Rainfall Intermittency Research Articles

Stochastic bias-correction of daily rainfall scenarios for hydrological applications

Abstract. The accuracy of rainfall predictions provided by climate models is crucial for the assessment of climate change impacts on hydrological processes. In fact, the presence of bias in downscaled precipitation may produce large bias in the assessment of soil moisture dynamics, river flows and groundwater recharge. In this study, a comparison between statistical properties of rainfall observations and model control simulations from a Regional Climate Model (RCM) was performed through a robust and meaningful representation of the precipitation process. The output of the adopted RCM was analysed and re-scaled exploiting the structure of a stochastic model of the point rainfall process. In particular, the stochastic model is able to adequately reproduce the rainfall intermittency at the synoptic scale, which is one of the crucial aspects for the Mediterranean environments. Possible alteration in the local rainfall regime was investigated by means of the historical daily time-series from a dense rain-gauge network, which were also used for the analysis of the RCM bias in terms of dry and wet periods and storm intensity. The result is a stochastic scheme for bias-correction at the RCM-cell scale, which produces a realistic representation of the daily rainfall intermittency and precipitation depths, though a residual bias in the storm intensity of longer storm events persists.

Rain event properties at the source of the Blue Nile River

Abstract. In the present study, spatial and temporal patterns of rain event properties are analysed. These event properties are rain event depth, event duration, mean event rain rate, peak rain rate and the time span between two consecutive rain events which is referred to as inter-event time (IET). In addition, we assessed how rain event properties change when the period over which rainfall data is aggregated changes from 1 to 6 min and when the minimum inter-event time (MIT) changes from 30 min to 8 h. Rainfall data is obtained from a field campaign in two wet seasons of June–August (JJA) of 2007 and 2008 in Gilgel Abbay watershed that is situated at the source basin of the Upper Blue Nile River in Ethiopia. The rainfall data was automatically recorded at eight stations. The results revealed that rain event depth is more related to peak rain rate than to event duration. At the start and towards the end of the wet season, the rain events have larger depth with longer duration and longer IET than those in mid-season. Event rain rate and IET are strongly related to terrain elevation. Sekela which is on a mountain area has the shortest IET while Bahir Dar which is at the south shore of Lake Tana has the longest IET. The period over which rainfall data is aggregated significantly affected the values of rain event properties that are estimated using relatively small value (30 min) of MIT but its effect diminished when the MIT is increased to 8 h. It is shown that increasing the value of MIT has the largest effect on rain event properties of mountain stations that are characterised by high rainfall intermittency.

From regional climate simulations to the hydrological information needed for basin scale impact studies

Abstract. The accuracy of local downscaling of rainfall predictions provided by climate models is crucial for the assessment of climate change impacts on hydrological processes because the presence of bias in downscaled precipitation may produce large bias in the assessment of soil moisture dynamics, river flows, and groundwater recharge. In this study, the output of a regional climate model (RCM) is downscaled using a stochastic modelling of the point rainfall process able to adequately reproduce the daily rainfall intermittency which is one of the crucial aspects for the hydrological processes characterizing Mediterranean environments. The historical time-series from a dense rain-gauge network were used for the analysis of the RCM bias in terms of dry and wet daily period and then to investigate the predicted alteration in the local rainfall regime. A Poisson Rectangular Pulse (PRP) model (Rodriguez-Iturbe et al., 1987) was finally adopted for the stochastic generation of local daily rainfall as a continuous-time point process with forcing parameters resulting from the bias correction of the RCM scenario.

Precipitation, dynamical intermittency, and sporadic randomness

While rainfall intermittency is a dynamical phenomenon, little progress has been made in the literature on the link between rainfall intermittency and atmospheric dynamics. We present the basic dynamical models of intermittency that are phenomenologically most similar to rainfall: Pomeau–Manneville Type-III and On–Off. We then illustrate each type with both a 1-D iterative map and a corresponding stochastic process stressing the appearance of these dynamics in high-dimensional (stochastic) systems as opposed to low-dimensional chaotic systems. We show that the pdf of rainfall intensities, the pdf of “laminar phases” (periods of zero rainfall intensity), and the spectrum of the rainfall series all have power-law behavior that is broadly consistent with intermittency in the classic types. Using a seasonal analysis, we find that summer convective rainfall at daily and sub-daily scales seems consistent with features of Type-III intermittency. The correspondence with Type-III intermittency and a preliminary entropic analysis further suggest that rainfall may be an example of sporadic randomness, blending deterministic and stochastic components.

Revisiting rainfall clustering and intermittency across different climatic regimes

One of the vexing questions in rainfall research is the role of intermittency and its nonuniversal signature in anomalous scaling functions. Whether this lack of universal behavior is due to the bursting patterns in rainfall intensity or the alternation between long dry periods and highly clustered wet phases (or both) remains an open issue. To progress on a narrower scope of this problem, the effects of intermittency originating from rainfall occurrence are first separated from rainfall intermittency induced by intensity variability. Across five climatic regimes considered here, it was shown that the rainfall occurrence process (OP) exhibits (1) a near‐constant spectral slope, (2) a near‐constant clustering exponent, and (3) a probability density function of dry phases displaying a power law behavior with an exponent β ≈ 1.5, consistent with other studies for timescales commensurate with frontal and storm systems. Also for the OP, the scaling exponents of the normalized higher order structure functions reveal an extensive monofractal scaling at all five climatic regimes. When taken together, these intersite results are suggestive that rainfall intensity modulations are the main cause of the nonuniversal anomalous scaling and not the clustering properties associated with the support. The nature of these modulations is markedly different when comparing rainfall to a familiar and often interrelated process such as scalar turbulence. In the case of turbulence, amplitude variability of scalar dissipation rates appear to mitigate the intermittency effects connected with anomalous scaling, while for rainfall series, intensity fluctuations seem to amplify them.

Using a furrow system for surface drainage under unsteady rain

Water excess during winter limits crop development on heavy clay soil conditions of the Gharb valley (Morocco). The furrow system to eliminate these negative effects is the adopted solution. This article focuses on the development of a water transfer model through a furrow system during unsteady rainfall event to evaluate the runoff volume resulting from a reference rainy event. This model contains a production function associated to a transfer function. The production function is based on the Green–Ampt infiltration equation. The latter has been adapted to account for unsteady rain conditions and rainfall intermittence. The transfer function is based on the kinematic wave model, the explicit solution of which is coupled with the water excess generated by the production function. Simulated runoff in the furrows is collected by a drainage ditch evacuating the flow outside a plot of 1.3 ha. The similarity between parameters of a furrow irrigation model and those of the production function is advantageously used for model calibration. The proposed modelling approach shows capabilities to predict water amount and peak discharges evacuated from a plot of around 1 ha by a furrow system under unsteady rainfall events. As an application, it is used to evaluate the ability of the surface drainage system to evacuate the excessive volumes of water under typical rainfalls.

Patterned vegetation and rainfall intermittency

We study a mathematical model for the dynamics of patterned dryland vegetation in the presence of rainfall intermittency, adopting a spatially explicit approach. We find that most results found for constant precipitation carry over to the case of intermittent rainfall, with a few important novelties. For intermittent precipitation, the functional forms of the water uptake and consequently of the vegetation growth rate play an important role. Nonlinear, concave-up forms of water uptake as a function of soil moisture lead to a beneficial effect of rainfall intermittency, with a stronger effect when vegetation feedbacks are absent. The results obtained with the explicit-space model employed here are in keeping with those provided by simpler, implicit-space approaches, and provide a more complete view of vegetation dynamics in arid ecosystems.

Modeling the effect of rainfall intermittency on the variability of solute persistence at the soil surface

A number of processes contribute to the decrease in solute concentration in the near surface; these include leaching and degradation. The combined effect of all loss processes is termed dissipation; its converse is termed persistence. In this paper we derive analytical expressions for the statistics of the time for a chemical, which undergoes linear equilibrium sorption and first‐order degradation under the influence of a random series of rainfall events, to dissipate to a specified concentration. Rainfall is characterized by a random time between events and a random storm depth, both of which are assumed to be exponentially distributed. Analytical expressions for the mean and variance of the time for dissipation to 50% (DT50) and 10% (DT90) of the applied concentration are derived. The model is in qualitative agreement with empirical observations of pesticide dissipation. The model results indicate that on average these solutes tend to dissipate exponentially with time. Rapid dissipation initially, followed by slower dissipation later, is more likely to occur. This may contribute to observations of apparent multirate dissipation which can be explained without recourse to sorption or biological kinetics. A larger average storm magnitude and/or more frequent storms reduce the mean and increase the coefficient of variation of dissipation times. A normal distribution, parameterized using the derived statistics, approximated the probability distribution of dissipation times reasonably well. On the basis of this we obtain a measure of the potential amount of chemical available for rapid off‐site leaching at a given time since application.

Investigating Error Metrics for Satellite Rainfall Data at Hydrologically Relevant Scales

Abstract This paper addresses the following open question: What set of error metrics for satellite rainfall data can advance the hydrologic application of new-generation, high-resolution rainfall products over land? The authors’ primary aim is to initiate a framework for building metrics that are mutually interpretable by hydrologists (users) and algorithm developers (data producers) and to provide more insightful information on the quality of the satellite estimates. In addition, hydrologists can use the framework to develop a space–time error model for simulating stochastic realizations of satellite estimates for quantification of the implication on hydrologic simulation uncertainty. First, the authors conceptualize the error metrics in three general dimensions: 1) spatial (how does the error vary in space?); 2) retrieval (how “off” is each rainfall estimate from the true value over rainy areas?); and 3) temporal (how does the error vary in time?). They suggest formulations for error metrics specific to each dimension, in addition to ones that are already widely used by the community. They then investigate the behavior of these metrics as a function of spatial scale ranging from 0.04° to 1.0° for the Precipitation Estimation from Remotely Sensed Information Using Artificial Neural Networks (PERSIANN) geostationary infrared-based algorithm. It is observed that moving to finer space–time scales for satellite rainfall estimation requires explicitly probabilistic measures that are mathematically amenable to space–time stochastic simulation of satellite rainfall data. The probability of detection of rain as a function of ground validation rainfall magnitude is found to be most sensitive to scale followed by the correlation length for detection of rain. Conventional metrics such as the correlation coefficient, frequency bias, false alarm ratio, and equitable threat score are found to be modestly sensitive to scales smaller than 0.24° latitude/longitude. Error metrics that account for an algorithm’s ability to capture rainfall intermittency as a function of space appear useful in identifying the useful spatial scales of application for the hydrologist. It is shown that metrics evolving from the proposed conceptual framework can identify seasonal and regional differences in reliability of four global satellite rainfall products over the United States more clearly than conventional metrics. The proposed framework for building such error metrics can lay a foundation for better interaction between the data-producing community and hydrologists in shaping the new generation of satellite-based, high-resolution rainfall products, including those being developed for the planned Global Precipitation Measurement (GPM) mission.

Rainfall intermittency and vegetation feedbacks in drylands

Abstract. We discuss a simplifed, conceptual model for the dynamics of the soil-vegetation system in drylands. The model considers the different dynamical processes taking place in vegetated and non-vegetated soil and it distinguishes between the upper soil layer, where rapid evaporation dominates, and the deeper root layer where only plant transpiration takes place. We explore the role of rainfall intermittency and of different plant colonization strategies, and discuss in detail the effect of two different vegetation feedbacks: reduced evaporation due to plant shading and increased infiltration in vegetated areas. The results of the analysis indicate that both temporal rainfall intermittency and the shading/infiltration feedbacks have a beneficial effect on vegetation. However, it turns out that in this model rainfall intermittency and vegetation feedbacks have almost a mutually exclusive role: whenever one of these two components is present, the addition of the other does not further affect vegetation dynamics in a significant way.

Evaluation of Rain Gauge Network Using Entropy Theory: Comparison of Mixed and Continuous Distribution Function Applications

In this study we compared applications of mixed and continuous distribution functions to the theory of entropy for the evaluation of rain gauge networks. The use of a mixed distribution function to evaluate rain gauge networks has an important advantage of considering rainfall intermittency in both time and space. Parameters of both mixed and continuous distribution functions were estimated using the daily rainfall data collected in the Choongju Dam Basin, Korea. The optimal number of rain gauge stations is, then, decided by using the entropy theory. The optimal number of rain gauge stations estimated by applying the mixed distribution function was found to be much smaller, but still reasonable, than that estimated by applying the continuous distribution function; mostly due to the small wet probability and the high coincidence of daily rainfall between rain gauge stations.

Analysis and Synthesis of Raindrop Size Distribution Time Series From Disdrometer Data

Hydrometeorological and radio propagation applications can benefit from the capability to model the time evolution of raindrop size distribution (RSD). A new stochastic vector autoregressive semi-Markov model is proposed to randomly synthesize (generate) the temporal series of the three driving parameters of a normalized gamma RSD. Rainfall intermittence is reproduced through a discrete semi-Markov process, modeled from disdrometer measurements using two-state analytical statistics of rain and dry period duration. The overall model is set up by means of a large set of disdrometer measurements, collected from 2003 to 2005 at Chilbolton, U.K. The driving parameters of the retrieved RSD are estimated using three approaches: the Gamma moment method and the 1D and 3D maximum-likelihood methods. Interestingly, these methodologies lead to quite different results, particularly when one is interested in evaluating RSD higher order moments such as the rain rate. The accuracy of the proposed RSD time-series generation technique is evaluated against available disdrometer measurements, providing excellent statistical scores.

Stochastic modelling of rainfall from satellite data

Summary Satellite-based rainfall monitoring is widely used for climatological studies because of its full global coverage but it is also of great importance for operational purposes especially in areas such as Africa where there is a lack of ground-based rainfall data. Satellite rainfall estimates have enormous potential benefits as input to hydrological and agricultural models because of their real time availability, low cost and full spatial coverage. One issue that needs to be addressed is the uncertainty on these estimates. This is particularly important in assessing the likely errors on the output from non-linear models (rainfall-runoff or crop yield) which make use of the rainfall estimates, aggregated over an area, as input. Correct assessment of the uncertainty on the rainfall is non-trivial as it must take account of • the difference in spatial support of the satellite information and independent data used for calibration • uncertainties on the independent calibration data • the non-Gaussian distribution of rainfall amount • the spatial intermittency of rainfall • the spatial correlation of the rainfall field This paper describes a method for estimating the uncertainty on satellite-based rainfall values taking account of these factors. The method involves firstly a stochastic calibration which completely describes the probability of rainfall occurrence and the pdf of rainfall amount for a given satellite value, and secondly the generation of ensemble of rainfall fields based on the stochastic calibration but with the correct spatial correlation structure within each ensemble member. This is achieved by the use of geostatistical sequential simulation. The ensemble generated in this way may be used to estimate uncertainty at larger spatial scales. A case study of daily rainfall monitoring in the Gambia, west Africa for the purpose of crop yield forecasting is presented to illustrate the method.

Temporal dynamics of hydrological threshold events

Abstract. The episodic nature of hydrological flows such as surface runoff and preferential flow is a result of the nonlinearity of their triggering and the intermittency of rainfall. In this paper we examine the temporal dynamics of threshold processes that are triggered by either an infiltration excess (IE) mechanism when rainfall intensity exceeds a specified threshold value, or a saturation excess (SE) mechanism governed by a storage threshold. We use existing and newly derived analytical results to describe probabilistic measures of the time between successive events in each case, and in the case of the SE triggering, we relate the statistics of the time between events (the inter-event time, denoted IET) to the statistics of storage and the underlying water balance. In the case of the IE mechanism, the temporal dynamics of flow events is found to be simply scaled statistics of rainfall timing. In the case of the SE mechanism the time between events becomes structured. With increasing climate aridity the mean and the variance of the time between SE events increases but temporal clustering, as measured by the coefficient of variation (CV) of the IET, reaches a maximum in deep stores when the climatic aridity index equals 1. In very humid and also very arid climates, the temporal clustering disappears, and the pattern of triggering is similar to that seen for the IE mechanism. In addition we show that the mean and variance of the magnitude of SE events decreases but the CV increases with increasing aridity. The CV of IETs is found to be approximately equal to the CV of the magnitude of SE events per storm only in very humid climates with the CV of event magnitude tending to be much larger than the CV of IETs in arid climates. In comparison to storage the maximum temporal clustering was found to be associated with a maximum in the variance of soil moisture. The CV of the time till the first saturation excess event was found to be greatest when the initial storage was at the threshold.

Basin‐scale soil moisture dynamics and the probabilistic characterization of carrier hydrologic flows: Slow, leaching‐prone components of the hydrologic response

This paper aims at a first theoretical step toward the probabilistic modeling of nutrient contents in catchment‐scale soil states. To this end, we wish to characterize probabilistically the slow, leaching‐prone component of the hydrologic response, which chiefly determines the export of dissolved nutrients from soil, as a result of interactions between mobile/immobile phases along the pathways of runoff formation. The influence of temporal fluctuations of soil moisture on slow components of the catchment‐scale runoff is thus investigated by means of a stochastic framework, where the intermittency of rainfall is modeled by a marked Poisson process with exponentially distributed intensities. The probability distribution of the relevant runoff components and its moment‐generating function are derived by coupling a stochastic description of soil moisture dynamics with a suitably simplified flow model. New exact solutions are achieved in two different cases, namely, when (1) infiltration rates are assumed proportional to the incoming rainfall depths, i.e., when surface runoff is negligible, and (2) infiltration rates are upwardly bounded by episodical soil saturations (e.g., for shallow soils). In both cases, the derived probability density functions of slow components of runoff are well described by a Gamma distribution, whose shape is controlled by the ratio between the runoff frequency and the inverse of the mean residence time of subsurface flow. The framework developed allows one to link the probabilistic structure of slow components of runoff with simple (pluviometric, soil, vegetation, and geomorphologic) macroscopic parameters, with implications for the ecohydrology of fluvial systems and for drought prediction in ungauged basins. Comparisons with Monte Carlo simulations of a more detailed rainfall‐runoff model to a real catchment located in northeastern Italy suggest the ability of the approach proposed to capture the main features of runoff probability distributions in heterogeneous catchments.

Vegetation response to rainfall intermittency in drylands: Results from a simple ecohydrological box model

Vegetation in arid and semi-arid regions is affected by intermittent water availability. We discuss a simple stochastic model describing the coupled dynamics of soil moisture and vegetation, and study the effects of rainfall intermittency. Soil moisture dynamics is described by a ecohydrological box model, while vegetation is represented by site occupancy dynamics in a spatially-implicit model. We show that temporal rainfall intermittency allows for vegetation persistence at low values of annual rainfall volume, whereas it would go extinct if rainfall were constant. Rainfall intermittency also generates long-term fluctuations in vegetation cover, even in the absence of significant inter-annual variations in the statistical properties of precipitation.

Estimation of areal reduction factors using a mixed gamma distribution

Summary This study proposes a methodology to estimate the areal reduction factor (ARF) using mixed probability density functions. In practice, ARFs are estimated by using contemporaneous rainfall data from a high density rain gauge network over the basin, which are rarely available. The methodology proposed in this study uses all rainfall data available, whether or not they are concurrent with others. A mixed probability density function is introduced to explain both the rainfall intermittency and variability, and is applied to both point rainfall and areal average rainfall, which are compared to estimate the ARFs. In this study, a mixed gamma distribution was applied for the Geum river basin, Korea, to estimate the ARFs of a one-day (or, 24 h) rainfall as an example. The new method was easier to apply and provided similar results with those of other studies.

Assessing the water balance in the Sahel: Impact of small scale rainfall variability on runoff

Summary The mesoscale variability of the Sahelian rainfall is analysed from a series of 30 high time resolution rainfall series covering 13 years and a 110 × 160 km2 area in the region of Niamey. It is shown that the convective scale variability is strongly influencing the spatial pattern of rainfields at larger time and spatial scales. This means that a proper assessment of the rainfall patterns at the mesoscale in the Sahel requires raingauge networks with a sufficient density to allow sampling this convective scale variability. This is usually not the case with operational networks whose density is in the order of 1–2 gauges per 10,000 km2. Computations carried out here show that the areal rainfall estimation error increases from 3% to 16% at the annual scale and from 21% to 113% at the event scale when the number of stations over a 100 × 100 km2 area decreases from 12 to 1. While being highly variable in space, the Sahelian rainfall is also highly intermittent in time. An analysis of the series of tipping bucket times leads to compute that 50% of the annual rain falls in less than 4 h with intensities larger than 35 mm/h. Areal rainfall statistics are compared to point rainfall statistics for event accumulated rainfall. The implication of these results for studying the influence of rainfall intermittency on runoff is discussed in a final section, as an introduction to a companion paper.

An entropic‐stochastic representation of rainfall intermittency: The origin of clustering and persistence

The well‐established physical and mathematical principle of maximum entropy, interpreted as maximum uncertainty, is used to explain the observed dependence properties of the rainfall occurrence process, including the clustering behavior and persistence. The conditions used for the maximization of entropy are as simple as possible, i.e., that the rainfall processes is intermittent with dependent occurrences. Intermittency is quantified by the probability that a time interval is dry, and dependence is quantified by the probability that two consecutive intervals are dry. These two probabilities are used as constraints in a multiple‐scale entropy maximization framework, which determines any conditional or unconditional probability of any sequence of dry and wet intervals at any timescale. Thus the rainfall occurrence process including its dependence structure is described by only two parameters. This dependence structure appears to be non‐Markovian. Application of this theoretical framework to the rainfall data set of Athens indicates good agreement of theoretical predictions and empirical data at the entire range of scales for which probabilities dry and wet can be estimated (from one hour to several months).

Rainfall Estimation in the Sahel. Part II: Evaluation of Rain Gauge Networks in the CILSS Countries and Objective Intercomparison of Rainfall Products

Abstract This study investigates the accuracy of various precipitation products for the Sahel. A first set of products is made of three ground-based precipitation estimates elaborated regionally from the gauge data collected by Centre Regional Agrometeorologie–Hydrologie–Meteorologie (AGRHYMET). The second set is made of four global products elaborated by various international data centers. The comparison between these two sets covers the period of 1986–2000. The evaluation of the entire operational network of the Sahelian countries indicates that on average the monthly estimation error for the July–September period is around 12% at a spatial scale of 2.5° × 2.5°. The estimation error increases from south to north and remains below 10% for the area south of 15°N and west of 11°E (representing 42% of the region studied). In the southern Sahel (south of 15°N), the rain gauge density needs to be at least 10 gauges per 2.5° × 2.5° grid cell for a monthly error of less than 10%. In the northern Sahel, this density increases to more than 20 gauges because of the large intermittency of rainfall. In contrast, for other continental regions outside Africa, some authors have found that only five gauges per 2.5° × 2.5° grid cell are needed to give a monthly error of less than 10%. The global products considered in this comparison are the Climate Prediction Center (CPC) merged analysis of precipitation (CMAP), Global Precipitation Climatology Project (GPCP), Global Precipitation Climatology Center (GPCC), and Geostationary Operational Environmental Satellite (GOES) precipitation index (GPI). Several methods (scatterplots, distribution comparisons, root-mean-square error, bias, Nash index, significance test for the mean, variance, and distribution function, and the standard deviation approach for the kriging interval) are first used for the intercomparison. All of these methods lead to the same conclusion that CMAP is slightly the better product overall, followed by GPCC, GPCP, and GPI, with large errors for GPI. However, based on the root-mean-square error, it is found that the regional rainfall product obtained from the synoptic network is better than the four global products. Based on the error function developed in a companion paper, an approach is proposed to take into account the uncertainty resulting from the fact that the reference values are not the real ground truth. This method was applied to the most densely sampled region in the Sahel and led to a significant decrease of the raw evaluation errors. The reevaluated error is independent of the gauge references.