet-following-dry events occurred during the 30-year calculation period. More reliable Pw/w estimation could be achieved if longer calculation periods are used to augment the sample size. This conclusion was substantiated by the fact that better linear responses of Pw/w were exhibited at the wettest Idabel site than at the driest Hooker site. Transition probabilities Pw/w and Pw/d interpolated linearly with the connection lines between the two endpoints for the validation periods agreed well with those directly calculated using observed daily data of the periods. The ME for transition probabilities ranged from 0.722 to 0.887 for all five sites. As a result of the good estimation of Pw/w and Pw/d, cumulative frequency distributions of dry and wet spell lengths generated using Pw/w and Pw/d interpolated for the validation periods agreed reasonably well with those directly calculated from the observed data of the same period. The degree of agreement between the disaggregated and observed daily amounts was comparable to the agreement between CLIGEN-reproduced and observed daily amounts in the study region. Overall, results showed that linear relationships between transition probabilities and mean monthly precipitation were adequate for interpolating Pw/w and Pw/d in a wide range of precipitation conditions. A period of longer than 30 years is preferred to obtain more accurate Pw/w estimation. For the two endpoints, the non-stationarity should not have any effect on estimated Pw/w even if a duration of longer than 30 years is used because only sorted dry and wet months are included in each group. Once established with confidence, the linear relationships, which span the driest and wettest hypothetical climatic conditions and embody the non-stationarity of climate states are suitable for interpolating Pw/w and Pw/d to climatic conditions within the entire range. It should be pointed out that although the linear interpolation model is preferably calibrated over longer durations, once calibrated it can be used to generate daily series at any desirable timescales. However, a short timescale of a decade may result in large uncertainty in estimating monthly precipitation variance, but a long timescale of four decades may likely violate the generator's stationarity assumptions if a non-stationary change occurs in future. Thus, an optimal range seems to be between two and three decades, depending on the size of climate change and research objectives.
The mean absolute RE between disaggregated and observed wet-day precipitation amounts for all five sites was 11.9% for the mean and 7.1% for the standard deviation. The mean daily precipitation amounts were consistently greater for the disaggregated values than for the observed data for all five sites. The overestimation of the means was compensated by the consistent underestimation of the numbers of wet days for all five sites. This compensation was also reported by Zhang and Garbrecht (2003) when CLIGEN-reproduced and observed daily sequences were compared. In their study, mean absolute RE was 4.7% for means and 3.7% for standard deviations across four sites in the same region. The differences in absolute RE between this study and that of Zhang and Garbrecht indicated the potential errors that could be introduced by the disaggregation method. However, the K-S test results were similar between the two studies, suggesting that errors introduced by the disaggregation method are inconsequential. Interestingly, annual maximum daily precipitation amounts were better simulated by CLIGEN with the disaggregation method than were the daily precipitation amounts. The good estimation of the maxima may be attributable to the suitability of Equation (1), which uses a skewed normal distribution and has been tested over 200 stations dispersed over US. The K-S tests showed that the disaggregated and observed annual maximum daily precipitation amounts were very likely from the same population for all five sites. The adequacy for simulating extreme precipitation events is critical for many impact assessment studies under climate change.
The temporal disaggregation method preserved the statistics of target monthly precipitation amountsextremely well, with a mean absolute RE of 0.8% for the mean and 1.6% for the standard deviation across all five sites. The critical P values of all tests (t, F, and K-S) were greater than 0.3 for all sites. Overall results demonstrate that the linear interpolation of Pw/w and Pw/d with mean monthly precipitation using either the direct connection method or the linear regression method in conjunction with Equations (2–5) is suitable for disaggregating monthly precipitation data to daily series for the study region using stochastic weather generators such as CLIGEN.