# Abstract

The mathematical representation of soil water movement exhibits uncertainties in all model components. Data assimilation methods, like the ensemble Kalman filter (EnKF), combine models and measurements into an improved representation and can – at least in principle – account for all uncertainties. However, a proper description of the uncertainties is required, which is particularly difficult in soil hydrology, where model errors typically vary rapidly in space and time. Inflation methods can account for unrepresented model errors. To improve the EnKF performance, I designed an inflation method specifically for soil hydrology, that is capable of adjusting inflation factors to spatiotemporally varying model errors. For the application on a real-world case, I assessed the key uncertainties for the specific hydraulic situation of a 1-D soil profile with TDR (time domain reflectometry)-measured water contents. With the EnKF, I directly represented and reduced all key uncertainties (initial condition, soil hydraulic parameters, small-scale heterogeneity, and upper boundary condition), except for an intermittent violation of the local equilibrium assumption by the Richards equation. To bridge this time, I introduced a closed-eye period, which ensures constant parameters and improves the EnKF towards the goal of knowledge fusion – the consistent aggregation of all information pertinent to some observed reality.

# Setting

In this project data from the** Grenzhof test site** are used, which comprises a weather station and a soil profile equipped with 11 TDR probes to measure soil water content. The surface is kept free of vegetation. The figure shows the upper boundary condition at the site from 1 October (day 1) until 29 November (day 60). Precipitation is measured with a tipping bucket rain gauge, while evaporation is calculated using the FAO Penman-Monteith equation. The green line shows the response of the topmost TDR probe.

The largest uncertainties when modelling this specific situation are the initial condition, soil hydraulic parameters, small-scale heterogeneity, upper boundary condition and a possible violation of the local equilibrium assumption during the rain events.

# Closed-eye period

With a standard EnKF all key uncertainties are directly represented and reduced by incorporating them into an augmented state, except for the intermittent violation of the local equilibrium assumption. This estimation is performed during time period C (day 18-22) and iterated 10 times over this period.

The figure shows an example of the development of one parameter, the saturated hydraulic conductivity of the topmost layer, during this time. The parameter is corrected towards higher values during the rain event (to compensate the non-represented violation of the local equilibrium assumption) and is slowly decreased afterwards again.

The closed-eye period is defined to account for the violation of the local equilibrium assumption. It extends from the minimum to the maximum value of the parameter. Another 10 iterations are preformed, but now the parameter estimation is paused during the closed-eye period. The EnKF is only used to guide the water content state during this time, when the local equilibrium assumption is violated. This leads to a new estimate for the saturated hydraulic conductivity. The parameter is shifted towards another value and varies less, indicating a more consistent description.

# Inflation

During the rain event the local equilibrium assumption of the Richards equation is violated. This is not directly represented in the model, leading to too small uncertainties. To compensate this a spatiotemporal adaptive inflation method was developed. The method increases the model uncertainty, depending on the difference of model and measurements and their corresponding uncertainties.

The figure shows the inflation for the standard EnKF and closed-eye EnKF. With the standard EnKF the non-represented model error is partly compensated through biased parameters, leading to a smaller inflation than with the closed-eye EnKF. There, the parameter bias is prevented, leading to larger deviations of model and measurement during times when the local equilibrium assumption is violated. Consequently, a stronger inflation is required during the closed-eye period to effectively guide the water content state through this time.

# Results

Predictions based on parameters estimated with the closed-eye EnKF outperform the standard EnKF during times when the local equilibrium assumption is met, but consequently worsens predictions when the assumption is violated. The standard EnKF compensates the model errors partly through biased parameters. The closed-eye EnKF prevents the incorporation of the model structural errors in the parameters, which then are closer to the believed true material properties. A description of the dynamics during local non-equilibrium phases remains an open challenge.

The approach shows a way to limit the incorporation of errors into parameters and is one step towards the goal of knowledge fusion – the consistent aggregation of all information pertinent to some observed reality.

# References

PhD thesis:

- Bauser, H. H.: Knowledge Fusion in Soil Hydrology, PhD thesis, Ruperto-Carola University Heidelberg, Heidelberg, Germany, doi:10.11588/heidok.00024713, 2018.

Peer reviewed articles:

- Bauser, H. H., D. Berg, O. Klein, and K. Roth, Inflation method for ensemble Kalman filter in soil hydrology,
*Hydrology and Earth System Sciences*, 22(9), 4921–4934, doi:10.5194/hess-22-4921-2018, 2018. - Bauser, H. H., S. Jaumann, D. Berg, and K. Roth, EnKF with closed-eye period – towards a consistent aggregation of information in soil hydrology,
*Hydrology and Earth System Sciences*, 20(12), 4999–5014, doi:10.5194/hess-20-4999-2016, 2016.