## 1. Introduction

[2] Land surface hydrology involves the study of the exchanges of water and energy between the land and the atmosphere and the movement of water within and over the land surface. How space-time variability in precipitation interacts with spatial heterogeneity of soils, topography, and vegetation and is partitioned into spatiotemporal variability of runoff, evaporation, and soil moisture storage are fundamental questions that underpin hydrologic predictions of all kinds. This hydrologic partitioning is usually expressed in terms of a dynamic water balance, which can be manifested in various characteristic signatures of catchment responses representing variability at a range of time (and also space) scales [*Wagener et al.*, 2007].

[3] An important concern for sustainable water management is how climate variability and/or change at annual to decadal time scales, and likewise hydrologic effects of land use and land cover changes, propagate through the landscape and result in changes to spatiotemporal patterns of variability of water balance over a range of time and space scales. Over the past 5 decades, considerable effort has been invested into exploring the roles of climate, soil, vegetation, and topography interactions in controlling water balance variability (both in time and in space) at the annual scale in order to generate the understanding needed to make predictions of the resulting changes to the water balance. Much more effort has been expended, however, in studying mean annual water balance than on interannual variability.

[4] This paper develops a framework for analyzing the space and time variability of annual water balances. In a companion paper [*Harman et al.*, 2011], this framework is used to understand the controls on sensitivity of water balance to interannual variations in precipitation. The framework used by both papers is built on previous work by *L'vovich* [1979] and *Ponce and Shetty* [1995a, 1995b]. Here we review previous approaches to water-balance modeling in order to motivate the (re)introduction of this framework. Past research into annual water balances has followed essentially two diametrically opposite and yet complementary approaches: empirical and process based.

### 1.1. Mean Annual Water Balance

[5] In the empirical approach, annual water balance is assessed on the basis of systematic analysis of long-term data sets of observed rainfall and runoff in different climatic or ecoregions of the world and empirical analyses of how the water balance is governed by climatic and landscape properties. A classic example of the empirical approach to analysis of mean annual water balance is the work of *Budyko* [1974], who viewed annual water balance, to first order, as a manifestation of the competition between available water and available energy. Budyko quantified mean annual water balance in terms of the ratio of mean annual evaporation to mean annual precipitation, *E/P*. On the basis of worldwide data sets from a large number of catchments, he demonstrated that *E/P* is determined, to first order, by the ratio of mean annual potential evaporation (i.e., measure of energy available) to mean annual precipitation (i.e., measure of water available), *E _{p}/P*, which he defined as a (climatic) aridity index. In recent times,

*Zhang et al.*[2001] extended the analysis of mean annual water balances, still working within the Budyko framework, to include the effects of the conversion of native (woody) vegetation to pasture and presented empirical relationships for

*E/P*as a function of not only

*E*but also a parameter

_{p}/P*W*, which is a coefficient that reflected plant available water.

*Yang et al.*[2007] carried out theoretical studies (based on dimensional analysis) leading to the analytical derivation of the Budyko curve, implemented this theory in over 100 catchments in China, and derived expressions for

*E/P*as a function of

*E*, the fraction of forest cover and a measure of soil moisture storage capacity (which can be partly related to soil depth).

_{p}/P[6] The value of these empirical approaches is that they involved deriving simple relationships governing long-term mean annual water balance, which has emerged through coevolution and self-organization of climate, soil, topography, and vegetation in different natural settings. However, it is not clear that they are sufficient to capture the climatic and landscape controls on more transient responses, including natural interannual variability, and catchment responses to nonstationary variations of both climate and changes in landscape properties through human actions. The latter will require a combination of both process-based and empirical approaches.

[7] In the process-based approach, the mean annual water balance is explored on the basis of explicit representations of the various hydrologic processes (e.g., infiltration, storage, drainage, evaporation, and transpiration) that constitute the water balance equation and the interacting roles of climate, soil, topography, and vegetation that govern these. The best example of the process-based approach to annual water balance is the classic work of *Eagleson* [1978] and *Eagleson and Tellers* [1982], who explored the cascading of a series of precipitation events through the landscape, in terms of alternative wetting and drying cycles, and explored their manifestation as mean annual water balance. In subsequent work, *Milly* [1994a] investigated the climate-landscape interactions governing the mean annual water balance through the use of a simple stochastic bucket model and used it to explore the physical basis of the *Budyko* [1974] curve, including competition between precipitation and atmospheric demand (i.e., water and energy) and the regulating influence of soil moisture storage. *Salvucci and Entekhabi* [1995] extended the mathematical formulations of *Eagleson*'s [1978] and *Eagleson and Tellers*'s [1982] work to explore the role of lateral flow processes in hillslopes. *Reggiani et al.* [2000] explored the physical basis of annual water balance by utilizing a lumped numerical model at the hillslope scale, which involved solution of the coupled mass and momentum balance equations that underpin the partitioning of precipitation into evaporation and both surface and subsurface runoff. They found that while total annual evaporation and runoff were predominantly governed by the Budyko-type aridity index, topographic slope and soil hydraulic conductivity were important factors in the partitioning of total runoff into surface and subsurface runoff.

[8] A number of studies utilized process-based models to explicitly investigate the effects of climatic seasonality on mean annual water balance. *Milly* [1994a, 1994b] and *Potter et al.* [2005] used conceptual (bucket) models to investigate the effects of climatic seasonality, through inclusion of interactions of seasonal climatic fluctuations with soil moisture storage. In this case, the effects of topography and subsurface drainage can become important determinants of intra-annual variability of water balance, which can be expected to have significant impacts also on mean annual water balance and interannual variability. *Yokoo et al.* [2008] explored the effects of seasonal variability of climatic inputs on mean annual water balances and the roles of climate, soil properties, and topography in modulating these impacts. Their numerical experiments with the use of a physically based water balance model [*Reggiani et al.*, 2000] showed that (1) the effects of seasonality are likely to be most important when the seasonal variabilities of precipitation and potential evaporation are out of phase and in arid climates and (2) the seasonality effects can be high in catchments with fine-grained soils and flat topographies, in which case surface runoff dominates, and also high in basins with coarse-grained soils and steep topographies, where subsurface runoff dominates.

### 1.2. Between-Year Variability of Annual Water Balance

[9] Interannual variability of the water balance response of natural (i.e., nonhuman impacted) catchments can arise, in a straightforward manner, from interannual variability of precipitation inputs or energy inputs. If catchment ecosystems can be assumed to respond, and adapt, rapidly to such variability, one can expect that empirical models of mean annual water balance [e.g., *Budyko*, 1974] can continue to reflect such interannual variability as well. A number of authors have exploited this space-time symmetry for quantifying interannual variability of annual water balance. *Yang et al.* [2007] extended the use of the Budyko framework to predict interannual variability of runoff and demonstrated reasonable accuracy in these predictions. Indeed, they extended it even further so as to predict, with reasonable accuracy, intra-annual (monthly) variability of runoff by explicitly accounting for the carryover of soil moisture storage between months. *Wang et al.* [2009] also adopted the Budyko framework to explore regional water balances in the Sand Dune Hills region of Nebraska and identified soil texture and groundwater as major controls on mean annual water balance and interannual variability. In a much earlier classic work, *Dooge* [1992] proposed that interannual variability of annual water balance can be predicted by working within the Budyko framework through derivation of a general expression for the elasticity of annual runoff to changes in annual precipitation or potential evaporation.

[10] However, in addition, interactions of intra-annual variability of climate inputs with the landscape processes (e.g., soil moisture storage and subsurface drainage) can also manifest in additional contributions to interannual variability. Process controls of interannual variability are therefore complex. Empirical methods based on annual water balance data alone are not sufficient to fully capture interannual variability; more process studies of intra-annual variability are therefore required to develop predictive understanding of the causes of interannual variability. *Jothityangkoon and Sivapalan* [2009], through analysis of data from a small number of catchments in Australia and New Zealand, with the use of diagnostic analysis of models of increasing complexity, highlighted the roles of intra-annual precipitation variability (seasonality versus storminess) on the interannual variability of runoff. *Potter and Zhang* [2009] confirmed the validity of these approaches using data from a large number of pristine catchments in Australia and in the process demonstrated the role of intra-annual (seasonal) variability of climate variables (precipitation and potential evaporation) on interannual variability of annual water balance. Recently, *Peel et al.* [2010] showed that differences in intra-annual variations of evapotranspiration between deciduous and evergreen vegetation types can also lead to interannual variability of runoff.

### 1.3. Functional Approach: Horton, L'vovich, and Ponce and Shetty

[11] Because interannual variability of annual water balance is affected significantly by intra-annual variability of climatic inputs and their interactions with the processes of soil moisture storage, plant water uptake, and subsurface drainage, analyses of interannual variability will not be complete until these catchment functions are explicitly considered. Both the empirical and process-based approaches must be generalized to account for the intra-annual variability and associated process interactions. A step forward in this direction is the work of *Horton* [1933] and of *L'vovich* [1979].

[12] As far back as 1933, Horton investigated interannual variability of water balance by exploring the capacity of catchments to store infiltrated water, as a function of soil type, and evaporate it in return, as controlled by vegetation. As part of his now classic paper, he introduced the ratio of catchment vaporization (transpiration plus evaporation) *V* to catchment wetting by precipitation (i.e., that part of the precipitation that does not run off immediately; this includes interception, surface ponding, and soil wetting), denoted as *W*. This ratio *V/W* (hereinafter referred to as the Horton index *H* [*Troch et al.*, 2009]) was found to remain remarkably constant from year to year when applied to only the growing season in a catchment located in New York (west branch of the Delaware; mean of 0.78 and standard deviation of 0.06), despite the large interannual variability of growing season precipitation (minimum of 406 mm, maximum of 812 mm). On the basis of these observations, *Horton* [1933, p. 456] hypothesized that the “natural vegetation of a region tends to develop to such an extent that it can utilize the largest possible proportion of the available soil moisture supplied by infiltration.”

[13] Motivated by Horton's stated hypothesis, *Troch et al.* [2009] estimated the Horton index for 89 catchments around the continental United States and confirmed that the Horton index does, indeed, remain remarkably constant between years in these catchments, with a relatively small variance. For example, Troch et al.'s analysis showed that for 50% of the catchments analyzed, the standard deviation of the Horton index is less than 0.06, while in about 90% of the cases the standard deviation is less than 0.10. Troch et al. also showed that as the aridity index *E _{p}/P* increased, the mean Horton index also increased, while its variability decreased significantly. In addition, the marked symmetry that the observed data exhibited between regional (between-catchment) variability of the Horton index (reflecting climate variability between catchments) and interannual (between-year) variability (a reflection of interannual variability of climatic inputs) suggested to them that vegetation across different ecoregions may be adapting in very similar ways to spatiotemporal variability of water and energy availability.

[14] In this paper we adopt an extension of the Hortonian approach to interannual variability, pioneered by *L'vovich* [1979], who presented an empirical theory for the two-stage water balance partitioning at the land surface. According to this theory, in the first stage, precipitation is partitioned into wetting (of canopy, litter, and soil) and quick flow (e.g., surface runoff), whereas in the second stage the wetting is further partitioned into vaporization (interception loss plus evaporation plus transpiration) and slow flow (e.g., subsurface runoff). The original work of *L'vovich* [1979] and subsequent work by *Ponce and Shetty* [1995a, 1995b] theorized that the partitioning at each stage is in the form of a competition between different catchment functions, which are expressible in terms of common mathematical forms that can be extracted from data. *L'vovich* [1979] implemented this approach in a large number of catchments located in a variety of ecoregions of the world and presented his results in the form of nomographs. In spite of the increased global interest in annual water balances and concerns about the effects of global change, apart from the theoretical analyses of *Ponce and Shetty* [1995a, 1995b], the work of *L'vovich* [1979] has not been subsequently followed up, despite at least 40 years of additional data that have become available. The *Horton* [1933] and *L'vovich* [1979] approaches are partly empirical since they are based on empirical analyses of rainfall-runoff data for the characterization of annual water balances. On the other hand, since they go further than Budyko and explicitly include the partitioning of annual precipitation into its major components of storage, release by quick flow and slow flow, and evapotranspiration (combining bare soil evaporation, interception loss, and plant water uptake), the Horton and L'vovich approaches can be deemed to characterize the functioning of the catchments at the annual scale. For these reasons, following *Wagener et al.* [2007], we define their approach as a *functional* approach.

### 1.4. Aims of This Paper

[15] This paper is aimed at exploring the variability of annual water balances, (1) regional, between-catchment variability of mean annual water balances and (2) the between-year (interannual) variability in a small number of selected catchments and the possible symmetry between these two kinds of variability, within the theoretical framework provided by the *L'vovich* [1979] and *Ponce and Shetty* [1995a, 1995b] formulations. We repeat the empirical analyses of *L'vovich* [1979] for 431 catchments in the continental United States and quantify the two-stage partitioning of precipitation at the catchment scale in all catchments. We fit the theoretical relationships of *Ponce and Shetty* [1995a, 1995b] to the resulting partitioning and explore regional patterns of the associated parameter values. Using nondimensional forms of the L'vovich–Ponce-Shetty relations, we investigate similarity of annual water balance responses between catchments and between years. In particular, we derive analytical expressions for several metrics of annual water balances: Horton index, a vaporization fraction (reflecting the Budyko curve), a base flow fraction, and a runoff fraction (equivalent to the traditional annual runoff coefficient). Finally, we use the same formulations to quantify the degree of damping of interannual variability of annual precipitation as it cascades through the catchment system and as manifested in the vaporization fraction (Budyko curve) and the Horton index to shed more light on Horton's hypothesis about the role of vegetation.

[16] The work presented in this paper complements parallel work being presented in two companion papers by S. Zanardo et al. (A stochastic, analytical model of the Horton index and implications for its physical controls, submitted to *Water Resources Research*, 2011) and H. Voepel et al. (Climate and landscape controls on catchment-scale vegetation water use, submitted to *Water Resources Research*, 2011) based on analysis of the same 431 Model Parameter Estimation Experiment (MOPEX) catchments as used in this study. Voepel et al. (submitted manuscript, 2011) adopt an empirical approach to interannual variability of annual water balances. They use rainfall-runoff data from the MOPEX catchments to develop empirical relationships between the Horton index and associated climatic and landscape properties. They also test the hypothesis that the Horton index can more reliably (than precipitation or aridity index) predict vegetation response quantified using annual maximum normalized difference vegetation index (NDVI). Zanardo et al. (submitted manuscript, 2011) adopt a process-based approach to evaluation of interannual variability and explore the climatic and landscape controls on the Horton index. For this they use a parsimonious catchment model driven by stochastic precipitation inputs, following the work of *Milly* [1994a, 1994b], *Porporato et al.* [2004], and *Botter et al.* [2007]. In this sense, the present paper and those of Voepel et al. (submitted manuscript, 2011) and Zanardo et al. (submitted manuscript, 2011) represent three alternative perspectives on the nature of annual water balance variability, both regional and interannual. Furthermore, another companion paper by *Harman et al.* [2011] adopts the same functional approach but extends the analysis to include the sensitivity and possible resilience of regional and interannual variability of water balance to changes in the climate drivers, in this case annual precipitation.

[17] The paper begins, in section 2, with a presentation of the *L'vovich* [1979] theory and the analytical formulations of *Ponce and Shetty* [1995a, 1995b]. Section 3 presents the empirical water balance analysis of the 431 MOPEX catchments to quantify the two-stage partitioning of annual precipitation into its three components, the fitting of the Ponce-Shetty functional forms to these empirical relationships, and the estimation of the associated parameters. Section 4 reformulates the Ponce and Shetty relationships into nondimensional forms in order to compactly represent the idea of “competition” and the possible symmetry between interannual and intercatchment variability and to assess these through analysis of the MOPEX data set. Section 5 uses this nondimensional formulation to derive analytical expressions for several metrics of annual water balance, including the Horton index, the vaporization fraction, and the base flow fraction, to demonstrate the unity of these formulations for characterizing between-catchment and between-year variability of annual water balance. Section 6 presents and implements the theory governing propagation of interannual variability through the catchment system and a quantification of the damping of variability produced during the two-stage partitioning. Finally, the paper concludes with a discussion of the results, their implications, and recommendations for further research.