## Runoff generation dynamics within a humid river basin

The present paper introduces an analytical approach for the description of the soil water balance and runoff production within a schematic river basin. The model is based on a stochastic differential equation where the rainfall is interpreted as an additive noise in the soil water balance and is assumed uniform over the basin, the basin heterogeneity is characterized by a parabolic distribution of the soil water storage capacity and the runoff production occurs for saturation excess. The model allowed to derive the probability density function of the produced surface runoff highlighting the role played by climate and physical characteristics of a basin on runoff dynamics. Finally, the model have been tested over a humid basin of Southern Italy proposing also a strategy for the parameters estimation.

How to cite: Manfreda, S., Runoff Generation Dynamics within a Humid River BasinNatural Hazard and Earth System Sciences, 8, 1349-1357, (doi:10.5194/nhess-8-1349-2008), 2008. [pdf]

## A stochastic approach for the description of the water balance dynamics in a river basin

The present paper introduces an analytical approach for the description of the soil water balance dynamics over a schematic river basin. The model is based on a stochastic diﬀerential equation where the rainfall forcing is interpreted as an additive noise in the soil water balance. This equation can be solved assuming known the spatial distribution of the soil moisture over the basin transforming the two dimensional problem in a one dimensional one. This assumption is particularly true in the case of humid and semihumid environments, where spatial redistribution of soil moisture becomes dominant producing a well deﬁned pattern. The model allowed to derive the probability density function of the saturated portion of a basin and of its relative saturation. This theory is based on the assumption that the water storage capacity varies across the basin following a parabolic distribution and the basin has homogeneous soil texture and vegetation cover. The methodology outlined the role played by the basin shape in the soil water balance. In particular, the resulting probability density functions of the relative basin saturation were found to be strongly controlled by the maximum water storage capacity of the basin, while the probability density functions of the relative saturated portion of the basin are strongly inﬂuenced by the spatial heterogeneity of the soil water storage capacity. Moreover, the saturated areas reach their maximum variability when the mean rainfall rate is almost equal to the soil water loss coefﬁcient given by the sum of the maximum rate of evapotranspiration and leakage loss in the soil water balance. The model was tested using the results of a continuous numerical simulation performed with a semi-distributed model in order to validate the proposed theoretical distributions.

How to cite: Manfreda, S., M. Fiorentino, A Stochastic Approach for the Description of the Water Balance Dynamics in a River BasinHydrology and Earth System Sciences, 12, 1189-1200, (doi:10.5194/hess-12-1189-2008), 2008.  [pdf]

## Reply to comment by S. Nadarajah on ‘‘Space-time modeling of soil moisture: Stochastic rainfall forcing with heterogeneous vegetation’’

The comment by Nadarajah [2007] focuses on the spatial correlation function of the rainfall forcing adopted in the theoretical analysis of the soil water balance addressed by Isham et al. [2005] and Rodrıguez-Iturbe et al. [2006].

How to cite: Manfreda, S., D.R. Cox, V. Isham, A. Porporato, I. Rodríguez-Iturbe, Reply to the Comment by S. Nadarajah on “Space-time modeling of soil moisture: Stochastic rainfall forcing with heterogeneous vegetation”Water Resources Research, 43, W10602, (doi:10.1029/2007WR006378), 2007. [pdf]

## On the spatial and temporal sampling of soil moisture fields

Recent work by Isham et al. and Rodrìguez-Iturbe et al. has characterized the space- time variability of soil moisture through its analytically derived covariance function which depends on soil properties, vegetation structure, and rainfall patterns typical of a region. This paper uses such characterization to address the strategies and methodologies for the sampling of soil moisture fields. The focus is on the estimation of the long-term mean soil moisture and the daily soil moisture averaged over a given area as a function of the network geometry, number of stations, number of sampling days and landscape heterogeneity. It is found that the spatial geometry of the network has a significant impact on the sampling of the average soil moisture over an area in any particular day, while it is much less relevant for the sampling of the long-term mean daily soil moisture over the region. In the latter case, the length of the record is a commanding factor in what concerns the variance of estimation, specially for soils with shallow rooted vegetation. Spatial vegetation heterogeneity plays an important role on the variance of estimation of the soil moisture, being particularly critical for the sampling of the average soil moisture over an area for a given day.

How to cite: Manfreda, S. and I. Rodrìguez-Iturbe, On the Spatial and Temporal Sampling of Soil Moisture FieldsWater Resources Research, 42, W05409, (doi:10.1029/2005WR004548), 2006. [pdf]

## Space-time modeling of soil moisture: Stochastic rainfall forcing with heterogeneous vegetation

The present paper complements that of Isham et al. (2005), who introduced a space-time soil moisture model driven by stochastic space-time rainfall forcing with homogeneous vegetation and in the absence of topographical landscape effects. However, the spatial variability of vegetation may significantly modify the soil moisture dynamics with important implications for hydrological modeling. In the present paper, vegetation heterogeneity is incorporated through a two dimensional Poisson process representing the coexistence of two functionally different types of plants (e.g., trees and grasses). The space-time statistical structure of relative soil moisture is characterized through its covariance function which depends on soil, vegetation, and rainfall patterns. The statistical properties of the soil moisture process averaged in space and time are also investigated. These properties are especially important for any modeling that aggregates soil moisture characteristics over a range of spatial and temporal scales. It is found that particularly at small scales, vegetation heterogeneity has a significant impact on the averaged process as compared with the uniform vegetation case. Also, averaging in space considerably smoothes the soil moisture process, but in contrast, averaging in time up to 1 week leads to little change in the variance of the averaged process.

How to cite: Rodríguez-Iturbe, I., V. Isham, D.R. Cox, S. Manfreda, A. Porporato, Space-time modeling of soil moisture: stochastic rainfall forcing with heterogeneous vegetationWater Resources Research, 42, W06D05, (doi:10.1029/2005WR004497), 2006. [pdf]

## Representation of space–time variability of soil moisture

A simpliﬁed spatial-temporal soil moisture model driven by stochastic spatial rainfall forcing is proposed. The model is mathematically tractable, and allows the spatial and temporal structure of soil moisture ﬁelds, induced by the spatial-temporal variability of rainfall and the spatial variability of vegetation, to be explored analytically. The inﬂuence of the main model parameters, reﬂecting the spatial scale of rain cells, the soil storage capacity, the rainfall interception and the soil water loss rate (representing evaporation and deep inﬁltration) is investigated. The variabilities of the spatially averaged soil moisture process, and that averaged in both space and time, are derived. The present analysis focuses on spatially uniform vegetation conditions; a follow-up paper will incorporate stochastically heterogeneous vegetation.

How to cite: Isham, V., D.R. Cox, I. Rodríguez-Iturbe, A. Porporato, S. Manfreda, Representation of Space-Time Variability of Soil MoistureProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461(2064), 4035 – 4055, (doi:10.1098/rspa.2005.1568), 2005. [pdf]