the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 20 Nov 2025
Spectral Analysis of Groundwater Level Time Series for Robust Estimation of Aquifer Response Times
Abstract. Groundwater resources represent Germany's most important source of freshwater but they are increasingly under pressure. Climate change, societal developments, and rising abstraction rates are impacting subsurface storage in ways that are currently difficult to predict, affecting both the quantity and quality of groundwater. To ensure sustainable groundwater management, it is crucial to evaluate the intrinsic and spatially variable vulnerability of groundwater systems, especially to prepare for the effects of hydrological extremes. In this context, the groundwater response time, defined as the timescale over which a groundwater system responds or adjusts to changes in external or internal conditions, serves as a valuable indicator for vulnerability assessments. Unlike traditional methods, we propose estimating response times through spectral analysis of groundwater level data. Time series from nearly 200 selected observation wells across Bavaria in Southern Germany were processed and transformed into the spectral domain. Corresponding recharge time series were extracted from high-resolution hydrological model outputs. By integrating these data with hydrogeomorphic information, we fitted a semi-analytical model to the groundwater level spectra to obtain aquifer response times. The semi-analytical solution for the spectral domain accurately reproduced the majority of observed groundwater level spectra. Most estimated response times fall between roughly 50 and 300 days. Significant correlation were found between the response time and the depth of the groundwater table. Groundwater systems exhibiting longer response times are interpreted as more resilient to drought conditions and therefore potentially better suited for groundwater abstraction than aquifers with shorter response times.
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Status: final response (author comments only)
- RC1: 'Comment on egusphere-2025-5666', Anonymous Referee #1, 27 Dec 2025
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RC2: 'Comment on egusphere-2025-5666', Gunnar Lischeid, 09 Feb 2026
Assessing the vulnerability of groundwater systems is one of the key challenges of groundwater research, not the least due to presumed increasing probability of extreme events. Most of the work done so far is based on modelling studies. Houben et al. followed a different approach based on earlier work by Zhang and Schilling. I agree that spectrum analysis has great and still massively underrated potential in this regard. It is to the credit of this work that it draws the attention of the scientific community to this. However, the devil is in the details.
- My major criticism is, that this study completely ignores the major source of low-pass filtering. The approach by Liang and Zhang (2013) works under the ideal conditions of tailored numeric experiments. Even then the effects on power spectra are fairly small and are restricted to small distances from the constant head boundary. In real world settings, though, these effects are hardly discernible due to the much stronger impact of vadose zone low-pass filtering. In contrast to the effect described by Liang and Zhang, however, the latter starts to manifest in the high frequency part. This phenomenon is closely related to the second law of thermodynamics (dissipation is the strongest for the high energy part of the spectrum). It applies likewise to soil temperature, electromagnetic waves, etc. Consequently, it doesn’t come as a surprise that the spectra are consistently underestimated for shallow groundwater sites and overestimated for deep groundwater sites (Fig. 7).
- 58-62: What is perceived here as an interference signal is actually the main driver of the low-pass filtering effect: It is the vadose zone rather than the aquifer – have a look at Fig. 4, 5, and 6 in Tsypin et al. (2025, https://doi.org/10.1016/j.jhydrol.2025.133193) or at Liesch and Wunsch (2019, https://doi.org/10.1016/j.jhydrol.2019.02.060). The thicker the vadose zone, the more pronounced is the low-pass filtering effect. The pore space in the aquifer is fully saturated, and water is hardly compressible. Thus, how could the aquifer buffer any input (except by discharging to the constant head boundary)?
- 98-100, 220-223: Cut-off frequencies, that is, the change of slope is a nice index to compare different time series. But it comes with two major problems: Firstly, deep groundwater head dynamics usually exhibits pronounced persistency with Hurst coefficients exceeding 0.5. Thus, legacy effects will last for very long (Koutsoyiannis 2013, http://dx.doi.org/10.1080/02626667.2013.804626). Actually, there is no upper limit of response time at all for very fundamental reasons. Secondly, such clear breaks of slope are limited to time series with rather low degrees of low-pass filtering like stream discharge. In contrast, power spectra of deep groundwater head time series often do not show any clear cut-off frequencies. See, e.g., observed groundwater head spectra (Fig. 4) in Zhang and Schilling (2004, https://doi.org/10.1029/2003WR002094) or Fig. 4 in Tsypin et al. (2025, https://doi.org/10.1016/j.jhydrol.2025.133193). There is no physical reason why low-pass filtering should stop at a certain frequency. It is just a matter of the intensity of the low-pass filtering process. Please comment.
- 152-156: Removing linear trends distorts the power spectrum. Trends do not develop independent from the spectral signatures but are an inevitable consequence of the low-pass filtering process which results in increasingly more smooth time series. Consequently, the length of periods of monotonic rise or fall increases and thus the probability to identify significant trends (Koutsoyiannis 2006, https://doi.org/10.1016/j.jhydrol.2005.02.030; Lischeid et al. 2021, https://doi.org/10.1016/j.jhydrol.2021.126096). This holds even for random white noise time series.
- 273-276: It is indicative that the paper does neither report on transmissivity nor on storage values. Other than the authors I would ascribe these uncertainties to disregarding the vadose zone effect on low-pass filtering (see above) rather than to uncertainties of flow line length determination. Please comment.
- 12-15, 300-306: The statement “Groundwater systems exhibiting longer response times are interpreted as more resilient to drought conditions and therefore potentially better suited for groundwater abstraction than aquifers with shorter response times” holds only in the short-term when the response time is long compared to the timescale of dry spells. But long-term buffering comes with the price of very slow and delayed recovery. Increasingly longer dry spells and extended periods of a monotonic decrease of groundwater head and total water storage which are now abundant in Central Europe (Xanke and Liesch 2022, https://doi.org/10.1007/s10040-021-02448-3; Wunsch et al. 2022, https://doi.org/10.1038/s41467-022-28770-2) question such recommendations. Please comment.
- Figure 1: Please explain the b) and c) panels. It is a little bit unusual and confusing to have two panels b) and c) each.
- Figure 2: Correct “lenth” in the figure caption.
- A3: I guess the term “spectral analysis” refers to the whole procedure rather than to the power spectra of groundwater head time series in particular, is that right? Otherwise, frequencies should always be < 1. Please clarify.
- A8: This spectrum seems to be indicative for a confined aquifer where short-term fluctuations are induced by atmospheric pressure variation, resulting in a remarkable atypical large share of variance in the high-frequency range.
Citation: https://doi.org/10.5194/egusphere-2025-5666-RC2
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Summary
The manuscript by Houben et al presents a methodology that combines time series analysis, GIS and analytical modelling to estimate the average response time of aquifers in a large, regional data set of groundwater level time series from observation wells in the upper Danube river basin. The approach rests on existing findings by (i.a. Houben et al., 2022; Liang & Zhang; Zhang & Schilling, 2004) regarding temporal scaling of groundwater head and relation to aquifer geometry, properties and recharge. Here, the focus was on estimating at each location the characteristic time scale, a single value that quantifies the rate at which an aquifer responds to an external stress. This value is then briefly framed as a characteristic to quantify aquifer vulnerability to drought and criteria for selection of aquifers for abstraction. While the approach generally is sound, my primary concern with this paper is that the value of the analysis is not apparent when compared to other studies that require fewer data, assumptions, and less effort, such as the referenced study by Kumar et al. (2016) or Ebeling et al. (2025). What benefit does the characteristic time scale provide versus the current standard method in groundwater drought analysis using correlation times of SP(E)I vs SGI for example or e.g. cross-correlations (e.g., Bloomfield & Marchant, 2013; Ebeling et al., 2025)? I think the authors need to reflect on - and clarify this before the manuscript can be considered for publication. Below find specific comments:
Technical comments
References
Bloomfield, J. P., & Marchant, B. P. (2013). Analysis of groundwater drought building on the standardised precipitation index approach. Hydrol. Earth Syst. Sci., 17(12), 4769-4787. https://doi.org/10.5194/hess-17-4769-2013
Changnon, S. A. (1987). Detecting drought conditions in Illinois. Circular no. 169.
Ebeling, P., Musolff, A., Kumar, R., Hartmann, A., & Fleckenstein, J. H. (2025). Groundwater head responses to droughts across Germany. Hydrol. Earth Syst. Sci., 29(13), 2925-2950. https://doi.org/10.5194/hess-29-2925-2025
Houben, T., Pujades, E., Kalbacher, T., Dietrich, P., & Attinger, S. (2022). From Dynamic Groundwater Level Measurements to Regional Aquifer Parameters— Assessing the Power of Spectral Analysis. Water Resources Research, 58(5). https://doi.org/10.1029/2021wr031289
Kumar, R., Musuuza, J. L., Van Loon, A. F., Teuling, A. J., Barthel, R., Ten Broek, J., Mai, J., Samaniego, L., & Attinger, S. (2016). Multiscale evaluation of the Standardized Precipitation Index as a groundwater drought indicator. Hydrol. Earth Syst. Sci., 20(3), 1117-1131. https://doi.org/10.5194/hess-20-1117-2016
Liang, X., & Zhang, Y.-K. (2013). Temporal and spatial variation and scaling of groundwater levels in a bounded unconfined aquifer. Journal of Hydrology, 479, 139-145. https://doi.org/10.1016/j.jhydrol.2012.11.044
Zhang, Y. K., & Schilling, K. (2004). Temporal scaling of hydraulic head and river base flow and its implication for groundwater recharge. Water Resources Research, 40(3), W035041-W035049.