Two study areas in Denmark are used in this study. The 440 km2 Karup catchment is located in the centre of Jutland (left in Fig. 1). The land use is mainly agriculture, and topographical elevation is between 20 and 100 m. The catchment lies in an alluvial plain with coarse sandy soils and a strongly groundwater dominated hydrological regime. The Ahlergaarde catchment is located in one of the most irrigated areas of Denmark (right in Fig. 1). Of the total catchment area of 1044 km2, 61 % is covered by agricultural crops. The surface geology consists mostly of sand and, also in this catchment, the streamflow is dominated by groundwater inflow.

The Karup catchment is a well-studied catchment in terms of model parameterization and model calibration (Refsgaard, 1997; Madsen, 2003; Zhang et al., 2015). A relatively simple model with a fast computation time was developed for this catchment to test and verify various DA methods. The Ahlergaarde catchment is the research catchment of the Danish Hydrological Observatory (www.hobe.dk, Jensen and Illangasekare, 2001). This study area is ideal for further testing DA methods using real measurements.

The MIKE SHE hydrological modelling system is used for developing models for the above two catchments. As a physically based distributed hydrological model, MIKE SHE simulates the major processes in the water cycle, including evapotranspiration, overland flow, unsaturated flow, groundwater flow, river flow and the interactions between them. MIKE SHE also has the flexibility of modelling each process at given spatial and temporal resolutions with different complexity. The complexity can be chosen according to the model purpose and data availability (Graham and Butts, 2005).

In the Karup catchment, the modelling is based on the following process descriptions: 2-D groundwater flow is assumed and modelled by one computational layer in the saturated zone, drain flow (pipes/ditches) is described by a simple conceptual relationship and occurs when the groundwater table exceeds the drain level, 1-D unsaturated flow is assumed and based on a simplified gravity-based flow equation, 1-D channel flow is assumed and based on kinematic routing, 2-D overland flow routing is based on the diffusive wave approximation of the Saint-Venant equations, and evapotranspiration is described, including interception, soil evaporation and transpiration by vegetation (DHI, 2015). The numerical discretization in the horizontal plane is a 1000×1000 m2 grid size. The model is forced by station-based daily precipitation and uniform daily values for reference evaporation. In the MIKE SHE model, the temporal resolution is dynamic and differs between the modules. For the maximum allowed time step, 6 h is specified for overland flow, 6 h for unsaturated flow and 12 h for saturated flow respectively.

For the Ahlergaarde catchment, the same model components are included as for the Karup catchment. For computational efficiency, and due to the fact that the exact irrigation information in terms of both location and amount is not known, the irrigation module is not activated in the model. The modelling approaches are the same as for Karup, except that 3-D groundwater flow is considered with six numerical layers defined according to geological stratigraphy. Another main difference is that the model uses a smaller grid size (200×200 m2). The finer model discretization enables the model to utilize finer-resolution system data such as geological stratigraphy, soil type and land use. The model is forced with grid-based daily precipitation, temperature and reference evaporation. In both catchments no-flow boundaries are defined along the catchment borders. The temporal resolution in the model is constrained by maximum time steps of 2 h for overland flow, 2 h for unsaturated flow and 6 h for saturated flow respectively. The model parameterization and model calibration are introduced in Sect. 2.3.

The finer model resolution and increased complexity for the Ahlergaarde catchment increase the simulation time significantly. For example, the average model time step in the groundwater zone decreases from 7.5 h in the Karup model to 1.3 h in the Ahlergaarde model. In consequence 1-year model simulation takes less than 1 min for Karup and around 1 h for Ahlergaarde. The differences in model resolution and simulation time for the two catchments are summarized in Table 1.

For both catchments, the model parameterization is kept relatively simple yet able to represent the overall spatial patterns of key model parameters. When specifying the parameter values for each property class (e.g. geological unit, vegetation type and soil type), most of the parameters cannot be estimated empirically or directly inferred from data. Thus model calibration is usually required using an optimization algorithm like AUTOCAL (Madsen, 2003) or PEST (Doherty, 2010).

For the Karup model, the most sensitive parameters describing the hydraulic properties of the river, unsaturated zone, saturated zone, and river–aquifer interaction are calibrated using AUTOCAL (Zhang et al., 2015). As calibration data we use 35 biweekly groundwater head observations and daily observations of stream discharge for a 6-year period (1969–1974) (Fig. 1).

The Ahlergaarde model is calibrated using PEST version 11.8 (Doherty, 2010). The data used in the calibration are groundwater head observations (466 in total) scattered over the catchment (not shown in Fig. 1) and river discharge observations from the period of 2006–2009. In most of the groundwater wells only one observation is available for the entire calibration period and only a few wells have time series. Discharge data comprise time series of daily values from five stations (Fig. 1). Similar to the Karup catchment, the most sensitive parameters (7 parameters) are selected for calibration, with 13 parameters tied to those 7 parameters. The calibrated values for those 7 parameters are listed in Table 2 (first 7 parameters) together with the confidence intervals obtained from the inversion process. The remaining parameters in Table 2 are not included for calibration, but are only selected for perturbation, with a detailed explanation given in Sect. 5. The original calibrated model uses a simplified two-layer approach to simulate unsaturated flow and evapotranspiration, where the average soil moisture is calculated for the root zone and the layer below the root zone. In order to assimilate in situ soil moisture data at different depths, the gravity flow module is used as a replacement for the two-layer approach in the unsaturated zone. By doing so, soil moisture can be calculated at different depths. The overall modelling performance in terms of water balance and discharge dynamics becomes marginally reduced compared to the original calibration results.

The assimilation algorithm used in this study is the ETKF, which is a popular variation of the EnKF (Evensen, 2003). Similar to the EnKF, the ETKF is a Monte Carlo implementation of the Kalman filter, which approximates the posterior probability distribution conditioned on a series of observations, and is able to deal with non-linear models. In comparison to the EnKF, the ETKF is a deterministic filter, as it does not require additional observation perturbations. The ETKF was originally introduced by Bishop et al. (2001) and later modified to be unbiased (Wang et al., 2004). As an ensemble-based deterministic filter, it has the advantage of calculating the forecast error covariance efficiently. It is also computationally faster than the ensemble square root filter (EnSRF) (Whitaker and Hamill, 2002).

To develop the DA algorithm, a state–space formulation is needed: Xt+1=MXt,Ut,θ≈MdXt,Ut̃,θ̃, where M is the stochastic model operator based on the numerical solution to the MIKE SHE equations, Md is the deterministic MIKE SHE model operator, Xt and Ut are the state vector and model forcing respectively at time step t, and θ stands for the model parameters. Ut̃ and θ̃ are the perturbed forcing and parameters respectively. Note that the stochastic model operator M is approximated by the deterministic MIKE SHE model, taking both model forcing uncertainty and model parameter uncertainty into account (Zhang et al., 2015). In both models, precipitation and potential evapotranspiration are perturbed by adding a random Gaussian noise to the actual value. The parameter uncertainty is described mainly using the covariance estimated from calibration. The selected parameters are assumed to be multivariate normal/lognormal distributed and perturbed using Latin hypercube sampling based on the associated parameter covariance. Additional post-processing steps are used to ensure that the perturbed parameters are still within realistic parameter ranges.

At time t+1, the observations can be written as Yt+1=HXt+1+εt+1,εt+1∼N0,Rt+1, where Y denotes the observation vector, and H is the linear mapping operator specifying the deterministic relationship between observations and model state X. In this study, the observations are either groundwater head, soil moisture, or both. Similarly, the state vector consists of groundwater head, soil moisture, or both. When two variables are assimilated, the state vector is augmented to accommodate both variables at all computational cells, and the observation operator H is revised to select the correct model equivalent and compare it with the corresponding observation. The observation noise is assumed to be Gaussian, temporally uncorrelated, and spatially uncorrelated, with the zero mean and a prescribed constant standard deviation σr for each observation type. Therefore, Rt+1 is a diagonal matrix with constant values for each observation along the diagonal (i.e. Rt+1=diagσr12,…,σr12,σr22,…,σr22,…,σro2,…,σro2) for total o observation types.

The forecast state distribution can be estimated by a finite number m of model realizations from Eq. (1) as follows: Xf=xf1,xf2,…,xfm, where the superscript f stands for “forecast”.

The forecast error covariance can be written as Pf=X′fX′fT/(m-1), where X′f is the forecast ensemble perturbation X′f=xf1-Xf‾,xf2-Xf‾,…,xfm-Xf‾ and Xf‾ is the ensemble mean. After assimilation, both the analysed state mean and the analysed error covariance can be calculated: Xa‾=Xf‾+KY-HXf‾,Pa=(I-KH)Pf, where the superscript a stands for “analysed”, and K is the Kalman gain defined as K=PfHTHPfHT+R-1.

In practise, Pa is never explicitly calculated, and only the ensemble mean and ensemble anomalies are updated. Based on factorizing Eq. (7) on both sides, the following equation is obtained: X′a=X′fTwhereT=I+HX′fTR-1HX′f/(m-1)-1/2U and U is an arbitrary orthonormal matrix UUT=I.

The MIKE SHE model is coupled with a generic DA library that handles the time propagation and update of the model ensemble based on the ETKF (Ridler et al., 2014b).

In ensemble-based Kalman filter systems, the forecast state and its associated uncertainty are represented by a limited ensemble of realizations. The undersampling can lead to filter inbreeding and spurious correlations in the error covariance matrix, which potentially can lead to filter divergence. Localization is a commonly used technique when applying ensemble-based Kalman filters to overcome this problem. By artificially reducing the impacted spatial domain of observations, the spurious correlation between two remote locations can be avoided. For each element in the state vector, local analysis (LA, Sakov and Bertino, 2010) is used to approximate the state error covariance within the local window. The ensemble anomalies outside this local window will be unchanged during the filter updates. However, LA is usually applied to a single state variable for which certain spatial correlations exist. When the state vector contains two or more variables, specifying the localization degree for each variable is not straightforward. More importantly, correlations between variables are not clear, because physical distances between variables may not exist. Similar to the approach by Kang et al. (2011), we introduced different variable localization schemes based on whether the correction of one variable can impact the update of other variables. In this section, the distance localization will be introduced first, followed by the variable localization.

When a single variable is assimilated (groundwater head or soil moisture), the state vector only consists of the corresponding observed variable at all model grid cells. Therefore, the remaining variables will not be changed directly from the filter. However, as both the groundwater component and unsaturated zone are fully coupled with surface water and other model components, the whole model state will be affected from updating a single variable. Different experiments are carried out using an ensemble size of 60:

NoDA: deterministic model without DA;

As the experiment names indicate, H stands for groundwater head and SM stands for soil moisture. Loc indicates that localization is added to the experiment.

Results from the DA experiments are shown in Fig. 4. When head is assimilated (DA_H), the RMSE for head improves significantly from 0.21 to 0.08 m. However, soil moistures at the three depths are basically not influenced. When localization is used (DA_HLoc), the corrections are localized around the head observations and the overall performance is slightly degraded.

When soil moisture at 5 cm depth is assimilated alone without localization (DA_SM5), the soil moisture profile clearly improves at all three depths. However, for head the performance is almost the same as in the deterministic model. Different localization scales have been tested with assimilating soil moisture at 5 cm depth (DA_SM5Loc and DA_SM5LocSmall). The result indicates that the overall assimilation performance decreases with a smaller localization scale.

When soil moisture at both 5 and 25 cm depths is assimilated (DA_SMBoth and DA_SMBothLoc), the performances are similar regardless of localization. Compared to the result from DA_SM5, the soil moisture estimate improves at 25 cm but slightly worsens at 5 cm. Compared to DA_SM5Loc, the results show some improvements at 25 and 50 cm. Again, groundwater head is hardly influenced by assimilating soil moisture. In the following experiments, we include observations at both 5 and 25 cm when soil moisture is assimilated.

As we can see from Fig. 4, univariate assimilation with localization improves the estimate of the assimilated variable albeit the results are slightly worse compared to the experiment without localization in the case of assimilating head or soil moisture at 5 cm. This could be explained as follows. Firstly, spatial correlations are affected by the catchment size and the relatively large grid size used. Pronounced correlations exist even between remote locations, and therefore localization may cut off true correlations, which leads to a worse result overall. Secondly, there are a relatively large number of observations compared to the size of the state vector, which reduces the problem of spurious correlation. Study shows that there is a strong relationship between the significance of spurious correlation and the number of observations (Rasmussen et al., 2015). Localization is more effective to reduce spurious correlation when the number of observations is relatively small. We also notice that the 50 cm depth soil moisture has an overall larger error compared to the surface layer; this is due to the fact that the soil moisture cell saturation in the deeper layer is more sensitive to the parameter uncertainty, which makes the deeper layer more difficult to reproduce.

In this section, several experiments assimilating both groundwater head and soil moisture are carried out with a focus to test different localization schemes. The abbreviations D and V indicate distance localization and variable localization respectively.

DA_HSM: assimilating both head and soil moisture (at both 5 and 25 cm depths) without localization to any variable.

Results from the DA experiments are shown in Fig. 5. When neither distance localization nor variable localization is used, all observations are used to update the state in all grid cells for each variable (DA_HSM). In this case the estimated correlations between groundwater head and soil moisture are used in the update. The DA results show improved performance for soil moisture at 5 and 25 cm, but much worse performance at 50 cm as well as for groundwater head. In the current filter settings the full state covariance matrix contains unrealistic, spurious correlations, which eventually degrade the update in the deeper soil layers.

In experiment DA_HSMLoc_DV, both distance localization and variable localization are used. Therefore, the state updates are spatially localized for each variable and the correlation between the two variables is neglected. Particularly in this case, when there is only soil moisture observation assimilated, the updates are limited to the upper 1 m soil moisture profile, while no correction is made for head. When both types of observation are assimilated, the corrections are made for each variable using its own error information. We can see from Fig. 5 that the experiment shows an overall improved result.

In experiment DA_HSMLoc_D, distance localization is applied to head and soil moisture, but variable localization is not included. In this case, regardless of observation type, the soil moisture is corrected within 1 m depth together with head. The result from this experiment shows an improved estimate for soil moisture at 5 and 25 cm, together with groundwater head. However, the soil moisture at 50 cm is slightly worsened. This indicates that the correlation between surface soil moisture and groundwater head estimated from the ensemble is valid and improves the assimilation performance. Compared to DA_HSM, the result shows that excluding the error information from deeper soils (below 1 m to saturation) reduces spurious correlations and improves the performance. However, compared to DA_HSMLoc_DV, the result is slightly worse for head and deeper soil moisture.

In experiment DA_HSMLoc_V, distance localization is off and variable localization is applied. This means that the error information from one variable is used to update the entire domain of its own variable but does not affect the other variable. The result indicates worse assimilation performance for soil moisture at 50 cm and for groundwater head. One potential reason is that the lower layers of the unsaturated zone are usually fully saturated but in this experiment corrected by the surface soil moisture observation, while the groundwater head is corrected by the head observation. Potential inconsistencies may exist with these two updates.

As mentioned in Sect. 3.2, localization allows the ensemble filters to work properly with a limited ensemble size. The above experiments are based on an ensemble size of 60, which is determined by balancing both assimilation performance and computational time. Some of the experiments are repeated for ensemble sizes of 30 and 90 respectively to analyse how the assimilation performance and the choice of localization are affected by the ensemble size. The results are shown in Fig. 6.

As can be seen from Fig. 6, in the experiment assimilating head without localization (DA_H), increasing the ensemble size (from 30 to 90) slightly improves the head estimation. However, the performance difference between ensemble sizes of 60 and 90 is small. When localization is used, the performances with all ensemble sizes are very similar (DA_Hloc).

In the experiment assimilating soil moisture at 5 cm depth without localization (DA_SM5), increasing the ensemble size also improves the soil moisture at deeper depths. This indicates that using only an ensemble size of 30 introduces a spurious correlation between surface soil and deeper soil, which is reduced with larger ensemble sizes. An ensemble size of 30 also leads to a much worse result for groundwater head compared to ensemble sizes of 60 or 90. When localization is used (DA_SM5Loc), the assimilation performance is similar using the three ensemble sizes. Compared to the DA_SM5, there is a large improvement in groundwater head when using an ensemble size of 30.

When both soil moisture (at 5 and 25 cm depths) and head are assimilated without localization (DA_HSM), the performance is generally improved when increasing ensemble size. However, increasing the ensemble size to 90 still leads to a worse performance for soil moisture at 50 cm and groundwater head compared to the deterministic model. When localization is used (DA_HSMLoc_DV), the soil moisture at 50 cm and the head improves as the ensemble size increases. Overall, the assimilation performance increases in DA_HSMLoc_DV when increasing the ensemble size.

Using an integrated model where the various hydrological processes are coupled, assimilation of head and soil moisture may also affect other model variables. The effects on evapotranspiration and river discharge are examined in this section. For actual evapotranspiration, we calculated average RMSE with respect to the true model of actual evapotranspiration over all 35 soil moisture observation locations during the DA period, and for discharge the performance at the catchment outlet for the entire assimilation period is evaluated using the Nash–Sutcliffe efficiency score. The results are summarized in Table 3.

The differences in RMSE for actual evapotranspiration among all experiments are small. When H is assimilated alone (DA_H and DA_H_Loc), actual evapotranspiration is basically unchanged, while when soil moisture is assimilated, RMSE is marginally reduced compared to the deterministic model.

The performance of discharge is slightly improved by assimilating head (DA_H and DA_H_Loc). The improvement is mainly with respect to low flow, which is underestimated by the deterministic model. This is expected as the baseflow is corrected by updating groundwater levels. When soil moisture is assimilated with localization (DA_SM5Loc and DA_SMBothLoc), the discharge is also slightly better. However, when both variables are assimilated without localization (DA_HSM), the discharge is significantly worse, with unrealistic peak flows during spring. This is a result of the poorer head estimations in the entire domain. When localization is used for soil moisture and groundwater head (DA_HSMLoc_DV), discharge is improved significantly and comparable with the deterministic model. This also demonstrates the necessity of using localization to constrain the spatial updates.

Groundwater heads are measured bi-hourly in nine wells (Fig. 1) using Eijkelkamp mini divers. The divers were installed in these wells in November 2012, and thus the length of the time series is limited. Moreover, due to occasional instrument failure, the data coverages are further constrained and vary among the wells. In the groundwater model six numerical layers are defined (layer 1 in the bottom and layer 6 in the top). The nine wells are screened at different depths. Wells M5398, M5637, M5353, and L8008 are screened in layer 5, while wells M5373, M5647, M5844, M5393 and M5366 are screened in layer 4. When comparing in situ head measurements with model-predicted equivalents, large-level differences usually occur due to scale disparities, and are sometimes also accompanied by dynamic differences. Therefore, we calculated the average difference between observations and model simulations, and subtracted this difference from the original data. By doing so, we can avoid introducing observation bias into the assimilation system. An example of the processed observations and the open loop ensemble for well 5737 (1 November 2012 to 31 December 2013) is shown in Fig. 7 (top panel).

Soil moisture is measured at 30 sites across the catchment according to representative combinations of topography, land cover, and soil type using Decagon 5TE sensors. The dominant land uses are heath, agriculture and forest. At each site, sensors are installed at three depths, 2.5, 22.5 and 52.5 cm, corresponding to measurement depth intervals of 0–5, 20–25 and 50–55 cm. Measurements are taken with 30 min intervals.

Most of the agriculture sites are irrigated in May and June, and the soil moisture is greatly influenced, with several sudden increases during that period. However, in the model irrigation is not considered because detailed information on irrigation at the local sites is not available. Therefore, the sites where irrigation is evident from the soil moisture recordings are excluded for assimilation. In addition, a quality control to correct for systematic biases and to filter out unrealistic values has been carried out for the remaining sites. Although measurements are carried out at three depths at each site, we only use measurements at 2.5 and 22.5 cm depths for assimilation, as the surface/near-surface moisture is of the most importance for the exchange of water and energy between land and the atmosphere. After processing, 18 out of 30 sites are used for assimilation (Fig. 1). As an example, Fig. 7 (middle and bottom panels) shows the processed soil moisture observations and the open loop ensemble at site nw1.1 (1 November 2012 to 31 December 2013).

In addition to groundwater and soil moisture observations, discharge observations are available in the Ahlergaarde catchment at the outlet and at tributaries (right side of Fig. 1). Evapotranspiration data-based eddy covariance measurements are available from a flux station (Voulund station) located in the catchment.

Similar to the experiment settings in the Karup catchment, the observation noise for each variable is assumed to be white Gaussian, with a homogeneous and constant standard deviation of 0.2 m for head and 5 % for soil volumetric water content. The head and soil moisture data are interpolated to weekly and daily frequencies respectively for assimilation. Due to the larger model domain, more complex process descriptions and finer spatial resolution compared to the Karup catchment set-up, the computational time for the Ahlergaarde catchment is substantial. This implies that a larger ensemble size is unaffordable. Furthermore, the more frequent data assimilation contributes to a longer simulation time. From these considerations, an ensemble size of 50 is adopted. With a 1-year assimilation period, the simulation time is around 3–7 days, depending on the experiment settings.

With the purpose of assimilating head and soil moisture, different experiments have been carried out to investigate the assimilation performance. Considering the large model domain and fine grid, localization becomes more important here than in the previous example. Distance localization is added to both variables separately, and variable localization is used when both variables are assimilated. For groundwater head, we allow for updates in all layers over the vertical. Horizontally, we use a localization radius of 5 km for all layers. For soil moisture, we use a horizontal localization radius of 1 km and a vertical localization depth of 0.5 m (top eight layers in the unsaturated zone). The following experiments are carried out:

NoDA: deterministic model without DA;

The assimilation performance is evaluated by comparing the model output with the observations (18 sites) using the average RMSE over the assimilation period. The result is summarized in Table 4. In the experiment with assimilating head only (DA_HLoc), the RMSE of the head decreases from 0.34 to 0.21 m. However, the soil moisture predictions at both depths do not improve compared to the deterministic model. In the experiment which assimilates only soil moisture (DA_SMLoc), the RMSE of soil moisture at both depths decreases, especially at depth 22.5 cm. The head estimate, however, shows a similar performance to the deterministic model. When both variables are assimilated (DA_HSMLoc_DV), the RMSE of the head decreases from 0.34 to 0.21 m. The RMSE of soil moisture decreases from 0.044 to 0.040 m3m-3 at 2.5 cm depth, and from 0.034 to 0.028 m3m-3 at 22.5 cm depth.

Figure 8 shows the assimilated results for the same sites as shown in Fig. 7. Clearly, after 1 November 2012 when the DA period starts, the ensemble mean is approaching the observations, especially for the head and soil moisture at 22.5 cm depth. Although limited observations are assimilated, corrections are made for a large area within the model domain. Figure 9 shows spatial root mean squared differences (RMSD) of soil moisture and head at corresponding observation layers between the assimilation result and the deterministic model, which illustrates the corrections made by DA spatially. For each grid cell, the variables' time series values from the assimilated model and the deterministic model are used to calculate the RMSD.

From Fig. 9, we can clearly see the effect of the assimilation in the model domain. For soil moisture relatively large corrections are made at 22.5 cm depth compared to the surface layer. Compared to groundwater head, however, the soil moisture corrections are more localized. For both soil moisture and groundwater head, most of the large corrections are made at places near the locations of observations. For groundwater head in the western and south-eastern regions where no head observations are available, the corrections are generally small.

In this section, the effect of assimilation on actual evapotranspiration and river discharge is evaluated by comparing model predictions and observations. Figure 10 compares discharge at the catchment outlet and evapotranspiration at the flux station for the different experiments. The flux station is located in the central–northern part of the catchment, with several soil moisture stations around. In both graphs in Fig. 10, only small differences are seen between different simulations. This is further substantiated by the performance measures listed in Table 5.

As shown in Table 5, RMSE for actual evapotranspiration is similar in all three assimilation experiments. There is a small improvement for discharge when head is assimilated (DA_HLoc). The experiment DA_HSMLoc_DV with both variables being assimilated provides better results overall.