Definition – why we need models

A model is a representation of one or more concepts that may be realized in the physical world (Friedenthal et al., 2014). Consequently, a model always describes the basic and most important components of a complex system. Modeling supports the conceptual exploration of the behavior of object or process and their interaction. Modeling is a mean of better understanding and generating hypotheses; it also supports the experiments in which hypotheses can be tested, and the outcomes can be predicted (Gregory, 1998). Modeling is also a good way to reproduce data which are difficult to measure (e.g. floods) from data which are more easily available (rainfalls, catchment geography).

The hydrologic real-word system can be simplified and characterized using a RR model.

By modeling runoff, we can better understand the hydrologic phenomena, as well as how those phenomena affect the hydrological cycle (Xu, 2002). RR models also help to reproduce floods in any situation which cannot be observed, or which is difficult to observe: for example, extrapolation to extreme floods, floods in ungauged catchments, forecast of floods.

A runoff model can be defined as a set of equations which helps estimate the amount of rainfall that turns into runoffs (Devia et al., 2015). These equations involve input variable (e.g. rainfall, slopes, etc.), output variable (here, various characteristics of the flood: runoff volume, peak flow, rising time, etc.), and parameters which describe the watershed. A variable in a hydrological system is understood to be a characteristic which may be measured, which assumes different values when measured at different times. Daily rainfall, runoff, evaporation, temperature, infiltration, soil moisture, etc. are some of the examples (Xu, 2002). A parameter is a quantity characterizing a system. The parameter is internal to the model and it can be estimated from data. It may or may not remain constant in time.

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Since the first runoff prediction model appeared in the 19^th century, RR models continued to develop until now. The major development of RR models is made thank to the availability of the learning data sets which were used to calibrate the nonlinear behavior of those models. Currently, a wide range of RR models are used, which applications highly depend on the purpose of modeling. Those models can be helpful in decision making by providing a means of quantitative extrapolation or prediction. RR models can be classified and divided into different types based on spatial resolution, input/output type, model simplicity, etc.

Despite the different classification, not all models fit into a single category because they are normally developed for different purposes (Singh, 2012). The selection of an appropriate RR models needs to be based on the modeling purposes, for instance, to understand and answer a specific question about hydrological processes, to assess the frequency of runoff events, to estimate the runoff generated for management (Vaze et al., 2011). Choosing the most appropriate models also based on the data availability, time and budget for modeling.

The first way to classify RR models is based on the model structure, which determines how runoff is calculated, by which RR models are categorized into empirical, conceptual and physical models. Some RR models can involve a few variables while others may require a large number of interconnecting variables. The model can have the structure varying simple to complex depending on the equations. The empirical model is the simplest type, while the physical mechanistic model is the most complex type of RR model. The application of both physical and conceptual models requires an understanding of the physical process involving in the movement of surface water in the hydrological cycle (Srinivasulu and Jain, 2008). The differences between the three types of models are explained in detail in Table 1.1.

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Table 1.1: Comparison of the basic structure for rainfall-runoff models (Sitterson et al., 2017)

Empirical Conceptual Physical interpretation of the model’s catchment area. The spatial interpretation of the catchment area is based on input data and the way in which runoff is generated and routed over the catchment (Figure 1.8). Following this classification, RR models are categorized into lumped, semi-distributed and distributed models. These models differ in consideration of variability in geology, soils, vegetation, and topography of a catchment which affect the relationship between rainfall and runoff (Beven, 2012)

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Figure 1.8: Visualization of the spatial structure in runoff models. A: Lumped model, B: Semi-Distributed model by sub-catchment, C: Semi-Distributed model by grid cell (Sitterson et al., 2017)

In particular, the lumped model does not consider the spatial variability within a catchment, while the semi-distributed model does consider some spatial variability, and distributed models consider and process spatial variability via the grid cells. In other words, different types of model consider different spatial processes and produce different output type. The detailed differences between those three types of models are presented in Table 1.2.

Table 1.2: Comparison of the spatial structures in rainfall-runoff models (Sitterson et al., 2017)

Lumped Semi-Distributed Distributed

sub-catchment All specific data by cell

Strengths

Fast computational time, good at simulating average conditions

Represents important

features in catchment Physically related to hydrological processes

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machine learning TOPMODEL[a], SWAT[b] VELMA[d]

[a] (Devia et al., 2015) [b] (Beven, 2012) [c] (Singh, 2012)

[d] (McKane, R. et al., 2014)

The final classification that is mentioned in this report is continuous/event-based RR models. Depending on how the model chooses the required initial condition, hydrological models will be categorized as continuous or event-based. The continuous approach requires an initialization, or warm-up period, in which the model is run until its state reaches a value that is no longer dependent on an arbitrarily selected initial value. The duration of this initialization period depends on the memory of previous conditions of the catchment and on the model. The duration of this period may last for a few months (Kitanidis and Bras, 1980b) or for one climatic cycle, however, for certain catchments with large aquifers feed streamflow, this period needs up to several years (Le Moine, 2008). This data requirement leads to one major drawback of continuous approach in operational forecasting perspective, as it is often difficult to provide the long continuous precipitation time series up to the day of interest due to difficulties in real-time data repatriation. Thus, it is necessary to gather a sufficiently long data series before making the first forecast in a new location if the continuous approach is used.

On the contrary, event-based models use a different method to obtain the initial value of the model states. There are different methods that are used in the event-based approach.

The values can be defined based on the climatology if the model states reliably represent measurable physical quantities. For instance, the event-based SCS-CN model can be integrated with soil moisture measurements for flow simulation on the small catchment (Brocca et al., 2009), however, the results should be generalized. Although continuous models have been applied a rigorous approach to estimate the initial conditions for a long time (Kitanidis and Bras, 1980a), event-based models are often preferred in real-time operational applications (Lamb and Kay, 2004). This is partly because event-based models are simpler, as they do not require all the necessary process as in continuous models, thus this type of models is more suitable to limited data. Besides, it is often challenging to look for

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high time resolution series, as it is difficult to maintain and validate automatic measurements networks over a long period of time in many countries. Nalbantis once suggested the use of coarser data series (e.g. daily) to estimate fine initial conditions to overcome this challenge (Nalbantis, 1995). Furthermore, the event-based approach may be culturally favored by some people who are traditional users of hydraulic propagation methods. Due to the reasons mentioned above, event-based models are becoming more widely used by practitioners.

Moreover, event-based models are useful for other purposes rather than flood forecasting, such as torrential flood modeling.

As opposed to continuous modeling which simulates streamflow over a long period (Stephens et al., 2018), the event-based approach considers each individual rainfall-runoff event separately. As precipitation and streamflow data are rarely collected at hourly or finer scales for a long period due to limited data availability, continuous hydrologic models are usually applied at daily scale. Meanwhile, event-based hydrologic models can be applied at sub-daily time scale (Yao et al., 2014), such as hourly or finer scale.

The continuous approach considers some flow components, including for example evapotranspiration and subsurface flow. Meanwhile, these flow components might be excluded to some extent or become less important in event-based approach, in which their underlying processes are not active for the considered rainfall-runoff events (Huang et al., 2016). The event-based models have advantages such as using limited data or reducing the complexity of the model and the number of parameters. However, they need to be initialized for each event, and the initial condition of the model has to be derived from an external variable, which expresses the state of the catchment at the beginning of the rain event. A comparison of event-based and continuous models was performed elsewhere (Berthet et al., 2009). It showed that both types of model were more or less equivalent. Recently, another study also compared continuous and event-based models and showed that event-based models are suitable for climate change impact (Stephens et al., 2018). However, event-based models are often the only option in many cases, when a complete series of data are not available.

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