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The significance of spatial variability of rainfall on simulated runoff: an evaluation based on the Upper Lee catchment, UK

The significance of spatial variability of rainfall on runoff is explored as a function of catchment scale and type, and antecedent conditions via the continuous time, semi-distributed probability distributed model (PDM) hydrological model applied to the Upper Lee catchment, UK. The impact of catchment scale and type is assessed using 11 nested catchments, and further assessed by artificially changing the catchment characteristics and translating these to model parameters (MPs) with uncertainty using model regionalisation. Dry and wet antecedent conditions are represented by ‘warming up’ the model under different rainfall time series. Synthetic rainfall events are introduced to directly relate the change in simulated runoff to the spatial variability of rainfall. Results show that runoff volume and peak are more sensitive to the spatial rainfall for more impermeable catchments; however, this sensitivity is significantly undermined under wet antecedent conditions. Although there is indication that the impact of spatial rainfall on runoff varies as a function of catchment scale, the variability of antecedent conditions between the synthetic catchments seems to mask this significance. Parameter uncertainty analysis highlights the importance of accurately representing the spatial variability of the catchment properties and their translation to MPs when investigating the effects of spatial properties of rainfall on runoff. This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (CC BY 4.0), which permits copying, adaptation and redistribution, provided the original work is properly cited (http://creativecommons.org/licenses/by/4.0/). doi: 10.2166/nh.2016.038 om https://iwaponline.com/hr/article-pdf/48/4/1118/365248/nh0481118.pdf 2020 I. G. Pechlivanidis (corresponding author) Swedish Meteorological and Hydrological Institute, Folkborgsvägen 1, Norrkoping S-601 76, Sweden E-mail: ilias.pechlivanidis@smhi.se N. McIntyre Centre for Water in the Minerals Industry, Sustainable Minerals Institute, University of Queensland, Brisbane, 4072 QLD, Australia H. S. Wheater School of Environment and Sustainability, University of Saskatchewan, Saskatoon, Canada S7N 5C8 and Department of Civil and Environmental Engineering, Imperial College London, London SW7 2AZ, UK


INTRODUCTION
Rainfall is a highly heterogeneous process over a wide range of scales in space, and lack of its spatial information can affect understanding of catchment-scale hydrological response and affect accuracy of hydrological model predictions (Singh ). For hydrological operational purposes, e.g., flood forecasting (in particular, model state initialisation and short lead-time forecasts) and management (catchment water balance), and design (e.g., flood frequency curves), a good knowledge of rainfall is essential (Lamb et al. ).
Consequently, one question that arises in rainfall-runoff studies is: 'where and when is the spatial nature of rainfall important to runoff response?' The rainfall-runoff relationship can be expected to depend on complex interactions between rainfall dynamics, physical properties and the spatial scale (i.e., catchment area) of the problem. Common methods to assess the effects of spatial rainfall are sought through comparison of rainfall-runoff responses between events within a catchment (e.g., Bell & Moore ; Cole & Moore ) or between catchments covering different hydrological regimes (Smith et al. ). Other studies have investigated these effects by running realisations of synthetic rainfall patterns through a calibrated hydrological model or running observed rainfall data through models representing idealised catchments (e.g., Arnaud et al. ). Overall, despite the extensive literature on the relationship between spatial rainfall and runoff, the picture is still complex in terms of the tools adopted, studied catchments and conclusions drawn from the analyses, with several studies concluding that spatial rainfall distribution is significant for runoff (Gabellani et  In addition, catchment perviousness and antecedent catchment conditions can affect the spatial rainfall-runoff relationship. Permeability can substantially control the impact of spatial rainfall on runoff, e.g., noting the high importance of spatial rainfall in urban areas (Tetzlaff & Uhlenbrook ). Moreover, the runoff prediction errors are generally considerably higher for dry than for wet conditions (Zehe et al. ); however, the spatial variability of soil moisture can also exert a strong control on runoff, with the dominant flow path varying according to the soil moisture conditions and consequently affecting the runoff peak and response time (Penna et al. ; Patil et al. ).
Most numerical studies are based on (semi-)distributed hydrological models, since this type of model has the potential to represent the effects of spatially variable inputs.
However, (semi-)distributed models can easily become over-parameterised and subsequently ill-posed with respect to the information content of available input-output data (Pechlivanidis et al. ), resulting in large uncertainties in the parameters and simulated runoff (Younger et al. ).
Regionalisation may be used to spatially generalise parameter estimates from well-gauged areas; however, this introduces new sources of uncertainty (Wagener & Wheater ). Therefore, investigation of the influence of spatial rainfall on simulated runoff should include testing the sensitivity of conclusions to uncertainty in spatial distributions of model parameters (MPs). This paper uses a semi-distributed rainfall-runoff model to explore the significance of spatial variability of rainfall on runoff properties (hydrograph shape, peak and volume) as a function of catchment scale, catchment type, degree of spatial rainfall variability and antecedent catchment conditions. We also address how uncertainty arising from MP regionalisation affects the results. This investigation is based on the spatial characteristics (i.e., shape, size) of the Upper Lee system, to the north of London, UK, consisting of 11 nested catchments (varying between 26 and 1,040 km 2 ), while the synthesis of physical properties is based on the regional relationships developed by Pechlivanidis et al. (). The hydrological variability of the catchment, and availability of extensive data sets, semidistributed models and regionalisation results make the Upper Lee a suitable case study.

THE STUDY CATCHMENT
The Upper Lee catchment is located to the north of London, UK, and classified as relatively large, draining an area of 1,040 km 2 (at Feildes Weir) to the River Thames within Greater London. The catchment's location, its physical and hydro-climatic characteristics, and the available data sets (i.e., hourly rainfall, runoff and potential evaporation (PE)) can be found in Figure 1 and Table 1 in the paper by Pechlivanidis et al. (); to avoid redundant information we do not repeat them here. In brief, the catchment is divided based on the available river gauge network. Thus, 11 subcatchments are identified varying in area from 26 to 1,040 km 2 : Gupsy Lane (26 km 2 ), Stevenage (31) We chose this model structure because its simplicity has allowed successful regionalisation via parameter regression against physical characteristics (Wagener & Wheater ).
In this study, in order to investigate the significance of spatial rainfall on runoff, we use the semi-distributed version of the  Table 4 in Pechlivanidis et al. ()) were produced using a stepwise regression procedure and relate the C max , K q , q and TP parameters of the PDM model with a set of physical catchment descriptors (CDs) taken from the UK's Flood Estimation Handbook, including drainage area, land use, geology, elevation, soil characteristics as well as climate variables (see Table 1). Based on parameter identification analysis, the b and K s parameter values are fixed for all subunits equal to 0.065 and 4,900 (hours), respectively (see Pechlivanidis et al. ). Note that due to the distinct nature of the porosity of the chalk, which is not straightforward to represent, the identified b and K s values may not be transferable at the Upper Lee chalk rivers (i.e., the river Mimram in Hertfordshire) and hence this parsimonious model structure could perform poorly.

Generation of synthetic physical characteristics
The physical properties of the Upper Lee at the subunit scale are artificially modified to represent two simplified catchment types (impermeable and permeable). The geology and land cover properties of the catchment, as described by URBEXT, BFIHOST and SPRHOST CDs, are controlled and are uniform across the whole catchment (yet their values depend on the synthetic catchment type), while the other descriptors (i.e., AREA, DPSBAR, DPLBAR and SAAR as defined in Table 1) are allowed to remain at the real values, since they represent geomorphological characteristics of the Upper Lee system. AREA, and hence DPLBAR, due to their high correlation (correlation coefficient is equal to 0.82) do not vary much among subunits (subunit areas vary between 15 and 25 km 2 ). This approach will allow parameters to vary in space; hence the catchments could be assumed semi-synthetic (based on artificially uniform soil, geology and land cover properties but maintaining the geomorphological characteristics).
This allows a more controlled experiment than trying to interpret differences between complex real catchments.
The relationships between MPs and CDs are physically explicable. For instance, C max has positive correlation with BFIHOST, which indicates that permeable chalk catchments (high BFIHOST) have high moisture storage capacity. C max is negatively correlated with URBEXT; more urbanised catchments tend to have low C max values while the more rural catchments have a higher range of values. As expected, BFI-HOST has a positive correlation with K q (fast routing residence time), while TP has negative correlation with URBEXT. More details can be found in Pechlivanidis et al.  Table 2. These are the same ranges used to develop the parameter regionalisation and hence parameter estimates are assumed to be reasonable within these ranges. Here we distinguish between the effects of two types of catchment -'permeable' and 'impermeable' within the context of the Upper Lee, whose properties are defined by the extremes of the ranges. In other words, the experiment assumes that impermeable catchments are heavily urbanised and with high percentage runoff (and hence low baseflow index); whereas the permeable catchments are mainly rural with high baseflow index and with the potential to infiltrate most of the effective rainfall (low percentage runoff).

Synthetic rainfall events and spatial deviation index
The analysis introduces a numerical experiment in which the rainfall-runoff model runs under synthetic rainfall events. Synthetic rainfall analyses have the potential to investigate events with high range of spatial variability, aiming to unambiguously relate the change in simulated discharge to the spatial variability of rainfall (assuming that the antecedent catchment conditions and time variability of rainfall are adequately controlledsee the section 'Representing the effect of antecedent conditions on the results').
To address the latter, during the generation of the synthetic events, rainfall volume, location and duration are preserved at the (sub)catchment scale.
In total, three synthetic rainfall events are generated at a 1 × 1 km 2 grid for the catchment and each gauged subcatchment. A rainfall core (defining the area in which rainfall occurs) is generated covering an area of three different diameters (see a schematic example in Figure A1(a), available   Figure A1(b)-1(d) (available with the online version of this paper).
A spatial deviation index (SDI) was defined for each rainfall event and each (sub)catchment as a measure of spatial variability (Segond et al. ). The deviation between the 1 × 1 km 2 gridded rainfall and the subunit averaged areal rainfall (SDI G ) is defined by: where r i represents the precipitation in the ith grid, r k is the subunit averaged areal precipitation, and N the number of grids for a given subunit. The catchment's SDI over the entire rainfall event can be estimated as a weighted value (high SDI values indicate high spatial variability) according to: where K is the number of subunits for a given catchment.

Representing the effect of antecedent conditions on the results
Here, we investigate the impact of antecedent conditions (wet and dry) on the spatial rainfall-runoff relationship. In addition, we look at the effect of increased variability in antecedent conditions. To do so, the synthetic rainfall events are repeated after a synthetic event has already been run to generate the antecedent soil moisture conditions. An event of low spatial variability is applied prior to the synthetic rainfall event of interest to generate almost homogeneous antecedent soil moisture conditions (named uniform), whereas a high spatially variable event is applied to represent high spatial variability of antecedent soil moisture conditions (named variable). The spatial variability of the antecedent conditions is presented in Figure A2 (available with the online version of this paper). This requires that a single event significantly affects the simulated soil moisture. To achieve this, dry conditions were created using the warm-up period 01/01/1991 to 07/06/1993 for all conditions, and then the event was applied (whereas in wet conditions the simulated soil would remain saturated irrespective of the nature of a subsequent event). Consequently, we achieve spatial smoothing of the dry antecedent conditions by considering their long-term history prior to any further investigation.

Analysis of simulated hydrological response
The simulated hydrological response is analysed for each subcatchment and each synthetic rainfall event. This is evaluated measuring the percentage change in the peak runoff and runoff volume of a hydrological response relative to the 'reference' simulated runoff, and also a modified definition of the that compares simulated runoff to the reference: where Q Ci and Q Ri are the calculated and 'reference' discharge at hour i, and n is the length (time-steps) of the time series. The hydrograph simulated from the high spatially variable event is the 'reference' runoff. These AFs downscale the parameter estimates to the subunit scale from the gauged subcatchments, however they maintain the prior spatial variability of the parameter values; in other words, they compensate for the uncertainty of regional relationships under the constraint that the spatial pattern must not change. We therefore investigate two sources of uncertainty: (a) parameter uncertainty due to the presence of equifinal AFs and (b) uncertainty in the generation of the regional relationships.
The experimental analysis evaluates the simulated hydrographs generated from the behavioural parameter sets derived from the product of regional estimates and AFs. The calibration of the AFs is based on a uniform random search, whereby 100,000 sets of the AFs are randomly sampled from pre-defined ranges and the 'optimum' 100 sets based on the Nash-Sutcliffe efficiency for each (sub)catchment individually are adopted; these behavioural sets correspond to Nash-Sutcliffe greater than 0.7 on average for all catchments.
Finally, two sets of MP-CD relationships are applied to assess uncertainty in the regionalisation models (see Table 3). The first set of MP-CD relationships considers that there may be many equally optimal sets of regression coefficients arising from MP equifinality. Therefore, here the original CDs are maintained but their coefficients are allowed to vary; hence this set is consistent with the physically meaningful relationships of Table 4 in Pechlivanidis et al. (). These relationships were generated using randomly selected equifinal MP sets (in here the 10th optimum parameter set). The second set omits the constraint of generating physically meaningful relationships and is solely based on statistical significance. Therefore, the empirical regression model aimed at maximising its performance (i.e., coefficient

Impact of regionalisation on reproducing runoff
We first summarise the model's ability to represent the spatial heterogeneity of physical properties (as described by the parameters) and reproduce the observed runoff at different locations; note that here the hydrological model is forced with observed rainfall time-series (see the section 'The study catchment'). Figure 1(a) shows the spatial variability of C max , K q , q and TP parameters derived from regional relationships in Table 4 in Pechlivanidis et al. (). Parameter estimates are variable and within their reasonable ranges.
AFs were next calibrated using the NSE objective function and applied to parameters C max , b, K q , K s and TP, while the q parameter is kept equal to SPRHOST due to its assumed direct physical correspondence (Figure 1(b)).

Effect of scale
Results in Figure 4 are similar to those presented in Figure 3, but with the subcatchments ordered along the x-axis from smallest to largest area. Similar patterns were observed for runoff peak and volume and therefore we only focus on peak runoff. For each catchment there are four boxplots corresponding to the percentage change of a hydrograph characteristic based on the 100 optimum MP sets for each of the two events (low and medium) and the two catchment types (permeable and impermeable).
Peak runoff and volume were sensitive to spatial rainfall at small-size impermeable catchments (up to 80 km 2 ) under dry antecedent conditions. However, results show that Sheering does not follow the trend, possibly due to the spatial variability of C max ; the spatial C max pattern (and hence the spatial pattern of saturated areas) can affect the effective rainfall generated at each time step.
The significance seems to slightly decrease at medium and large catchment scales. Here, this result is not due to the channel routing dampening effect which occurs at the large scales, since it was found that results were relatively insensitive to the channel routing; the simplistic channel model being used is not capable of representing the smoothing effect due, for example, to over-bank flow  As explained earlier, the uniform catchment wetness and hillslope routing parameters seem to undermine the relationship between hydrograph characteristics and spatial rainfall. However, it is interesting to note that peak runoff slightly changes (5% change on peak runoff on average) at the large catchment scale (i.e., Feildes Weir). It seems that for large catchments, the residence time in the channel increases significantly and tends to be closer to the hillslope travel time; and hence the spatial distribution of rainfall becomes a main controlling factor. Results based on runoff volume also showed a relationship to spatial rainfall variability. Overall, runoff volume results mirrored peak runoff results.

Impact of spatial variability of antecedent catchment conditions
Here, we assess the impact of spatial variability of antecedent catchment conditions (uniform and variable) for each (sub)catchment. Uniform is used to represent less spatially variable antecedent conditions, as generated from a prior low spatially variable synthetic rainfall event, whereas variable is used to represent more spatially variable antecedent conditions as generated from a prior high spatially variable synthetic rainfall event. We present the NSE ref measure of simulated runoff (simulations from the low event) to the 'reference' runoff for uniform and variable antecedent conditions, respectively.
Results show that the spatial antecedent conditions play a significant role on the impact of spatial rainfall on the NSE ref measure (see Figure A3, available with the online version of this paper). Although results for both permeable and impermeable scenarios are sensitive to the spatial variability of antecedent conditions, results show that this sensitivity is higher for permeable catchments, probably due to higher C max values and hence more scope for variability in initial wetness. Overall, it is interesting to note the variability of the results based on the MP set used, particularly for the permeable catchments, due to variability in initial conditions. This is related to the existence of fully or partially saturated areas over the catchments. Results based on peak runoff and volume also showed the same relationship to spatial rainfall under the two spatially variable antecedent catchment conditions. Impact of uncertainty in regional relationships on the

significance of results
Results based on the first set of MP-CD relationships (developed using equifinal parameter sets; however, the relationships maintained their physical meaning but allowed CD coefficients to vary) again highlighted the significance of spatial variability of rainfall on peak runoff and volume.
Trends and conclusions were similar to those derived in the previous sections.
Results based on the second set of relationships (regression equations were not constrained to link MP with CDs which are expected to be hydrologically related, but were optimised to maximise the performance of the regional model) showed that the spatial variability of rainfall can influence the hydrograph properties (peak runoff and volume). The same was found for impermeable catchment types under dry antecedent conditions. However, unlike the previous analysis, results did not show any clear relationship with catchment scale. In addition, the uncertainty of the results, as described by the variability of the results using the behavioural parameter sets, was higher than before. It should be noted that results were influenced by the introduction of different CDs, resulting in different spatial variability of MPs over the catchment (i.e., Kq was allowed to vary in space using the second set of relationships in Table 3). This highlights the sensitivity of the results to the structure of the MP-CD relationships; the more physically sensible CDs used in Table 4 in Pechlivanidis et al. () produce more reliable results.

DISCUSSION
In this paper, we build upon previous studies on the effects of spatial variability of rainfall on runoff generation, and in several parts we contradict and/or complement previous Caution is required when the spatial rainfall-runoff relationship is investigated numerically via rainfall-runoff models; simulated runoff variability does not only depend on spatial rainfall variability but also on the rainfall-runoff model itself and its spatial heterogeneity, and hence sources of error/uncertainty should be addressed (Younger et al.  and assessed the sensitivity of the results to these relationships. In addition, in this paper we propagate the uncertainty associated with the regional relationships (repeating the analysis using different plausible regional relationships) to the results,

SUMMARY AND CONCLUSIONS
The motivation of this study is the need to improve understanding about the significance of spatial properties of rainfall on the hydrological response as a function of catchment scale and type, and antecedent catchment conditions.
We investigate three synthetic rainfall-runoff events at 11 The spatial discretisation of the rainfall-runoff model affects the results particularly at small catchments. Results show that the spatial variability of antecedent catchment conditions has a significant effect on runoff response. Highly spatially variable antecedent conditions can significantly affect the hydrological response due to the existence of fully or partially fully saturated areas over the catchment during the event. Finally, hydrograph responses to different spatial patterns in rainfall depend on assumptions used for MP estimation and also the spatial variation in MPs indicating the need of an uncertainty framework in such investigation.