武汉九峰站地下水变化对重力场观测的影响

本文从地下水渗透过程的物理机制出发,采用一维水动力学模拟,利用井水位和降雨数据模拟计算了武汉九峰站附近的土壤含水率变化,在此基础上估计了地下水变化导致的重力效应.其峰对峰变化幅度达到15.94μGal,说明当利用精密重力观测研究长周期效应时实施台站地下水改正的必要性;频域分析表明,地下水重力效应在周年频段上的振幅最大,说明地下水变化对重力的最大影响来自季节性变化.对比模拟计算的地下水重力效应和经过潮汐、大气、极移等改正后的高精度超导重力残差(峰对峰变化幅度为12.73μGal),发现两者在时域和频域均具有良好的一致性,说明超导重力残差信号主要来源于局部地区地下水的变化,同时也验证了本文使用的水动力学模拟方法的正确性.     

Tackling mass redistribution phenomena by time-dependent GRACE- and terrestrial gravity observations

Time variable gravity field models derived from the satellite mission GRACE have been demonstrated to be consistent with water mass variations in the global hydrological cycle. Independent observations are provided by terrestrial measurements. In order to achieve a maximum of reliability and information gain, ground-based gravity observations may be deployed for comparison with the gravity field variations derived from the GRACE satellite mission. In this context, the data of the network of superconducting gravimeters (SG) of the ‘Global Geodynamics Project’ (GGP) are of particular interest. This study is focused on the dense SG network in Central Europe with its long-term gravity observations. It is shown that after the separation and reduction of local hydrological effects in the SG observations especially for subsurface stations, the time-variable gravity signals from GRACE agree well with the terrestrial observations from the SG station cluster.Station stability of the SG sites with respect to vertical deformations was checked by GNSS based observations. Most of the variability can be explained by loading effects due to changes in continental water storage, and, in general, the stability of all stations has been confirmed.From comparisons based on correlation and coherence analyses in combination with the root mean square (RMS) variability of the time series emerges, that the maximum correspondence between the SG and GRACE time series is achieved when filtering the GRACE data with Gaussian filters of about 1000 km filter length, which is in accordance with previous publications.Empirical Orthogonal Functions (EOF) analysis was applied to the gravity time series in order to identify common characteristic spatial and temporal patterns. The high correspondence of the first modes for GRACE and SG data implies that the first EOF mode represents a large-scale (Central European) time-variable gravity signal seen by both the GRACE satellites and the SG cluster.     

Gravity field variations from superconducting gravimeters for GRACE validation

The network of superconducting gravimeters (SG) of the ‘Global Geodynamics Project’ (GGP) offers the unique opportunity to supplement and validate the gravity field variations derived from the GRACE satellite mission. Because of the different spatial and temporal resolutions of the gravity data a combination of all datasets can be used to retrieve a maximum of information regarding mass transfers especially related to hydrology which is deployable as constraint for hydrological modelling.For a consistent combination of the datasets the gap between terrestrial data of superconducting and absolute gravimeters (AG) and from satellite data has to be bridged. A successful combination of SG and AG data could be realized for several stations which resulted in time series of the highest accuracy and long-term stability.In principle, the same reductions applied to GRACE data have to be taken into account for the terrestrial data. The separation of local hydrological effects in SG observations is crucial for the comparison with satellite-derived gravity data. It is shown that even for stations with a hydrological challenging situation such as Moxa/Germany local hydrology-induced effects can be successfully modelled.Currently, the study focuses on Europe with its dense and long-term observation network. Regarding the consistency of the SG gravity variations they are representative for a larger region. From a comparison with GRACE-derived gravity field changes, and the variations due to hydrological models a principle good agreement emerges.     

A comparison of methods to estimate groundwater recharge from bare soil based on data observed by a large-scale lysimeter

Effectively estimating groundwater recharge is critical to manage water resources, especially in arid and semi-arid regions as impacted by intensive human activities and climate changes. Rare insights have been gained into groundwater recharge since direct observation is hard to carry out. Although several methods are currently available to estimate groundwater recharge, the estimated results may cover noticeable bias. The behaviours of different methods based on different conceptual frameworks and exhibiting different levels of complexity should be examined to estimate actual groundwater recharge. This study aims to assess the performance of four common methods to estimate groundwater recharge. For this end, large-scale lysimeters equipped with soil water content sensors and water table sensors were set up at a research site established in Guanzhong Basin of China. The data achieved by 1-year observation were employed to compare four estimation methods. As revealed from the results, the following findings are drawn. (a) Groundwater level fluctuation (GLF) method is simple, whereas its accuracy is determined by specific yield, and adopting a water balance method to estimate specific yield can considerably enhance the accuracy of GLF. (b) The calibrated numerical model can obtain the optimal result compared with the other methods, whereas long-term observation data are required for parameter calibration. (c) In the water balance method, the maximum entropy production (MEP) model and a practical method (estimating evaporation between two rainfall events) were used to calculate evaporation. As indicated by the results, water balance method combined with MEP is capable of obtaining more reliable results of groundwater recharge compared with the practical method. (d) With an analytical model based on linearized Richards' equation, accurate results can be achieved. What is more, the analytical model only needs the measurement of soil moisture near the surface. The limitation of this method is that it is difficult to determine the maximal water flux. The mentioned findings are of critical implications to the management and sustainable development of groundwater.     

Hydrological modelling for assessing spatio-temporal groundwater recharge variations in the water-stressed Amathole Water Supply System,Eastern Cape,South Africa

To increase the resilience of regional water supply systems in South Africa in the face of anticipated climatic changes and a constant increase in water demand, water supply sources require diversification. Many water-stressed metropolitan regions in South Africa depend largely on surface water to cover their water demand. While climatic and river discharge data is widely available in these regions, information on groundwater resources – which could support supply source diversification – is scarce. Groundwater recharge is a key parameter that is used to estimate groundwater amounts that can be sustainably exploited at a sub-watershed level. Therefore, the objective of this study was to develop a reliable hydrological modelling routine that enables the assessment of regional spatio-temporal variations of groundwater recharge to discern the most promising areas for groundwater development. Accordingly, we present a semi-distributed hydrological modelling approach that incorporates water balance routines coupled with baseflow modelling techniques to yield spatio-temporal variations of groundwater recharge on a regional level. The approach is demonstrated for the actively managed catchment areas of the Amathole Water Supply System situated in a semi-arid part of the Eastern Cape of South Africa. In the investigated study area, annual groundwater recharge exhibits a high spatio-temporal heterogeneity and is estimated to vary between ~0.5% and 8% of annual rainfall. Despite some uncertainties induced by limited data availability, calibration and validation of the model were found to be satisfactory and yielded model results similar to (point) data of annual groundwater recharge reported in earlier studies. Our approach is therefore found to derive crucial information for efficiently targeting more detailed groundwater exploration studies and could work as a blueprint for orientating groundwater potential exploration in similar environments.     

Local and global hydrological contributions to gravity variations observed in Strasbourg

We investigate the contribution of local and global hydrology to the superconducting gravimeter (SG) installed in the Strasbourg observatory. A deterministic approach is presented to account for the contribution of water storage variations in the soils in the vicinity of the gravimeter: both amount and distribution of water masses are determined before calculating Newtonian attraction. No adjustment is performed on gravity time series.Two multi-depth Frequency Domain Reflectometer (FDR) probes have been installed to monitor the amount of water stored in the soil layer above the gravimeter. Since August 2005, they have been monitoring the variation of the water content of the entire soil thickness. Several investigations have been undertaken in order to estimate the distribution of water masses: a precise local DEM (Digital Elevation Model) has been determined using differential GPS. The geometry and heterogeneity of the soil layer have been evaluated thanks to geophysical and geomechanical prospections. The comparison between observed and modelled gravity variations shows that daily up to seasonal variations are in good agreement. For long-term variations, deep water storage and other processes have to be modelled to explain recorded gravity variations.     

Estimation of groundwater recharge in sandy till with two different methods using groundwater level fluctuations

Two methods for estimation of groundwater recharge, both based on groundwater level fluctuations, were applied in a moraine area in southeastern Sweden. The first method utilized a onedimensional soil water model which was tested against observed groundwater levels. The boundary conditions were defined by using standard meteorological data and submodels for precipitation, snow dynamics, interception, evapotranspiration and horizontal groundwater outflow. The second method directly transformed groundwater level fluctuations to equivalent amounts of water from a constructed recession curve and the specific yield concept. Conceptually the two methods could be characterized as inflow and response methods respectively.

A good fit between observed and simulated groundwater levels was obtained by the soil water modelling. The results were, however, shown to be rather insensitive to displacement in the water balance between evapotranspiration and groundwater outflow, giving a good fit for a simulated net groundwater recharge ranging between 134 and 197 mm. The results from the attempts to use groundwater level fluctuations directly were discouraging. Compared to the soil water simulations the results were unstable and quite different for different years. It was impossible to use a constant specific yield or even different specific yields depending on depth.

The conclusion was that the possibilities to use groundwater level data for quantitative water balance studies are limited under the studied climatical and hydrogeological conditions. The modelling effort clearly demonstrated the need for a better quantitative knowledge on soil properties if water balance information is to be deduced. The soil water model though, could be a valuable tool studying variations within and between different years as well as processes and single events.     

Evaluation of formation pore pressure behind the casing using borehole gravity data

Reliable estimates of the fluid pressure in the pore space of rocks are critical for different aspects of petroleum exploration and production including injection operations and scenarios of water flooding. Numerous approaches are available for formation pore pressure evaluation, however, these measurements become a challenge inside a cased borehole, and a list of possible options is short: either the casing is to be perforated, or the production tubing needs to be disconnected to perform the pressure tests. We present a method for through-casing evaluation of formation pore pressure without shutting down production. We suggest equipping an observation well with a borehole gravimeter and acquiring time variations of the vertical component of the gravity field. Changes in gravity occur during gas production and are related to time variations of formation pore pressure. Gravity changes obtained in the observation well are supposed to be inverted for time-dependent formation pore pressure variations beyond the casing. Our results and recommendations are based on numerical modeling of pore pressure spatial distribution during gas field exploitation and relevant changes in borehole gravity. Benchmark models were elaborated in order to consider a dynamic process of pressure changes in time and space under conditions similar to those in the Medvezhye gas field (Russia). Different modeling scenarios are considered for early and late stages of gas field exploitation. The sensitivity analysis was performed to estimate quantitatively a sensitivity of borehole temporal gravity changes to variations in formation pore pressure behind the casing. Based on resolution analysis we justify the possibility to extract the gravity measurements directly related to changes in pore pressure from the total changes in the gravity field due to reservoir exploitation. The impact of pore pressure on the gravity field measured in boreholes during the water flooding is also evaluated, and obtained results are discussed.     

A semi‐analytical solution for groundwater responses to stream‐stage variations and tidal fluctuations in a coastal aquifer

A two‐dimensional semi‐analytical solution to analyse stream–aquifer interactions in a coastal aquifer where groundwater level responds to tidal effects is presented. The conceptual model considered is a two‐dimensional subsurface system with stream and coastline boundaries at right angles. The dimensional and non‐dimensional boundary value problems were solved for water level in the aquifer by successive application of Laplace and Fourier transform techniques, and the results were obtained by numerical inversion of the transformed solution. The solution was then verified by reducing the solutions to one‐dimensional known problems and comparing the results with those from previous studies. Hypothetical examples were used to examine the characteristics of water‐level variations due to the variations in stream stage and the fluctuations in tide level. Sensitivity analysis indicated that streambed leakance has no influence over the amplitude of groundwater fluctuations, but that the effect of stream stage increases with increasing leakance. Little difference was observed in the water level for different aquifer penetration ratios with narrow stream width. Increases in streambed leakance caused increases in the effect of aquifer penetration by the stream on the water level. An increased specific yield value resulted in decreased amplitude of water fluctuations and mean water level, and showed that water‐level variations due to stream and tidal boundaries are sensitive to specific yield. Copyright © 2006 John Wiley & Sons, Ltd.     

Improving Groundwater Model Calibration with Repeat Microgravity Measurements

Groundwater-flow models depend on hydraulic head and flux observations for evaluation and calibration. A different type of observation—change in storage measured using repeat microgravity—can also be used for parameter estimation by simulating the expected change in gravity from a groundwater model and including the observation misfit in the objective function. The method is demonstrated using new software linked to MODFLOW input and output files and field data from the vicinity of the All American Canal in southeast California, USA. Over a 10-year period following lining of the previously highly permeable canal with concrete, gravity decreased by over 100 μGal (equivalent to about 2.5 m of free-standing water) at some locations as seepage decreased and the remnant groundwater mound dissipated into the aquifer or was removed by groundwater pumping. Simulated gravity from a MODFLOW model closely matched observations, and repeat microgravity data proved useful for constraining both hydraulic conductivity and specific yield estimates. Specific yield estimated using the infinite-horizontal slab approximation agreed well with model-derived values, and the departure from the linear, flat-water-table approximation was small, less than 2%, despite relatively large and dynamic water-table slope. First-order second-moment parameter uncertainty analysis shows reduction in uncertainty for all hydraulic conductivity and specific yield parameter estimates with the addition of repeat microgravity data, as compared to drawdown data alone.     

Gravitational effect of water circulation in the northwest Yunnan

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地球重力场的精细频谱结构及其应用

综述了近年内在全球地球动力学合作观测和研究网络框架下开展的重力场观测、频谱结构和应用研究方面的成果. 内容涉及精密大气、海潮负荷信号检测, 重力潮汐和自由核章动参数测定, 海潮模型和重力固体潮模型有效性检验, 重力潮汐实验模型构制, 地球球型基频和低阶震型谱峰分裂现象和地球Chandler摆动等方面. 文章还介绍了综合现代大地测量技术, 全球超导重力仪的长期、连续观测在地表水循环、同震和震后形变、地球慢形变和地壳垂直运动等方面将发挥重要作用的情况.     

Long-term crustal deformation monitored by gravity and space techniques at Medicina, Italy and Wettzell, Germany

Series of gravity recordings at the stations Medicina (Italy) and Wettzell (Germany) are investigated to separate seasonal gravity variations from long-term trends in gravity. The findings are compared to height variations monitored by continuous GPS observations. To study the origin of these variations in height and gravity the environmental parameters at the stations are included in the fact finding. In Medicina, a clear seasonal signal is visible in the gravity and height data series, caused by seasonal fluctuations in the atmosphere including mass redistribution, the ocean, groundwater but also by geo-mechanical effects such as soil consolidation and thermal expansion of the structure supporting the GPS antenna. In Wettzell, no seasonal effect could be clearly identified, and the long-term trend in gravity is mainly caused by ground water variations. The successful combination of height and gravity series with the derived ratio of gravity to height changes indicates that the long-term trends in height and gravity are most likely due to mass changes rather than to tectonic movements.     

Long-term and seasonal gravity changes at the Strasbourg station and their relation to crustal deformation and hydrology

The time-varying gravity is observed at the Strasbourg station with super-conducting gravimeters (SG) since 1987, with a first record from 1987 to 1996 (GWR T005) and a second one in continuity from 1996 till now (GWR C026). The long-term behaviour of the SG is constrained by regular absolute gravity (AG) measurements, which are performed in parallel since 1989, first with the JILAg-5 instrument and later on with the FG5#206. Moreover, a permanent GPS station, which belongs to the French geodetic network, has been installed at the end of 1999. We will show that the AG measurements suggest that the gravity is slowly increasing in time at a rate of about 1.6 μGal/year, and besides, exhibits a quasi-annual component of several μGal variable amplitude. We present an analysis of the GPS data obtained at the SG station in the Rhine graben as well as at another regional station in the Vosges mountains distant by about 40 km in order to constrain the gravity contribution due to the vertical displacement of the station in the earth's gravity field (geometrical part). The tectonic context of the region is described and our first results from our two data sets of limited duration suggest a small subsidence of our station in the graben; however, this fact needs further confirmation when more geodetic data will be available. We also analyze the water table changes beneath the station (local scale) and in the Alsatian Plain (regional scale) to estimate the hydrological contributions from ground water to the gravity residual signal, and we show that some similarities exist between the gravity residuals and the hydrological contributions, especially for the seasonal terms. Other missing contributions of annual period (air mass motion in the atmosphere, ocean circulation, continental hydrology) have to be considered.     

基于GRACE的空间约束方法监测华北平原地下水储量变化

华北平原作为我国重要的工农业基地和政治经济中心,面临着严重的水资源危机.因此,开展对华北平原地下水储量变化的监测工作具有重要现实意义与科学价值.本文基于GRACE重力卫星的空间约束方法,研究了华北平原地下水储量变化的时空分布规律,并与地面水井实测与地下水模型结果进行了综合比较和分析.结果表明:2002-2014年,华北平原地下水存在明显的长期亏损,GRACE估计的亏损速率为-7.4±0.9 km~3·a~(-1),而地面水井资料估计的浅层地下水亏损速率为-1.2 km~3·a~1,对比两者之间的差异可以发现,华北平原的地下水亏损以深层地下水为主.2002-2008年,GRACE估计的华北平原地下水亏损速率为-5.3±2.2 km~3·a~(-1),这与华北平原两个地下水模型得到的平均亏损速率-5.4 km~3·a~(-1)十分吻合.通过华北平原区域地下水模型的独立验证,说明GRACE可以有效评估华北平原的地下水储量变化趋势.除了长期亏损的趋势项之外,华北平原地下水还存在明显的年际变化特征,并与该地区年降雨量变化特征一致.在降雨偏少的2002年、2005-2009年和2014年,华北平原地下水储量显著减少.在空间分布上,GRACE结果表明,华北平原的地下水储量减少主要发生在山前平原和中部平原区,这也与水井实测资料和区域地下水模型结果较为吻合.与GRACE和区域地下水模型相比,目前的全球水文模型仍无法准确估计华北平原地下水变化的空间分布和亏损速率.上述研究表明,GRACE提供了评估华北平原地下水储量变化的重要监测手段.     

Determination of specific yield for the Biscayne Aquifer with a canal-drawdown test

Data from a large-scale canal-drawdown test were used to estimate the specific yield (sy) of the Biscayne Aquifer, an unconfined limestone aquifer in southeast Florida. The drawdown test involved dropping the water level in a canal by about 30 cm and monitoring the response of hydraulic head in the surrounding aquifer. Specific yield was determined by analyzing data from the unsteady portion of the drawdown test using an analytical stream-aquifer interaction model (Zlotnik and Huang 1999). Specific yield values computed from drawdown at individual piezometers ranged from 0.050 to 0.57, most likely indicating heterogeneity of specific yield within the aquifer (small-scale variation in hydraulic conductivity may also have contributed to the differences in sy among piezometers). A value of 0.15 (our best estimate) was computed based on all drawdown data from all piezometers. We incorporated our best estimate of specific yield into a large-scale two-dimensional numerical MODFLOW-based ground water flow model and made predictions of head during a 183-day period at four wells located 337 to 2546 m from the canal. We found good agreement between observed and predicted heads, indicating our estimate of specific yield is representative of the large portion of the Biscayne Aquifer studied here. This work represents a practical and novel approach to the determination of a key hydrogeological parameter (the storage parameter needed for simulation and calculation of transient unconfined ground water flow), at a large spatial scale (a common scale for water resource modeling), for a highly transmissive limestone aquifer (in which execution of a traditional pump test would be impractical and would likely yield ambiguous results). Accurate estimates of specific yield and other hydrogeological parameters are critical for management of water supply, Everglades environmental restoration, flood control, and other issues related to the ground water hydrology of the Biscayne Aquifer.     

Estimation of areal specific yield in sands using the central limit theorem

A new approach is demonstrated that permits a reliable estimate of specific yield using published values of the van Genuchten water retention parameters and effective grain sizes and the measured effective grain sizes of soil samples. The specific yield distribution of the soil texture was computed using the published values of the van Genuchten parameters. The specific yield values and the published values of effective grain sizes were then used to construct a specific yield–effective grain size curve, which estimates the ‘point’ specific yield of the soil samples. Applying the central limit theorem, the point specific yields could be transformed into an ‘areal’ specific yield for a study area. Compared with other commonly used approaches, the present procedure requires relatively low computational efforts and readily obtainable data. It is cost effective and does not depend on soil texture classification. More importantly, it incorporates the depth to water table and the variations in grain sizes inherent in natural soil conditions in the estimation. The approach developed was applied for estimating the specific yield of an unconfined sandy aquifer created by land reclamation in the equatorial region. The values obtained were compared with field measurements and the typical ranges of specific yield from the literature. Instead of a single estimate of the specific yield, the method yields a confidence interval with a high confidence level of 95% and with a narrower range than the typical ranges from the literature. In addition, the estimated values are close to the field measurements; hence, the procedure provides a cost‐effective alternative to field measurement. The applicability of the present approach could be extended to sites with heterogeneity in the horizontal direction. Nevertheless, the applicability of the present approach for layered soil profiles requires further evaluations. Copyright © 2006 John Wiley & Sons, Ltd.     

2005-2008年北京地区典型重力变化跟踪与分析

自2006年起,北京地区一些流动重力测点测值在趋势变化的基础上加速变化,昌平和顺义等测点附近重力场非均匀性变化增强。以北京昌平重力观测点为中心增加2条重力剖面观测线,并进行3期观测,分析重力场演化特征,采用地下水重力效应经验公式、自由空气改正关系式,分别改正地下水位、地壳形变对重力典型变化点的影响。研究表明：重力变化是地下水位、地壳形变影响的结果,重力剖面观测区及附近重力变化比较平稳,变化均在误差范围内,排除该异常是地震前兆异常的可能。     

Contribution of magnetic resonance sounding to aquifer characterization and recharge estimate in semiarid Niger

To improve the knowledge of the regionally important Continental Terminal 3 (CT3) aquifer in south-western Niger, fifteen magnetic resonance soundings (MRS) were carried out in December 2005 in the vicinity of wells and boreholes. The output MRS geophysical parameters, i.e. water content and decay constants versus depth, were compared to hydrogeological characteristics, i.e. water table depth, total porosity, specific yield and transmissivity estimated from direct measurements, pumping tests and transient groundwater modelling. The MRS-determined parameters were then used to estimate the rates of groundwater recharge.Contained in poorly consolidated Tertiary sandstones, the CT3 aquifer's water table has continuously risen by 4 m in total over the past four decades. Additionally, a significant portion of this increase has occurred in the past decade alone, with an annual rise now ranging between 0.1 and 0.3 m depending on the monitored well. Increase in groundwater recharge due to land clearance and deforestation explains this situation. According to previous estimations, the pre-clearing recharge ranged from 1 to 5 mm per year in 1950–60 s, while more recent recharge rates (1990s–2000s) range from 20 to 50 mm per year. These recharge values are directly affected by estimated aquifer specific yield value, while the spatial variation of rates of water table rise can be attributed to large scale hydrodynamic heterogeneities in the aquifer. However, few field measurements were available to confirm these assumptions.The main results of this study are: (1) The water table depth and aquifer transmissivity are estimated from MRS output parameters with an average accuracy of ± 10% and ± 9% respectively. (2) The MRS-determined water content is linked to both the total porosity and the specific yield of the aquifer, but no quantitative formulation can be proposed as yet. (3) Using the average MRS-determined water content over the investigated area, i.e. 13%, the groundwater recharge rates can be estimated to be ~ 2 mm per year in the 1950–1960s (pre-clearing period), and ~ 23 mm per year for the last decade. (4) The variations in specific yield and transmissivity cannot explain by themselves the spatial variability of the rise of the water table. (5) The ranges in transmissivity and water content obtained from MRS are more realistic than the groundwater modelling outputs. Therefore, MRS could be used to better constrain the aquifer parameters in groundwater modelling with a dense site network.Finally, this work illustrates how MRS can successfully improve characterisation and transient multi-year groundwater balance of commonly found sedimentary aquifers, particularly when integrated with well observations and pumping tests.     

Gravity Changes and Internal Processes: Some Results Obtained from Observations at Three Volcanoes

Temporal gravity changes provide information about mass and/or density variations within and below the volcano edifice. Three active volcanoes have been under investigation; each of them related to a plate boundary: Mayon/Luzon/Philippines, Merapi/Java/Indonesia, and Galeras/Colombia. The observed gravity changes are smaller than previously expected but significant. For the three volcanoes under investigation, and within the observation period, mainly the increase of gravity is observed, ranging from 1,000 nm^–2 to 1,600 nms^–2. Unexpectedly, the gravity increase is confined to a rather small area with radii of 5 to 8 km around the summit. At Mayon and Merapi the parallel GPS measurements yield no significant elevation changes. This is crucial for the interpretation, as the internal pressure variations do not lead to significant deformation at the surface. Thus the classical Mogi-model for a shallow extending magma reservoir cannot apply. To confine the possible models, the attraction due to changes of groundwater level or soil moisture is estimated along the slope of Merapi exemplarily by 2-D modelling. Mass redistribution or density changes were evaluated within the vent as well as deeper fluid processes to explain the gravity variations; the results are compared to the model incorporating the additional effect of elastic deformation.