Corresponding author: L. Brocca, Research Institute for Geo-Hydrological Protection, National Research Council, Perugia, Italy. (email@example.com)
 Rainfall and soil moisture, SM, are two important quantities for modeling the interaction between the land surface and the atmosphere. Usually, rainfall observations are used as input data for modeling the time evolution of SM within hydrological and land surface models. In this study, by inverting the soil-water balance equation, a simple analytical relationship for estimating rainfall accumulations from the knowledge of SM time series is obtained. In situ and satellite SM observations from three different sites in Italy, Spain, and France are used to test the reliability of the proposed approach in contrasting climatic conditions. The results show that the model is able to satisfactorily reproduce daily rainfall data when in situ SM observations are employed (correlation coefficient, R, nearly equal to 0.9). Furthermore, also by using satellite data reasonable results are obtained in reproducing 4 day rainfall accumulations with R-values close to 0.8. Based on these preliminary results, the proposed approach can be adopted conveniently to improve rainfall estimation at a catchment scale and as a supplementary source of data to estimate rainfall at a global scale.
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 The relevance of rainfall and soil moisture, SM, data to understand the global climate system [Seneviratne et al., 2010] has been underlined by the Global Climate Observing System (GCOS), which listed them among the “Essential Climate Variables.” As regards hydrological applications, rainfall and antecedent SM are the two key factors influencing the runoff generation process during floods [Crow et al., 2009].
 The monitoring of rainfall by ground stations (rain gauges and meteorological radars) and satellite sensors has been well established for many years even though it still suffers from several limitations [Crow et al., 2011], i.e., the spatial representativeness of rain gauge stations and the quantitative accuracy of radar and satellite sensors. On the other hand, SM monitoring has advanced considerably in the last two decades through the development of in situ networks [Dorigo et al., 2011], remote sensing sensors and advanced retrieval algorithms [Wagner et al., 2007; Kerr et al., 2012]. Usually, SM data sets are employed within hydrological and meteorological models to improve the simulation of their internal state(s) by using data assimilation techniques [Dharssi et al., 2011; Brocca et al., 2012; de Rosnay et al., 2012]. Conversely, rainfall observations represent the main input data needed for SM modeling [Famiglietti and Wood, 1994].
 However, SM dynamics and rainfall share an obvious physical connection thus offering the chance to invert the rainfall-SM feedback, i.e., to improve rainfall estimation by using SM observations [Pellarin et al., 2008]. With this concept in mind, Crow et al.  developed a data assimilation approach, enhanced by Crow et al. , to correct rainfall estimates using remotely sensed surface SM retrievals. The same approach has also been applied in order to evaluate the skill of different satellite SM products at a global scale [Crow and Zhan, 2007; Crow et al., 2010; Parinussa et al., 2011].
 In a rather different context, Kirchner  demonstrated that under specific conditions, it is possible “doing hydrology backwards,” i.e., to infer rainfall (and evaporation) time series from discharge data. Kirchner's approach is based on the definition of a storage-discharge relationship allowing SM storage to be inferred from streamflow fluctuations. Teuling et al.  and Krier et al.  applied this approach in two different regions (Switzerland and Luxembourg) and found it more suitable during wet conditions.
 In this study, similarly to Kirchner , a simple approach is proposed for estimating rainfall accumulations through the knowledge of SM observations. Specifically, by inverting the soil-water balance equation, a direct analyticrelationship between rainfall and SM, not including discharge, is derived. Therefore, this approach significantly differs from Pellarin et al.  and Crow et al.  methods who used SM observations to correct and not to directly estimate rainfall. In situ SM and rainfall observations from three different sites in Italy, Spain, and France are used to test the reliability of the proposed approach. Moreover, a preliminary test by using satellite SM data derived from the Advanced SCATterometer (ASCAT) is also carried out in order to potentially apply the procedure to enhance rainfall estimation, or for satellite SM validation, at a global scale [Crow et al., 2010].
 The soil water balance for a layer depth Z [L] can be described by the following expression:
where s(t) [−] is the relative saturation of the soil, t [T] is the time, and p(t), r(t), e(t), and g(t) [L/T] are the precipitation, runoff, evapotranspiration, and drainage rate, respectively.
 By rearranging equation (1), precipitation rates can be inferred from the knowledge of SM, runoff, evapotranspiration, and drainage rates:
 According to Kirchner , whenever it rains we can assume that the evaporation rate is relatively low and, hence, negligible (e(t) = 0). Moreover, a stronger assumption may be made considering that all precipitation infiltrates into the soil and, hence, the runoff rate is zero (r(t) = 0). The latter assumption will have an impact mainly when the soil is close to saturation and it is expected that it will provide an underestimation of rainfall [Crow et al., 2011].
 For the drainage rate, the following relation may be adopted [Famiglietti and Wood, 1994]:
where a [L/T] and b [−] are two parameters expressing the nonlinearity between drainage rate and soil saturation.
 Combining equations (2) and (3) with the assumptions made above yields:
 Therefore, equation (4) can be used for estimating the precipitation rate from the knowledge of relative SM, s(t), its fluctuations in time, ds(t)/dt, and three parameters (Z, a, and b) to be estimated through calibration. Equation (4) is applied here in a discrete form by using a fourth-order Runge-Kutta scheme for its solution. Moreover, negative rainfall values that might occur during some dry-down cycles are set equal to zero.
3 Study Area and Data Sets
 Three sites in Europe are considered: Umbria (hereinafter named UMB) in Italy, Remedhus (REM) in Spain, and Valescure (VOB) in France (see Brocca et al.  for a map with their location). The sites are characterized by contrasting climatic and topographic conditions and they have been selected for their quality-checked hourly rainfall and SM observations. Table 1 summarizes the main characteristics of each site along with the information about the SM data set used in this study. Note that a different year is used for each site in accordance with the availability of rainfall and SM data. In Italy and Spain, rainfall data for the period 2007–2010 and 2008–2011, respectively, are also used for the preliminary analysis employing ASCAT-derived SM retrievals.
Table 1. Main Characteristics of the Experimental Sites and the Soil Moisture Data set Used for This Studya
Elevation (m a.s.l.)
Sensor Depth (mm)
MAR: mean annual rainfall, MAT: mean annual temperature.
 The UMB SM network, located in central Italy, is currently composed of 15 Frequency Domain Reflectometry (FDR) probes measuring volumetric SM at three depths. The REM network is located in the central sector of the Duero basin (Spain) and has 23 SM stations [Martinez-Fernandez and Ceballos, 2005]. Each station has been equipped with capacitance probes installed horizontally at a depth of 5 cm. The VOB site is located in a small experimental catchment in France [Tramblay et al., 2010]. In 2005, 12 Time Domain Reflectometry (TDR) SM probes were installed at different depths in five plots. In this preliminary study, hourly SM data collected by one representative station for each network [Brocca et al., 2011] are used (see Table 1).
3.1 ASCAT Soil Moisture Products
 The Advanced SCATterometer (ASCAT) is a real-aperture radar instrument launched on board the MetOp satellite in 2006 measuring radar backscatter at C-band (5.255 GHz) in VV polarization. The spatial resolution of ASCAT is 25 km and measurements for central Europe are generally obtained once a day. The surface SM product is retrieved from the ASCAT backscatter measurements, using a time series-based change detection approach previously developed for the ERS-1/2 scatterometers by Wagner et al. . The derived Surface Soil Moisture (SSM) product (corresponding to a depth of 2–3 cm) ranges between 0% (dry) and 100% (wet) and represents the relative soil saturation. Additionally, in order to obtain a root-zone SM product, SWI (Soil Water Index), the exponential filter approach proposed by Wagner et al.  is adopted which depends on a single parameter, T, the characteristic time length. Usually, the SWI is found to be a more robust product representative of a deeper soil layer and less affected by measurement noise. The reader is referred to Wagner et al.  for a detailed description of the exponential filter. Validation studies of the ASCAT SM products (SSM and SWI) assessed their accuracy by using both in situ and modeled SM observations across different regions in Europe [Brocca et al., 2011] and worldwide [Albergel et al., 2012].
4.1 In Situ Observations
 The first analysis is carried out by using hourly in situ SM observations. In order to obtain a robust set of parameters, the model is calibrated to reproduce daily rainfall observations. The maximization of the Nash-Sutcliffe efficiency index, NS, is selected as objective function and the correlation coefficient, R, and the root mean square error, RMSE, are also used for model evaluation. Figure 1 shows the observed versus simulated daily rainfall for the three sites along with the observed SM time series. As it can be seen, the simple analytical equation proposed here is found to be capable of satisfactorily reproducing the observed daily rainfall data for all the investigated sites. In terms of performance scores (see Table 2), the NS and R-values are always higher than 0.8 and 0.9, respectively, with the best scores obtained for REM site (NS = 0.893 and R = 0.945). The parameter values are also consistent with the expected behavior: the highest Z value is obtained for VOB site for which the SM sensor is installed on a deeper layer (300 mm); the a value for REM site, characterized by the coarser soil texture (see Table 1), is found to be the highest. A closer look at the results displayed in Figure 1 shows that the model tends to underestimate rainfall values for rainfall events that overwhelms the storage capacity of the soil layer (e.g., April 2011 at UMB site and November 2008 at VOB site).
Table 2. Summary of the Performance of the Comparison Between Observed and Simulated Rainfall Data by Using in Situ Soil Moisture Observations for the Three Investigated Sites and the Different Configurations for the Parametersa
 With the parameters calibrated on daily rainfall values, the model reliability for different aggregation intervals of rainfall, from 3 to 72 h, is also examined in Figure 2. It is shown that a quick decrease in the performance, both for NS and R, occurs for aggregation intervals of less than 12 h while the performance remains quite similar for intervals of more than 24 h. Therefore, by using hourly observations, the model is found to be reliable for estimating rainfall data cumulated on a period of more than 12 h. The lower performance for high-resolution rainfall estimation must be attributed not only to model deficiencies but also to the accuracy of rainfall and SM observations that is expected to be lower for high-resolution data.
 Two additional configurations, for further simplifying equation (4), are taken into account by assuming: (1) a = 0, i.e., rainfall is estimated with the knowledge of only SM fluctuations, ds(t)/dt, and (2) b = 1, i.e., the drainage rate is linearly related to soil saturation. The results of these simulations are reported in Table 2. For UMB site, the two simplifications do not provide a significant reduction of model performance with NS-values still higher than 0.77 (for 24 h). On the other hand, for the other two sites and mainly for REM, the performance significantly decreases (NS < 0.6). Comparing the two model simplifications, the one assuming b = 1 provides better results with average NS and R-values equal to 0.711 and 0.843, respectively. Therefore, this formulation can be adopted to reduce the number of parameters to be calibrated and, hence, the uncertainties related to their estimation.
4.2 Satellite Observations
 By using satellite data, it may be possible to estimate rainfall at a global scale through equation (4). In this study, a preliminary analysis considering 4 years of data for two ASCAT pixels in Italy and Spain is conducted. The two pixels are characterized by low vegetation density and are located in areas where ASCAT SM retrievals are found to be satisfactorily [Brocca et al., 2011]. The model is calibrated to reproduce 4 day rainfall accumulations and the SWI product computed with T equal to 5 days is selected as the basic ASCAT-derived product. Figure 3 shows the comparison between observed and simulated rainfall for the two pixels. Generally, the model is still found to be fairly reliable with NS (R) values equal to 0.602 and 0.638 (0.800 and 0.799) for Italy and Spain, respectively. As expected, the model tends to underestimate rainfall in very wet conditions (e.g., January 2009 and 2011 in Spain) but is generally able to reproduce the observed rainfall pattern for both areas with performance scores in accordance with those obtained with state-of-art satellite rainfall products [Stampoulis and Anagnostou, 2012]. Moreover, the model is tested in reproducing rainfall accumulations from 1 to 10 days and by using different values for the parameter T. Note that if T = 0, the analysis is addressed for the SSM product. Figure 4 summarizes the results of this analysis and shows that the model performance is quite good for rainfall data cumulated over a period of more than 3 days while poor scores are obtained in reproducing daily rainfall data. These results must be attributed to the daily temporal resolution of satellite data and the low reliability of equation (4) in estimating rainfall with the same aggregation interval of the input SM data (see also Figure 2). Moreover, Figure 4 highlights that the NS-values, on average, increase by 30% using the SWI (with T = 3 days) product with respect to SSM. The lower performance obtained with T = 0 must be attributed to the ASCAT retrievals noise that is strongly reduced by applying the exponential filter.
 Finally, in order to test the applicability of equation (4) at a global scale, average model parameters have been computed for the two regions. In particular, by using T = 5 days, Z = 90 mm, a = 15 mm/d, and b = 6, an average R and NS value equal to ~0.77 and 0.46 is obtained for the two regions. This initial result can be considered adequate even though additional analyses for different regions at a global scale are certainly required in order to derive findings that are more robust.
 A simple analytical relation, equation (4), is found to be capable of estimating rainfall observations from SM data. Specifically, satisfactory results are obtained in reproducing 1 day and 4 day rainfall accumulations when in situ and satellite observations, respectively, are employed. Based on these preliminary results, the application of the proposed approach opens new opportunities on both a basin and global scale.
 Indeed, today, by using dense SM networks or new monitoring techniques [e.g., Rosenbaum et al., 2012], catchment-scale SM data can be derived. Rainfall data estimated from this SM time series can be employed within hydrological models to address improvement in runoff prediction.
 By using remote sensing data, the procedure can be applied at a global scale thus providing a supplementary source of information for rainfall estimation, mainly in poorly gauged regions. The rainfall estimates obtained with this procedure should be compared with the standard satellite-derived rainfall products to assess the possible benefits (if any) that can be obtained. Moreover, through the same approach, the validation of different satellite SM products can also be carried out [Parinussa et al., 2011]. It is worth noting that nowadays long-term satellite SM products have been derived [e.g., Dorigo et al., 2012] thus potentially allowing at the one hand long-term rainfall estimates and, on the other hand, of reconstructing past precipitations in case where SM records are available but precipitations are not [Kirchner, 2009]. Moreover, as both the temporal resolution and the accuracy of satellite SM retrievals have been steadily increasing over time, more accurate rainfall estimates are also expected.
 We would like to acknowledge the Umbria Region (Italy), José Martinez-Fernandez from the Centro Hispano Luso de Investigaciones Agrarias, Universidad de Salamanca (Spain), and Sandra Perez from the Département de Géographie, Université de Nice-Sophia-Antipolis (France) for providing the in situ SM and rainfall data. The ASCAT surface SM data have been generated within the framework of the Satellite Application Facility in Support to Operational Hydrology and Water Management (H-SAF). We are also grateful to Crow, one anonymous reviewer, and the Editor for their comments and suggestions. We dedicate this work to the memory of our co-author Florisa Melone, who passed away on 28 October 2012. The only consolation we have is that we had the honor and privilege of working with her and had her in our life as colleague and mainly as friend.