Advances in Water Resources 109 (2017) 253–266 Contents lists available at ScienceDirect Advances in Water Resources journal homepage: www.elsevier.com/locate/advwatres Decomposing the satellite precipitation error propagation through the rainfall-runoff processes Yiwen Mei a, Emmanouil N. Anagnostou a,∗, Xinyi Shen a, Efthymios I. Nikolopoulos a,b a b Civil and Environmental Engineering, University of Connecticut, Storrs, CT, USA Innovation Technologies Center, ITC, Athens, Greece a r t i c l e i n f o Article history: Received 9 February 2017 Revised 25 July 2017 Accepted 13 September 2017 Available online 21 September 2017 Keyword: Satellite precipitation Error propagation Runoff generation Runoff routing Hydrological model a b s t r a c t This study uses an analytical hydrological framework to investigate the error propagation from satellite precipitation products to hydrological simulations. Specifically, the analytical formulation of the framework allows linking the error in hydrograph properties (i.e., cumulative volume, centroid, and dispersion) to the space-time characteristics of error in satellite-precipitation, runoff generation, and routing. Main finding from this study are that (i) the error in spatial and temporal covariance between rainfall and runoff generation is not contributing significantly to the error in cumulative volume of flood events; (ii) errors in runoff generation and routing time are of equal importance in terms of the overall error in the arrival of flood event centroid; and (iii) errors in the variability of runoff generation time is the main contributor to the error in dispersion of flood event hydrograph. Furthermore, sensitivity tests show that errors in hydrograph properties are strongly correlated with errors in the space-time characteristics of precipitation, runoff generation and routing parameters estimated by the analytical framework. © 2017 Published by Elsevier Ltd. 1. Introduction The evaluation of satellite precipitation error propagation in hydrological simulation provides information on how the precipitation error characteristics interact with the rainfall-runoff transformation process resulting in error of flow simulation. Such information can help facilitate a number of flood risk and water management applications. Many past studies have focused on the assessment of the error propagation of satellite precipitation products through the rainfall-runoff process based on statistical error analyses. A common finding has been that the properties of error propagation (magnification vs. dampening and linear vs. nonlinear) vary with respect to different factors. For instance, wet initial soil moisture conditions are associated with a more linear error transformation from precipitation to runoff (Mei et al., 2016a; Shah and Mishra, 2016; Nikolopoulos et al., 2011); larger basins are often observed to be more tolerant of random error in precipitation estimation (Mei et al., 2016a; Vergara et al., 2013). Other factors, such as regional climate pattern, land cover types, seasonality, and the choice of hydrological model or modeling complexity, also affect the error propagation properties (Kim et al., 2016; Gebregiorgis and Hossain, 2013; Xue et al., 2013; ∗ Corresponding author. E-mail addresses: manos@uconn.edu, manos@engr.uconn.edu (E.N. Anagnostou). https://doi.org/10.1016/j.advwatres.2017.09.012 0309-1708/© 2017 Published by Elsevier Ltd. Gebregiorgis et al., 2012; Beighley et al., 2011). However, little knowledge exists as to how the spatial and temporal factors of the rainfall-runoff generation and routing processes interact with the satellite precipitation error to manifest in runoff simulation uncertainty. Researchers have recently developed an analytical framework to explore the dependence of the catchment flood response characteristics (that is, cumulative volume, centroid, and dispersion) on the spatial and temporal variability of the rainfall patterns, runoff generation and routing across hillslopes and channel network (Mei et al., 2017; Viglione et al., 2010a; Woods and Sivapalan, 1999). The analytical framework can be used as a diagnostic tool to assess the relative importance of these space and time terms of catchment flood response on the properties of the flood hydrograph. Past work on this topic has concluded that the spacetime information on runoff generation and routing is important for short-duration events occurring over small mountainous basins (Viglione et al., 2010b). The framework has also been utilized to understand the effect of neglecting the spatial information on rainfall in runoff generation. Zoccatelli et al. (2011) proposed the concept of spatial moment of catchment rainfall, built upon the framework. They assessed the difference in timing of hydrographs if the spatial information of rainfall is neglected (lumped rain) and found that the timing of event-based hydrographs is sensitive to the location of the rainfall mass center over the catchment. Following a similar 254 Y. Mei et al. / Advances in Water Resources 109 (2017) 253–266 methodology, others have investigated the effects of the secondorder moment of catchment rainfall and catchment-scale storm velocity on runoff generation (Mei et al., 2014; Nikolopoulos et al., 2014). Nikolopoulos et al. (2014) compared the variability of flood event hydrographs generated from rainfall fields without a spatial pattern and those with a fixed spatial pattern to hydrographs derived from the original rainfall field. Their results highlight the important role of spatial information in shaping the variability of flood event hydrographs. Among these studies, the analytical framework has been used as a tool to formulate the differences in the space-time characteristics of rainfall, with the corresponding flow simulations. In this study, we demonstrate the use of the framework for linking errors in satellite precipitation with errors in flow simulations driven by satellite precipitation products. This allows us to decompose the precipitation error propagation into the runoff generation and routing processes that manifest in the overall flood hydrograph error. This offers information on the relative importance of those error terms with respect to their contribution to error in satellitedriven flow simulations, providing an understanding of the level of space and time complexity required to model the error propagation process at the scale of catchment and event. We believe this approach is beneficial to algorithm developers and hydrologic modelers who use satellite precipitation as forcing dataset. Furthermore, it can guide the selection of an optimal spatiotemporal resolution (a tradeoff between the space and time variability of observations) for hydrological applications of satellite precipitation products, which will help end users to choose the optimal space and time complexity of satellite-driven hydrological modeling for their applications. The paper is structured as follows: in Section 2, we formulate the analytical satellite precipitation error propagation framework and its connection to the hydrological analytical framework. Section 3 describes the datasets and the distributed hydrological model used for the implementation and parameterization of the analytical error modeling framework. Section 4 presents the results regarding the analytical framework predictions of volumetric error, error in timing and shape of the hydrograph. Last (Section 5), we present our conclusions and discuss limitations and future directions of this study. 2. Error modeling framework 2.1. A framework for catchment flood response Viglione et al. (2010a) (V2010 hereafter) proposed a hydrologic analytical framework generalized from the framework studied in Woods and Sivapalan (1999). Recently, Mei et al. (2017) (M2017 hereafter) further developed V2010 framework to include multiple flow components. This study follows the main characteristics of the V2010 framework for runoff generation while the runoff routing process are based on M2017. V2010 conceptualize the catchment flood response process in three stages. In stage one, rain falls on the catchment surface and is converted to rainfall excess. A parsimonious runoff generation function (that is, a runoff coefficient) is adopted to conceptualize the complex space-time interactions of precipitation in runoff generation. The rainfall excess generated from the first stage is subjected to two subsequent routing stages, namely hillslope and channel routing, referring to the processes by which the rainfall excess is routed from the point it was generated to the entrance of the channel network and then to the catchment outlet. The hillslope and channel routing stage are often lumped as one holistic runoff routing stage based on different definitions of the routing parameters (Mei et al., 2017; Zoccatelli et al., 2011; Mei et al., 2014; Zoccatelli et al., 2010). Each of the stages is associated with a “holding time,” treated as a random variable (Rodríguez-Iturbe and Valdés, 1979). The framework focuses on three quantities: the storm-average catchment rainfall excess, the expectation and variance of catchment response time. These are used to estimate the cumulative volume, the centroid, and the dispersion of the event flow hydrograph. The storm-average catchment rainfall excess, [R]at in mm/h, is the intensity of excess rainwater generated during the storm event period over the catchment area. Following the work of V2010, [R]at is written as [R]at = [P ]at [W ]at + {[P ]a , [W ]a }t + {[P ]t , [W ]t }a        R1 R2   + {P − [P ]t , W − [W ]t }a  t   R3  (1) R4 where P, W, and R are, respectively, the space-time variable precipitation, runoff coefficient, and rainfall excess; [] and {} with subscript a and/or t stand for the expectation and covariance (variance if the variables are the same) operators applied to the dimensions of catchment area and/or storm period. The definitions of R1, R2, R3, and R4 are listed in Table 1. The storm-cumulative catchment rainfall excess (the product of storm-average catchment rainfall excess and the duration of storm) is used to estimate the cumulative volume of the event flow hydrograph (V in mm), defined as V =  (2) Q (t )dt TF where Q(t) is the event flow hydrograph and TF is the period of flow event. The catchment response time is a random variable composed of the runoff generation time and runoff routing time. Thus, the expectation of catchment response time, E(Tq ), is the summation of the expectation of holding times of these two stages, (3) E (Tq ) = E (Tr ) + E (Tn )       Stage 1 Stage 2 where Tq , Tr , and Tn are variables representing the holding times of catchment response, runoff generation and routing processes. The expectation of catchment response time is an estimator of the centroid of event flow hydrograph (C in hour). C is calculated as the instantaneous time weighted by the flow rate: C= TF t · Q (t )dt TF (4) Q (t )dt C is a measure of time from the beginning of the flow event to the mass center of the event; it can be used as a surrogate for the concept of time to peak for a mono-modal flow event. For the runoff generation stage, we follow V2010 to model the holding Tr by the instantaneous time (T), which follows a uniform distribution of the storm period (Mei et al., 2017; Viglione et al., 2010a; Woods and Sivapalan, 1999). The holding time for the runoff routing stage, Tn , is modeled by a spatially distributed network routing time (θ n ), as it is in M2017. The holding time for E(Tr ) and E(Tn ) are written as E (Tr ) = |TP | 2  + E1 E (Tn ) = [θn ]a +  E3 {T , [R]a }t (5) [R ]  at  E2 {θn , [R]t }a  [R]at  E4  (6) where TP is the storm period and |TP | is the duration of the storm period. The definition of term E1 through E4 are given in Table 1. 255 Y. Mei et al. / Advances in Water Resources 109 (2017) 253–266 Table 1 Definition of quantities in the hydrologic analytical framework. Term Meaning R1 R2 R3 R4 Product between spatiotemporal-average rainfall and runoff coefficient Temporal covariance between the catchment-average rainfall and runoff coefficient Spatial covariance between storm-average rainfall and runoff coefficient Temporal mean of spatial covariance between temporal variation of precipitation and runoff coefficient E1 E2 E(Tr ) E3 E4 E(Tn ) Half duration of the rainfall event Time distance from the event midpoint to the temporal mass center of catchment-average rainfall excess Temporal mass center of the catchment-average rainfall excess Spatial mean of the network routing time Distance from the geomorphologic center of catchment to the spatial mass center of the storm-average rainfall excess Expectation of runoff routing time v1 v2 var(Tr ) v3 v4 var(Tn ) c cov(Tr ,Tn ) Variance in time generated by a temporal invariant catchment-average rainfall excess Additional variance causes by the temporal variation in catchment-average rainfall excess Variance of time with respect to the catchment-average rainfall excess Spatial variance of the network routing time Additional variance causes by the spatial variation in storm-average rainfall excess with respect to the network routing time Variance of runoff routing time Time evolution of rainfall excess over the catchment network Covariance between the runoff generation time and the runoff routing time The variance of catchment response time, var(Tq ) in h2 , for the two-stage framework is contributed by the variance of the holding times from the stages and the covariance between the holding times of the two stages: var (Tq ) = var (Tr ) + var (Tn ) + 2cov(Tr , Tn )          Stage 1 Stage 2 (7) Stage 1 & 2 The square root of the variance of catchment response time is used to estimate the dispersion of event flow hydrograph (S in hour). S is defined as the square root of the second moment of time with respect to flow rate: S= TF (t − C )2 Q (t )dt TF Q (t )dt (8) S reflects the distribution of mass with respect to the mass center of flow. For example, a flow event with most of its mass located around the mass center (that is, a flow event with a sharp peak) takes small S; a flow event with its mass located farther away from the mass center (that is, a flow event with peaks located at its beginning and end) is characterized by large S. Following the V2010 and M2017 frameworks, the two variance terms of Eq. (7) can be written as: 2 2 var (Tr ) = |TP |    12 v1 var (Tn ) = {θn }a +    v3 T , [R]a +  t − [R]at |TP |{T , [R]a }t [R]at −  { T , [R]a t2 2 [R]at }   v2 θn2 , [R]t  a [R]at 2[θn ]a {θn , [R]t }a − − [R]at  v4 2 (9)   (10) Refer to Table 1 for the definitions of these terms. A covariance term, cov(Tr ,Tn ), also appear in Eq. (7), which reflects the movement of rainfall excess mass center over the isochrones of the network routing time during the storm period. It is thus often used to interpret the movement of storm (Zoccatelli et al., 2011; Mei et al., 2014; Nikolopoulos et al., 2014; Zoccatelli et al., 2015). The term cov(Tr ,Tn ) may be written as cov(Tr , Tn ) = {T , {θn , R}a }t  [R]at − {T , [R]a }t {θn , [R]t }a  c 2 [R]at  2.2. Error in catchment flood response A scheme for representing satellite precipitation error propagation via the hydrological analytical framework is shown in Fig. 1. The property pairs (satellite vs. reference) of catchment flood response, namely [R]at , E(Tq ), and var(Tq ), are calculated based on the analytical framework (the sign “ˆ” represents the satellite-derived properties). The discrepancies among these property pairs are defined by the differences, that is, [R]at , E(Tq ), and var(Tq ) in Fig. 1. To test the sensitivity of the analytical error modeling framework, [R]at , E(Tq ), and var(Tq ) are compared with the error in cumulative volume, centroid, and dispersion (ε V , ε C , and ε S ) of a flood event hydrograph, based on the simulations from a numerical model (described later in Section 3.1). Given the additive relationships of the terms in Eqs. (1), (3), and (7) to the properties, the error in properties of catchment flood response are decomposed into terms quantifying the different spacetime interactions among rainfall, runoff generation, and routing. The error in storm-average catchment rainfall excess, [R]at , may be written as ˆ − R1 + Rˆ2 − R2 + Rˆ3 − R3 + Rˆ4 − R4, [R]at = R            1  R 1 {θn , [R]t }2a [R]at Definitions of the term in Eq. (11) are provided in Table 1. (11) R 2 R 3 (12) R 4 where the terms with/without a “ˆ” are derived based on the satellite/reference precipitation. Four error terms exist in Eq. (12). R1 represents the difference in satellite precipitation product’s spatiotemporal-average precipitation and runoff coefficient. It is a quantification of the systematic error in runoff generation derived between the satellite and reference rainfall. R2 and R3 represent the differences in temporal and spatial covariance between the spatial- and temporal-average precipitation and runoff coefficient, respectively. R4 stands for the difference in temporal correlation between the spatial variation of precipitation and the runoff coefficient. The terms R2, R3 and R4 represent the correlation in space and time patterns between the runoff generations from different rainfall fields. Eq. (12) indicates that the error in storm-average catchment rainfall excess can be decomposed into terms representing the lumped (R1), temporal (R2), spatial (R3), and spatiotemporal (R4) information. This means that if only the space-time aggregated estimate of rainfall is available for the two rainfall fields, the total error in the storm-average catchment rainfall excess is R1. If the available information is at the level of time series/spatial maps, then the total difference is a sum of R1 and R2/R3. 256 Y. Mei et al. / Advances in Water Resources 109 (2017) 253–266 Fig. 1. Schematic of the precipitation error propagation framework. The error in expectation of catchment response time, E(Tq ), is contributed from the runoff generation stage and the runoff routing stage. Therefore, E(Tq ) is the sum of error from the two stages:     E (Tq ) = E Tr − E (Tr ) + E Tn − E (Tn )       E (Tr ) (13) E (Tn ) Given the additive relationship of E(Tr ) and E(Tn ) to E(Tq ), we can rewrite Eq. (13) as following: E (Tq ) = Eˆ2  − E2 + Eˆ 3 − E3 + Eˆ 4 − E4        E2 E3 (14) E4 Note that no E1 term appears in the equation because the satellite- and reference-suggested rainfall event pairs are always taking the same event period, and the subtraction between the two E1–the half-length of events–is always zero. E(Tq ) can be decomposed into three error terms from the equation. E2 represents the difference in temporal covariance between catchment-average rainfall excess and time. E3 represents the error in mean runoff routing time between runoff generated by the two rainfall fields. E4 is the difference in spatial covariance between storm-average rainfall excess and runoff routing time. Eq. (14) dictates that the error in expectation of catchment response time can be categorized into error related to the space-time average magnitude of routing time (E3), the temporal and spatial correlation between holding times and rainfall excess (E2 and E4). Analogously, with knowledge only of the means of the variables–the cumulative amount of runoff generated during the event period and the mean values of runoff routing time— E(Tq ) equals E3. With the temporal distribution of rainfall excess, E(Tq ) returns to the sum of E2 and E3. The available information on spatial pattern of rainfall and runoff routing times results in a total error equal to E3 plus E4. The error in variance of catchment response time, var(Tq ), comes from three aspects. They are the error in variance arising from the runoff generation stage and the runoff routing stage and the covariance between these two stages. Thus,     var (Tq ) = var Tr − var (Tr ) + var Tn − var (Tn )        var (Tr )   var (Tn )  +2 cov Tr , Tn − cov(Tr , Tn )    cov(Tr ,Tn ) (15) Each of the terms in Eq. (15) may be further expressed using, (10), and (11):   var (Tq ) = vˆ 2  − v2 + vˆ 3 − v3 + vˆ 4 − v4 +2 cˆ − c           v 2 v 3 v 4 (16) c Again, there is no v1 term because, for any event, the difference of v1 derived between the satellite and reference rainfall event pair is always zero. var(Tq ) is described by four error terms. v2 stands for the difference in additional variance caused by the temporal variation in catchment-average rainfall excess. v3 represents the difference in variance of runoff routing time. v4 is the difference in additional variance caused by the spatial variation in stormaverage rainfall excess. c is the difference in covariance between the runoff generation time and runoff routing time. It can also be interpreted as the difference in motion of runoff generations over the catchment (Zoccatelli et al., 2011; Mei et al., 2014; Nikolopoulos et al., 2014; Zoccatelli et al., 2015). Therefore, the difference in variance of catchment response time is separated into terms standing for the lumped (v3), temporal (v2), spatial (v4), and spatiotemporal (2c) information. var(Tq ) equals v3 if only the amount of cumulative rainfall excess is provided. The availability of rainfall as a time series results in a total difference of variation equal to v2 plus v3. The information on distributed spatial rainfall excess returns a total error as the sum of v3 and v4. 257 Y. Mei et al. / Advances in Water Resources 109 (2017) 253–266 The three error quantities in catchment flood response–[R]at , E(Tq ), and var(Tq )–are used to estimate the error in flood event properties (that is, ε V , ε C , and ε S ). ε V , ε C , and ε S are similarly determined by the differences between V, C, and S derived from the satellite-driven flow simulations to those from reference precipitation, defined as: εV = Vˆ − V εC = Cˆ − C εS = Sˆ − S (17) (18) Table 2 Error metrics of the hydrograph properties between radar-rainfall driven event flow simulations and observed flow events (values are in percentage). Basin MRE (%) Volume Centroid Dispersion NCRMS (%) Swift Fishing Tar Swift Fishing Tar 2.4 −2.1 −6.9 −4.0 −3.1 −10.5 −8.4 −2.3 3.2 41.6 6.6 7.8 38.3 7.8 10.7 36.6 5.9 7.6 (19) where Vˆ (V), Cˆ (C), and Sˆ (S) are derived from the satellite(reference-) driven flow simulations. The term ε V is the error in cumulative volume of flow and is estimated by |TP |[R]at . ε C is the error in flood event centroid, estimated by E(Tq ); it can be used as a surrogate for the error in time to peak for events with one peak. ε S is the error in dispersion of the flood event hydrograph; it is a measure of difference in degree of dispersion between the hydrographs. The square root of var(Tq ) is used to estimate ε S . 3. Implementation of the framework The implementation of the analytical error modeling framework requires four variables, rainfall (P), runoff coefficient (W), runoff generation time (Tr ), and runoff routing time (Tn ). To retrieve the detailed spatiotemporal variability of the five parameters over the study catchments and events, we used a numerical distributed hydrologic model. This section describes the distributed hydrologic model and the integration of a radar-based precipitation product (considered the reference) and twelve different satellite precipitation products with the model for the flow simulations at the three study catchments (Sections 3.1 and 3.2). Section 3.3 discusses the use of the distributed hydrologic model for retrieving parameters for the framework. 3.1. Hydrologic model setup The Coupled Routing and Excess Storage (CREST) distributed hydrologic model version 2.1 (Wang et al., 2011; Shen et al., 2016a) is used in this study. CREST consists of a land surface module and a runoff routing module. The land surface module takes into account four processes in the precipitation redistribution by the soilvegetation-atmosphere structure—canopy interception, infiltration, evapotranspiration (ET), and runoff generation. Canopy interception can be estimated using the leaf area index data or, more empirically, by applying a multiplier to the precipitation data. The infiltration rate is calculated based on the variable infiltration curve originally developed in the Xin’ anjiang Model (Zhao, 1992), and the actual ET (AET) is determined in terms of water and energy budget using precipitation, soil water availability, and PET. For runoff generation, the excess rainfall is separated into two components—the surface and subsurface runoff, modeled by the overland and interflow reservoirs, respectively. The runoff routing process of CREST consists of the subgrid scale and grid-to-grid routing. The subgrid scale routing is modeled by the overland and interflow reservoirs with different response times, and the grid-togrid routing is implemented by a spatially distributed concentration time. The model was set up for three nested catchments (Swift, Fishing and Tar) of the Tar River basin in North Carolina, USA, with a drainage area of 426 km2 , 1374 km2 and 2406 km2 respectively (Fig. 2). We adopted in our study the same model setup as described by Mei et al. (2017). Specifically, we generated the catchment areas from the Hydrological data and maps based on Shuttle Elevation Derivatives at multiple Scales (HydroSHEDS) (Lehner et al., 2008), and the spatiotemporal resolution of the model was 1 km and hourly. We used the Stage IV (STIV) radarbased multisensor precipitation estimates (Lin and Mitchell, 2005) and the PET data available from the North American Regional Reanalysis (NARR) (Mesinger et al., 2006) as input meteorological forcing datasets. Space-time resolution for the STIV precipitation and NARR PET products were 4 km hourly and 32 km 3-hourly, respectively. For simplicity, the vegetation interception process was conceptually computed by applying a constant multiplier to the precipitation fields; the percentage of impervious surface and the hillslope response time of the catchment were modeled by constants which we optimized through model calibration. We calibrated CREST with respect to the hourly flow rates from United States Geological Survey (USGS) station observations for the period 20 04–20 06, while 20 02–20 03 was used for model spin up. Results exhibited reasonable model performance, with hourly Nash-Sutcliff coefficient efficiency (NSCE) being 0.69, 0.62, and 0.66 for Swift, Fishing, and Tar, respectively. Mei et al. (2017) selected 180 rainfall-runoff events over the catchments from 2003–2012, based on the performance of the hydrological simulations. In this study, we utilized events of the inventory from 2003 to 2010, which is the overlapping period of the twelve satellite products. This results in 160 rainfall-runoff events where 55, 50, and 55, respectively, are from the Swift, Fishing, and Tar basins. The STIV-based CREST simulations of these events were used as reference. To evaluate the radar-based event flow simulations against the observed flow events, comparisons based on the three hydrograph properties (V, C, and S, defined in Eqs. (2), (4) & (8)) are illustrated in Fig. 3. The mean relative error (MRE) and normalized centered root mean square error (NCRMS) are also shown in Table 2. MRE and NCRMS are defined as: MRE = N NCRMS = (Xs − Xo ) N i=1 Xo i=1  1 N N i=1  (20) Xs − Xo − N1 1 N N i=1 N i=1 Xo 2 (Xs − Xo ) (21) where X (Xo ) can be one of the hydrograph properties calculated from simulated (observed) flow events and N is the total number of events for a basin. Results from Fig. 3 display a clear linear relationship of the hydrograph properties derived from the radar-based simulations to the ones obtained from the observed flow. This is especially noted for the centroid and dispersion properties. Values of the MRE and NCRMS are listed in Table 2. As shown in the table, most of the MRE and NCRMS values for the centroid and dispersion parameters are within 10%. This indicates the timing and shape of the radar-derived hydrograph simulations were in good agreement with the observed flow event hydrographs. MRE values of the volume parameter are also within 10%, but exhibit larger NCRMS (up to 40%), indicating high degree of random error in the estimations. Based on these results we consider that the radar-based CREST simulations can adequately represent the characteristics of eventhydrographs. 258 Y. Mei et al. / Advances in Water Resources 109 (2017) 253–266 Fig. 2. Geolocation and elevation map of the study area. Fig. 3. Scatter plots between observed and STIV-based flow event properties. 3.2. Satellite-based flow simulations This study evaluated twelve quasi-global satellite precipitation products. The first three were the 3B42 products from the Tropical Rainfall Measuring Mission Multi-Satellite Precipitation Analysis. The 3B42 products are available in real time (Trt ), in adjustment from the climatological correction algorithm (Tcca ), and in post processing using gauge adjustment (Tg ) (Duan et al., 1992; Huffman et al., 2007). These 3-hourly products are in 0.25° spatial resolution. Another three products evaluated were the Precipitation Estimation from Remotely Sensed Information using Artificial Neural Networks (PERSIANN) product (Huffman et al., 2010), the PERSIANN Cloud Classification System product (Sorooshian et al., 20 0 0), and the gauge-adjusted version of PERSIANN (Hong et al., 2004; Adler et al., 2003). These three products are abbreviated as Prt , Pccs , and Pg , respectively. Product Prt and its gauge-adjusted version, Pg , are in 0.25°/3-hourly resolution, while the Pccs is 0.04°/hourly. Also used were the National Oceanic and Atmospheric Administration Climate Prediction Center morphing technique (CMORPH) product available at resolutions of 0.25°/3-hourly and 0.072°/hourly, abbreviated as Crt and HCrt , respectively (Huffman et al., 2009). The gauge-corrected versions of these two products (Joyce et al., 2004), denoted as Cg and HCg , were also included. Finally, we considered the Global Satellite Mapping of Precipitation (GSMaP) version 5 Microwave-IR Combined product and the 259 Y. Mei et al. / Advances in Water Resources 109 (2017) 253–266 Motion Vector Kalman Filter GSMaP (Xie et al., 2011) and its gaugeadjusted counterpart (Ushio et al., 2009), abbreviated as Grt and Gg , respectively. The two GSMaP products are hourly and in 0.1° spatial resolution. We used the twelve satellite precipitation products to drive CREST for flow simulations of the 160 rainfall-runoff events, with the optimum parameters calibrated for the STIV precipitation. For each event, we forced the simulation of each satellite product to take the same initial condition outputted by the STIV simulation. This ensured the same initial condition for each of the event pairs simulated by the different products. 2011; Shen et al., 2016a). The two rainfall excess components entering the overland and interflow reservoirs are routed by different concentration times (i.e., routing time from current grid to next downstream grid, defined by Eq. (23) of Wang et al. (2011)). For each component, a sum of concentration times along the flow paths for the grid cells yields the runoff routing time. Runoff routing time for the overland flow and the interflow component, θ O and θ I , are defined as: 3.3. Framework variables θI ( a ) = The four analytical error framework variables were the precipitation (P), runoff coefficient (W), runoff generation time (Tr ), and runoff routing time (Tn ). It is very important to note that the analytical formulations of these variables varied based on the physical structure of the hydrologic model and the use of different model may result in different analytical formulations. The runoff generation time, Tr , is assumed equal to the rainfall event time (i.e. runoff is generated instantaneously). The precipitation variable is considered as the net amount of precipitation that reaches the catchment surface after actual evaporation loss and vegetation intercepted rainfall is subtracted (Mei et al., 2017): where l is the length of the flow path from a grid to its adjacent downstream grid and Lh /Lc represents the space of the hillslope/channel flow path from a grid-cell to the catchment outlet; s is the local slope and β is the flow speed exponent. K1 (greater than one) is the runoff velocity coefficient used to distinguish the channel routing velocity to the hillslope routing one. Eq. (24) has two terms, accounting for the time consumptions over the hillslope path and the channel path. Therefore, the use of this network routing time to represent the runoff routing stage take into consideration the delay due to both the hillslope and channel routing. The coefficient K2 in Eq. (25) is a coefficient used to distinguish the overland flow velocity to the interflow one; it takes value smaller than 1 so that θ I is always larger than θ O , implying longer routing time for the interflow component. To calculate the equivalent runoff routing time for the total rainfall excess, we followed the analytical method proposed in M2017. That is a linear combination of the runoff routing time of each component: P (a, t ) = CI P ′ (a, t ) − Ea (a, t ) (22) where P’ is the original precipitation data (i.e. either the STIV data or the satellite products); Ea is the actual evapotranspiration calculated from the hydrologic model. Scalar CI is the multiplier used to simplify the canopy interception. The precipitation, P, is converted to runoff according to the fraction of impervious area and soil moisture capacity. That is, over the impervious area, 100% of rainfall is converted to runoff; over the pervious area, runoff generation follows the tension water capacity curve assumption developed in the Xin’ anjiang Model; that is runoff production at a point, occurs only on repletion of the tension water storage (Zhao, 1992). These two portions of rainfall excess enter an overland and an interflow linear reservoir, respectively, participating in the subsequent runoff routing processes. We conveniently treated both impervious area and max water storage parameters as spatially uniform (optimized through model calibration). According to these assumptions the total runoff coefficient is formulated as, W (a, t ) = IM + (1 − IM )  SM a, t β ( ) WM (23) where SM is the space-time variant soil moisture (mm). Scalar IM and WM are the impervious ratio and maximum water storage (mm) parameter; β is the shape parameter of the variable infiltration curve. For grid cells covered by 100% of impervious surface (IM is 1), only the first term in Eq. (23) preserved and the runoff coefficient is 1. For grid cells with no impervious surface (IM is 0), the runoff coefficient is controlled by the dynamics of soil moisture and the maximum water storage capacity of the basin (i.e., the second term of Eq. (23)); and when the SM values go up to the WM value of the catchment, the gird cells are dominated by infiltration excess and the runoff coefficient reaches unity. For grid cells consisting of both pervious and impervious surface, the runoff generation can be varied between saturation excess to infiltration excess based on the soil moisture. The runoff routing stage of the analytical framework is implemented by the downstream routing of CREST. The downstream routing in CREST is based on a two-layer scheme describing overland flow and interflow from one cell to the neighboring downstream one, with consideration of open channel flow (Wang et al., θO ( a ) =  l (a ) β Lh s (a ) +  Lc l (a ) K1 s(a )β θO ( a ) K2 θ ( a ) = ψ O θO ( a ) + ψ I θI ( a ) (24) (25) (26) where parameters ψ O and ψ I are the weights of rainfall excess, defined as the cumulative amount of a rainfall excess component over the amount of total rainfall excess. That means the sum of ψ O and ψ I equals 1. Eq. (26) ensures θ does not go above/below the quickest/slowest responses. 4. Results 4.1. Error in storm-average catchment rainfall To provide an overall understanding of the discrepancies between each of the satellite precipitation products and the STIV reference rainfall data, the error in storm-average catchment rainfall, normalized to the reference-derived storm-average catchment rainfall, [P]at /[P]at , for all of the rainfall-runoff events over the three basins is plotted in Fig. 4. At basin scale, it can be seen that the patterns of [P]at /[P]at distribution are fairly similar. Generally, most of the satellite precipitation products, especially the near-real-time products, tend to underestimate the rainfall intensity of the events. The gauge-adjusted products show medians of error much closer to zero for all the basins. The higher resolution CMORPH and PERSIANN are characterized by medians slightly closer to zero than their coarser resolution counterparts. Among all the 0.25°/3-hourly products, those with medians of [P]at /[P]at closest to zero are the gauge-adjusted CMORPH and GSMaP. For the near-real-time TMPA products, the climatological correction algorithm can reduce the error in the estimation of event rainfall intensity from the real-time 3B42. 4.2. Error in storm-average catchment rainfall excess The error in storm-average catchment rainfall excess is normalized with respect to the radar rainfall-derived rainfall excess 260 Y. Mei et al. / Advances in Water Resources 109 (2017) 253–266 Fig. 4. Error in storm-average catchment rainfall for all satellite precipitation products. and the quantity [R]at /[R]at , as illustrated in Fig. 5. The distributions of [R]at /[R]at share quite similar patterns with those of [P]at /[P]at in Fig. 4. Most of the satellite product–driven flow simulations underestimate the STIV-driven ones, with negative medians of the [R]at /[R]at quantity. [R]at /[R]at derived from the gauge-adjusted products are characterized by medians closer to zero than the corresponding near-real-time ones. The medians of [R]at /[R]at derived from the gauge-adjusted 3B42 and the high resolution CMORPH are closer to zero compared to the other gauge-adjusted products. Among all the near-real-time products, the Tcca product outperforms the others with regard to the relative error in storm-average catchment rainfall excess for the three basins. The high resolution products (HCrt , HCg , and Pccs ) outperform the corresponding coarse resolution ones (Crt , Cg , and Prt ) in terms of the medians of [R]at /[R]at . Altogether, these observations demonstrate the importance of higher resolution estimates and the inclusion of gauge-adjustment to the accuracy of satellite-driven flow simulations. To further understand the distribution of total error in the different space-time catchment flood response processes, we illustrate the magnitudes of the terms in Eq. (12), using the HCg product as an example in Fig. 6 (the error distributions of stormaverage catchment rainfall excess for the other products are provided in Figure S1 of the supplemental material). Note that the terms in Eq. (12) have been normalized by [R]at , derived from the radar rainfall, to provide the relative magnitudes. Underestimation of the storm-average catchment rainfall excess prevails for most of the events, as indicated by the negative medians for nearly all of the terms. The figure also shows R1 is the main contributor to the total error in [R]at , indicated by the widest value ranges for all basins. This is expected, since R1 is the main contributor to [R]at as demonstrated by Mei et al. (2017). The value ranges of R2 also demonstrate that the error in temporal covariance is more significant than the errors in spatial covariance (R3) or the movement of rainfall (R4). Overall, the systematic component of error outweighs the error in the space-time covariance and movement in the smooth topography of the study area. A basin-wise comparison indicates the variations of the error terms are generally highest for the smallest basin (and vice versa for the largest basin). This is explained by the decrease in magnitude of the respective rainfall excess components (terms R1, R2, R3, and R4), as the increase in basin areas points to the dampening effect of basin area on the magnitudes of precipitation (Mei et al., 2017). Another observation on the basin scale is that the magnitudes of R2, R3, and R4 are closest to R1 for the smallest basin and farthest away for the largest. This implies a decrease in the relative importance of the space and time information with the increase in basin area. 4.3. Error in expectation of catchment response time The error in expectation of catchment response time (normalized by the radar-derived expected catchment response time) for the twelve satellite precipitation products is shown in Fig. 7. Overall, the distributions of E(Tq )/E(Tq ) for all the products are symmetrical with respect to zero, indicating no preferences for either advance or delay in arrival of the flow event mass center estimated using the satellite precipitation forcing. The gauge-adjusted GSMaP product is characterized by medians of E(Tq )/E(Tq ) closest to zero and the narrowest value ranges of all products. This Y. Mei et al. / Advances in Water Resources 109 (2017) 253–266 261 Fig. 5. Same as in Fig. 4, but for error in storm-average catchment rainfall excess. Fig. 6. Magnitudes of error terms in Eq. (12) derived for the HCg product. implies the Gg product provides generally the most consistent estimates on timing of the flow events. By comparing values of E(Tq )/E(Tq ) derived by the gauge-adjusted products and high resolution products, we concluded that the benefits of finer spacetime resolution and inclusion of rain gauge information are not obvious. Mei et al. (2016a) similarly conclude that the gauge-adjusted products cannot adjust properties related to the shape of the hydrograph. A focus evaluation on magnitudes of the normalized error terms in Eq. (14) for the HCg product is rendered in Fig. 8. Note that E1 (not shown) is always zero, and, thus, E(Tr ) is contributed by E2 solely. The distributions of error in the different space- time terms for the rest of the products are provided in Figure S2 of the supplemental materials, which shows the medians of the terms very close to zero, consistent with the observations from Fig. 7. A comparison of E(Tr )(term E2 in the figure) and E(Tn ) indicates that the contribution to the error in expectation of catchment response time from the runoff generation stage is larger than that from the runoff routing stages. Overall, the errors, particularly in temporal (E2), but also in spatial covariance (E4), are the main contributors to E(Tq ); the errors in mean of runoff routing time (E3) are secondary. No clear basin-scale dependencies are illustrated for the magnitude of either E(Tr ) or E(Tn ), except for E3. 262 Y. Mei et al. / Advances in Water Resources 109 (2017) 253–266 Fig. 7. Same as in Fig. 4, but for the expectation of catchment response time. Fig. 8. Same as in Fig. 6, but for Eq. (14). 4.4. Error in variance of catchment response time The normalized error in variance of catchment response time, var(Tq )/var(Tq ), is evaluated in Fig. 9. The satellite-derived var(Tq ) slightly underestimates the radar-derived in most of the cases, as indicated by the negative medians. This is particularly seen for the estimates of the Swift basin. Since var(Tq ) is a measure of dispersion of the hydrograph, this observation implies that the reference flow time series are more dispersed than the satellite-derived ones. The HCg product for Swift and Fishing basin and the Tg products for the Tar basin generally outperform the others in estimating the variance of catchment response time exhibiting medians closer to zero and narrower value ranges. In the case of the Swift catchment, nearly all the gauge-adjusted products are characterized by medians of var(Tq )/var(Tq ) closer to zero. But this is less pronounced for the other two larger catchments. This again refers to the relatively weak effects exerted by satellite precipitation gauge adjustment on the shape of the hydrograph (Mei et al., 2016a). The benefit of higher resolution is not clear in terms of estimating the shape of the hydrograph. The magnitudes of the different error terms in Eq. (16) derived from the HCg product are illustrated as boxplots in Fig. 10 (results for the other products are provided in Figure S3 of the supplemental material). Note that v1 is not shown because it is always zero, Y. Mei et al. / Advances in Water Resources 109 (2017) 253–266 263 Fig. 9. Same as in Fig. 4, but for the variance of catchment response time. Fig. 10. Same as in Fig. 6, but for Eq. (16). and, therefore, var(Tr ) equals v2. Overall, patterns of the distribution of the different terms suggest no obvious basin-scale dependency. Value ranges of var(Tr ) (v2 in the figure) are about five times and two times those of var(Tn ) and cov(Tr ,Tn ) (c in the figure), respectively, pointing to that most of the error in shape of the hydrograph is contributed by the runoff generation stage. It is also noted that the movement of rainfall excess with respect to the catchment channel network accounts for a considerable amount of error contribution, highlighting the importance of accurate estimates on the space-time variability of rainfall. To sum up, these observations indicate var(Tq ) is mainly caused by the error in temporal variations of rainfall excess (v2 and c), while contributions from the spatial variations (v3 and v4) are less relevant. The relatively low contribution from the spatial variability is again ascribed to the smooth topographical setup of the study area (Mei et al., 2017). 4.5. Analytical vs. numerical-based error evaluation In the previous sections, we showed how the V2010 framework (with modifications based on M2017) can be used for decomposing the error in catchment flood response into the error in spatial 264 Y. Mei et al. / Advances in Water Resources 109 (2017) 253–266 Fig. 11. Scatterplot of the framework-derived error quantities vs. the error in hydrograph properties. Table 3 Sensitivity of the error in catchment flood response and the error in hydrograph properties. Error quantity Volume Centroid Dispersion ME (%) CRMS (%) Swift Fishing Tar Swift Fishing Tar −1.7 −5.3 −7.2 −0.5 −0.7 −2.5 −0.8 −1.1 −3.5 12.6 11.9 19.1 8.3 6.0 11.7 5.7 5.9 10.0 and temporal variability of the rainfall patterns, runoff generation and runoff routing across hillslopes and channel network. The error in catchment flood response analyzed are: the error in rainfall excess intensity, the expected catchment response time, and the variance of the response time (i.e., [R]at , E(Tq ), and var(Tq )). These three error quantities are used to estimate the error in hydrograph properties. To test the consistency of those estimations, we compared the three error quantities with those calculated from the hydrographs simulated by the numerical hydrologic model. Fig. 11 renders scatter plots between the normalized errors for the different types of satellite precipitation products, with the mean error (ME) and centered root mean square error (CRMS) statistics reported in Table 3. ME and CRMS are defined as the nominators of Eqs. (20) and (21) by replacing Xs and Xo as the normalized error estimated by the framework and by the hydrologic model, respectively. The first row of Fig. 11 shows a fairly strong linear relationship following the one-to-one line between the normalized error in rainfall excess intensity, [R]at /[R]at , and that of the cumulative flow volume, V/V. This is confirmed by the low magnitudes of mean error (ME, less than 7%) and centered root mean square error (CRMS, less than 18%) shown in Table 3. According to these results the framework error quantity [R]at slightly underestimates V calculated from the distributed hydrologic model (CREST in this case); the random component of error outweighs the systematic one by more than five times. Basin-wise comparison shows the magnitudes of CRMS generally decreasing with basin scale while ME for the Fishing basin case is the lowest. The correlation between normalized error in expectation of catchment response time and normalized error in hydrograph centroid is illustrated in the second row of Fig. 11. The systematic and random error metrics (ME and CRMS) are listed in Table 3 for the three basins. Overall, the correlation is lower than that of rainfall excess, but the scatterplots of the analytical- vs. numericalderived error generally follows the one-to-one line. The Fishing and Tar basin cases show higher correlation than the Swift basin, confirmed by the lower ME and CRMS in the table. Values of ME are all negative, indicating underestimations of C by E(Tq ); the Y. Mei et al. / Advances in Water Resources 109 (2017) 253–266 magnitudes of ME are lower than those of CRMS (for at least two times), revealing that the random error is the main source of uncertainty in the estimation of C. The last row of Fig. 11 and Table 3 illustrate the sensitivity test √ √ on  var(Tq )/ var(Tq ) to S/S. The plots show good correlation but the scatter deviates from the one-to-one line. The linear relationship is more pronounced for the cases of the Fishing and Tar basins, as revealed by the lower magnitudes of ME and CRMS in the table. Values of ME are all negative, referring underestimation √ of S by  var(Tq ). The magnitudes of CRMS are more than two times as the ME ones, which again points to the fact that the random component of error is the main issue for the estimations. This may be attributed to the high random error for the estimation of the dispersion parameter as illustrated in M2017. Similar to the estimations of C, the Fishing and Tar basin cases show higher correlation than the Swift basin. 5. Conclusions In this study, we presented the use of an analytical framework for catchment flood response to assess the error propagation process of satellite precipitation through the rainfall-runoff translation. This framework allowed to decompose the error in hydrograph properties into terms representing the different space-time interactions among rainfall, runoff generation and routing. Specifically, by using the framework the error in three hydrograph properties (that is, cumulative volume, centroid, and dispersion of the hydrograph) were estimated by the error in three corresponding framework quantities (i.e., storm-average catchment rainfall excess, and expectation and variance of catchment response time). Error in the three framework quantities were broken into terms reflecting the error related to the mean, the spatial and temporal variation and the movement of rainfall/rainfall excess. The framework was demonstrated based on 160 rainfall-runoff events simulated with a distributed hydrologic model driven by twelve satellite precipitation products and the Stage IV radar-based precipitation product as reference. We found that the satellite precipitation products underestimated the intensity of rainfall and rainfall excess of the flood events examined in this study. Overall, the gauge-adjusted and high-resolution products yielded lower error magnitudes than the corresponding near-real-time and coarse resolution counterparts. The error in the product of space-time average rainfall and runoff coefficient constitute the main contributor to the total error in the storm-average catchment rainfall excess. Errors in representing the movement of rainfall and the temporal and spatial correlation between precipitation and runoff coefficient are of secondary significance. The mean error of catchment response time was low. Gauge adjustment and spatiotemporal resolution of the precipitation product revealed no clear effects on the estimation of event timing. Additionally, the error components through the runoff generation and runoff routing stages shared comparable magnitudes. Error due to the differences in temporal/spatial coevolution/colocation of rainfall excess and the holding time contributed significantly to the total error in the mean catchment response time. For the error in catchment response time variance, the gaugeadjusted and high-resolution products did not consistently exhibit lower magnitudes. The runoff generation stage contributed a larger amount of error to the variance of catchment response time than the runoff routing stage and the movement of rainfall excess. The error terms related to the temporal variation of rainfall excess and the storm movement were the most significant contributors to the total error of catchment response time variance. Results of the sensitivity tests suggested strong correlation for the estimations on the three error quantities, especially for the 265 error in rainfall excess vs. error in cumulative flow volume estimation, which was shown to follow the one-to-one relationship. Slight systematic underestimation was common to the estimations of the error quantities in hydrograph properties, while the random components of error outweighed the systematic components in terms of magnitude. We acknowledge certain limitations of the implementation of the hydrologic analytical framework, as well as its use in analyzing the error propagation of precipitation in this study. First, we would like to note the dependence of the error analysis on the hydrologic model used to derive values for the analytical framework’s parameters (Mei et al., 2017). This means the calculation of these variables could vary across models, or be unavailable for certain hydrological models that cannot associate with the analytical formulation. To circumvent this issue, we suggest use of independent information, such as that obtained from remote sensing, to retrieve the variables. For instance, the space-time variability of runoff coefficient may be estimated based on the soil moisture stage and percentage of impervious area (Mega et al., 2014; Massari et al., 2014); the runoff routing-related parameters could be estimated based on the hydrologic and geomorphologic properties of catchments (Shen et al., 2016b; Penna et al., 2011). Moreover, our results revealed that the error in runoff generation due to the differences in space and time covariance between rainfall and runoff coefficient are of minor importance. This may be a finding specific to the moderate to high flow generation events used in the study and the lack of orographic effects on the triggering storms of the study area. Future studies should focus on flash flood–scale events and more complex terrain catchments to investigate the relative importance of error terms related to the spatial and temporal distribution of rainfall and runoff generation. Acknowledgements The current study was supported by a research grant from the Connecticut Institute for Resilience and Climate Adaptation. Efthymios Nikolopoulos was supported by the European Union– funded EartH2Observe (ENVE.2013.6.3–3) project. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.advwatres.2017.09.012. 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