Reversing hydrology: quantifying the temporal aggregation effect of catchment rainfall estimation using sub-hourly data

Inferred rainfall sequences generated by a novel method of inverting a continuous time transfer function show a smoothed profile when compared to the observed rainfall, however, streamflow generated using the inferred catchment rainfall is almost identical to observed streamflow (Rt > 97%). This paper compares the effective rainfall inferred by the regularised inversion process (termed inferred effective rainfall (IER)) proposed by the authors with effective rainfall derived from the observed catchment rainfall (termed observed effective rainfall (OER)) in both time and frequency domains in order to confirm that, by using the dominant catchment dynamics in the inversion process, the main characteristics of catchment rainfall are being captured by the IER estimates. Estimates of the resolution of the IER are found in the time domain by comparison with aggregated sequences of OER, and in the frequency domain by comparing the amplitude spectra of observed and IER. The temporal resolution of the rainfall estimates is affected by the slow time constant of the catchment, reflecting the presence of slow hydrological pathways, for example, aquifers, and by the rainfall regime, for example, dominance of convective or frontal rainfall. It is also affected by the goodness-of-fit of the original forward rainfall–streamflow model. doi: 10.2166/nh.2015.076 om http://iwaponline.com/hr/article-pdf/47/3/630/368207/nh0470630.pdf er 2020 A. Kretzschmar W. Tych (corresponding author) N. A. Chappell K. J. Beven Lancaster Environment Centre, Lancaster University, Lancaster LA1 4YQ, UK E-mail: w.tych@lancaster.ac.uk K. J. Beven Department of Earth Sciences, Uppsala University, Uppsala, Sweden


INTRODUCTION
Rainfall is the key driver of catchment processes and is usually the main input to rainfall-streamflow models. If the rainfall and/or streamflow data used to identify or calibrate a model are wrong or disinformative, the model will be wrong and cannot be used to predict the future with any certainty. Blöschl et al. () state that if the dominant pathways, storage and time-scales of a catchment are well defined then a model should potentially reproduce the catchment dynamics under a range of conditions. It is often the case that hydrological variables, such as rainfall and streamflow, are measured at hourly or sub-hourly intervals then aggregated up to a coarser resolution before being used as input to rainfall-streamflow models resulting in the loss of information about the finer detail of the catchment processes (Littlewood & Croke , ; Littlewood et al. ). Kretzschmar et al. () have proposed a method for inferring catchment rainfall using sub-hourly streamflow data. The resulting rainfall record is smoothed to a coarser resolution than the original data but should still retain the most pertinent information.
This paper investigates the implications of the reduced resolution and the potential loss of information introduced by the regularisation process in both the time and frequency domains. Both temporal and spatial aggregation are incorporated in the transfer function model, however, only the temporal aspect is considered here. The effect of spatial rainfall distribution using sub-catchments will be the subject of a future publication.
The method developed and tested by Kretzschmar et al. ()termed the RegDer methodinverts a continuous time transfer function (CT-TF) model using a regularised derivative technique to infer catchment effective rainfall from streamflow with the aim of improving estimates of catchment rainfall arguing that a model that is well-fitting and invertible is likely to be robust in terms of replicating the catchment system. In the context of this study, observed catchment rainfall (may be derived from one or more raingauges by any suitable method, e.g., Thiessen polygons) is converted to observed effective rainfall (OER) by a nonlinear transform designed to render the relationship between the rainfall input and streamflow output (via a CT-TF) linear.
The inversion process takes the catchment streamflow and, using a regularisation process, infers effective rainfall (IER), which is then converted to inferred catchment rainfall (ICR) by the reverse of the non-linear transform.
The effective rainfall (both OER and IER) may be termed scaled rainfall (related to Andrews et al. ()) as it is derived from the overall catchment rainfall.
The classical approach to inverse (as opposed to reverse) modelling involves the estimation of non-linearity (rainfall or baseflow separation) and the unit hydrograph (UH), which is an approximation to the impulse response of the catchment. Boorman () and Chapman () use sets of event hydrographs to estimate the catchment UH. Boorman () superimposed event data before applying a separation technique and concluded that the data required may be more coarsely sampled than might be expected because one rain-gauge is unlikely to be representative of the whole catchment. Chapman () used an iterative procedure to infer rainfall patterns for individual events before applying baseflow separation. The resultant UHs had higher peaks and shorter rise times and durations than those obtained by conventional methods. He viewed the effective rainfall as the output from a non-linear store. They suggested further investigation using data-based mechanistic (DBM) modelling methods as described by Young & Romanowicz () and Young & Garnier () for estimating CT models from discrete input data.
Such models generate parameter values independent of the input sampling rateas long as the sampling rate is sufficiently high in comparison to the dominant dynamics of the system. Advantages of using the CT formulation include allowing a much larger range of system dynamics to be modelled, e.g., 'stiff' systems that have a wide range of time constants (TC), typical of many hydrological systems. The outputs from such a model can be sampled at any timestep, including non-integer, and the parameters have a direct physical interpretation (Young ). () demonstrated that models calibrated using a smoothed rainfall signal (due to coarse sampling) may result in underestimation of streamflow. Further calibration, required to compensate, leads to the loss of physical meaning of parameters. They also concluded that parameters estimated at one sampling interval were not transferable to other intervals; a conclusion echoed by Littlewood () and Littlewood & Croke ().
Studies by Clark & Kavetski () showed that in some cases, numerical errors due to the time-step are larger than model structural errors and can even balance them out to produce good results. The follow-up study by Kavetski & Clark () looked at its impact on sensitivity analysis, parameter optimisation and Monte Carlo uncertainty analysis.
They concluded that use of an inappropriate time-step can lead to erroneous and inconsistent estimates of model parameters and obscure the identification of hydrological processes and catchment behaviour. Littlewood & Croke () found that a discrete model using daily data overestimated TC for the River Wye gauged at Cefn Brwyn when compared to those estimated from hourly data confirming that parameter values were dependent on the time-step.
They discussed the loss of information due to the effect of used 'black box' modelling and frequency analysis to study the behaviour of a karst system (located at Fuenmajor, Huesca, Spain). They concluded that the method works well for a linear system and that Fuenmajor has a linear hydrological response to rainfall at all except high frequencies. They suggest that the non-linearity issues might be addressed using appropriate techniques such as wavelets or neural networks. Szolgayová et al. () utilised wavelets to deseasonalise a hydrological time-series and suggested that the technique had potential for modelling series showing long-term dependency (interpreted as containing low frequency components).
The method introduced by Kretzschmar et al. () showed that given that the rainfall-streamflow model captures the dynamics of the catchment system, the high frequency detail of the rainfall distribution is not necessary for the prediction of streamflow due to the damping (or low-pass filter) effect of the catchment response. The numerical properties of the regularisation as applied to the inversion process place a mathematical constraint of smoothness balanced against a loss of some temporal resolution in the inferred rainfall time-series. The regularisation and therefore smoothing level is controlled through the Noise Variance Ratio (NVR), optimised as part of the process and is only applied when necessary, i.e., when the analytically inverted catchment transfer function model is improper (has a numerator order higher than the denominator order).

APPLICATION CATCHMENTS
RegDer has been tested on two headwater catchments with widely differing rainfall and response characteristics -Baru in humid, tropical Borneo and Blind Beck, in humid temperate UK.
Barutropical catchment The choice of these two experimental catchments, therefore, allowed the initial evaluation of the estimation of catchment rainfall from streamflow for the end-member extremes of a basin with tropical convective rainfall and shallow flow pathways to a basin with temperate frontal rainfall (i. e., much lower intensity) and deep flow pathways (i.e., much greater basin damping or temporal integration).  where P is the observed rainfall, Q the observed streamflow and α is a parameter, estimated from the data. P e is the effective observed rainfall (ERER) and Q is used as a surrogate for catchment wetness. Both catchments used in this study proved to be predominantly linear in their behaviour so transformation (Equation (1)) was not used. In the initial study, a wide range of possible models was identified using algorithms from the Captain Toolbox for Matlab (Taylor et al. ). The models selected were a good fit to the data and were suitable for inversion. The Nash-Sutcliffe efficiency (NSE or R t 2 ) is commonly used to compare the performance of hydrological models. Often, several models can be identified that fit the data well (the equifinality concept of Beven ()). From these, models with few parameters to be estimated that inverted well were selected. In this study, a second-order linear model was found to fit both catchments. The output from the RegDer process is an IER series to which the inverse of the power law is then applied, if necessary, to construct an ICR sequence. The process is illustrated in Figure 3.

MODEL FORMULATION AND PHYSICAL INTERPRETATION
The transfer function model inversion process has been described in Kretzschmar et al. (). It involves transition from the transfer function catchment model: to the direct inverse (in general non-realisable): which is then implemented using regularised streamflow derivatives in the form of: where d s n Q is the Laplace transform of the optimised regularised estimate of the nth time derivative of Q: d n =dt n Q. The regularised derivative estimates replace the higher order derivatives in Equation (3), which otherwise make Equation (3)   In order to investigate this, the IER is compared to aggregated effective observed rainfall sequences with increasing levels of aggregation until a good match is found (high value of R t 2 or R). Two methods of aggregation have been used: (1) averaging over a range of time-series and (2) moving average over varying time-scales. Two measures are used to assess the correspondence between the IR and the aggregated effective rain: (1) R t 2 , the coefficient of determination and (2) R, the instantaneous Pearson correlation coefficient. They are given by: where ER indicates a value from the aggregated effective rainfall sequence with mean ER and IER is the corresponding value from the IER sequence with mean IER. Both R t 2 and R values tend towards a maximum value as aggregation increases. The aggregation level at which the maximum is reached is identified and taken as an estimate of the resolution of the inferred effective series. This value is then compared to the system fast time constant (TC q ) and the N-S sampling limit.

Continuous model formulation
One of the advantages of using a CT model formulation is that the parameters have a direct physical interpretation independent of the model's sampling rate (Young ).
The continuous time model formulation for a second-order model is given by: where y is the measured streamflow at time t, δ is the transport delay and u is the effective rainfall at time t À δ. If the denominator can be factorised and has real roots, Equation (6) can be rewritten as: where TC q and TC s are the system time constants and are often significantly differenta 'stiff' system. Decomposing the model into a parallel form gives: given by: so the fraction of the total streamflow along each pathway can be calculated from: The fraction of streamflow attributed to the slow response component is sometimes termed the slow flow index (SFI) (Littlewood et al. ). The example shown here uses a second-order model but the general principle can be extended to higher order models. Details of the inversion and regularisation processes can be found in Kretzschmar et al. ().

Sampling frequency
The N-S frequency gives the upper limit on the size of the sampling interval, Δt, that will enable the system dynamics to be represented without distortion (aliasing -Bloomfield , p. 21). Aliasing occurs when a system is measured at an insufficient sampling rate to adequately define the signal from the data.
The N-S theorem states that the longest sampling step for a signal with bandwidth Ω (maximum frequency, where Ω ¼ 2πf in cycles per time unit) to be represented is: in order to completely define the system in absence of observation disturbance (Young ). If the sampling interval is small enough to uniquely define the system, the estimated CT model should be independent of the rate of sampling.
Conversely, if the frequency of the inferred output is less than the N-S limit, then the system dynamics should be adequately captured. Other estimates of the sufficient sampling interval, designed to avoid proximity to the Nyquist limit, have been made by Ljung () and Young (). In terms of system TCs, these limits are given by:    Figure 4 illustrates the smoothed rainfall distribution of the IER sequence obtained using the RegDer method. Similar    Table 1 and compared with the fast time constant (TC q ) and the N-S sampling limit.  (hr), time constants (TC qfast and TC sslow), SFI (the percentage of the flow taking the slow pathway), the N-S safe sampling limit (hr) and the time resolution of the IER estimated by resampling and moving average methods. Also shown is the frequency domain estimate of the resolutionthe cut-off point beyond which the signal carries very little information (illustrated in Figure 8) and can be considered unimportant. The estimated time resolution of the IER sequences is less than the dominant (fast) mode of the catchments and considerably less than the 'safe' N-S limit  the cut-off point, shaded in Figure 8, carries a very small proportion of the power of the signal and can be considered non-significant. The processes and characteristics limiting the IER accuracy include the slow components of the catchment dynamics and the rainfall regime. These can be seen as the 'usual suspects' affecting the inversion process. The general goodness-of-fit of the initial catchment model (rainfallstreamflow) appears to be a factor as well (see Figure 9), indicating that the IER estimation method presented here can be used to assess the quality of available data and the degree to which the data characterise the catchment. Further work is required using a range of catchment and rainfall regimes to confirm these results and explain them in terms of rainfall and catchment characteristics, as well as investigating spatial relationships. The latter will be evaluated using catchment data with multiple rain-gauges, and are the subject of the forthcoming work. Rainfall is the key driver of streamflow with the pattern varying from event to event, however, the underlying catchment characteristics, for example, soil, geology, topography, may modify this. A combination of inversion and spectral analysis may provide a method for untangling the effects of catchment characteristics and rainfall regime on streamflow generation and has the potential for characterising the effects of future changes in catchment and/or rainfall characteristics due to, for example, climate change.