HYDROLOGICAL PROCESSES Hydrol. Process. 16, 531–556 (2002) DOI: 10.1002/hyp.352 Tikhonov regularization as a tool for assimilating soil moisture data in distributed hydrological models E. E. van Loon1* and P. A. Troch2 1 Erosion and Soil & Water Conservation Group, Wageningen University, Nieuwe Kanaal 11, 6709 PA Wageningen, The Netherlands 2 Sub-department of Water Resources, Wageningen University, Nieuwe Kanaal 11, 6709 PA Wageningen, The Netherlands Abstract: Discharge, water table depth, and soil moisture content have been observed at a high spatial and temporal resolution in a 44 ha catchment in Costa Rica over a period of 5 months. On the basis of the observations in the first 3 months (period A), two distinct soil moisture models are identified and calibrated: a linear stochastic time-varying state-space model, and a geo-statistical model. Both models are defined at various spatial and temporal resolutions. For the subsequent period of 2 months (period B), four different ways to predict the soil moisture dynamics in the catchment are compared: (1) the application of the dynamic models in open-loop form; (2) a re-calibration of the dynamic models with soil moisture data in period B, and subsequent prediction in open-loop form; (3) prediction with the geo- statistical models, using the soil moisture data in period B; (4) prediction by combining the outcomes of (1) and (3) via generalized cross-validation. The last method, which is a form of data assimilation, compares favourably with the three alternatives. Over a range of resolutions, the predictions by data assimilation have overall uncertainties that are approximately half that of the other prediction methods and have a favourable error structure (i.e. close to Gaussian) over space as well as time. In addition, data assimilation gives optimal predictions at finer resolutions compared with the other methods. Compared with prediction with the models in open-loop form, both re-calibration with soil moisture observations and data assimilation result in enhanced discharge predictions, whereas the prediction of ground water depths is not improved. Copyright 2002 John Wiley & Sons, Ltd. KEY WORDS soil moisture prediction; catchment scale; data assimilation; regularization; generalized cross-validation INTRODUCTION The ability to predict soil water storage and movement in a heterogeneous landscape is important to manage water resources. Many studies have addressed spatial variability of soils and soil water by recognizing the stochastic nature of local variability (e.g. Greminger et al., 1985; Yeh et al., 1986; Unlu et al., 1990). Also, systematic components have been identified and linked to topographic characteristics (e.g. Hanna et al., 1982; Moore et al., 1991; Hairston and Grigal, 1991), soil morphological features (e.g. Kreznor et al., 1989), or chemical and physical attributes (Brubaker et al., 1993). The integration of both systematic and stochastic components has partially been achieved by conditioning geo-statistical techniques with secondary data such as topographic indices via (indicator) co-kriging (e.g. Lehmann et al., 1995; Western et al., 1998, 1999). However, geo-statistical techniques do not explicitly incorporate the knowledge about system dynamics and cannot easily take advantage of additional conditioning information at various scales, such as catchment discharge or evapotranspiration from different vegetation patches. Such an integration of observations from various sources with a dynamic model is known as data assimilation. Examples of data assimilation techniques applied to soil moisture estimation are found in Callies et al. (1998), Calvet et al. (1998), Galantowicz et al. (1999), Hoeben and Troch (2000), Houser et al. (1998), Katul et al. (1993) and Mahfouf (1991). All * Correspondence to: E. E. van Loon, Sub-department of Water Resources, Wageningen University, Nieuwe Kanaal 11, 6709 PA Wageningen, The Netherlands. E-mail: [email protected] Received 20 June 2000 Copyright 2002 John Wiley & Sons, Ltd. Accepted 16 May 2001 532 E. E. VAN LOON AND P. A. TROCH these data assimilation studies have in common that they consider one-dimensional soil water movement (i.e. in the vertical direction), while utilizing only rain and remotely sensed soil moisture estimations as observations. In summary, we can say that there are two distinct types of soil moisture study: those that focus on the lateral soil moisture distribution and use static models, and those that consider the vertical distribution of soil moisture while using dynamic models. The first type of study uses mainly field observations of soil moisture in combination with soil and terrain properties, whereas the second type of study almost exclusively uses remote-sensing observations. It is the aim of this study to combine elements from both areas, as a first step towards an integration of the two approaches. More precisely, in this study a data assimilation technique is applied to a system where both lateral and vertical soil water movement take place and for which only ground- based observations are available. First a distributed hydrological model (which will be called the d-model) and a geo-statistical soil moisture model (s-model) are developed. Then a data assimilation algorithm (the da-model) is developed to combine the results from both models. The relative efficiency of the da-model is compared with the alternative methods of state estimation through the straightforward application of the d-model, the re-calibrated distributed hydrological model (the dc-model) or the s-model. Each model is parameterized for a number of resolutions. The reason for considering different resolutions is that apriori it is unclear at which resolutions the different methods will perform best. Different model parameterizations are required at different resolutions because the models for the hydrological system under consideration can generally not be defined at multiple resolutions (van Loon and Keesman, 2000). MATERIAL AND METHODS Description of data The data for this study have been collected in a 44 ha catchment in north-west Costa Rica. A series of low-resolution measurements is available from 20 July 1997 till 21 December 1997 and within this period a set of high-resolution measurements is available for 4 October till 21 December. At four locations rain has been measured using tipping buckets. At two locations discharge has been measured at 1 min time instants, using v-crest weirs. Ground water depth has been observed manually in 20 piezometers at hourly instants during and just after rain, and daily between rainfall events. During the period 20 July–4 October, volumetric soil moisture was measured at 40 locations once every 4 days, and during the period 4 October–21 December at 60 locations once every 2 days. For the soil moisture measurements a Trime time domain reflectometry (TDR) system (in plastic tubes) was used, enabling the measurement of soil moisture over 20 cm layers down to 80 cm. The location of the various instruments is shown in Figure 1. Close to the Trime tubes about 150 soil moisture measurements were taken within the 0–10 cm topsoil every 10 days, using a TDR system that was directly inserted into the ground. These measurements were used to correlate soil moisture observations with soil and terrain properties and for checking the other measurements. Terrain and soil were mapped in detail. The terrain was measured using a kinematic GPS technique in combination with a conventional ground-based survey. Soil colour, soil depth, the dimensions of cracks, stability of soil aggregates, organic matter content, and texture were determined at 90 locations, and the hydraulic permeability was determined at 30 of these locations, using a Guelph permeameter (both at 10 and 20 cm depths). In addition, the soil colour, the dimension of cracks, the areal density of cracks and the texture (field-determined) were observed at a regular spacing of 20 ð 20 m2. The purpose of the soil data is either to relate these directly to soil moisture behaviour or group them into a limited number of model units. A contour map and the distribution of upstream area A and the lnA/ tan ˇ index are shown in Figures 2 and 3. The data collected at this site have been extensively described and analysed with regard to overland flow in van Loon and Stroosnijder (2001) and van Loon et al. (submitted). In this study the emphasis will be on soil moisture. Copyright 2002 John Wiley & Sons, Ltd. Hydrol. Process. 16, 531–556 (2002) SOIL MOISTURE PREDICTION BY DATA ASSIMILATION 533 piezometers soil moisture measurements rain gauge c b a weir N 0 300 m Figure 1. Location of observations 172 168 174 170 178 170 168 172 174 172 174 172 178 N elevation (masl) 0 300 m 164 173 181 Figure 2. Contour map of the study catchment Formulating the distributed hydrological model The soil moisture distribution in the research catchment is described by a model that considers three horizontally distributed state variables: surface water q, moisture in the top 0Ð4 m of the soil w and water in the deeper soil layer (0Ð4 m down to parent material, which varies in depth from 0Ð4to3m)d.Inthe model w represents the unsaturated zone and d represents the saturated zone, which implies that soil moisture observations are related to the value of w and piezometric observations to the ground water level in d. Horizontally the terrain is subdivided by a regular grid. The deeper soil layer is assumed to be impermeable at the bottom, and there is assumed to be no flow at the catchment boundaries. Both surface flow and flow in the upper soil are assumed to be one-dimensional, following the direction of steepest descent as defined Copyright 2002 John Wiley & Sons, Ltd. Hydrol. Process. 16, 531–556 (2002) 534 E. E. VAN LOON AND P. A. TROCH Upstream area (A, ha) 40 30 20 N 10 0 300 m Topographic index: ln(A/tanβ) 15 10 5 Figure 3.