Gravity changes associated with variations in local landwater distributions: Observations and hydrological modeling at Isawa Fan, northern Japan
 Takahito Kazama^{1, 3}Email author,
 Yoshiaki Tamura^{2},
 Kazuyoshi Asari^{2},
 Seiji Manabe^{2} and
 Shuhei Okubo^{3}
https://doi.org/10.5047/eps.2011.11.003
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2012
Received: 2 September 2011
Accepted: 17 November 2011
Published: 25 May 2012
Abstract
Gravity changes associated with variations in local landwater distributions have been observed at Isawa Fan in northern Japan, and modeled by hydrological equations. We solve the Richards equation numerically for the time variation in the vertical soil moisture distribution, which is then spatially integrated to estimate gravity changes due to the soil moisture distribution. In modeling Isawa Fan, we assume a simple hydrological model: a horizontally homogeneous soil in an infinite halfspace. The estimated gravity is consistent with the observed gravity during a 50day period within about 0.4 µgal root mean square, owing to both observed soil parameter values and the observation building geometry being incorporated into the hydrological model. However, the estimated gravity cannot fully reproduce annual gravity changes observed during a 2year time frame, because the boundary conditions in the modeling determine only local water distributions and the resultant shortperiod gravity changes. Instead, the observed gravity over these 2 years can be reproduced within about 1.0 µgal root mean square, if the additional parameters of the annual gravity change (A_{ac} and A_{as}) and the snowfall effect (A_{s}) are calculated by the function regression to the observed gravity with the leastsquares method. The hydrological modeling techniques presented here can be utilized at all gravity sites in flat areas similar to Isawa Fan, such that hydrological effects in gravity data can be corrected and mass transfers associated with earthquakes and volcanoes can be monitored.
Key words
1. Introduction
Observing changes in gravity is one of the most powerful methods for detecting mass redistributions associated with solidearth tectonics, such as those resulting from earthquakes and volcanism (Furuya et al., 2003; Imanishi et al., 2004). However, gravity measurements are also sensitive to changes in landwater distributions in the vicinity of gravimeters. These changes originate from precipitation, snow cover, soil water infiltration, groundwater flow, river runoff and variations in the water level of oceans and lakes (e.g., Sato et al., 2001; Van Camp et al., 2006; Kazama and Okubo, 2009; Christiansen et al., 2011). To detect the minute gravity changes associated with solidearth tectonics, water distribution effects in the original gravity data must be corrected to a high accuracy.
Many scientists have modeled the effects of landwater distribution on the observed gravity (e.g., Francis et al., 2004; Abe et al., 2006; Boy and Hinderer, 2006; Meurers et al., 2007; Jahr et al., 2009; Leirião et al., 2009; Longuevergne et al., 2009; Nawa et al., 2009; Pfeffer et al., 2011). For example, Hanada et al. (1990) found a simple proportional relationship between water level and absolute gravity (coefficient: +16 µgal/m) observed at Isawa Fan, northern Japan. Imanishi et al. (2006) reproduced landwater distribution effects observed at Matsushiro, central Japan, with an empirical tank model, assuming an instant gravity response to precipitation of −0.040 µgal/mm and a linear gravity recovery after precipitation of +3 × 10^{−6}µgal/s. Wziontek et al. (2009) explained a seasonal gravity change of ∼10 µgal peaktopeak at superconducting gravity stations in Europe by attraction and loading effects attributable to global landwater distributions. Creutzfeldt et al. (2010a, b) reproduced a nontidal gravity change of ∼10 µgal in amplitude at Wettzell, Germany, by using spatial integrations of local water distributions observed with a lysimeter and borehole moisture meters.
However, the following three shortcomings exist in previous studies when estimating the effect on gravity due to landwater distributions. Firstly, the majority of models, including the tank model, assume a linear gravity response to precipitation (e.g., Hanada et al., 1990; Imanishi et al., 2006; Nawa et al., 2009; Lampitelli and Francis, 2010). Since water flow is governed by nonlinear equations (as demonstrated in this study), there are limitations in applying linear theory to gravity responses (Kazama et al., 2005). Secondly, empirical methods can over or underestimate gravity amplitudes resulting from landwater distribution, because empirical methods only fit simple regression curves to observed gravity data (e.g., Bower and Courtier, 1998;
Harnisch and Harnisch, 2006). To reproduce the gravity response to water distribution, the distribution should be estimated by applying hydrological modeling, independently of observed gravity data. Thirdly, regional and global water storages have often been reproduced rather than local water storage (e.g., Sato et al., 2001; Boy and Hinderer, 2006; Neumeyer et al., 2006). Meanwhile, a number of studies report that the dominant area for land gravity observation is within a several hundred meter radius surrounding a gravimeter (e.g., Hasan et al., 2006; Kazama and Okubo, 2009), since gravity (i.e., the integral of the gravitational force) has a high sensitivity to mass in the vicinity of gravimeters (see Eq. (17)). Therefore, it is important to observe and model water distributions near these gravimeters, assuming a local hydrological model.
Taking these aspects into account, hydrological nonlinear equations should be solved directly for local water distributions and the consequent gravity change. Previous studies have utilized the hydrological equations (e.g., Abe et al., 2006; Hasan et al., 2008; Kazama and Okubo, 2009; Krause et al., 2009; Leirião et al., 2009; Naujoks et al., 2010), although all still show problems relating to the reproducibility of changes in gravity. For example, Abe et al. (2006) found that soilwater effects, estimated by a soil moisture retention function, corresponded to 80% of the observed instant gravity change during rainfall events in Bandung, Indonesia. However, the length of time for estimating the gravity was within only a single day during the rainfall events, because Abe et al. focused on shortperiod soil water infiltration within several meters from the gravimeter. To reproduce gravity responses over a longer period (i.e., more than one day), both a largescale computation area and a longperiod water distribution change must be taken into account. Kazama and Okubo (2009) found that gradual decreases in gravity after rainfall events, estimated through hydrological nonlinear equations were, within the observation error, consistent with the gravity observed at the Asama volcano (in central Japan). However, the estimated amplitude for the instant gravity increase during rainfall corresponded to only 75% of the observed gravity, possibly owing to poor reproducibility of water distributions in the undulating volcanic region. Therefore, the method proposed by Kazama and Okubo (2009) should first be applied to an area having a simple topography (i.e., a plain), in order to test the correctness of their model.
In addition, for an accurate estimation of local water distributions and consequent gravity changes, hydrological and meteorological data must be collected near gravimeters. Researchers have carried out electrical resistivity surveys (Van Camp et al., 2006), lysimeter measurements (Creutzfeldt et al., 2010a), and boreholecore sampling (Creutzfeldt et al., 2010b), to understand, in detail, depthdependent water distributions below the ground surface; however, such depth profile data is not available for all gravity observation sites, mainly because of financial limitations. A different approach is to realize that the bulk of the water mass distribution occurs in the uppermost ground layers, because soil pores and free spaces at the ground surface can preserve significant amounts of water in the form of soil water and snow (e.g., Sato et al., 2006). Therefore, it is vital to collect hydrological and meteorological data, such as precipitation, soil moisture and unconfined groundwater height, at the shallowest part of the ground in order to understand the gravity changes due to landwater distributions.
Thus, we are motivated to observe and model local landwater distributions near gravimeters in flat areas, with the aim of reproducing hydrological gravity changes accurately by applying hydrological modeling. Hydrological and meteorological data, in addition to gravity, are observed continuously within a radius of 50 m at Isawa Fan (Iwate Prefecture, northern Japan). In this research, a nonlinear soil water diffusion equation (the socalled Richards equation; Richards, 1931; Kazama and Okubo, 2009) is solved to find the spatiotemporal soil water distribution, and this is spatially integrated to estimate the gravity change. The estimated change in gravity is shown to be consistent with a 50day gravity change observed by a superconducting gravimeter (Crossley et al., 1999; Goodkind, 1999), within about 0.4 µgal root mean square (RMS).
2. Observations at Isawa Fan
Finally, the gray bars in Fig. 3(b) denote the snow depth at Otemachi (∼2.4 km northeast from the gravity observatory; triangle in Fig. 1), observed by the Iwate prefectural government (Iwate Prefecture, 2011). We utilized this data because the snow gauge at the weather station in Mizusawa Observatory was out of action during the measurement period. Snow accumulates in the region around Mizusawa Observatory from November to March every year, and its maximum depth during the three observation years was 27 cm on 21 February 2009.
Figure 3(c) shows the moisture content variation in m^{3}/m^{3} near the weather station at Mizusawa Observatory, measured every 5 min with a soil moisture meter (Profile Probe PR2; DeltaT Devices Ltd.). This meter observes voltage changes at six sensors, and converts these voltages into an overall moisture change by using the conversion equation for organic soil (Miller and Gaskin, 1999). In this figure, θ_{obs}(t) (solid line) signifies the average moisture from the six observations at 0–1m depth below the ground surface, and Δθ_{obs}(t) (gray envelope) is the 1σ error range for the observed moisture. The absolute moisture value is 0.40 ∼ 0.55, and is consistent with the value found by a soil test (Grossman and Reinsch, 2002) of 0.51±0.04 on 8 November, 2008. θ_{obs}(t) increases sharply at rainfall events dependent on the amount of rain, and decreases exponentially after rainfall because of evapotranspiration and infiltration. However, the moisture level is approximately constant during the winter seasons (from December to April), suggesting that the covering snow blocks evapotranspiration from the soil surface and the covering snow provides moisture to the soil at a constant rate.
Figure 3(d) shows the average water table height, h(t), from the ground surface level, z_{s}; estimated from the water level variations of four shallow wells within 50 m of the gravity observatory. After 2 October, 2008, (solid line), the water height was measured with newlyinstalled water pressure gauges (Mini Diver DI501; Schlumberger Ltd.), since the original data taken by floating water gauges (gray line) was often saturated when the change in water level exceeded the wire length between the float and pulley (as seen between June and July, 2008). The water table is located at a depth of 0.5–3 m below the ground surface, and rose steeply at precipitation events in synchronization with soil moisture increases. After precipitation events the water table reduced at an almost linear rate, because the unconfined groundwater was regularly pumped for domestic and agricultural use at Isawa Fan, and the groundwater flowed out eastward to the edge of Isawa Fan. (Water electrical conductivity at Suginodo Spring (circle in Fig. 1) is 220 µS/cm, about four times larger than that at Mizusawa wells (∼50 µS/cm), showing that the groundwater flows eastward at Isawa Fan under conditions of domestic and agricultural water pollution.)
Figure 3(e) plots the superconducting gravity data during the period 23 January, 2009, to 5 December, 2010. Gravity data was calculated from measured output voltages with a conversion coefficient of −56.082 [µgal/V] (Tamura et al., 2005), and divided into tidal, barometric, trend and irregular components by using the BAYTAPG software package (Tamura et al., 1991). In addition, effects from the longperiod tide, the polar motion and the arbitrary linear drift (−0.095 µgal/day) were corrected in the trend components. The solid and gray lines in Fig. 3(e) are the corrected trend component, g_{obs}(t), and the irregular component (i.e., the gravity observation noise), respectively. g_{obs}(t) is not shown between June and July, 2009, owing to a controller failure causing the noise amplitude and the g_{obs}(t) drift rate to be elevated. This failure was repaired by changing the controller on 27 July, 2009. g_{obs}(t) decreases by ∼10 µgal in two years, implying that instrumental drift was not fully removed from the observations. Furthermore, g_{obs}(t) increases quickly by a few microgals during precipitation events, in line with the soil moisture increase and water height rise (as seen for March 2009, October 2009, March 2010 and July 2010), suggesting that changes in the gravity signal are associated with corresponding changes in the landwater distribution.
Note that the 2008 IwateMiyagi Nairiku Earthquake (M_{w} 6.9) occurred in a southwestern area outside the Isawa Fan on 14 June, 2008. The coseismic slip at two main faults (labeled MF1 and MF2 in Fig. 1) reached a maximum of ∼3.5 m (Ohta et al., 2008), which triggered aseismic slip at the Detana Fault (DF in Fig. 1) of about 30 cm in the few months after the mainshock (Iinuma et al., 2009). The superconducting gravimeter at Mizusawa Observatory, however, did not detect any gravity change associated with these fault slips, because the gravimeter was installed about a half year after the earthquake event, and the aseismic slip effect cannot be separated from the instrumental drift in the superconducting gravity data.
3. Numerical Methods
Hydrologically, the nearsurface crust consists of an unsaturated layer (or soil water layer), saturated layers (or groundwater layers, aquifers) and impermeable layers. In general, precipitated water: (1) falls onto the ground surface, (2) infiltrates vertically into the unsaturated layer as soil water, (3) reaches the water table of the unconfined aquifer, (4) flows horizontally into the unconfined aquifer as groundwater, and (5) discharges into rivers, lakes and the sea. If, for gravity changes, the water leakage at impermeable layers, and groundwater pumping from confined aquifers, are neglected, then the gravity change must be calculated from the spatiotemporal distributions of the soil water and the unconfined groundwater (Fig. 2).

Vertical soil water velocity at the ground surface is set to be equal to the effective precipitation: where z_{s} and p are the ground surface height above sea level (Fig. 2) and the infiltration capacity for the ground, respectively. The infiltration capacity p was set as 1.0 for the soil ground surface for this study, because all precipitation is considered to infiltrate into the soil unless the precipitation intensity exceeds 100 mm/h (Murai and Iwasaki, 1975).

Water content at the saturated layer (z ≤ h(t))is fixed to be the maximum water content (i.e., effective ^{porosity)}, θ_{max}: Here, d is the depth of the computed region and was set equal to 10.0 m to satisfy the condition z_{s} − d ≤ h(t) for all times. In this study, the observed water height change, h(t) (Fig. 3(d)), was utilized to decide the position where the moisture is fixed to θ_{max}, and θ_{max} was observed with a test on soil sampled at the observatory (see Eq. (16)). Note that the horizontal groundwater flow in the unconfined aquifer is not modeled numerically in this study, because the effects of groundwater flow and its temporal change will be incorporated as this boundary condition into our hydrological modeling.

A steadystate soil water distribution was assumed for the initial distribution of θ(z, t): where θ_{s}(Z) is the steadystate soil water distribution at a vertical height Z above the water table (Fig. 2). θ_{s}(Z) was estimated from the FDM calculations by continuing to apply an average effective precipitation P_{0} onto the ground surface before the soil water distribution converges. Here, the average value of the effective precipitation was given by for the average annual precipitation, R_{0}, and the annual evapotranspiration estimated from average monthly air temperatures, E_{0} (Thornthwaite, 1948), taken over the period from 1979 to 2000, by using observed data from the Esashi AMeDAS observatory (Japan Meteorological Agency, 2011).

We assumed the following exponential dependencies for the permeability K(θ) and the diffusivity D(θ) on the water content θ (Gardner and Mayhugh, 1958; Davidson et al., 1969; Olsson and Rose, 1978): where K_{s} is the saturated permeability, D_{s} is the saturated diffusivity, a and b are positive constants and θ_{max} and θ_{min} are the maximum and minimum water content, respectively. For Isawa Fan, the soil parameter values were taken to be K_{s} and θ_{max} were derived from tests on soil sampled near the gravity observatory (Klute and Dirksen, 1986; Grossman and Reinsch, 2002), whereas D_{s}, a, b and θ_{min} were estimated by trialanderror fitting of the theoretical results to the observed moisture variation. Note that these soil parameter values are consistent with those for typical silt soil (Olsson and Rose, 1978; Amemiya et al., 1991; Nishigaki, 1991). We used these soil parameter values for the entire computational region, assuming homogeneous soil around the observatory.
Hence, in the manner described, we estimated the soil moisture (θ(z, t)) and the gravity (g_{w}(t) and g_{d}(t))at Mizusawa during the three years from 2008 to 2010. In the next section, we compare the estimated results and the observed data (Fig. 3) during only two years, from January 2009 to December 2010, owing to preliminary computations suggesting that about half a year is required for the moisture distribution θ(z, t) to undergo a transition from the steady state to the unsteady state.
4. Modeled Results
In contrast, the moisture response to precipitation at deeper layers (Fig. 5(b)–(f)) has greater attenuation and shows a larger lag because of soil moisture diffusion (Eq. (5)). For example, the estimated soil moisture increase from April to May 2010 is attenuated by a half at depths between 25–30 cm and by a quarter between 95–100 cm, and the time of the first moisture peak is delayed by 1.3 and 6.5 days, respectively. On occasion, a number of precipitation events concatenate into one moisture peak, as seen in April 2009, July 2009 and July 2010. Thus, nonlinear characteristics in soil water equations can be seen; the proceeding infiltration fronts draw level with the initial front, since the initial infiltration increases soil moisture, θ; permeability, K(θ) (Eq. (14)), and diffusivity, D(θ) (Eq. (15)); and soil moisture speed, υ (Eq. (4)), in sequence (figure 2.17 after Nakano, 1991).
The solid line in Fig. 5(g) represents the average estimated soil moisture 0–1 m below the ground surface, θ(t). The average soil moisture changes between 0.37 and 0.50, and correlates to the observed soil moisture at Mizusawa to within onesigma error range (gray area in Fig. 5(g)). It must be emphasized here that the use of adequate soil parameters (Eq. (16)) is essential for obtaining accurate water distributions in soil. Even though we equate snowfall to be equivalent to rainfall on the ground surface, the similar precision can be estimated for the moisture content even in winter. We attribute this result to less precipitation in winter resulting in small changes in soil moisture distribution, and most of the covering snow infiltrating as snowmelt water, even if there is a lag between snowfall and melting.
The thin line in Fig. 6(b) plots the linear gravity change associated with the instrumental drift, g_{d}(t), calculated from the leastsquares method. The slopes before and after the instrumental maintenance in June 2009 are −0.009 and −0.018 µgal/day, respectively. With −0.095 µgal/day already subtracted from the gravity data (see Section 2), the total drift rates are therefore −0.104 and −0.113 µgal/day, respectively.
The thick line in Fig. 6(b) shows the predicted total change in gravity, Although g_{cal}(t) qualitatively reproduces gravity increases during precipitation events, the amount of the increase for the calculated gravity is about twice that of the observed gravity (gray line), as seen in April 2009, July 2010 and September 2010. Additionally, the estimated gravity decay after each precipitation is faster than that observed, and the difference between g_{obs}(t) and g_{cal}(t) is larger than 3 µgal in August 2009 and April 2010.
To summarize these results: (1) Our model can reproduce the water distribution in soil within the observation error range (Fig. 5(g)) owing to the choosing of adequate soil parameters. (2) The estimated gravity value, g_{cal}(t) (Eqs. (17) and (20)), is not consistent with that observed, in terms of both the increased gravity amounts during precipitation and the rates of gravity decrease after precipitation. A potential cause of this inconsistency is the horizontal heterogeneity of the water distribution in the vicinity of the gravimeter; for the case of Mizusawa, the gravity observation building itself could affect the horizontal water heterogeneity, and the subsequent gravity disturbance (e.g., Creutzfeldt et al., 2010a). Thus, in the following subsection, an effect of the building will also be taken into account when estimating the water distribution and the gravity change.
4.1 Building effect
The gray line in Fig. 4 shows the steadystate distribution of the soil moisture for the ground inside the building, is a maximum and is saturated at the water table, and decreases at a faster rate than for the steadystate moisture distribution outside the building, θ_{s}(Z; p = 1) (the black line in Fig. 4). Moreover, θ_{s}(Z; p = 0) converges to a minimum value θ_{min} (= 0.28) at Z > 4 [m], since water is not supplied from the upper layers.
In Fig. 9(b) and (c), the estimated gravity, g_{cal}(t), composed of the moisture effect and the drift effect (Eq. (28)) is compared with the observed gravity, g_{obs}(t). Figure 9(c) shows gravity changes for 50 days (from 26 April to 15 June, 2010). The amplitude of g_{cal}(t) during these 50 days is ∼3.8 µgal, comparable with that of g_{obs}(t) (4.4 µgal). Variations of g_{cal}(t) over this period are also consistent with g_{obs}(t), both in gravity increases during each precipitation event and in peaktime delays, which are about 3–4 days after the beginning of each event. This peaktime delay in gravity is most likely a consequence of the delay in the water level increase around the gravimeter, since the unconfined groundwater must flow from the upper part of Isawa Fan to Mizusawa Observatory (cf. detailed discussion in Section 5.2). The hydrological model presented here can reproduce the delayed gravity response to precipitation, because the observed water level, h(t), is utilized as one of the boundary conditions when estimating the moisture distributions. In addition, g_{cal}(t) is consistent with g_{obs}(t) for gravity reduction rates after precipitation, originating from the instrumental drift ( [µgal/day]) and water mass loss in the soil owing to evapotranspiration and groundwater flow. Hence, in the above respects, the hydrological model reproduced the observed gravity changes during the 50 days both qualitatively and quantitatively, with a small RMS (0.44 µgal) and a high correlation coefficient (CC: 0.89).
Conversely, the variation of over a longer period is in part contradictory with g_{obs}(t). The gray and thick black lines in Fig. 9(b) show the observed and estimated changes in gravity (g_{obs}(t) and g_{cal}(t)), respectively. When compared with (the thick black line in Fig. 6(b)), the increase in amplitude of g_{cal}(t) during each precipitation event agrees with g_{obs}(t), especially in April 2009, October 2009 and January 2010. In addition, the RMS and CC values in Fig. 9(b) show that there is better agreement between g_{cal}(t) and g_{obs}(t) than equivalent values resulting from Fig. 6(b), suggesting the significance of considering the building effect in estimating accurate gravity changes. However, g_{cal}(t) still differs from g_{obs}(t) by up to 4 µgal, as seen in July 2009, March–May 2010 and July–August 2010. The gravity difference, , seems to be positive in summer (June–August) and negative in winter and spring (February–May), implying the existence of an annual gravity signal in g_{obs}(t). (Note that the negative gravity difference in winter 2009 might be compensated for by the large decreasing trend, A_{ds} = −0.026 [µgal/day].)
To summarize the results in this section: (1) The hydrological model that considers building effects can reproduce the observed gravity change at Mizusawa during a 50day period with an accuracy of about 0.4 µgal (Fig. 9(c)). (2) The variation over the two years for the estimated gravity is not still consistent with observed annual gravity changes. The latter might be caused by the landwater scale covered in the hydrological model; since the model uses the locallyobserved water level and precipitation in the boundary conditions, only local moisture distributions and their consequent shortperiod (∼50 days) gravity changes are accurately predicted. In contrast, the model proposed here does not take into account the regional/global horizontal water distribution, and therefore cannot reproduce longperiod (∼2 years) changes in gravity, including the annual component of the gravity. Although, in principle, two or threedimensional landwater modeling (e.g., Kazama and Okubo, 2009) is required to address the highlighted concern, this study focuses on a simple hydrological model to estimate accurate landwater distributions and gravity changes, as indicated in Introduction. Thus, in the next section, the amplitude of the annual change in gravity will be calculated by using the leastsquares method. Furthermore, the effects resulting from a covering of snow will be considered to estimate more accurate changes in gravity in winter, because, compared with rain, snow remaining on the ground and building roofs can cause a significant gravity change over a longer duration.
4.2 Annual gravity change and snow effect
The thick line in Fig. 10(b) shows the g_{cal}(t), including the effects of soil moisture , the instrumental drift (g_{d}(t)), the annual gravity change (g_{a}(t)) and the covering snow (g_{s}(t)). Note that the instrumental drift rates are A_{ds} − 0.095 = −0.108 [µgal/day] (before June 2009) and [µgal/day] (after June 2009), because −0.095 µgal/day was already subtracted from the original gravity data (see Section 2). It can be seen that g_{cal}(t), including the annual term, matches g_{obs}(t) (gray line) more closely, especially for August 2009, March–May 2010 and July–August 2010, when compared with the thick line in Fig. 9(b) (not including the annual gravity term). g_{cal}(t) also agrees with g_{obs}(t) during winter (such as in February 2009 and January 2010), owing to the snowcovering effect applied in the model. In addition, the RMS and the CC values are 1.03 µgal and 0.97, respectively, showing a good correlation between g_{obs}(t) and g_{cal}(t). From these results, it can be seen that, by adding both the annual gravity term and the snow effects (Eq. (31)), our proposed hydrological modeling can now reproduce longterm gravity changes within about 1 µgal RMS.
In contrast, a large discrepancy remains between g_{obs}(t) and g_{cal}(t) of about 2 µgal in March 2009, August–September 2009 and April 2010. The cause of this discrepancy may lie in the low dimension of the hydrological modeling; here, we have supposed a simple vertical soil water flow (Eq. (5)) and two or threedimensional modeling of the water flow (e.g., Kazama and Okubo, 2009) would match the observed gravity including the annual component, with a higher accuracy. Moreover, to precisely predict gravity changes in winter, snow physics must be added into our hydrological model, such as water interaction between covering snow and soil water (Pomeroy et al., 2007), water sublimation at the top of the covering snow (Strasser et al., 2008), and spatiotemporal heterogeneity of the snow depth and the density (Heki, 2004; Doi et al., 2010).
5. Discussion
5.1 Dependency on soil parameters
The central panel (b) in Figs. 11–15 shows the variation of the moisture over time at 50 cm below the ground surface for p = 1.0; θ(z_{s} − 0.5, t; p = 1). The moisture does not change even if the value of P_{0} is changed (Fig. 15(b)), since P_{0} relates only to the steadystate moisture profile, and not to the timevariable moisture, and the contribution from the difference in the steadystate moisture profiles (Fig. 15(a)) is small for the timevariable moisture; the steadystate profiles up to only 52 cm are utilized for the initial moisture distribution for the case of Isawa Fan (see beginning of Section 4). In addition, the moisture changes are slight (0.02 at most) when the values of a and b are altered (Figs. 13(b) and 14(b)), implying that the effect of soil parameter nonlinearity (Eqs. (14) and (15)) is small when estimating the timedependent moisture for the case of Mizusawa. Conversely, θ(z_{s} − 0.5, t; p = 1) changes considerably for changes in K_{s} or D_{s} (Figs. 11(b) and 12(b)). For example, when K_{s} is varied from 5.0 × 10^{−8} down to 3.0 × 10^{−8} and up to 7.0 × 10^{−8} [m/s], the moisture value increases and reduces by ∼0.04, respectively (Fig. 11(b)). Furthermore, when D_{s} is changed from 1.0 × 10^{−6} down to 3.0 × 10^{−7} and up to 3.0 × 10^{−6} [m^{2}/s], the moisture value falls by about 0.08 and rises by about 0.07, respectively (Fig. 12(b)). These characteristics are mainly due to variations in K_{s} and D_{s} giving rise to changes in the soil water velocity v (Eq. (4)), and a balance change between the diffusion (the first term on the righthand side of Eq. (5)) and the infiltration (the second term on the righthand side of Eq. (5)) (e.g., Jury and Horton, 2004).
The lower panel (c) in Figs. 11–15 shows the gravity change, g_{w}(t) (Eq. (17)). Since g_{w}(t) is calculated from the integration of the moisture distribution, θ(z, t), the gravity dependence on each parameter is approximately equal to that of the moisture (Figs. 11(b)–15(b)). For example, the gravity amplitude sensitively changes according to variations in K_{s} and D_{s} by about 2.5 and 4 µgal, respectively. Moreover, the gravity difference is within 1 µgal for the changes in a (Fig. 13(c)) and b (Fig. 14(c)), and is exactly zero for changes of P_{0} (Fig. 15(c)).
Thus, moisture and gravity variations are dependent on the soil parameter values, especially K_{s} and D_{s} for the case of Mizusawa presented here. That is, appropriate values for the soil parameters are essential for the realistic estimation of the moisture variation and the consequent gravity changes by the hydrological model. As described in Section 3, the soil parameter values in this study were derived from soil tests and the curve fitting to the observed moisture variation. Therefore, our hydrological model technique can be applied to any gravity stations, if reasonable parameter values are determined through observations of the soil parameters themselves and spatiotemporal moisture distributions.
5.2 Comparison between the physical model and empirical models

The proportional coefficient between the water level and the gravity is A_{1} = 1.68 [µgal/m] (Fig. 16(a)). This value may be interpreted as the soil moisture in the unsaturated layer increased by about 4 volumetric % in an infinite halfspace, assuming constant moisture values in the saturated and unsaturated layers (θ_{sat} and θ_{unsat}, respectively) as follows: (θ_{sat} − θ_{unsat}) is close to the observed moisture increase during precipitation events (Fig. 3(c)), though this model makes too many simplifying assumptions for the hydrological structure in soil. (The actual moisture value in the unsaturated layer is not constant, but increases in the lowermost part of the unsaturated layer owing to the capillary effect, as shown in Fig. 4.)

The black lines in Figs. 16(b)–(c) show the empirical gravity models, g_{2}(t) and g_{3}(t). The proportional coefficients between the precipitations in the empirical models and the gravity are 28.4 and 25.3 µgal, respectively, both about twice as large as the expected instant gravity response for a 1m precipitation outside the observation building, This deviation might be a result of precipitation on the building infiltrating into the ground outside of the building through the guttering downpipes, suggesting the necessity for consideration of a horizontally heterogeneous precipitation (i.e., P = P(x, y, t)) in our hydrological model. Another reason could be that the observed gravity, g_{obs}(t) (gray line in Figs. 16(a)–(c)), cannot be fully explained by an instant response to precipitation, such as R(t) and P(t), because g_{obs}(t) is highly dominated by the water level change, h(t), for the case of Mizusawa (as demonstrated later).

The difference in the gravity reduction rate (A_{ds}) between g_{2}(t) and g_{3}(t) is This rate difference is exactly derived from the evapotranspiration, because the evapotranspiration effect is included in the second term in Eq. (40) and in the first term in Eq. (41). Therefore, the mass loss rate converted from Δ A_{ds} is which is approximately equal to the evapotranspiration rate in summer (gray bar in Fig. 3(a)).

A further point of interest is in the increase timings of the calculated gravity during precipitation events. Although g_{2}(t) and g_{3}(t) increase sharply in synchronization with precipitation (Fig. 16(d)), the timing of these increases is about 1 day earlier than for the observed gravity. In contrast, the gravity g_{1}(t) is consistent with the observed gravity, even for the increase timings, since g_{1}(t) is estimated from the water level change, h(t), whose peak time is delayed due to horizontal groundwater flow from the upper part of Isawa Fan. These facts suggest that, in the case of Isawa Fan, the primary source of the observed gravity is water level changes in the saturated layer, rather than instant mass increase near the ground surface from precipitation. (Our proposed hydrological model can also reproduce the peaktime delay of the observed gravity, as shown in Fig. 9(c), because the water level data is employed in one of the boundary conditions.)
From the above, the empirical models can reproduce the observed gravity in rapid increases during precipitation and gradual decreases after precipitation, with smaller RMS values and larger CCs (right side of Figs. 16(a)–(c)) than those of our hydrological modeling (right side of Fig. 10(b)). However, the empirical models only fit the observed gravity with arbitrary functions; they do not account for the nonlinear flow of the soil moisture. As described in Introduction, the hydrological gravity changes should be modeled and corrected independently of the observed gravity data, since the empirical models mostly tend to even subtract the solidearth gravity signal from the observed gravity. Therefore, we emphasize that the physical hydrology model presented here is important for generating realistic water distributions and estimating the consequent changes in gravity.
6. Conclusions
We gathered hydrological, meteorological and superconducting gravity data at Isawa Fan in northern Japan, and modeled the spatiotemporal soil moisture distribution and the resultant gravity changes by using hydrological equations. In the hydrological modeling, the Richards equation is numerically solved for soil moisture θ(z, t), and spatial integration of θ(z, t) is performed, in order to calculate changes in gravity due to soil moisture. Estimated θ(z, t) values are consistent with those observed at Isawa Fan within the limits of observation error. When adequate soil parameters and building effects are included in the model, the estimated gravity g_{cal}(t) (Eq. (28)) agrees with the observed superconducting gravity g_{obs}(t) during a 50day period within 0.4 µgal RMS. However, over a longer period (∼2 years), g_{cal}(t) cannot fully reproduce observed annual gravity changes, because the proposed model only considered variations in the local water distribution and the associated shortperiod (∼50 days) gravity change. Although, in principle, highdimensional physical modeling must be applied to a larger region including Isawa Fan (e.g., Kazama and Okubo, 2009), here we added two effects— annual gravity change and snowfall—into g_{cal}(t), and calculated proportionality coefficients (A_{ac}, A_{as} and A_{s})by the leastsquares method. The reformulated gravity equation (Eq. (31)) matched the observed gravity g_{obs}(t), including the annual gravity change, with a low RMS value (∼1.0 µgal) and high CC (0.97). The calculated amplitudes of the annual gravity (∼1.5 µgal) and snowfall effect (∼ −0.08 µgal/cm) can be explained by water mass loading (Heki, 2004) and snow accumulation on the ground and the building roof, respectively.
Finally, we remark that gravity corrections for hydrological effects by physical models are essential for accurately detecting the mass transfers associated with earthquakes and volcanoes from gravity data. Our physical modeling can be utilized for hydrological gravity corrections at all gravity stations in plain areas, if sufficient soil parameter values and certain observation data (e.g., precipitation and water level) are available. In the future, we will release the Fortran coding used for this hydrological modeling as the “Gwater 1D” program package, which will again be applied to the superconducting gravity data at Mizusawa in order to monitor the postseismic gravity changes associated with the 2011 M_{w} 9. 0 Tohoku Earthquake (Ide et al., 2011;Ozawa et al., 2011).
Declarations
Acknowledgments
The authors thank Y. Imanishi, K. Nawa and Y. Fukuda for their helpful discussions regarding our hydrological modeling. We also acknowledge Peter BauerGottwein and another anonymous reviewer for their help in improving the manuscript. This work was supported in part by GrantinAid for Scientific Research (20244071) and JSPS Fellows (197267 and 22225). This paper fulfills partial requirements for the PhD degree of T. Kazama at the University of Tokyo. GMT (Wessel and Smith, 1998) is used to create some of the figures in this paper.
Authors’ Affiliations
References
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