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Environmental Modelling & Software journal homepage: www.elsevier.com/locate/envsoft

An R package for assessment of statistical downscaling methods for hydrological climate change impact studies Martin Hanel*, Roman Ko zín, Martin Hermanovský, Radek Roub 1176, Prague 6, Czech Republic Czech University of Life Sciences, Kamýcka

a r t i c l e i n f o

a b s t r a c t

Article history: Received 7 September 2016 Received in revised form 23 March 2017 Accepted 28 March 2017

Due to inherent bias the climate model simulated precipitation and temperature cannot be used to drive a hydrological model without pre-processing e statistical downscaling. This often consists of reducing the bias in the climate model simulations (bias correction) and/or transformation of the observed data in order to match the projected changes (delta change). The validation of the statistical downscaling methods is typically limited to the scale for which the transformation was calibrated and the driving variables (precipitation and temperature) of the hydrological model. The paper introduces an R package ”musica” which provides ready to use tools for routine validation of statistical downscaling methods at multiple time scales as well as several advanced methods for statistical downscaling. The musica package is used to validate simulated runoff. It is shown that using conventional methods for downscaling of precipitation and temperature often leads to substantial biases in simulated runoff at all time scales. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Bias correction Time scale Runoff Multiscale bias correction Multiscale delta change

Software availability Name of Software: musica Developer: Martin Hanel, Faculty of Environmental Sciences, Czech University of Life Sciences, Prague, Czech Republic Contact: [email protected] Year ﬁrst available: 2016 (current version 0.1.3) Program language: R Availability: Freely available through CRAN (cran.r-project.org), development version at github.com/hanel/musica

1. Introduction Due to inherent bias and relatively coarse resolution, the climate model simulated precipitation and temperature cannot be used to drive a hydrological model without pre-processing e statistical downscaling. This often consists of reducing the bias in climate model simulations (bias correction) and/or transformation of the observed data in order to match the projected changes (change factor methods). Numerous methods have therefore been recently proposed. These methods range from simple scaling, considering

* Corresponding author. E-mail address: [email protected] (M. Hanel). http://dx.doi.org/10.1016/j.envsoft.2017.03.036 1364-8152/© 2017 Elsevier Ltd. All rights reserved.

mean, through nonlinear transformations affecting mean and variability, to more sophisticated methods transforming the whole distribution of variables of interest, spatial or inter-variable dependence, etc. An overview of bias correction methods has been provided by Maraun et al. (2010), various aspects of change factor methods are discussed e.g. by Anandhi et al. (2011). Despite their wide use in climate change impact assessment, some serious concerns about common approaches as well as their fundamental assumptions have been raised in the literature in recent years. These relate especially to stationarity of bias (Chen et al., 2015), potential alteration of climate change signal (Hagemann et al., 2011; Muerth et al., 2013) as well as inter-variable (Ehret et al., 2012; Teng et al., 2015) and spatial (Hnilica et al., 2016) dependence. Simulations of regional and global climate models are usually provided at daily time step, and this is also the temporal scale at which the simulated variables are downscaled and statistical downscaling methods are evaluated, although many applications (e.g. reservoir storage-yield assessment or drought analysis) often consider monthly data (see e.g. Hanel et al., 2013; Turner and Galelli, 2016) or integrate data on different scales. Example for the latter is the river basin management under climate change conditions with typical scales ranging from daily (e.g. weather and climate) and monthly (water demand and use, reservoir management) to annual (land-use, crop choice) or longer (Van Delden et al., 2007; Efstratiadis et al., 2014).

M. Hanel et al. / Environmental Modelling & Software 95 (2017) 22e28

The state-of-the-art bias correction methods are able to transform the simulated data such that the corrected distribution perfectly matches that of observed data and also so that the dependence structure between variables can be reasonably considered. The correspondence of the distribution of corrected and observed variables at daily time scale does not, however, imply correspondence at longer (or shorter) temporal scales. This is due to the temporal structure of the simulated variable, which is typically unaffected by the correction. This has already been recognized by, for example, Haerter et al. (2011), Johnson and Sharma (2012), Ehret et al. (2012) or Mehrotra and Sharma (2016). Similarly, daily change factors applied to observed data do not necessarily result in the same monthly, seasonal and annual changes as expected from the climate model simulation used for the derivation of (daily) change factors. In addition, the evaluation of bias correction methods is often limited to those variables simulated by the climate model (e.g., precipitation and temperature) and does not consider the output variables of an impact model. In combination with uncorrected bias at multi-day and longer time scales, this may have serious consequences for assessing long-term hydrological balance of a catchment as well as of extreme hydrological events, because catchment dynamics is to a large extent related to temporal distribution of rainfall. Indeed Teng et al. (2015) demonstrated that some widely used bias correction methods cannot overcome the limitations of the climate models in simulating all important precipitation characteristics that inﬂuence runoff, in particular, daily precipitation sequence. While the scale dependence and alteration of bias through the impact model has already been described in the literature (see references above) together with complex statistical downscaling methods, the vast majority of climate change impact assessment studies still relies on standard approaches, without recognizing potential magnitude of the introduced errors. We argue, that this is, at least partly, due to the lack of ready-to-use tools for multiscale assessment of statistical downscaling methods, which is in general simple, yet might be technically unappealing. This was a motivation for development of the R package ”musica”, which is introduced in the present paper (Sect. 2). The main purpose of the package is to make the performance assessment of the statistical downscaling methods at multiple time scales comfortable in order to become standard part of climate change impact assessment. The focus is not only on driving variables of the impact model (typically precipitation and temperature) but also on the output of the impact model (e.g. runoff). For demonstration purposes and assessment of uncertainty due to statistical downscaling methods the package also includes several methods allowing for standard and multiscale bias correction and delta change transformation implementing the time-scale nesting procedure suggested by Pegram et al. (2009). Currently we do not implement any method for correction of the dependence between variables since the published methods, though in principle universal, often lead to spurious results when applied to different variables than those that were used for the development of the method. For instance the application of the method proposed by Hnilica et al. (2016) or Efstratiadis et al. (2014) for multi-site precipitation leads to negative precipitation amounts when used for correction of the dependence between precipitation and temperature at single site. The capabilities of the package are demonstrated (Sect. 3) considering assessment of bias and changes in runoff simulated with a hydrological model driven by statistically downscaled precipitation for a small catchment in the Czech Republic (Sect. 2.2). Paper is closed by concluding remarks (Sect. 4).

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2. The musica package 2.1. Summary of climate change scenario development strategies Two popular strategies for derivation of input data (corresponding to future climate) into the hydrological model are bias correction and delta change method (see Fig. 1). Bias correction derives transfer functions for transformation of climate model simulated data in the control period such that statistical properties of the corrected simulation resambles observed data as much as possible. These calibrated transfer functions are then used to correct climate model simulation for the scenario period. In the delta change method, the changes in statistical properties between climate model simulation for the control and scenario periods are used to derive change factors. These change factors are then used for transformation of the observed data such that the changes between transformed and original observed data are close to those from the climate model simulation. The transformed data (bias corrected or delta change) are then used to drive hydrological model in order to obtain runoff corresponding to the scenario period. Skill of the bias correction and the projected changes can be then assessed with respect to various statistical properties and time scales. The musica package includes functions supporting some of these steps, in particular (1) easy aggregation of multivariate time series into custom time scales, (2) comparison of statistical summaries between different data sets at multiple time scales (e.g., observed and bias-corrected data), (3) comparison of relationships between variables across different data sets at multiple time scales (e.g., correlation of precipitation and temperature in control and scenario simulation), and (4) transformation of time series at custom time scales implementing the time-scale nesting procedure suggested by Pegram et al. (2009). The overview of musica main functionalities is given in Table 1. Detailed demonstration of the capabilities of the package is provided in package vignette (https://cran.r-project.org/web/ packages/musica/vignettes/using_musica.html). 2.2. Data The functionality of the package is demonstrated through assessment of climate change impact on runoff at the Cucice basin (861 km2, mean annual precipitation 593 mm, mean temperature 7.23 C and mean annual runoff 129 mm) in the Czech Republic. Basin-average precipitation (PR) and temperature (TAS) for the period 1970e1999 interpolated from the station data and from a simulation of regional climate model (RCM) for the control (1970e1999) and scenario (2070e2099) periods were considered. The basin average PR and TAS were interpolated from the station data (for observation) or calculated as a weighted mean from grid boxes intersecting the basin, with weights proportional to the area of the intersection (for simulated data). The RCM simulation was conducted with the CLM model within the CORDEX project and was driven by CNRM-CM5 global climate model under RCP4.5. In addition, hydrological model Bilan calibrated (and validated) within previous studies (Hanel et al., 2013) was used to simulate runoff. Please note, that all runoff data assessed in the paper are simulated by the Bilan model. The case study presents daily data, however, the musica package accepts also monthly (in general any time scale) data. 2.3. Assessment of bias and changes Observed precipitation and temperature at 15-day (D15), monthly (M1), annual (Y1), and 5-year (Y5) time scales are given in

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Fig. 1. Schematic illustration showing the processing steps between input data (solid bold rectangles) and the impact (hydrological) model output (rounded shapes). Transfer functions based on input data (considering bias correction or delta change - solid gray rectangles) are derived in order to obtain transformed series (bias-corrected or transfored by change factors - dashed rectangles), which are then used to drive the hydrological model. Functions for time series transformation and assessment of residual biases and changes, which are provided by the musica package are given in parentheses.

Table 1 Overview of musica main functionalities. decomp compare vcompare msTrans_abs msTrans_dif

The original time series in daily (or monthly) time step is decomposed into series of averages over speciﬁed periods. Evaluates distance (difference or ratio) between statistical characteristics of speciﬁed data sets at custom time scales. Assess the relations between two decomposed variables. Applies standard quantile mapping at custom time scales. Transforms observed data such that the changes in summary statistics of variables at custom time scales are similar to those obtained from climate model simulation.

Fig. 2 using the decomp function from the musica package. Dn further denotes the n-day average, Mn the n-month average, Yn the n-year average and G1 the overall mean. In decomp nonoverlapping periods are considered. The monthly scales are based on calendar months and daily scales are calculated continuously throughout the year. Season or month is assigned to n-day periods based on centre of the window. Grouping of months into seasons can be controlled by setting the starting month of the year in the decomp function. Statistical summaries of the distribution of each variable at each time scale can be examined in order to compare different data sources (e.g., control simulation to observations) or different time periods (e.g., scenario to control period). It is rather common (though often overlooked) that the biases and changes differ substantially between different time scales. This is illustrated in Figs. 3 and 4 presenting relative (PR) and absolute (TAS) bias in mean observed and climate model simulated data for the control period (Fig. 3) and changes in standard deviation between the control and scenario periods (Fig. 4) for the CLM simulation (function compare is used). Development version of the musica package includes also dcompare function allowing for comparison of the whole distribution functions. Finally, the relationships between variables (e.g., correlation) are also typically varying with temporal aggregation and may differ considerably between observation and climate model simulation (Fig. 5, using vcompare function). The musica

package allows for arbitrary summary function (simply passed as an argument to compare or vcompare) for comparing the variables. 2.4. Multiscale transformations 2.4.1. Bias correction While the vast majority of bias correction methods considers daily data, other temporal scales should be taken into account also, as is clear from Figs. 3e4. Therefore musica package provides a ﬂexible interface for application of bias correction at custom time scale(s), based on the nesting procedure suggested e.g. by Haerter et al. (2011) or Pegram et al. (2009). Suppose we like to correct variable X at a number of time scales s ¼ s½0; s½1; s½2; …, with s½0 the original scale (typically D1 or M1). The time series Xs½i (i.e. the variable X aggregated to a scale s½i) is corrected by standard quantile mapping independently for each C . The quantile time scale, yielding the corrected time series Xs½i mapping implemented in the qmap package (Gudmundsson et al., 2012) is applied. Except for the original scale, the values of the time series at scale s½i can be also obtained by aggregation of the corrected variables A ¼ AðX C from the closest shorter time scale, i.e. Xs½i s½i1 Þ, with A the aggregation from s½i 1 to s½i. Each item of the time series at original scale Xs½0 can be speciﬁed by temporal index (e.g. speciﬁc

M. Hanel et al. / Environmental Modelling & Software 95 (2017) 22e28

M C Xs½0 ðt0 Þ ¼ Xs½0 ðt0 Þ

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C ðt Þ Y Xs½i i i>0

A ðt Þ Xs½i i

(1)

for precipitation and M C Xs½0 ðt0 Þ ¼ Xs½0 ðt0 Þ þ

C ðt Þ X Xs½i i i>0

Fig. 2. Basin-average precipitation and temperature at 5 year (Y5), 1 year (Y1), 1 month (M1), and 15-day (D15) time scales. The overall mean is indicated by a dotted line.

A ðt Þ Xs½i i

(2)

for temperature. Since each value at the original scale s½0 is corrected with respect to multiple scales, the multiscale-corrected series does not result in perfect match with observed data. To minimize the differences accross all considered time scales, the procedure has to be repeated several times (i.e. the multi-scale corrected series is again corrected at each time scale independently, then aggregated and passed to Eq. (1) or (2)). The number of iterations required is usually small (3e5), but depends on the number of considered time scales and structure of biases. The procedure is further referred to as multiscale bias correction. The same strategy has been adopted also within more complex methods (e.g. Johnson and Sharma, 2012; Mehrotra and Sharma, 2016). Let PRM , TASM be the ﬁnal corrected precipitation and temperature, respectively, and RM M the runoff simulated with hydrological model driven by PRM and TASM . The residual bias in characteristics g (e.g., mean) at temporal scale s½i with respect to the observed data (PRO , TASO , RM O ) is deﬁned as O M O gðTASM s½i Þ gðTASs½i Þ for temperature, gðPRs½i Þ=gðPRs½i Þ 1 for preM Þ=gðRM O Þ 1 for runoff. cipitation and gðRMs½i s½i

Fig. 3. Bias in mean basin-average precipitation and temperature at various time scales between observed and climate model simulated data for the control period. For sub-seasonal time scales, the biases are averaged over seasons (December-JanuaryFebruary - DJF, March-April-May - MAM, June-July-August - JJA, September-OctoberNovember - SON).

Fig. 4. Changes in standard deviation of basin-average precipitation and temperature between the control and scenario periods in the climate model simulation at various time scales. For sub-seasonal time scales, the changes are averaged over seasons (December-January-February - DJF, March-April-May - MAM, June-July-August - JJA, September-October-November - SON).

day) ts½0 but also belongs to some period ts½1 ; ts½2 ; … of the longer scales (e.g. speciﬁc month, season, year). For instance, for the second day of the scenario period and s ¼ ðD1; M1; Y1Þ we get tD1 ¼ 2070-01-02, tM1 ¼ 2070-01 and tY1 ¼ 2070. Let ti ¼ ts½i ; ts½iþ1 ; … denote the temporal index at scale s½i. The M ðt Þ is ﬁnally obtained as multiscale-corrected value Xs½0 0

2.4.2. Delta method Because the distribution of simulated runoff based on biascorrected precipitation and temperature possibly remains biased, as shown for example by Teng et al. (2015) and as is further demonstrated also in Section 3.1, the musica package provides tools for transformation of observed data (the delta change approach). This strategy uses observed data for the control period and transformed observed data for the scenario period, i.e. no biases originating from climate model simulation are introduced into the runoff series in the control period. Similarly as for the bias-correction, musica allows for transformation of observed data considering custom time scales. Let H and F be the control (historical) and scenario (future) simulation, respectively. The method consists in ﬁnding a change factor Cs½i for time series at time scale s½i as

Cs½i

g Fs½i ¼ g Hs½i

(3)

for precipitation and

Cs½i ¼ g Fs½i g Hs½i

(4)

for temperature, with g being a function providing statistical summary of time series, most often mean or, for example, empirical cumulative distribution function (in the latter case Cs½i is a vector of change factors corresponding to the considered quantiles of the cumulative distribution function). The transformation is then C as applied to observed data yielding the transformed value Xs½i

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M. Hanel et al. / Environmental Modelling & Software 95 (2017) 22e28

Fig. 5. Pearson correlation between precipitation and temperature in observed data (red) as well as control (blue) and scenario (green) RCM simulation.(For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

C Xs½i ¼ Os½i Cs½i

(5)

for precipitation and C Xs½i ¼ Os½i þ Cs½i

(6)

for temperature, with Os½i the observed value at scale s½i. In most applications the transformation is determined and applied for each month separately. To assure the consistence between temporal scales a nesting procedure similar to that described in Sect. 2.4.1 is used. Except for A ) at scale s½i the original scale the aggregated change factors (Cs½i can be obtained by comparing aggregated data from scale s½i 1 to A observations, i.e. by replacing Fs½i and Hs½i in Eqs. (3) and (4) by Xs½i and Os½i , respectively. Finally, the multiscale-transformed time seM ðt Þ for temporal index t is obtained from ries Xs½i 0 0

M C Xs½0 ðt0 Þ ¼ Xs½0 ðt0 Þ

Y Cs½i ðti Þ i>0

A ðt Þ Cs½i i

(7)

for precipitation and M C Xs½0 ðt0 Þ ¼ Xs½0 ðt0 Þ þ

X Cs½i ðti Þ i>0

A ðt Þ Cs½i i

(8)

for temperature. As in the case of multiscale bias correction, the procedure has to be repeated several times. The musica package currently implements number of choices for g, such as mean, empirical distribution function and linear and loess approximation of the empirical distribution function.

3.1. Bias correction and assessment Climate model simulated precipitation and temperature was bias corrected by two methods: standard quantile mapping calibrated at daily time scale and multiscale bias correction calibrated at daily (D1), monthly (M1), seasonal (M3) and annual (Y1) time scale. Time series of bias corrected precipitation and temperature were used as inputs to the Bilan hydrological model. Various statistical summaries were evaluated at D1, M1, M3, Y1 and G1 time scales and compared to observed data. Residual bias was then assessed. Fig. 6 shows residual bias in median precipitation, temperature, and simulated runoff. Bias for median (wet-day) precipitation in the uncorrected (original) RCM simulation (not shown) at D1 is less than 0.5 (i.e., 50%, 0.8 mm/day) in absolute values. Except for summer (JJA), the biases at longer temporal scales are larger (e.g., more than 100% in winter at M1, i.e. > 30 mm/month). The standard bias correction removes the bias at D1, while the residual bias at M1 reaches 20% ( 6 mm/month), which can still be considered marginal with respect to natural variability of monthly precipitation and measurement errors. The multiscale approach removes the bias nearly completely across the whole range of temporal scales. Similarly, huge bias in temperature (2.5 to 1.5 C) in the original simulation (not shown) is completely eliminated at D1 in the case of standard bias correction. The residual bias for M1 is in absolute value less than 0.5 C except for February with large negative bias (1 C). The multiscale approach reduces the residual bias to below 0.1 C for all scales. Despite the elimination of bias in precipitation and temperature at daily time scale for standard and multiscale bias correction

3. Applications In this section, we brieﬂy demonstrate use of musica package for assessing the residual bias in precipitation and temperature together with the residual bias in runoff simulated by hydrological model Bilan driven by corrected precipitation and temperature. Please note, that standard assessment of bias correction methods should consider independent validation period in which the corrected data are compared to observations in order to assess robustness of the correction method against non-stationarity of bias. Our objective here is, however, to demonstrate that considerable biases persist even in the calibration period (at time scales that were not used for calibration or in the output of an impact model). Therefore the validation for independent period is not presented here. Further, application of the multiscale delta change method is presented together with assessment of changes resulting from the standard and multiscale bias correction methods.

Fig. 6. Residual bias in median (wet-day) precipitation (left), temperature (middle), and simulated runoff (right) after standard (top) and after multiscale (bottom) bias correction for different temporal scales with respect to observed data.

M. Hanel et al. / Environmental Modelling & Software 95 (2017) 22e28

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Fig. 7. Changes in mean (wet-day) precipitation (top), temperature (middle), and simulated runoff (bottom) in original RCM simulation (left) after standard (STANDARD BC) and multiscale (MULTISCALE BC) bias correction and delta change considering mean change for each month (MEAN DC) and changes in the distribution function at multiple time scales (LOESS DC). For shorter than seasonal scales the lines correspond to averages over months forming the seasons with line color indicating the season (December-January-February DJF, March-April-May - MAM, June-July-August - JJA, September-October-November - SON).

methods, the corresponding residual biases in simulated runoff for D1 are as much as 50% (standard correction) and 20% (multiscale), respectively. For longer time scales, the residual bias is considerably larger - in general up to 100% (standard correction) and 40% (multiscale correction), respectively. Looking at absolute values (not shown) the residual bias for standard correction reaches 6.5 mm/month, 11 mm/season and 29 mm/year while the multiscale correction results in 3 mm/month, 11 mm/season and 15 mm/ year for M1, M3 and Y1, respectively. Please note that the residual biases are even larger for characteristics of extremes. For example, the residual bias in maximum runoff (not shown) at M1 is as much as 300% using standard bias correction and as much as 50% for the multiscale approach except in summer, when there is greater than 150% overestimation. The correction methods assessed in this section do not address the correction of the dependence structure between precipitation and temperature. Quantile mapping was shown to be close to linear transformation (Hnilica et al., 2016) therefore it is not likely that the correction would alter considerably the simulated correlation structure. Though expected to be relatively small (at least at short time scales - cf. Fig. 5 showing only a weak correlation between precipitation and temperature at shorter than annual scales) the contribution of the dependence structure to the residual bias in runoff is uncertain and its quantiﬁcation is beyond the scope of this paper. 3.2. Projected changes Standard and multiscale bias-corrected precipitation and temperature together with precipitation and temperature transformed by two variants of the delta method were used to drive hydrological model Bilan for the control and scenario periods. The ﬁrst delta change method (denoted MEAN DC further) considered constant change for each month, while the second method transformed the whole empirical distribution function (after smoothing with nonparametric regression, further denoted LOESS DC) at daily (D1), monthly (M1), seasonal (M3) and annual (Y1) time scales. The changes in precipitation, temperature and runoff between scenario and control periods at D1, M1, M3, Y1 and G1 time scales are given in Fig. 7. For the bias correction methods the changes correspond to the ratio/difference of the corrected variables between the scenario and control periods, for the delta methods the changes are calculated as the ratio/difference between the delta change scenario and observed data.

Changes in precipitation often differ by 10e20% between different methods. The effect of bias correction on the projected changes is clear. That is especially true in the case of multiscale bias correction, which shows a different pattern of changes than does any other method. The differences are less clear in the case of temperature, when the projected changes usually differ by as much as 0.5 C between the methods. In the case of runoff, however, the differences in seasonal distribution and magnitude of changes are strongly enhanced. 4. Summary Standard methods for developing climate change scenarios (bias correction and delta change methods) often ignore the properties of the transformed variables at longer than daily temporal scales, as well as the characteristics of the output of the impact (e.g., hydrological) model. In the present paper, we introduce the R package ’musica’, which provides tools for assessment of statistical downscaling methods at multiple temporal scales and also methods for multiscale bias correction and delta change transformation. We demonstrate that the correction of a RCM simulation at a particular time scale does not imply reasonable values at other scales and that the transformation through a hydrological model leads to biases even for those scales at which the inputs were corrected. Similarly, the estimated changes in runoff depend strongly on the method used in deriving the scenarios. We therefore emphasize the need to check the properties of climate change scenarios at multiple time scales as well as to examine their performance in the impact model, which can be done conveniently using the developed musica package. Acknowledgements This study was supported by the Czech Science Foundation (grant no. 16-16549S). We thank two anonymous reviewers and the editor for constructive comments. References Anandhi, A., Frei, A., Pierson, D.C., Schneiderman, E.M., Zion, M.S., Lounsbury, D., Matonse, A.H., 2011. Examination of change factor methodologies for climate change impact assessment. Water Resour. Res. 47, W03501. Chen, J., Brissette, F.P., Lucas-Picher, P., 2015. Assessing the limits of bias-correcting climate model outputs for climate change impact studies. J. Geophys. Res. Atmos. 120, 1123e1136. Efstratiadis, A., Dialynas, Y.G., Kozanis, S., Koutsoyiannis, D., 2014. A multivariate

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