WATER RESOURCES RESEARCH, VOL. 48, W12601, doi:10.1029/2012WR012446, 2012
An improved standardization procedure to remove systematic low frequency variability biases in GCM simulations Rajeshwar Mehrotra1 and Ashish Sharma1 Received 27 May 2012; revised 5 September 2012; accepted 1 November 2012; published 15 December 2012.
[1] The quality of the absolute estimates of general circulation models (GCMs) calls into
question the direct use of GCM outputs for climate change impact assessment studies, particularly at regional scales. Statistical correction of GCM output is often necessary when significant systematic biases occur between the modeled output and observations. A common procedure is to correct the GCM output by removing the systematic biases in low-order moments relative to observations or to reanalysis data at daily, monthly, or seasonal timescales. In this paper, we present an extension of a recently published nested bias correction (NBC) technique to correct for the low- as well as higher-order moments biases in the GCM-derived variables across selected multiple time-scales. The proposed recursive nested bias correction (RNBC) approach offers an improved basis for applying bias correction at multiple timescales over the original NBC procedure. The method ensures that the bias-corrected series exhibits improvements that are consistently spread over all of the timescales considered. Different variations of the approach starting from the standard NBC to the more complex recursive alternatives are tested to assess their impacts on a range of GCM-simulated atmospheric variables of interest in downscaling applications related to hydrology and water resources. Results of the study suggest that three to five iteration RNBCs are the most effective in removing distributional and persistence related biases across the timescales considered. Citation: Mehrotra, R., and A. Sharma (2012), An improved standardization procedure to remove systematic low frequency variability biases in GCM simulations, Water Resour. Res., 48, W12601, doi:10.1029/2012WR012446.
1.
Introduction
[2] Climate change impact assessment studies attempt to quantify the future risks to water resources systems, particularly at regional and catchment scales. However, there are a number of limitations that prevent the direct use of GCM output in studies dealing with the assessment of the impacts of climate change on water resources systems. One of the most important issues is the mismatch between the coarse resolution projections of global climate models (GCMs) and fine resolution data requirements of hydrologic models required in assessing the impacts at the catchment level [Intergovernmental Panel on Climate Change (IPCC), 2007; Vicuna et al., 2007; Fowler et al., 2007]. To bridge this gap, dynamical or statistical downscaling approaches are routinely applied to transfer the GCM output to finer spatial resolutions for use in the impact studies [Wilby et al., 2004]. Furthermore, the GCM simulations rely on the accurate modeling of the energy and moisture cycles as well as the simulation of clouds, which has been identified as one of the 1 School of Civil and Environmental Engineering, Water Research Centre, University of New South Wales, Sydney, Australia.
Corresponding author: R. Mehrotra, School of Civil and Environmental Engineering, University of New South Wales, Kensington, Sydney 2052, Australia. (
[email protected]) Published in 2012 by the American Geophysical Union.
remaining significant uncertainties in GCM outputs [Solomon et al., 2007]. Because of these assumptions, a GCM may not simulate climate variables accurately, and there is a difference between the observed and simulated climate variable known as bias. This limits the direct application of GCM simulations in downscaling and hydrological modeling studies [Liang et al., 2008; Wood et al., 2004]. To circumvent this, raw output of GCMs or regional climate models (RCMs) is bias-corrected to improve the quality of GCM simulations for use in either downscaling or as a direct input in the hydrological modeling studies [Maurer and Hidalgo, 2008]. The reliability of the downscaled outputs depends significantly on the reliability of the large-scale climate model simulations that serve as boundary conditions or forcings to the downscaling model used. [3] Another related issue (which is also linked to biases in GCM variables as discussed later) is the misrepresentation of observed year-to-year (or low-frequency) variability in the raw GCM simulations of rainfall and other variables [Bates et al., 2008; Lin, 2007]. The large-scale climate modes such as the El Niño Southern Oscillation (ENSO), the Indian Ocean Dipole (IOD), and the Interdecadal Pacific Oscillation (IPO) largely influence the interannual and interdecadal variability in rainfall. Climate change impact studies dealing with water resource assessment, such as reservoir yields, flood, and drought, often require that the year-to-year persistence in a range of climate variables be properly simulated. The low frequency variability has significant implications in
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assessing water security for a future climate, with an undersimulation of sustained droughts leading to overestimated reservoir yields, artificially enhancing the reliability associated with our existing water supply infrastructure to sustain future demands [Milly et al., 2008; Johnson and Sharma, 2011]. In this paper we focus on the latter problem of biases in GCM/RCM derived outputs. [4] Commonly used bias correction procedures eliminate the systematic biases in mean and variance of GCM predictor variables relative to observations (or reanalysis data) [Wilby et al., 2004]. The procedure usually involves first standardizing the GCM-simulated variables by subtracting the mean and dividing by the standard deviation, and then multiplying by the standard deviation and adding the mean of the corresponding observed or reanalysis data for a predefined baseline period at a timescale of interest. Other bias correction approaches mainly concerning the direct use of GCM/RCM output (primarily precipitation and temperature) use scaling, quantile matching, correction factors, and transfer functions [e.g., Chiew and McMahon, 2002; Mpelasoka and Chiew, 2009; Ines and Hansen, 2006; Li et al., 2010; Piani et al., 2010; Wood et al., 2004]. One of the main limitations of the delta change or scaling approaches is that variability is assumed to remain the same in the future. The quantile-based bias correction approach is a relatively simple approach that has been successfully used in hydrologic and many other climate impact studies [e.g., Cayan et al., 2008; Li et al., 2010; Maurer and Hidalgo, 2008]. The approach removes biases across the entire frequency distribution of a given variable. Fowler et al. [2007] and Johnson and Sharma [2012] provide a detailed review of various bias correction approaches. [5] Johnson and Sharma [2012] have recently proposed a nesting logic to postprocess GCM simulations for biases in mean, standard deviation, and LAG1 correlation at multiple time-scales, referring to the approach as the nested bias correction (NBC) method. In this paper we examine the impact of this bias correction procedure at multiple timescales and propose a refinement to the method that further improves the representation of variability at multiple timescales. [6] The remainder of this paper is structured as follows: Section 2 presents an overview of the NBC and the modification proposed. Section 3 provides a description of the atmospheric variables considered in the study. Section 4 presents the results and discussion and the conclusions drawn from the study are presented in section 5.
that the bias correction formulation proposed in the paper and that illustrated in the work of Johnson and Sharma [2012] can correct for biases in LAG1 autocorrelation only. If the data exhibits higher-order autocorrelations, then the approach needs to be modified by adopting a more complex model structure. The general structure of the model to be used (with possibly higher order lags) is detailed by Johnson and Sharma [2012] and can be implemented fairly easily as needed. [8] We illustrate the applicability of the approach to a daily time series. Let Pi,j,k,l represent the GCM-simulated daily output of a variable, Pj,k,l the sum/average of the daily values for each month j, Pk,l the sum/average of the monthly values for each season k, and Pl represent the sum/ average of the seasonal values for each year l. In our notation, subscript i is used for day, j for month, k for season, and l for year. We use Ph(.) for observations and Pm(.) for the bias-corrected outputs. Please note that while for precipitation higher aggregated timescale values are formed by summing the daily values, for a majority of other variables, such as pressure and temperature, it is the average of daily values of all days within that time-scale.
2.
where g() and () represent the model and associated parameters for each of the timescales of interest (day, month, season, annual, and tri-annual [3-yr] in our illustration above) used in the transformation. The following explains in brief the stepwise procedure required in the transformation. [10] 1. Calculate the reanalysis and GCM series daily mean, standard deviation, and LAG1 autocorrelation. To ensure the stability of the parameters, calculate them using the observations falling within a moving window of 31 d centered on the current day of interest. [11] 2. Correct the daily GCM series for bias in mean by removing the GCM series mean and adding the reanalysis mean.
Methodology
[7] The NBC method can be used to correct for distributional and persistence attributes on multiple timescales of interest. For this paper, we have represented the distributional attributes by order 1 and 2 moments and persistence by the LAG1 autocorrelation coefficient, all modeled at five selected bias correction timescales: daily, monthly, seasonal, annual, and tri-annual. As noted by Johnson and Sharma [2012], these choices are made for convenience and simplicity of presentation, and the outlined approach could accommodate more generic representations of both distributional and persistence attributes, as well as low frequency influences that may not be captured at the 3-yr timescale. It may be noted
2.1. Nested Bias Correction Procedure [9] The NBC procedure is described in detail by Johnson and Sharma [2012] and is briefly reproduced here. The approach transforms the GCM daily time series, Pi; j;k;l to ~ m j, where represents a parameters set, such that gP i; j;k;l ~m P ðÞ exhibits the same distributional and persistence attributes as Ph(.) across the multiple nesting timescales used. For the general implementation of this nesting algorithm using daily, monthly, seasonal, annual, and tri-annual timescales, the bias-corrected time series for the different temporal scales can be expressed as shown in (1). m h m m m ~m P ¼ g P i i; j;k;l;3l i; j;k;l;3l Pj;k;l;3l ; Pk;l;3l ; Pl;3l ; P3l ; i ; m h m m ~m P j;k;l;3l ¼ gj Pj;k;l;3l Pk;l;3l ; Pl;3l ; P3l ; j ; m h m ~m ; P ¼ g P ; P ; P j k k;l;3l k;l;3l l;3l 3l ~ m ¼ gj Ph Pm ; l ; P l;3l l;3l 3l m ~ ¼ g3l Ph 3l ; P 3l 3l
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[12] 3. Calculate the mean of the daily GCM-meancorrected series, subtract the mean, and bias correct the residuals for standard deviation, and add the mean thereafter. [13] 4. Calculate the mean and standard deviations of the daily GCM mean and standard deviation-corrected series and find residuals by subtracting the mean and dividing by the standard deviation. Bias correct the residuals Rm t for a day t LAG1 correlation, where the form of the correction is based on a standard autoregressive lag 1 model [Johnson and Sharma, 2012], ~m R t
¼
~m rh R t1
0 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m m m R r R B t t1 C ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi A; þ 1 ðrh Þ2 @ q 1 ½rm 2
(2)
m
~ is the bias-corrected residual for day t, rh is the where R t reanalysis, and rm is the GCM time series LAG1 correlation for that calendar day. [14] 5. Multiply the bias-corrected residuals by the standard deviation and add the mean thereafter. [15] 6. Aggregate the daily series to higher timescale and follow the same steps for other bias correction timescales as well. [16] It may be noted the three-step procedure adopted here to correct for the biases in the mean, standard deviation, and LAG1 autocorrelation at each preselected bias-correction timescale is somewhat different from that mentioned by Johnson and Sharma [2012]. The modified procedure ensures that the biases in the future climate series are corrected without affecting the future climate change signal. This aspect is covered in more detail in the next section. 2.2. Three Step Bias-Correction Procedure [17] The commonly used bias-correction procedure may not be sufficiently rigorous in a future climate setting. We explore the applicability of the standard bias correction procedure as described by Wilby et al. [2004] and adopted in NBC [Johnson and Sharma, 2012] in a changed climate setting, by considering a simple example. Let Xh be the reanalysis atmospheric data, X1 be a present day climate GCM simulation, and X2 be a changed climate GCM simulation. We denote the mean and standard deviation of reanalysis time series ðXh Þ as, h and Sh , and of the current climate GCM series ðX1 Þ as, 1 and S1 . Then the present day climate GCM simulation can be recentered around reanalysis mean using the transformation X10 ¼ X1 1 þ h to give a mean of X10 as h , or recentered and rescaled using X10 ¼ Sh ðX1 1 þ h Þ=S1 to give the mean as h and the standard deviation as Sh . [18] Similarly, for the changed climate simulation, assume that the mean and standard deviation of X2 series are 1 þ and S1 2, respectively (expressed as a function of current climate parameters). Applying the same recentering transformation as used for the present day climate, gives, X20 ¼ X2 1 þ h with a mean of X20 as h þ preserving the climate change signal. The combined recentering and rescaling using the same transformation as used for the present day climate gives, X20 ¼ Sh ðX2 1 þ h Þ=S1 . The mean and standard deviation of this transformed series are Sh ðh þ Þ=S1 and Sh 2, producing an (unverifiable) rescaling of the climate change signal.
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[19] In the three-step procedure outlined earlier, the present day climate GCM simulations are recentered around the reanalysis mean using the transformation, X10 ¼ X1 1 þ h , to give a mean of X10 as h . Now, recentering the X10 series around the series mean (which is h ), rescaling and adding the mean thereafter yields, X100 ¼ Sh ðX1 h Þ=S1 þ h , to give a mean and standard deviation of X100 as h and Sh , which is similar to the first case. [20] Applying the same recentering transformation as used for the present day climate, for the changed climate simulation, gives X20 ¼ X2 1 þ h , with the mean as h þ , preserving the climate change signal. Now, recentering the X20 series around the series mean (which is h þ ), rescaling and adding the mean thereafter yields, X200 ¼ Sh ðX2 h Þ=S1 þ h þ , with the mean as h þ and the standard deviation as Sh , producing an appropriate rescaling of the climate change signal.
3.
Data Used
[21] In order to illustrate the applicability of the NBC logic, not only for the precipitation series (as used by Johnson and Sharma [2012]), but also for other atmospheric predictor variables commonly used in statistical/dynamical downscaling, we proceeded as follows. We extracted the daily times series of geopotential height, relative and specific humidity, air temperature, dew-point depression (the difference of the air temperature from the dew-point temperature [Charles et al., 1999]), and equivalent potential temperature [Evans et al., 2004] at 500, 700, and 850 hPa, and mean sea level pressure at 25 grid points over Sydney, Australia. Reanalysis data are from 1960 to 2002 from the National Center for Environmental Prediction (NCEP) and provided by the NOAA-CIRES Climate Diagnostics Centre, Boulder, Colorado (available at http://www.cdc.noaa.gov/). Thus, a total of 19 variables (six atmospheric variables at three pressure levels and the solo mean sea level pressure) are considered. Similarly, daily output of CSIRO’s Mk3.0 GCM for these variables, for the same time period, for nine grid points over Sydney is obtained. The 25 reanalysis grid-point values are interpolated to obtain the daily series at nine CSIRO grid points. For simplicity, a single daily series of these variables (reanalysis as well as GCM) is formed by taking the inverse distance-weighted average of nine grid point values over Sydney. Further analysis is carried out using this areaaveraged series only. The bias correction procedure is applied in leave-one-out year cross-validation mode, i.e., leaving out 1-yr of data at a time and calculating the statistics of the remaining series and applying bias correction to the observations of the left out year. [22] Although, it is possible to first bias-correct the time series of air temperature and specific humidity subsequently deriving the relative humidity, dew-point depression, and equivalent potential temperature variables from these bias-corrected series, the derived series will not have the desired standard deviation and LAG1 autocorrelation statistics at higher timescales of month, season, and year.
4.
Results
[23] Figure 1 presents scatterplots of reanalysis and raw GCM-simulated values of atmospheric variables at multiple timescales. As expected, the statistics of GCM-derived
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Figure 1. Scatterplots of daily, monthly, seasonal, and annual means, standard deviations, and LAG1 correlations of reanalysis and raw GCM data. Points on the plots denote variables. For annual plots 19, for seasonal 19 4, for monthly 19 12, and for daily 19 365, points are shown. Mean values of all variables are rescaled to lie between 0 and 100.
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Table 1. Bias-Correction Options Considered Option
Description
RAW Dms Dmsc DM DMS DMA DSA DMSA DMSA3Y 5DMSA3Y
Raw GCM data Bias correction in mean and SD applied at daily timescale only. Bias correction in mean, SD, and LAG1 correlation applied at daily timescale only. Bias correction in mean, SD, and LAG1 correlation applied at daily and monthly timescales. Bias correction in mean, SD, and LAG1 correlation applied at daily, monthly, and seasonal timescales. Bias correction in mean, SD, and LAG1 correlation applied at daily, monthly, and annual timescales. Bias correction in mean, SD, and LAG1 correlation applied at daily, seasonal, and annual timescales. Bias correction in mean, SD, and LAG1 correlation applied at daily, monthly, seasonal, and annual timescales. Bias correction in mean, SD, and LAG1 correlation applied at daily, monthly, seasonal, annual, and 3-yr timescales. Five iterations of nesting bias correction in mean, SD, and LAG1 correlation applied at daily, monthly, seasonal, annual, and 3-yr timescales.
Figure 2. Average absolute percentage biases (AAPB) in mean and standard deviation and Average absolute bias (AAB) in LAG1 correlation at daily, monthly, seasonal, and annual timescales calculated across all variables for different bias correction options. 5 of 8
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variables exhibit biases in comparison to reanalysis data at all timescales. More specifically, significant biases are observed at higher timescales and in the LAG1 autocorrelation statistic. Commonly used bias-correction procedures correct the GCM time series only for mean and standard deviation at a prespecified time-scale (i.e., daily, monthly, seasonal, or annual), an approach that would not impart the lower-frequency attributes needed in the series presented here. [24] In order to understand the implications of the biascorrection procedure applied at a single or a few predefined timescales, we apply nested bias correction (in cross validation) by varying the combinations of timescale being nested. Given the five bias correction timescales, nesting is performed over and including the case of no bias correction and order 1 and 2 moments and persistence biases, a total of 10 bias correction alternatives (Table 1) are assessed. To facilitate the presentation of results, average absolute percentage bias (AAPB) for mean and standard deviation and average absolute bias (AAB) for LAG1 autocorrelation is evaluated across all of the variables and plotted for all of the combinations considered (Figure 2). Some interesting conclusions can be drawn from Figure 2. By correcting for mean at the daily timescale, biases in means at all timescales are removed. However, biases in standard deviation and LAG1 autocorrelation remains largely unaffected at monthly and higher timescales. As expected, the nesting bias-correction procedure reduces the biases in standard deviation and LAG1 autocorrelation at higher timescales. However, nesting at daily, monthly, and annual timescales is not able to reduce significantly the biases in seasonal standard deviation and LAG1 autocorrelation statistics. Also, correcting for biases at seasonal and annual timescales, to some extent, increases the biases in standard deviation and LAG1 correlation at daily and monthly timescales. The inclusion of a bias correction at 3-yr aggregated time series introduces biases in all statistics at daily, monthly seasonal, and annual timescales. Perhaps, the limited length of data (only 13 data points for 43 yr of record used in cross-validation) used may be the cause behind the biases in the statistics at lower timescales. Based on this result and keeping in mind the fact that bias correction at higher time-scales introduces some biases at lower timescales, we propose a recursive procedure to correct for the biases in the nesting procedure across all timescales. The recursive nesting procedure operates as follows: [25] 1. Similar to the NBC procedure, calculate the reanalysis and GCM series daily mean, standard deviation, and LAG1 autocorrelation using the observations falling within a moving window of 31 d centered on the current day of interest. [26] 2. Correct the daily GCM series for biases in mean, standard deviation, and LAG1 correlation by following a three-step procedure of NBC. [27] 3. Aggregate the daily series to a higher-selected bias correction timescale and follow the same steps for other timescales as well. [28] 4. Go back to step 2 and repeat steps 2 and 3 ITR times, where ITR represents the number of required iterations. [29] Figure 3 illustrates the results obtained using the above described recursive NBC procedure. Shown in the figure are the variations of AAPB and AAB in mean, standard deviation, and LAG1 correlation statistics at daily, monthly, and annual timescales as a function of the number of iterations, again in a cross-validation setting. It is important to
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Figure 3. Average absolute percentage biases (AAPB) in mean and standard deviation and average absolute bias (AAB) in LAG1 correlation at daily, monthly, and annual timescales calculated across all variables. note that a distinct improvement in the representation of variability is there even with a single iteration being performed. The maximum improvement in the daily series is achieved after three iterations. For both monthly and annual series, some improvement is possible with more iterations but this improvement does not translate to the results at the daily timescale. Considering three to five iterations across scales is suggested as a viable configuration to adopt when applying the recursive NBC procedure to remove low-frequency biases in GCM-simulated variables for the data and region analyzed. [30] Finally, Figure 4 presents the scatterplots of means, standard deviation, and LAG1 correlation statistics of reanalysis and bias-corrected series at multiple timescales
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Figure 4. The same as Figure 1 but for bias-corrected series in cross validation (using nesting at all timescales and five iterations).
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(similar to Figure 1) obtained using the recursive nesting bias-correction procedure with five iterations. As can be seen, the recursive NBC procedure provides substantial improvements in these statistics at all of the timescales considered.
5.
Conclusions
[31] In this paper we have demonstrated the effectiveness and utility of the nesting bias-correction procedure to correct for variability and persistence in GCM-derived variables on multiple timescales. Properly bias-corrected GCM outputs, when used to drive the downscaling models, can substantially improve the downscaled results and reduce ensuing prediction errors. A range of nested and non-nested bias correction formulations were assessed. Results of the study suggested that a nested bias-correction procedure with iterative steps and applications across multiple timescales (termed here as the recursive NBC procedure) can significantly reduce the biases in mean, variability, and persistence-related attributes in GCM simulations.
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