Intra-annual rainfall variability control on interannual
variability of catchment water balance: A stochastic analysis
S. Zanardo,1,2C. J. Harman,3P. A. Troch,4P. S. C. Rao,1and M. Sivapalan5,6
Received 10 August 2010; revised 7 November 2011; accepted 26 November 2011; published 18 January 2012. [1] We evaluate the extent to which within-year rainfall variability controls interannual
variability of catchment water balance. To this end, we analytically derive the probability density function of the annual Budyko evaporation index, B (i.e., the ratio of annual actual evapotranspiration to annual precipitation), by accounting for the stochastic nature of intra-annual rainfall fluctuation and neglecting all other sources of variability. We apply our analytical model to 424 catchments located in different climatic regions across the
conterminous United States to perform this assessment. In general, we found that the model is capable of explaining mean B but is less accurate in predicting its coefficient of variation. Nonetheless, in a significant number of catchments the model can provide adequate
predictions of the probability density function of B. Clear geographic patterns can be distinguished in the residuals between observed and predicted statistics of B. Interannual variability is thus not always associated with random intra-annual rainfall fluctuations. In some regions, other controls, such as seasonality and vegetation adaptations, are possibly more important. A sensitivity analysis of model parameters helped characterize the dominant controls on the distribution of B in terms of three dimensionless ratios that include climatic and soil characteristics. This study represents the first step in a diagnostic, data-driven analysis of the climatic controls on the interannual variability of catchment water balance.
Citation: Zanardo, S., C. J. Harman, P. A. Troch, P. S. C. Rao, and M. Sivapalan (2012), Intra-annual rainfall variability control on interannual variability of catchment water balance: A stochastic analysis, Water Resour. Res., 48, W00J16, doi:10.1029/ 2010WR009869.
1. Introduction
[2] Catchments can be considered as space-time filters, within which the variability of soil moisture, runoff and evaporation reflects the temporal variability of climatic inputs as well as the heterogeneity of soils, topography and vegetation [Wagener et al., 2007]. Understanding the rela-tive roles of climate, soil, topography and vegetation in controlling the water balance in a catchment is important for assessing the effects of climate and landscape changes on water resources and ecosystem services.
[3] At the annual scale, water balance variability may arise straightforwardly from the interannual fluctuation of cli-mate drivers, possibly altered by long-term changes in land-scape properties. However, intra-annual (i.e., within-year)
variability of climate drivers and their nonlinear interac-tions with the landscape can also affect the interannual water balance variability. At the catchment scale, the an-nual partitioning of precipitation into runoff, water vapor (i.e., evaporation and transpiration) and soil moisture stor-age can be the result of runoff generation and vegetation-soil-topography interactions operating at short time scales. For example, the soil moisture storage prior to a storm con-trols the amount of runoff generated during a storm [Ye et al., 1997]. These antecedent soil moisture conditions are in turn controlled by the previous history of rainfall and evapotranspiration, and will exhibit spatial variations depending on soil, topography and vegetation [Teuling and Troch, 2005]. The cumulative effects of these interactions occurring during individual storms and during interstorm periods can lead to significant differences in the water bal-ance between years. Additionally, significant differences in interannual variability can be observed between catchments because of differences in climate, soils, topography, and vegetation cover. The overall aim of this work is to investi-gate the effect of short-term hydrologic fluctuations, repre-sented by the event-scale rainfall pattern, on the variability of annual water balance.
[4] We analyze the catchment-scale water balance fol-lowing the conceptualization provided by Budyko [1974]. Budyko’s hypothesis empirically describes the patterns of long-term water balance between catchments in different cli-mates. The Budyko curve expresses the long-term Budyko evaporation index (ratio of long-term mean evaporation to
1School of Civil Engineering, Purdue University, West Lafayette,
Indi-ana, USA.
2Dipartimento di Ingegneria Idraulica Marittima Ambientale e
Geotecn-ica, Universita` degli Studi di Padova, Padua, Italy.
3Department of Civil and Environmental Engineering, University of
Illi-nois at Urbana-Champaign, Urbana, IlliIlli-nois, USA.
4Department of Hydrology and Water Resources, University of Arizona,
Tucson, Arizona, USA.
5Department of Geography, University of Illinois at Urbana-Champaign,
Urbana, Illinois, USA.
6Water Resources Section, Faculty of Civil Engineering and
Geoscien-ces, Delft University of Technology, Delft, Netherlands.
Copyright 2012 by the American Geophysical Union 0043-1397/12/2010WR009869
long-term mean precipitation) as a function of the aridity index (ratio of long-term mean potential evaporation to long-term mean precipitation). There have been several studies aimed at elucidating the climate-soil-vegetation interactions that can explain Budyko’s curve [Milly, 1994; Farmer et al., 2003; Rodriguez-Iturbe and Porporato, 2004]. While previous focus has been on understanding the controls on the long-term Budyko evaporation index, less effort has been spent on investigating how the index varies, when it is calculated over a limited time period. For exam-ple, as shown in several recent studies [e.g., Zhang et al., 2008; Donohue et al., 2010], the variability of the annual Budyko evaporation index (i.e., calculated year by year) may in fact be considerable.
[5] In this paper we explore the effects of within-year rainfall fluctuations on the variability of the annual Budyko evaporation index, which from now on will be referred to as either B or the ‘‘evaporation index.’’ In order to isolate the effects of rainfall events from the other sources of vari-ability that influence fluctuations of annual water balance, a model is required. To this end, we use a well-known sto-chastic framework able to probabilistically characterize soil moisture dynamics [Rodriguez-Iturbe et al., 1999] as well as runoff production [Botter et al., 2007a] at some spa-tial scale. The model is based on event-scale rainfall prop-erties (i.e., mean intensity and mean frequency) and lumped climatic and soil parameters. In particular, we derive an analytical expression for the probability density function of B, which is solely based on rainfall fluctuations and purposely neglects all other sources of variability.
[6] Predictions produced by our stochastic model are then compared against estimates obtained from data of 424 catchments across the continental United States. This will provide a strong, data-driven evaluation of the effect of short-term rainfall fluctuations on annual water balance variability. Moreover, a sensitivity analysis on model pa-rameters will enable us to point out the main controls on the distribution of B.
[7] The use of a simple modeling approach, where only some (hypothetical) major controls are accounted for, in combination with an extensive data analysis of both inter-annual and inter-catchment variations, can be very useful in providing insights into the main causes of the observed differences between places. Within this framework, i.e., the so-called top-down approach [Sivapalan et al., 2003], inad-equate model predictions are not considered as failures of the model, but rather as diagnostics that reveal the strength of other controls that may have been left out. This approach, given the broad data set considered, will hope-fully lead to improved and parsimonious models of catchment responses to long-term climate variability and change.
2. Conceptual Framework
[8] At the catchment scale, the annual vaporization, V, can be estimated as the difference between annual precipi-tation, P, and annual runoff, Q. For our conceptual frame-work we will thus refer to the following definition of the evaporation index, B:
B ¼ V
V þ Q: (1)
It should be noted that this simple conceptualization requires the assumption that the carry-over of catchment water storage from one year to the next is negligible when compared against annual precipitation, evaporation and runoff (an assumption commonly made [e.g., Koster and Suarez, 1999]).
[9] In what follows we provide an analytical framework to derive the probability density function (pdf) of B, in which the only source of variability is the fluctuation of rainfall events. In order to achieve a probabilistic character-ization of B, we first derive an analytical expression for the pdf of Q. To this end, we follow the model framework pro-posed by Porporato et al. [2004], which has been recently extended by Botter et al. [2007a], to derive the pdf of daily streamflows. The catchment is conceptualized as a single lumped unit and rainfall is assumed to be uniformly distrib-uted over the drainage area, all spatial scale effects are neglected. The time step used to characterize within-year variability of rainfall is daily. Following Rodriguez-Iturbe et al. [1999], we assume that the occurrence of random rain-fall events can be modeled as a Poisson process with rate pa-rameter p. We then assume that depths of rainfall events are
exponentially distributed with a parameter representing the inverse of the mean rainfall depth. The active part of the soil, or root zone, is described by constant, spatially averaged properties (rooting depth, Zr½L, porosity, n (dimensionless), relative soil moisture, s (dimensionless)). A drainage event, q, which is the part of rainfall that leaves the catchment as runoff, is triggered in the source zone as soon as the soil moisture exceeds a critical threshold, s1. Although this
threshold is known to fall somewhere between field capacity and complete saturation, it is generally difficult to estimate its value. Therefore, following Botter et al. [2007a], we assume it to be equal to field capacity. The evapotranspira-tion rate is assumed to linearly decrease from a constant maximum value (which we will refer to as potential evapo-transpiration rate (PET)), when s ¼ s1, to 0 at the wilting point, s ¼ sw.
[10] Following Botter et al. [2007a, 2008, 2010], we fur-ther assume that the drainage events, q, also constitute a Poisson process with rate parameter p. With these
assumptions, the distribution of depths of drainage events q is not altered during the passage through the vadose zone and remains exponential with parameter [Rodriguez-Iturbe et al., 1999]. The pdf of the interarrival times, , between drainage events and the pdf of drainage depths, q, can then be expressed as follows:
pðÞ ¼ e; (2)
pqðqÞ ¼ eq; (3) where can be directly calculated as a function of p, ,
the water storage capacity, Zrnðs1swÞ, and PET [Laio
et al., 2001].
[11] The model, so far, probabilistically characterizes the frequency and the magnitude of drainage events. Total run-off, Q, over a year can be obtained as the sum of the runoff volumes, q, from all of the drainage events that occur over the year, provided that travel times of runoff through the catchment is much smaller than a year.
[12] On the basis of equations (2) and (3) it can be shown that the pdf of annual runoff, Q, is given by
pQðQÞ ¼X
1 ¼1
pQðQjÞ pðÞ; (4)
where pQðQjÞ is the pdf of Q conditional to the number of
drainage events, , and p is the Poisson distribution with rate . Equation (4) represents the interannual variability of runoff produced by rainfall event fluctuations alone. An analogous expression for annual rainfall totals is used by Porporato et al. [2006] and compared with the one obtained by explicitly including rainfall interannual variability. In this study, we are testing the hypothesis that the simple rain-fall event fluctuation is the dominant control on the variabil-ity of the annual water balance; to this end, we purposely neglect all other sources of variability, and therefore, we base the derivation of the pdf of B upon equation (4).
[13] In order to derive the pdf of the evaporation index (equation (1)), it would be necessary to fully take into account the variability in the annual vaporization V, including its possible covariation with Q, which poses se-rious mathematical difficulties. In this paper we pursue an analytical approach in order to benefit from the advantage that it could explicitly reveal the main climatic controls on the variability of B. Therefore, as a first step, we use a simplified approach, where we replace the vaporization V in equation (1) with its long-term climatic mean. Essen-tially, this assumption postulates that annual vaporization is a second-order control on the interannual variability of the evaporation index, relative to annual runoff, which is assumed to be the first-order control. This assumption is consistent with the work of Oishi et al. [2010], who pro-vided an explanation for the observed relative constancy of annual vaporization (i.e., relative to interannual variability of runoff), as well as Daly and Porporato [2006]. Our data set (i.e., the 424 Model Parameter Estimation Experiment (MOPEX) catchments) also shows that for half of the sites considered, the coefficient of variation (CV) of annual va-porization is at least 2.7 times lower than the CV of annual runoff. A detailed analysis of the relative variability of va-porization and runoff, as well as their correlation, is pre-sented in the auxiliary material.1 The effects of this
assumption are further discussed in section 3.2.
[14] On the basis of the conceptual framework proposed by
Rodriguez-Iturbe et al. [1999], Porporato et al. [2004] pro-vided an analytical expression for the mean daily evapotrans-piration rate, <ET>, as a function of the soil moisture dynamics and the stochastic rainfall process. Mean annual va-porization is then computed as the product of <ET> and the length, L, of the time period considered (i.e., one year in our case).
[15] The pdf of the evaporation index, pBðBÞ can finally
be derived from the pdf of the annual runoff, pQðQÞ as
pBðBÞ ¼ pQðQðBÞÞdQdB; (5)
where from equation (1), QðBÞ ¼ ðV VBÞ=B. Given equation (4), the pdf of the evaporation index is expressed as follows: pBðBÞ ¼ V eðB1ÞVB LððB 1ÞV Þþ ffiffi L p ffiffiffi p ffiffi p eðB1ÞV B LI1 2 ffiffiL p ffiffiffiffiffiffiffiffiffiffiffiffi1 B1 ð ÞV p ffiffiffi p ffiffi p ffiffiffiffiffiffiffiffiffiffiffiffi1 B1 ð ÞV p ! B2 ; (6)
where V ¼ <ET>L and is the Dirac delta function. 3. Application to the MOPEX Data Set
[16] The theoretical derivations and analysis presented thus far provide a framework for understanding the controls on interannual variability of the evaporation index. We will now apply this model to data from the MOPEX database (http://www.nws.noaa.gov/oh/mopex) in order to test the validity of the approach and, in particular, to test the hy-pothesis that rainfall event fluctuations represent a domi-nant control on the interannual variation of water balance. Note that this same database is also used in companion papers [Sivapalan et al., 2011; Harman et al., 2011], all of which approach the nature and causes of interannual vari-ability from different perspectives. The MOPEX data set includes approximately 50 years of rainfall and streamflow records, as well as estimates of soil properties (i.e., poros-ity, field capacity and wilting point), for 424 catchments located across the continental United States, belonging to a variety of ecoregions (Figure 1). The annual humidity index (i.e., the ratio of mean annual precipitation to mean annual potential evapotranspiration) for these catchments varies from 0.1 to 4 and the drainage areas vary from 66 to 10,329 km2.
3.1. Validation
[17] The stochastic model was applied to the MOPEX database in order to test whether it is able to reproduce
Figure 1. Locations of Model Parameter Estimation
Experiment (MOPEX) sites across the United States.
1Auxiliary materials are available in the HTML. doi:10.1029/
observed variations of the mean and the coefficient of vari-ation of the evaporvari-ation index, B. For each year of record and for all the 424 catchments we obtained B from rainfall and runoff records; then, for each catchment we calculated the distribution of B, as well as its mean and variance. The parameters used in the model were estimated from the MOPEX data set. The parameters p and represent the
mean rainfall event frequency and the inverse of the mean rainfall depth computed using the entire daily rainfall records; the potential evapotranspiration (PET) is obtained from the NOAA’s free water evaporation atlas [Farnsworth et al., 1982]. Soil properties are available within the MOPEX data set except for the catchment average rooting depth, Zr. In order to estimate the rooting depth we used the model pro-posed by Schenk [2008], which relates the expected value of rooting depths to long-term precipitation and evapotranspira-tion estimates, as well as to soil texture. We computed the distribution of rooting depths for all the MOPEX catchments and used, as an effective value for our model, the depths con-taining 95% (D95) of all roots in the profiles.
[18] Probability distributions of evaporation index obtained from the data and those based on model predictions are compared using the Cramer–Von Mises test, which pro-vides an integrated measure of the difference between two cumulative distributions. Figure 2a shows the exceedance probability of the test p-value across the MOPEX catchments as an integrated representation of the efficiency in all the sites considered. A common significance level used for this test is 0.05; Figure 2a shows how, referring to such value, the hypothesis that the predicted distribution fits the observed one should be rejected for approximately 50% of the sites.
[19] A further evaluation of the accuracy of the model can be carried out by comparing the observed means and coefficients of variation of B for all the 424 sites to the ones calculated with the model (Figures 2b and 2c). The range of variability of mean B is between 0.2 and 1, whereas observed coefficients of variation vary in the range 0 to 0.6. Figure 2b shows a good agreement between data and model output; the coefficient of determination is 0.57. As for the coefficient of variation of B, Figure 2c shows that, while for many catchments the model still gives adequate predictions, for a large fraction of the catchments the model does a relatively poor job; the coefficient of determination in this case is 0.15.
[20] Overall this analysis shows that in spite of the sim-plifying assumptions adopted, the model is able to repro-duce the mean evaporation index, and, in many of the catchments considered, even its interannual variability. In the remainder of the MOPEX catchments, which is still a considerable number, the model failed at predicting the
Figure 2
Figure 2. Evaluation of model predictions. (a) The
exceedance frequency of the p-value from the Cramer–Von Mises test obtained by comparing analytical and observed cumulative distribution of the evaporation index, B, for all the MOPEX catchments. (b) The comparison between modeled and observed means of B. (c) The comparison between modeled and observed coefficients of variation of B. R2 is the coefficient of determination, E
ns is the Nash-Sutcliffe efficiency coefficient, and RMSE is the root-mean-square error.
coefficient of variation of B. This implies that in those cases, the model is not able to capture some important mechanisms responsible for the interannual variability of B, which therefore turns the focus of the analysis on the assumptions used in the derivations. In fact, model failures were expected given the broad data set considered and it would be interesting to assess in which regions and/or under which climatic conditions the simple model fails. This will help to identify the sites where controls other than the ones considered in the model may play an important role. This will be the subject of discussion in section 5.
3.2. Evaluation of the Major Assumptions
[21] The discrepancies between model predictions and data-based estimates of the evaporation index may be due to an inadequacy of our model assumptions to represent the actual system. In the analytical derivation we made several simplifying assumptions; some of them, such as the expo-nential distribution of rainfall depths and the assumption of negligible carryover between years, are commonly used and accepted, therefore they will not be further discussed. On the other hand, the approximation of the drainage inputs as a Poisson process and the assumption of constant annual vaporization are somehow newer, or at least less used in previous studies, therefore we believe their effect is worth exploring. Given the large data set available it is possible to test whether, and to what extent, such assumptions are
responsible for model failures, which is the purpose of section 3.2.
[22] The effects of the assumptions are evaluated in the following way: first we identify an index that measures the impact of the assumption, then we evaluate such index for all the catchments, and finally we show to what extent pre-dictions improve as we progressively refine the data set by removing those catchments in which the index exceeds a certain threshold. If the quality of predictions, evaluated in terms of the coefficient of determination, R2, increases as
we progressively refine the data set, we can conclude that the assumption has an effect on the model efficiency.
[23] The assumption of random drainage events concep-tualized as a Poisson process is suitable when the memory involved in the drainage process is relatively small. Here we use the coefficient of variation (CV) of the drainage interarrival times as a measure of the memory of the drain-age process. Memoryless processes have a CV of their interarrival times equal to 1; however, for most of the catchments considered, this CV is found to be larger than 1, going up to a value of 2.2. The larger this value, the more dispersed the distribution of the interarrival times and therefore the larger the memory of the drainage process. CV of drainage interarrival times was numerically com-puted with the soil moisture model for each of the MOPEX catchments. In Figures 3a and 3b we show how predictions of the mean and the coefficient of variation of B improve as
Figure 3. Evaluation of the two main model assumptions: (a and b) Poissonian drainage and (c and d)
constant annual evapotranspiration. On the x axis the threshold values for the refining procedure is reported (see text), and the y axis shows the coefficient of determination, R2, for the predictions relative to each refined data set. Figures 3a and 3c show the comparison between modeled and observed mean of B after progressively refining the data set, and Figures 3b and 3d show the coefficient of variation of B.
we progressively refine the data set by removing the catch-ments in which the CV of drainage interarrival times is larger than a variable threshold (reported on the x axis). The graphs show how the efficiency (in terms of R2, y axis)
of the model in predicting the mean of B, clearly increases with the refining procedure. As for the coefficient of varia-tion of B, the efficiency increases up to a certain value and then starts decreasing; this indicates that for low values of the CV of drainage interarrivals, the Poisson assumption is no longer a limiting assumption for the prediction of the CV of B and there exist more important sources of error. There are few cases in which interarrival times have CV less than one, which implies that their distribution decays faster than for an exponential distribution and that the drainage arrivals are more equally spread out, which still indicates memory in the process. For the purpose of this evaluation we considered a positive difference between these CVs and 1.
[24] To evaluate the impact of the approximation of vari-able annual vaporization by its mean value we considered the ratio between the coefficient of variation of annual va-porization and the coefficient of variation of the annual run-off, assuming that the smaller this ratio the smaller the impact of the assumption. In Figures 3c and 3d we show the R2 for the predictions of mean (Figure 3c) and CV
(Figure 3d) of B, after progressively refining the data set. The efficiency of the model in predicting the mean of B does not seem to significantly increase as the data set is progressively refined. As for the CV of B, an improvement can be observed only for the lower values of the refining threshold, but there is no continuity with the R2values for
higher thresholds. Therefore, it is difficult to tell whether the sudden increase of efficiency is due to the suitability of the assumption of constant vaporization or the subsets of catchments happen to possess characteristics that increase the predictability of the CV of B. Overall, the results in Figure 3 suggest that the approximation adopted is less crit-ical than the assumption that drainage events can be con-ceptualized as a Poisson process.
[25] The effect of a variable annual vaporization can also be evaluated by simply comparing the two distributions of the evaporation index computed assuming a constant and a variable vaporization, respectively. This was done in two ways: first using the observed data (i.e., using the measured annual runoff and vaporization) and then using a numerical model applied to each catchment (i.e., computing annual runoff and vaporization with the model on the basis of the MOPEX data set). In both cases, the comparison between the pairs of pdfs of the evaporation index computed with variable and constant vaporization values yields further insights into the viability of this assumption. The result is shown in Figure 4, which reports the exceedance frequency across the catchments of the Cramer–Von Mises p-value, computed for each of the catchments and for both the meth-ods (using the data and using the model). The insets repre-sent two examples of the comparison of the pdf of the evaporation index obtained for variable annual vaporization and the corresponding pdf obtained with constant (mean) annual vaporization for the data case; the purpose is only to provide a qualitative evaluation of the goodness of fit associated with the p-value of the Cramer–Von Mises test. If we refer to a significance level of 0.05, only a few (2%
of the whole data set) of the pairs of distributions computed for each site can be considered statistically different. This holds for both the pairs of distributions obtained with the model and the pairs of distributions obtained with the data. It is interesting that the variability of annual vaporization does not seem to seriously impact the distribution of the an-nual evaporation index for most of the catchments, which confirms the adequacy of the assumption. A similar result was also presented by Sivapalan et al. [2011], although Figure 4 provides stronger and more definitive evidence. 4. Sensitivity Analysis on Physical Controls
[26] The mean and standard deviation of B can be expressed as functions of three dimensionless variables (see the auxiliary material): (1) (¼ PET=p), which
represents the Budyko aridity index, (2) the capacity index, (¼ nZrðs1swÞ), defined as the ratio of the storage
capacity of the soil to the mean precipitation depth, and (3) the mean number of storm events in the time period of in-terest, (¼ pL). While the first and the second groups
were already used in previous studies [e.g., Porporato et al., 2004], the third one stems from the fact that our analy-sis focuses on accumulated quantities, which thus depend on the accumulation period. The numbers and are pure climate indicators, representing the competition between energy and water availability and the frequency of storm events compared with the duration of the period consid-ered, respectively. On the other hand, the capacity ratio embeds a soil as well as a climate component.
[27] It needs to be pointed out that in the data analysis described in section 3 we always use L ¼ 365 days, there-fore in this case the parameter trivially represents the rainfall frequency, p. However, our analytical solution is
not limited to the case of a constant time length, L, which is in fact an important parameter. For this reason, in section 4 we evaluate the effect of all the derived dimensionless vari-ables (including ) by means of a sensitivity analysis. Results provided hold only in those cases where our frame-work does apply, i.e., when the main source of variability for the annual water balance is the storm fluctuation.
[28] In Figure 5 we show how mean (Figure 5a) and standard deviation (Figure 5b) of the evaporation index vary with and , whereas in Figure 6 the control of and on the mean (a) and the standard deviation (b) is explored. Figure 5 is drawn assuming ¼ 45, whereas in Figure 6 we assume ¼ 7. Figures 5a and 6a show that in humid sites (i.e., where is relatively low), the major con-trol on mean B is the aridity index, whereas as the aridity increases, becomes less important and is the dominant control. In other words, the capacity of the catchment to store water has more impact on the water balance in water limited climates than in energy limited climates. Moreover, as and increase, their effect on the mean evaporation index progressively decreases, and for high values of either one of them the mean of B tends to 1. As for the number of storm events, , Figure 6a shows that there is a relatively low effect on the mean of B when is low. However this effect can be observed only for low, and possibly unrealis-tic, values of , therefore we can effectively conclude that number of rainfall events does not control the mean evapo-ration index.
[29] Figures 5b and 6b give some insights into the vari-ability of the evaporation index, showing its standard devia-tion as a funcdevia-tion of , , and . Interestingly, as opposed to the plot relative to the mean, the standard deviation plot is not monotonic but there is a maximum line that divides the region of interest into two zones. As we move from the left side of the plot and gradually increase the aridity index, a relatively steep increase in the variability of B is observed; after the maximum is reached, the variability slightly decreases. Let us focus for a moment on the region of the graph where is greater than 2–2.5: we observe that the line of maximum variability closely follows the ¼ 1 line. This suggests that those situations in which the water availability approximately equals the atmospheric demand represent a sort of critical point for the water balance, which produces the highest variability in the response. This con-clusion is in agreement with the results obtained by other researchers who have found that stochastic soil moisture dy-namics show high levels of variability in intermediate cli-mates, producing intriguing behavior such as temporal clustering of macropore initiation [McGrath et al., 2007] and unconstrained optimal rooting depths [Guswa, 2008]. Let us now consider the lower part of the graph, as decreases the maximum variability line bends away from
the ¼ 1 vertical intercept, toward a horizontal asymptote. As with the mean, when aridity is low the main control on the variability of B is the capacity index. For < 1, its effect on the variability of B is significant and decreases as exceeds 1.
[30] Figure 6b represents the dependence of standard deviation of B on and , assuming ¼ 7. The graph shows that in humid sites the variability of B is driven by , whereas, as increases, the number of rainfall events becomes a more important factor for the variability of B. Overall, a large number of rainfall events in the year pro-duces stability in the annual water balance, even though for each value of the water balance may be more or less vari-able depending on the mutual relationship between and . [31] Figures 5a and 5b, is also consistent with one of the results of Milly [1994], whose case study showed that an increase in the soil storage capacity yields a smaller change in the water balance than does an equal decrease. Our anal-ysis actually suggests that an increase in yields a smaller change in the mean annual water balance and its interan-nual variability, than does an equal decrease. In fact, as increases, the annual water balance and its interannual vari-ability tend to a constant value that is solely driven by cli-mate variability.
Figure 4. Cramer–Von Mises test to evaluate the discrepancy between the distribution of B obtained
with variable annual vaporization and the one obtained with constant annual vaporization for all the MOPEX catchments. The evaluation is expressed in terms of the exceedance frequency of the p-values. Each of the points represent a pair of distributions computed at a single catchment. The analysis was done using only the data (blue points) and only the model (red points). The insets represent two examples (relative to a p-value P ¼ 0.6 and a p-value P ¼ 0.05) of the comparison between the distribution of B obtained with variable annual vaporization and the one obtained with constant annual vaporization (using data only).
5. Regional and Climatic Variations in Model Response
[32] As pointed out in section 3, for some of the catch-ments the model does not provide accurate predictions for the mean evaporation index and for many it poorly predicts its coefficient of variation. In fact, in many cases, model
errors show patterns related to geographic position and cli-mate. These results suggest that in those regions (or climates) where the model works well, the model conceptualization describes properly the system response. In regions where the model fails to reproduce the observed variability, there are likely other controls on catchment annual water balance that play an important role. In our model the only source of
Figure 5. Sensitivity analysis of the evaporation index on
climate and physical controls. (a) Mean and (b) standard deviation of the evaporation index as functions of the Budyko aridity index, ¼ PET=p, and the capacity index, ¼
nZrðs1swÞ. The plots are obtained for ¼ 45.
Figure 6. Sensitivity analysis of the evaporation index on
climate and physical controls. (a) Mean and (b) standard deviation of the evaporation index as functions of the Budyko aridity index, ¼ PET=p, and the mean number
of storm events, ¼ pL. The plots are obtained for
variability considered is the random fluctuation of rainfall events. Therefore, in those catchments where the model fails to predict the statistics of the evaporation index, there may be other controls that either amplify or damp the variability produced by rainfall. To limit the effect of the assumptions on the results, we excluded from this analysis 20% of the catchments, in which the two main assumptions (i.e., the drainage events as a Poisson process, and approximation of variable annual vaporization by its mean) seem to be ques-tionable, following the criteria explained in section 3.2.
[33] Figures 7a and 7c show the residuals between observed and predicted means and coefficients of variation of the evaporation index, respectively. Residuals are nor-malized by the observed values of mean and coefficient of variation. Positive values (warm colors) represent an over-estimation of the model predictions with respect to the observed data. A clear pattern can be observed in the resid-uals of the mean. In the eastern part of the United States, corresponding to the Appalachian chain, the mean of B is generally underestimated, whereas in the western part of the United States a general overestimation is observed. Good predictions are produced in the Midwest as well as in the central regions of the United States.
[34] The map of CV residuals is more scattered. How-ever, some clustering is still observed. In dry regions, such as Arizona, New Mexico, and Texas, an overall overesti-mation of the variability of B is observed. On the other hand, two clusters of variability amplification are observed in the western United States and in part of the Midwest. In the Midwest, the analysis may be affected by intensive ag-ricultural management (e.g., irrigation), which obviously is not taken into account by the model. In the western United
States it is unclear what could cause this amplification, which demands further investigation. Possibly, ocean-atmosphere teleconnections (such as ENSO and PDO), known to play a key role on water availability in this area, can help explain some of our results. In the eastern part of United States the variability of B is generally well pre-dicted, suggesting that in those regions the within-year fluc-tuation of water partitioning is the most important control on the distribution of annual water balance.
[35] Thompson et al. [2011], through an analysis on evap-otranspiration patterns across 13 FLUXNET sites, show how the effect of seasonality in energy and water supply plays an important role in the intra-annual fluctuations of evapotranspiration. It is still unclear to what extent such fluctuations affect the interannual variability of the evapora-tion index, but some of the spatial patterns observed in the residuals may arise from the seasonality of water balance.
[36] Figures 7b and 7d show the residuals for the mean and the coefficient of variation of B, normalized by the observed values, as functions of the annual humidity index. Figures show how the residuals on the mean, as well as their scatter, increase with the humidity index. In the driest catchments predictions of the mean are very good. In con-trast, the variability of B in water-limited catchments is largely overpredicted. In energy-limited catchments, the model is generally able to capture the observed values and the accuracy of predictions is more homogeneous (i.e., overpredictions and underpredictions more or less balance each other). It is very interesting that the largest errors are found exclusively in dry sites because this suggests that in water-limited environments, short-term rainfall variability does not constitute the major control on the interannual
Figure 7. Residuals between (a and b) predicted and observed means and (c and d) coefficients of
vari-ation of B as functions of the geographic position (Figures 7a and 7c) and of the annual humidity index (Figures 7b and 7d). Residuals are normalized by the observed values. Positive values (warm colors) characterize predictions higher than observations.
variability of water balance. Therefore, the causes of this damping effect are to be sought among other aspects of the annual water balance. For example, following Huxman et al. [2004], this effect may be produced by a long-term selection pressure on vegetation, arising from different ability of spe-cies to acclimate to resource variability in water-limited environments.
6. Conclusions
[37] The aim of this work was to explore the effects of rainfall event fluctuations on the interannual water balance variability at the catchment scale. To this end, we derived an analytical expression for the probability density function of the evaporation index, B, by means of a stochastic, pro-cess-based model. The model only accounts for random rainfall fluctuations and other significant controls, such as climate seasonality and vegetation dynamics, were pur-posely not included.
[38] The approach was tested with data from 424 catch-ments distributed across the continental United States. In general, good agreement was obtained between model pre-dictions of the mean evaporation index and estimates from the data. The model, however, was less accurate in repro-ducing the coefficient of variation of B.
[39] Deficiencies of the model were expected, given the high degree of diversity among the sites considered and the strong simplifying assumptions used. Nevertheless, it was surprising to observe that for a significant number of catch-ments, the simple model is able to adequately predict the pdf of B, and, in general, that modeling errors are region-ally clustered. This implies that there exist regions where the within-year fluctuation of water partitioning can be assumed to be the major control on interannual variability of water balance, but in other regions the simple upscaling of random event-scale processes cannot fully characterize the observed annual variation of water balance. In those regions, our modeling results suggest that other controls (e.g., seasonality or vegetation functioning) may be respon-sible for the observed interannual variability.
[40] Our analysis showed that this separation of regions is related to the competition between water and energy availability. A consistent overestimation of the variability of B was observed in arid regions. Overall, the geographi-cal clustering of model (in)efficiency, which implies the (in)validity of the model over regional rather than local scales, is intriguing and calls for further explorations of the effect of large scales dynamics over catchment-scale water balance.
[41] A sensitivity analysis on model parameters provided further insight on physical controls on water balance, although these results are believed to hold only in those cases where our framework holds. The analysis showed that the Budyko aridity index, (i.e., the ratio of long-term precipitation to potential evapotranspiration), is the main control on the mean of B in humid sites. A dimensionless soil storage capacity, (i.e., the ratio of soil storage capacity to mean storm depth), also has an important impact on the mean annual water balance, but only in water limited environments. The interannual variability of B is strongly related to both, , and the number of rainfall events in the year, whereas it is affected by only in dry
climates. The value ¼ 1 seems to represent a critical tran-sition in the competition between controls on interannual variability of water balance. This is in agreement with pre-vious studies that show that soil moisture dynamics exhibits maximum variability in intermediate climates [Knapp and Smith, 2001; McGrath et al., 2007].
[42] These results further suggest that in those regions where predictions are fairly accurate, our analytical result may be used as a first-order, parsimonious estimator for the variability of B when a detailed site description and an adequate data set are not available.
[43] Although our modeling framework has been widely used in previous hydrological studies, some caveats and limitations are worth pointing out. The analysis requires the use of lumped parameters which sometimes may not represent the actual characteristics of the catchment. This limitation cannot be easily overcome as the estimation of the required parameters is difficult at the catchment scale. Fortunately, previous studies [e.g., Milly, 1994; Laio et al., 2006 Botter et al., 2007b, 2008] have increased confidence in this kind of approach, showing how model predictions adequately matched the observations even at large scales.
[44] While our work has highlighted the effect of event-scale rainfall fluctuations on the interannual variability of water balance, it can be easily extended in a number of ways. The stochastic model can be extended to include: (1) seasonality of the climate drivers, (2) explicit consideration of processes that occur at hillslope or catchment scales, including the role of topography and organized heterogene-ity of soil and vegetation properties, and (3) vegetation dynamics in response to water and energy variability. Such model development can be guided by the results provided by the present study. This is left for further research.
[45] Acknowledgements. Work on this paper commenced during the summer institute organized at the University of British Columbia (UBC) during June–July 2009 as part of the NSF-funded project Water Cycle Dynamic in a Changing Environment: Advancing Hydrologic Science through Synthesis (NSF grant EAR-0636043, M. Sivapalan, principle in-vestigator). We acknowledge the support and advice of numerous partici-pants at the summer institute (students and faculty mentors). Special thanks are owed to Mateij Durcik for help with the background data preparation and analysis of the MOPEX catchments. Thanks also go to Marwan Hassan and the Department of Geography of UBC for hosting the summer institute and for providing outstanding facilities, without which this work would not have been possible. The funding and support of ‘‘Fondazione Aldo Gini’’ is gratefully acknowledged. The authors wish to thank the anonymous reviewers for providing extremely useful suggestions and feedback.
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C. J. Harman, Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61820, USA.
P. S. C. Rao and S. Zanardo, School of Civil Engineering, Purdue University, West Lafayette, IN 47907, USA. (szanardo@umn.edu)
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P. A. Troch, Department of Hydrology and Water Resources, University of Arizona, Tucson, AZ 85721, USA.
