Unrepresented model errors – effect on estimated soil hydraulic material properties

Unrepresented model errors influence the estimation of effective soil hydraulic material properties. As the required model complexity for a consistent description of the measurement data is application–dependent and unknown a priori, we implemented a structural error analysis based on the inversion of increasingly complex models. We show that the method can indicate unrepresented model errors and quantify their effects on the resulting materials properties. To this end, a complicated 2D subsurface architecture (ASSESS) was forced with a fluctuating groundwater table while Time Domain Reflectometry 5 (TDR) and hydraulic potential measurement devices monitored the hydraulic state. In this work, we analyze the quantitative effect of unrepresented (i) sensor position uncertainty, (ii) small scale-heterogeneity, and (iii) 2D flow phenomena on estimated soil hydraulic material properties with a 1D and a 2D study. The results of these studies demonstrate three main points: (i) The fewer sensors are available per material, the larger is the effect of unrepresented model errors on the resulting material properties. (ii) The 1D study is likely to yield biased parameters due to unrepresented lateral flow. (iii) Representing and 10 estimating sensor positions as well as small–scale heterogeneity decreased the mean absolute error of the water content data by more than a factor of 2 to 0.004.


General Comments
As a referee, but also as a reader of studies dealing, among various sources of uncertainties, also with those associated with the locations of sensors that monitor a flow process, there is always something causing me some concern.When setting up an experimental test, efforts are made reducing errors (especially the systematic errors) and, among other things, one measures the positions of the various sensors as accurately as possible.I also understand that this task can be a bit more complicated under field conditions, especially when inserting the sensors at the greatest soil depths.Therefore and to the benefit of a wider readership, the authors should justify more why they are interested in this type of uncertainty.Moreover, I have the feeling that the error in sensor location should be viewed more as a systematic error rather than a random error.I think that the method employed by the authors might not be adequate to treat the presence of systematic errors.Some clarifications and a discussion on this point seem deserving.Reply: We agree, that efforts are made to measure the positions of the various sensors as accurately as possible.Yet, the surface and/or the subsurface structure may change with time and requirements for accuracy and precision may change a posteriori.We clarified this in Sect.A1.4 (Page 23, Line 16).We agree that the uncertainty in the sensor position is a systematic or structural error.This is the reason why this uncertainty was represented and the parameter estimation algorithm was used to propose more consistent positions of the sensors minimizing this systematic error.Reply: Lacking direct measurements of the unsaturated hydraulic conductivity at the position of the TDR sensors, the presented method merely yields an estimate of the initial hydraulic state and an approximation of the soil water characteristic.The remaining parameters for the initial hydraulic conductivity function (K s and τ ) are taken from Carsel and Parrish (1988, 10.1029/WR024i005p00755) and are independent of the presented measurement data.Hence, the presented method is not applicable to approximate the hydraulic conductivity function.
P.1, L.20-23.On the topic of inverse modeling applied to Soil Hydrology, I suggest citing the more recent and comprehensive papers by Hopmans et al. ( 2002) and/or by Vrugt and Dane (2006).Concerning the lab-scale experiment, the paper by Romano and Santini (1999) also treat types of errors of interest for the present study.As for the in-situ applications, the paper by Romano (1993) can also be in line with some aspects of the present study.
Reply: We revised the introduction accordingly.Please also note the reply to comment 1 of SC1.Reply: Since the applications of the referred studies are diverse, we kept the formulation more general.One exemplary process, which is neglected in most of the studies, is small-scale heterogeneity.
As the major focus of the manuscript, we investigate the effect of neglected structural errors which lead to suboptimal results using the χ 2 penalty criterion.Therefore, we also use the χ 2 penalty criterion coupled with the Levenberg-Marquardt algorithm and quantify the effect of unrepresented model errors by resulting residuals and material properties of the different setups (Sect.2.3 and Sect.2.4).
In order to compare the best result of the different setups, we are rather interested in maximum likelihood instead of its distribution in this work.The former is more efficiently found with the Levenberg-Marquardt compared to the DREAM algorithm.Additionally, if the χ 2 is used as likelihood function in DREAM, the discussed problem of neglected processes and uncertainties will remain the same as we use a flat prior in this study.Also, adding additional material would make the already long manuscript even longer.
(*) P.4, L.8-10.Strictly speaking, the θ-based Richards equation describes the variations in space (x, y, and z coordinates) and time (t) of the volumetric soil water content.Then, due to the selected relationship between water content and matric pressure head, one can retrieve the corresponding variations in h.Reply: We changed the wording in Sect.2.2.1 (Page 4, Line 9).
(*) P.19, L.25-27.This is a quite common outcome when modeling of data with a maximum likelihood estimator and optimization techniques.I think that this problem should be addressed in another way.Namely, more in terms of the information content of the available input datasets.Does the initial information content increase when adding the additional data?Are the additional data not at all, or weakly, or strongly P.1, L.13.The authors claim that the approximated soil water retention function is reasonable close to the inversion results.Actually and allowing for the types of water flow processes investigated, it would have been more interesting and effective that the favorable outcome is in terms of the unsaturated hydraulic conductivity function.From the results depicted in the right plots of Fig.10and Fig.13, this does not seem C2

P
.1, L.22.The paper bySchneider et al. (2006)  was published in HESS, not in Hess-D.Reply: We corrected the reference.(*)P.2, L.10-13.It is not clear (at least to me) which processes the authors are talking about.For example, the sensor position is definitely not a process.Moreover, as far as I am aware, the previous studies refer to minimum unknown parameters to be estimated mainly because they employed the classic χ 2 penalty criterion coupled with the Levenberg-Marquardt (LM) algorithm.Why do not compare the present results with those ones whether you use, for example, the DREAM tool developed by Vrugt (2016)?By doing that way, the paper would be even more interesting since the authors C3 developed a modified LM algorithm.