Watersheds degrade and this makes river flow less predictable: bigger floods and lower dry season flow. A parsimonious null model of flow persistence links local knowledge to hard data.
In the analysis of watershed functions, we deal with a complex of factors that influence processes and patterns in the landscape that ultimately translate a temporal pattern of rainfall into a temporal pattern of stream flow, which aggregates up to a river. Downstream stakeholders start from what they want to see ('perfectly regular flow of clean water') and observe a pattern of stream and river flow that doesn't match their expectations. They search for interventions on the 'anthropogenic' groups of causes ('deforestation', 'degradation'), but need to understand the potential reach of such interventions in view of the geological and climatic background. In the absence of knowledge of what happens upstream, an observer of river flow can deduce a fair amount of information from a time series of river flow data.
Models of river flow, even relatively simple ones such as GenRiver, are over-parameterized relative to the information that we can use to check the statistical validity of the model. There are multiple ways of achieving a similar level of 'model fit' between measured and predicted river flow patterns, and the fit obtained may thus be 'right for the wrong reasons'. Using the 'validated model' outside of the calibration range may then be as risky as using a simple regression line. In testing the 'lack of fit' of a model we can benefit from having a 'null-model', a model that takes basic properties of the data into account, without specific hypotheses about the way rainfall translates into river flow.
The FlowPer.xls model provides a parsimonious null-model, that is based on temporal autocorrelation or an empirical 'flow persistence' in the river flow data. The basic form is a recursive relationship between river flow Q at subsequent days:
Qt+1 = fp Qt + Qadd
where Qt and Qt+1 represent the river flow on subsequent days, fp is the flow persistence factor ([0< fp <1]) and Qadd is a random variate that reflects inputs from recent rainfall.
Qadd and fp are related, as Sum Qadd1 = (1 – fp) Qadd. Thus, if fp = 1, Qadd = 0 and river flow is constant, regardless of rainfall (the ideally buffered system…). If fp = 0 there is no relation between river flow on subsequent days and the river is extremely 'flashy', alternating between high and low flows without temporal predictability within the frequency distribution of Qadd.
The term Qadd1 can be described as a statistical distribution with a probability of a non-zero value, a mean and a measure of variance, plus two parameters that describe a seasonal pattern (peak and shape of the distribution, e.g. Weibull). This makes for 5 parameters for Qadd1 (and six for the whole model) that are derived from the data. It leaves many degrees of freedom for more specific models that, for example, make use of measured rainfall.
Figure 1. Multiple influences on process and pattern of river flow and the downstream perceptions of 'ecosystem services' (modified from van Noordwijk et al. 2006)
Figure 2. Example of the type of ‘fit’ that can be achieved for the 6-parameter null-model
If we partition the total flow Qtot into water flow by three pathways (surface runoff, interflow and groundwaterflow), we can obtain Qtot = Qrunoff + Qinterflow + Qgwflow . Each type of flow pathway will typically have a different flow persistence, fp,runoff , fp,interflow and fp,gwflow , respectively.
Qtot,t+1 = (fp,runoff(Qrunoff,t/Qtot,t)+ fp,interflow(Qinterflow,t/Qtot,t)+ fp,gwflow (Qgwflow,t/Qtot,t))Qtot,t+ Qadd,t
As we can expect values for fp,runoff , fp,interflow and fp,gwflow of about 0, 0.5 and close to 1, respectively, we can interpret the relative contributions of the 3 flow pathways from the overall f value.
In a more detailed model, the daily value of f will shift according to the predicted contributions of the three types of flow, rather than being a constant. Together with the way Q relates to rainfall, this gives space for improved model fits.
Part of the 'flow persistence' may in fact derive from 'rainfall persistence', or the increased
probability of daily rainfall after a rainy day, and/or from the increased probability of dry days
to follow dry days, even after a monthly pattern ion rainfall is accounted for.
Data quality
When applying the GenRiver model to landscapes where at least some riverflow data are available, there is an opportunity to assess the 'lack of fit' between model and measurements. Lack of fit can be due to 1) inaccuracy or error in the data (e.g. with incomplete representation of spatial variability on rainfall, and/or errors in the data records), 2) suboptimal model parametrization, 3) error and/or oversimplification in the model process description. Component 3 can only be assessed if components 1 and 2 can be quantified. Tests of data consistency can be used to assess component 1, e.g. at seasonal aggregate level. Steps can include:
A. Sum P - Sum Q gives an estimate of total evapotranspiration. Values below 500 or above 1500 mm/year are suspect. These may indicate errors in P or Q registration, error in the area or deviation from the 'closed catchment' assumption (e.g. subsurface flows out of or into the catchment are non-negligible).
B. Cumulative Sum Q versus Sum P during the year: large jumps will require explanation.
C. Flow persistence Qi+1 versus Qi plots may indicate gaps in the data or 'outliers' that indicate errors.
Parametrization of FlowPer
Once flow data have passed minimum quality checks, we can use them to parameterize the null model, esp. the Fp parameter.
If an fp value of zero is used, the mean of Qadd will equal the mean flow, but the variance of daily Qadd estimates will be high. If an fp value of 1.0 is used, many of the Qadd estimates obtained will be negative, and the variance will be relatively high. In between these extremes we expect a local minimum for the variance of Qadd, at an intermediate fp value. The model uses this property to provide the final estimate.
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