The groundwater response to barometric pressure and gravity changes caused by Earth tides has long been observed and is a powerful yet underutilised tool to passively characterise subsurface systems . While atmospheric pressure changes act as a load on the subsurface, and its groundwater pressure response can be related to compressible properties of the formation e.g., Earth tides cause areal strain, resulting in small pore pressure changes e.g.. The main research focus has long been the removal of both signals from the groundwater pressure in order to better understand and quantify processes such as pumping tests or recharge e.g.. However, tidal forces are ubiquitous, and their groundwater response can therefore also be utilised to quantify in situ hydro-geomechanical properties . Tidal harmonic components have long been named depending on their frequency . A comprehensive list of the most common components found in groundwater head measurements is summarised in Table . reviewed the state of the science, highlighted the potential for such passive approaches and coined the term tidal subsurface analysis (TSA).

analysed the groundwater response to both atmospheric pressure changes and Earth tides to quantify hydraulic and elastic properties. They reported that the frequency component S2 of the groundwater response exhibits a reliable and strong response to Earth tides but could not be used because it is contaminated by atmospheric pressure influences. For this reason, they concentrated their analysis on the Earth tide frequency M2. and used the groundwater response to atmospheric tides to estimate barometric efficiency, infer system confinement and calculate compressible storage. They used the tide frequency S2 in their work but did not take into account the effects of the phase lags between the Earth, atmospheric and groundwater tides that have been found to introduce errors in the analysis for higher levels of barometric efficiency.

published a method which objectively quantifies barometric efficiency (BE) using the groundwater response to atmospheric tides. Their approach considered the impact of phase lags between S2AT and S2ET in the amplitude response of the groundwater system. Their work demonstrated that (1) the harmonic addition theorem could be used to quantitatively disentangle the groundwater response to both Earth and atmospheric tides (EATs) acting at the same frequency, and (2) a theoretical Earth tide record is sufficient for this purpose. Because Earth tide records can be calculated with very high accuracy for any location on Earth and for time periods of general interest e.g., this has opened the door for the widespread use of common barometric and groundwater pressure measurements (the latter in the form of standard well water levels) to characterise and quantify groundwater systems with little effort. compared different BE estimation methods and concluded that delivers robust results.

The method by is based on the assumption that the borehole water level is representative of subsurface pore pressure, i.e. there is an instantaneous and undamped response. Under such conditions, only the phase difference between the theoretical Earth and atmospheric tide drivers is required to correct the groundwater response amplitude. However, it has been established that the well water level response to Earth tide forces also depends on the hydraulic properties of the aquifer and well geometry and vadose zone air transport for conditions that are not confined . In fact, the amplitude and phase responses to Earth tides embedded in well water levels have been used to quantify subsurface hydraulic properties e.g.. reports an average phase shift of -12∘ between the M2 Earth tide potential and its well water level response. Consequently, an instantaneous and undamped groundwater response to Earth tides is not always a given and phase delays must be also considered when quantifying BES2 from the groundwater response to atmospheric tides.

In this technical note, we generalise the method by by more completely disentangling the groundwater response to EATs in the frequency domain. We then illustrate the interpretative value of this new approach, using an example atmospheric pressure and borehole water level record that is strongly affected by EATs, followed by a verification of our results, using the well's barometric response function calculated in the time domain.

The three BEAT examples illustrated in show a limited impact of Earth tides relative to those of the atmospheric tides, resulting in a small magnitude correction at S2. To illustrate our new tidal disentanglement approach, we deliberately use a well water level record in which the Earth tide influence exceeds that of the atmospheric tides. This record originates from the well of BLM-1 in Death Valley (California, USA; WGS84; -116.471360∘ longitude, 36.408130∘ latitude; height – 688 m; casing and screen radius – 0.127 m; screen length – 106 m), which provided data for a previous analysis . The data set used here was recorded in the same well but at a later time. The data set was sampled at 15 min intervals (96 samples per day).

Figure  shows the barometric pressure (BP; black line, converted to pressure head equivalent), groundwater pressure head (GW; blue line, as measured in the well) and Earth tide strains (ET; red line) for BLM-1 over a time period of almost 6 months (25 June to 16 December 2009). The Earth tide strains were calculated using the Python package PyGTide for the same time period and sampling frequency as the pressure measurements. It is interesting to note that both Earth tide and atmospheric pressure signatures are clearly visible in the groundwater response. In fact, the impact of both atmospheric pressure and Earth tide strains on the borehole water level is obvious just by looking at the raw data set.

As a next step, we extracted the tidal harmonic components from all three time series (BP, ET and GW in Fig. ) using the harmonic least squares estimation approach described in Sect. . Figure a shows the estimated amplitudes of the tidal components compared to a Fourier amplitude spectrum of the groundwater record. This example illustrates that separating tidal components with close-by frequencies is a clear challenge for the Fourier transform, even when the record duration is longer than that recommended by . It further highlights the well-known fact that spectral leakage can lead to errors in estimating the properties of the harmonic components e.g..

Figure b maps the amplitudes of the tidal components (Table ) extracted from the original time series depicted in Fig.  against their phases. As expected, the strongest impact stems from the Earth-tide-only component M2. It is interesting to observe the similarity in the groundwater response magnitudes for all other Earth tide components, e.g. K1, O1, N2, Q1 and M1 (in decreasing order of impact). It is apparent that, at frequencies for which the groundwater record is influenced by both Earth and atmospheric tides, there is a substantial misalignment in amplitudes and phases of these components in groundwater in comparison to the forcing signals, e.g. see S2 or S1.

To illustrate the tidal disentanglement, we use polar plots to visualise the key components. Figure a shows the amplitude and phase of the M2 component present in ET and GW. We apply the approach developed in Sect.  to infer the S2 response in the well that is caused by the Earth tides alone, using the theoretical Earth tide strains (note that the ET magnitude is scaled to improve comparison). Figure b shows the atmospheric tide driver at S2, the combined well water level response to EAT, the inferred groundwater response to S2 and the disentangled S2 response to atmospheric tides. All parameters and uncertainties calculated in this work are summarised in Table .

To account for the amplitude damping and phase shifting of the well in response to the harmonic pore pressure changes, we have used the dimensions of the well BLM-1 (see previous details) to calculate the solution space of the analytical solution for confined conditions (Appendix ) for the M2 frequency and for realistic limits of hydraulic conductivity (1×10-8<K<1×10-2 m/s) and specific storage (1×10-7<Ss<1⋅10-3 1/m). The aerial strain sensitivity and phase shift between Earth tides and well response to M2 is shown in Fig. . We further used Eqs. () and () to estimate the hydraulic conductivity as K≈4.2×10-6 m/s (ranging from 2.0×10-6 to impossibly high values) and specific storage as Ss≈6.7×10-7 1/m (ranging from 6.69×10-7 to 6.77×10-7 1/m), which is representative for the materials along the well screen from the groundwater response to Earth tides (note the black annotations in Fig. ). The Ss falls within the poroelastic limits determined by .

The damping of the amplitude in the well in this case is only AM2r≈0.998, which differs to the aquifer pore pressure by merely ≈0.2%. It is important to note the high sensitivity to phase differences, i.e. small changes in ΔϕM2 can cause large changes in the value of K (Eq.  and Fig. b). However, we note that as the sensitivity of the phase difference ΔϕM2 to K drastically reduces in lower permeability settings (Fig. b), the confidence in K values will increase. Conveniently, this is also the value range where the amplitude response is most affected (Fig. a), allowing confidence in the robustness of our new BES2 estimation approach for the investigated property ranges.

Using Eq. (), we calculate a BES2=0.60 from the disentangled groundwater response to atmospheric tides at S2 frequency. This is significantly different to the BES2=1.29 that results from using the method by . The latter is clearly erroneous, since it is larger than one, owing to the limited phase correction and leading to an incomplete disentanglement of the groundwater response to EATs. To verify our results, we independently calculated BEBRF for this location using the well-established barometric response function (BRF) approach developed and illustrated previously a brief summary of the theory is given in Appendix ;. The asymptotic value of the BRF at larger delay times represents confined conditions. The results further exhibit an exponential increase in the BRF over lag time (Fig. b). This is typical for water exchange controlled by the subsurface hydraulic conductivity where the shorter the lag time the stronger the deviation from BE , and it complies well with the frequency-dependent relationship shown in Fig.  .

The BRF-based BEBRF≈0.60 exactly matches that calculated using Eq. (), confirming the robustness of the tidal disentanglement methodology we present in this work. The BRF is able to adequately remove the Earth and atmospheric influences on the measured groundwater levels (Fig. a). While BRFs are capable of indicating system confinement and estimating BE, they have not been used for estimating hydraulic properties.

The negative phase shift between Earth tides and groundwater pressure (ΔϕM2GW.ET=-1.1∘; see Fig. a) can be interpreted as horizontal flow between the subsurface and the well, which occurs under confined conditions . Confined conditions can also be interpreted from the fact that the BRF time values reach a maximum value that is representative of the BE . Since this means that the well is screened in the confined zone, vertical loading due to atmospheric tides should also predominantly induce horizontal flow between the subsurface and the well with the same phase difference as given in response to Earth tides but considering the typical 180∘ (Fig. b) difference related to the subsurface stress balance . In fact, ΔϕS2GW.AT=-9.8∘, which is also negative and very close to the phase delay in response to Earth tides.

Previous works have reported that a phase difference of 180∘ between the atmospheric tides and the groundwater response at S2 can be used to indicate confinement . However, this did not consider the fully disentangled groundwater response to atmospheric tides. Furthermore, has illustrated that the well response to barometric pressure depends on a number of processes accounting for the pressure propagation between surface and well, resulting in a frequency-dependent response of the borehole water level to barometric forcing. We propose that a combination of ΔϕM2 and ΔϕS2 could be diagnostic of the subsurface conditions at the borehole location. For example, a negative ΔϕM2 is indicative of horizontal flow occurring under confined conditions for which ΔϕS2 should equally be shifted to account for the 180∘ phase difference. A positive ΔϕM2 has been attributed to vertical water movement under which the response of ΔϕS2 could become diagnostic of the vertical pressure propagation and vadose zone properties . However, further research is required to develop a robust approach for detecting the confinement status at a particular frequency.

We present a frequency domain method to disentangle the groundwater response to Earth and atmospheric tides. It is a more general solution than that previously presented by since it is also applicable to subsurface environments with lower hydraulic conductivities, where measurable damping and time lags between formation pressure changes and well water level responses may be present. The approach only requires simultaneous records of barometric and groundwater pressure head in combination with theoretical Earth tide potential, gravity or strain variations, which are either standard measurements or can be calculated for any borehole geo-position using readily available software. Our novel approach exploits the fact that the complex harmonic components can be determined for each variable, i.e. barometric pressure, borehole water level and Earth tides from which subsurface flow direction (horizontal vs. vertical) in response to stresses allows the inference of confinement, estimation of barometric efficiency (BE), hydraulic conductivity and specific storage. We show that BE calculation using the borehole water level response to atmospheric tides may be substantially influenced by the hydraulic conductivity of the materials surrounding the well screen where K<1×10-5 m/s.

Our method enables an improved and rapid estimation of BE in general but especially for cases where the borehole water level is strongly influenced by Earth tides. Under such conditions, the influence of the phase difference between Earth tides and its groundwater response also has to be considered when revealing the atmospheric tide embedded in the S2 component of groundwater head measurements. The generalised solution further improves existing approaches and provides a next step towards a more reliable quantification of subsurface hydro-geomechanical properties using the groundwater response to Earth and atmospheric tides .