Tiltmeter Data Research Articles

An integrated extended Kalman filter–implicit level set algorithm for monitoring planar hydraulic fractures

We describe a novel approach to the inversion of elasto-static tiltmeter measurements to monitor planar hydraulic fractures propagating within three-dimensional elastic media. The technique combines the extended Kalman filter (EKF), which predicts and updates state estimates using tiltmeter measurement time-series, with a novel implicit level set algorithm (ILSA), which solves the coupled elasto-hydrodynamic equations. The EKF and ILSA are integrated to produce an algorithm to locate the unknown fracture-free boundary. A scaling argument is used to derive a strategy to tune the algorithm parameters to enable measurement information to compensate for unmodeled dynamics. Synthetic tiltmeter data for three numerical experiments are generated by introducing significant changes to the fracture geometry by altering the confining geological stress field. Even though there is no confining stress field in the dynamic model used by the new EKF-ILSA scheme, it is able to use synthetic data to arrive at remarkably accurate predictions of the fracture widths and footprints. These experiments also explore the robustness of the algorithm to noise and to placement of tiltmeter arrays operating in the near-field and far-field regimes. In these experiments, the appropriate parameter choices and strategies to improve the robustness of the algorithm to significant measurement noise are explored.

Quantifying fractured crystalline-rock properties using well tests, earth tides and barometric effects

Summary Characterization of fractured rock aquifers often requires the acquisition and analysis of diverse datasets obtained from various instrumentation configurations. In this investigation at the fractured rock research site in Floyd County, Virginia, a high-precision borehole extensometer and tiltmeter were used during pumping to monitor deformation in the vicinity of fractures identified from borehole logging. Strain data obtained from earth tide analyses were used with the extensometer and tiltmeter data to quantify hydromechanical properties, including fracture volumetric specific storage, porosity, Poisson’s ratio, the drained formation elastic modulus, and the effective dip direction of the fracture. Borehole tiltmeter data were used to estimate deformation caused by an aquifer test consisting of three pumping and recovery periods performed in well EX-1. During each period of the aquifer test the extensometer, located in W-03 and 27.7 m from the pumping well, was anchored over 2-m-long sections of (1) a fracture in hydraulic communication with EX-1, (2) a fracture that is not hydraulically connected with EX-1, and (3) an unfractured portion of bedrock directly above the hydraulically connected fracture. Results from the pumping tests yielded compressibilities of 1.3 × 10 −11 Pa −1 , and 1.7 × 10 −11 Pa −1 for the lower and upper fractures, respectively. When coupled with areal strain calculated from earth tide analyses the volumetric specific storage values are 3.2 × 10 −11 Pa −1 , and 2.8 × 10 −11 Pa −1 and the Poisson’s ratios are 0.26 and 0.31, respectively. Using this with a calculated barometric efficiency of 0.45 allows for porosity calculations of 0.02 and 0.03, respectively for the vicinity of fractures in well W-03.

Repeated fluid-transfer episodes as a mechanism for the recent dynamics of Campi Flegrei caldera (1989–2010)

[1] We have analyzed a multiparametric data set of seismological, geodetic and geochemical data recorded at Campi Flegrei caldera since 1982. We focus here on the period 1989–2010 that followed the last bradyseismic crisis of 1982–1984. Since then, there have been at least five repeated minor episodes of ground uplift accompanied by seismicity. We have reanalyzed old paper and digital seismic data sets dating back to 1982. The paper recordings show evidence of long-period events in January 1982 and March 1989, and we have digitized some of these significant waveforms. Furthermore, the revision of digital seismograms dating back to 1994 shows a significant swarm of long-period events in August 1994. Volcano-tectonic and long-period events hypocenters have been relocated in a three-dimensional velocity model. Statistical analysis of volcano-tectonic seismicity shows many similarities and few differences between 1982–1984 and the following period 1989–2010. Long-period waveforms have been analyzed using spectral analysis, which shows a grouping into three macrofamilies. Similarities in the seismic signature of episodes of minor uplift suggest that they originate from the injection of fluids into the deep part of a geothermal reservoir (about 2.5 km depth) and in its transfer toward a shallower part (about 0.75 km depth). Most of the observed geophysical signals are related to this second phase. The evidence consists of spatial and temporal connections between the ground deformation, long-period and volcano-tectonic seismicity and changes in the geochemical parameters of fumaroles. In this study we focused our analysis on two uplift episodes observed in 2000 and 2006. The joint inversion of Differential Synthetic Aperture Radar (DInSAR) and tiltmeter data show that during these periods the ground deformation was generated by at least two distinct sources located at different depths, with the shallower activated in the later stages of the uplift episodes. Our interpretation of the recent dynamics of Campi Flegrei is that the deep part of the geothermal reservoir inflates in response to mass and heat input from a magmatic source. When the pressure exceeds a threshold, fluids starts to migrate into the shallower part. During this transfer, long-period sources are activated in response to the fluid motion. The gradual diffusion of fluids in the surrounding rocks lowers the resistance of a pervasive fracture system generating shallow microseismicity. Finally, fluids reach the surface, which gives a distinct geochemical signature to the overlying fumaroles.

Recurrence behavior of short‐term slow slip and correlated nonvolcanic tremor episodes in western Shikoku, southwest Japan

From the National Research Institute for Earth Science and Disaster Prevention Hi‐net tiltmeter data, we determine time‐dependent source processes of seven repeating short‐term slow slip events (SSE) in the western Shikoku region, southwest Japan, over the period from 2002 to 2007. The tiltmeters record clear signals of SSEs characterized by a slow rise, a week‐long duration, and a magnitude on the order of 10−2−10−1μrad. We apply a time‐dependent slip inversion method to the tilt records for the seven SSEs. For each SSE, a slip area migrates in the strike direction of the subducting slab at a rate of ∼10 km d−1. Comparing the slow slip area with epicenters of accompanying nonvolcanic tremors in subday time resolution, we find that the two phenomena are located on a smaller area than previously reported and migrate as a group. A moment rate function for each SSE exhibits good correlation with the temporal change in tremor activity. A comparison of the slip distributions of all of the analyzed SSEs shows that there is a patch of slow slip area (∼30 km in diameter) that is shared by all of the episodes. In addition, the location of the slip patch is in good agreement with the epicenters of tremors and very low frequency earthquakes. These lines of evidence suggest that the identified slip patch is one of the main slow slip patches in the region and a recurrence of SSEs may be controlled by the patch and that these “slow earthquakes” might be different manifestations of a single slip process on the deep plate interface.

Monitoring hydraulic fractures: state estimation using an extended Kalman filter

There is considerable interest in using remote elastostatic deformations to identify the evolving geometry of underground fractures that are forced to propagate by the injection of high pressure viscous fluids. These so-called hydraulic fractures are used to increase the permeability in oil and gas reservoirs as well as to pre-fracture ore-bodies for enhanced mineral extraction. The undesirable intrusion of these hydraulic fractures into environmentally sensitive areas or into regions in mines which might pose safety hazards has stimulated the search for techniques to enable the evolving hydraulic fracture geometries to be monitored. Previous approaches to this problem have involved the inversion of the elastostatic data at isolated time steps in the time series provided by tiltmeter measurements of the displacement gradient field at selected points in the elastic medium. At each time step, parameters in simple static models of the fracture (e.g. a single displacement discontinuity) are identified. The approach adopted in this paper is not to regard the sequence of sampled elastostatic data as independent, but rather to treat the data as linked by the coupled elastic-lubrication equations that govern the propagation of the evolving hydraulic fracture. We combine the Extended Kalman Filter (EKF) with features of a recently developed implicit numerical scheme to solve the coupled free boundary problem in order to form a novel algorithm to identify the evolving fracture geometry. Numerical experiments demonstrate that, despite excluding significant physical processes in the forward numerical model, the EKF-numerical algorithm is able to compensate for the un-modeled dynamics by using the information fed back from tiltmeter data. Indeed the proposed algorithm is able to provide reasonably faithful estimates of the fracture geometry, which are shown to converge to the actual hydraulic fracture geometry as the number of tiltmeters is increased. Since the location of tiltmeters can affect the resolution of the method, the algorithm can also be used to design the deployment of tiltmeters to optimize the resolution in regions of particular interest.

APPLICATION OF THE EMPIRICAL MODE DECOMPOSITION TO FIELD TILTMETER DATA FOR HYDRAULIC FRACTURE MAPPING

Empirical Mode Decomposition (EMD) is a fully data-driven, adaptive technique for analyzing time series from nonlinear and nonstationary processes. The starting point of EMD is to treat the signal as a superposition of different intrinsic modes of oscillations (fast oscillations superimposed on slow oscillations). The essence of this method is to empirically identify the intrinsic oscillatory modes by their characteristic time scale imbedded in a signal, and then decompose the signal into a collection of a finite and often small number of intrinsic mode functions (IMF) through a so-called sifting process. Each IMF component then represents only one mode of both amplitude and frequency modulated oscillation of the signal at a certain time scale or frequency band, and the sum of all the IMF components as well as a residual produces a perfect reconstruction of the original signal. Partial reconstruction can be achieved by selectively removing fast or slowly varying IMFs, which provides a method to remove unwanted (noise) parts of the signal. In this paper, the EMD is applied to quantitative analysis of field tiltmeter data collected to monitor and map hydraulic fractures. The fracture-related tilt components are extracted by identifying the relevant IMFs that contribute to them, which allows removing the noise and background trend components from the raw data. The extracted tilt data are then inverted to obtain the volume and orientation of the hydraulic fractures. Physically reasonable prediction of the hydraulic fracture volume is used to demonstrate that the application of EMD to field tiltmeter data analyzing can be successfully carried out.

Imaging Fractures Using Temperature Transients From Perturbation Analysis—A Novel Surveillance Technique Applied to the Belridge Diatomite

Summary Results of a temperature transient analysis of data from Shell's diatomite steamdrive pilots are used to image hydraulic injection fracture lengths, angles, and heat injectivities into the low-permeability formation. The Phase I pilot is a limited-interval injection test. In Phase Π, steam is injected into two 350-ft upper and lower zones through separate hydraulic fractures. Temperature response of both pilots is monitored with 16 logging-observation wells. A perturbation analysis of the nonlinear pressure diffusion and heat-transport equations indicates that at a permeability of approximately 0.1 md or less, heat transport in the diatomite tends to be dominated by thermal diffusivity, and pressure diffusion is dominated by the ratio of thermal expansion to fluid compressibility. Under these conditions, the temperature observed at a logging-observation well is governed by a dimensionless quantity that depends on the perpendicular distance between the observation well and the hydraulic fracture divided by the square root of time. Using this dependence, a novel method is developed for imaging hydraulic-fracture geometry and relative heat injectivity from the temperature history of the pilot. The azimuth of the Phase I hydraulic fracture is determined to be 14° ± 2 N-NE. The azimuth of the Phase II upper hydraulic fracture is determined to be 16° ± 2 N-NE in the northern half of the pilot, and estimated to increase to 21° ± 2 in the southern half of the pilot. The azimuth of the lower hydraulic fracture averages 19° ± 4. The hydraulic fractures are found to be symmetric around both injectors, with an estimated length of 200 ft. Increased steam injection after the first year of pilot operations caused what is interpreted to be horizontal fractures toward the west in the G cycle and the east in the M cycle. These features are imaged to be at least 100 and 160 ft, respectively, along the hydraulic fracture azimuth. These conclusions are compared to tiltmeter data, microseismic data, and a simulation history match of pilot performance. Microseismic events recorded in the pilot are apparently not diagnostic of heat delivery to the formation.

Extreme Multistage Fracturing Improves Vertical Coverage and Well Performance in the Lost Hills Field

Summary In an attempt to improve production response, fracturing designs in the Lost Hills field went from a standard three-to six-stage design to an extreme 15-to 18-stage design to stimulate approximately 1,000 ft of net pay. The previous standard designs were becoming borderline economic, and if development was to continue, then either production response had to improve or costs had to be reduced. Previous emphasis was placed on reducing fracture-treatment costs by pumping fewer stages and lower proppant volumes, but the cumulative production response decreased as a result. Previous tests using an increased number of fracture stages did not improve economics because of the increased fracturing costs and minimal production increases. When a new coiled-tubing fracturing technique was implemented to perform multistage jobs at reduced costs and the stage count per well was increased, production response and economics improved. This paper will discuss the differences in job execution, analysis of vertical fracture coverage using surface and downhole tiltmeter data, and cumulative production response from the different designs tested. Both treatment and stage design with this fracturing technique are being refined further as the performance and statistical analysis of previous design changes are completed.

Comparison of fluid tiltmeter data with long‐period seismograms: Surface waves and Earth's free oscillations

We compare observations of long‐period seismic surface waves and free oscillations recorded by high‐resolution long‐base fluid tube tiltmeters and by nearby broadband seismometers after large earthquakes. The quality of the tiltmeter data is comparable to that of the best horizontal component seismic data, recording some of the gravest free oscillations of the Earth, as well as successive passages of seismic surface waves circling the globe. We compare the observations with theoretical seismograms and with theoretical tilt. The predicted and observed surface wave tilt waveforms are very similar provided that we take into account horizontal acceleration effects on the tiltmeter. Phase and amplitude anomalies between the waveforms are well explained by the theoretical transfer function of the instrument. Likewise, observed horizontal seismograms converted into tilt match the tiltmeter data very well. Long‐base fluid tube tiltmeters could potentially contribute to obtain high‐quality measurements of the long‐period seismic spectrum.

Upper mantle imaging beneath the Japan Islands by Hi-net tiltmeter recordings

Abstract We present a novel receiver-function image of the upper mantle structure around the Japan subduction zone. To increase the amount of available waveform data containing the relatively lower frequency component, we examined whether the Hi-net tiltmeter recordings are usable for imaging the upper mantle discontinuities by comparing them with broadband seismograms in different frequency bands. We found that the two are comparable at a frequency band between 0.02 and 0.16 Hz. To make receiver functions from tiltmeter data, stacked vertical components of broadband seismograms were used as source-time functions. Since such source-time functions may include biases from local structure, we also produced regional stacked source-time functions. The receiver function with the above frequency band does not seem to be affected by local structure. In the images derived from the receiver-function gathers, we were able to visualize both the oceanic Moho and the lower slab boundary, which could be traced down to depths of 400 km and 600 km, respectively. These images also show an uplift of the 410-km discontinuity and a depression of the 660-km discontinuity in the regions that are probably affected by the cold subducting Pacific slab.

Time-dependent basal stress conditions beneath Black Rapids Glacier, Alaska, USA, inferred from measurements of ice deformation and surface motion

AbstractObservations of surface motion and ice deformation from 2002–03 were used to infer mean stress fields in a cross-section of Black Rapids Glacier, Alaska, USA, over seasonal timescales. Basal shear stresses in a well-defined zone north of the center line (orographic left) were approximately 7% and 16% lower in spring and summer, respectively, than in winter. Correspondingly higher stresses were found near the margins. These changes in the basal shear stress distribution were sufficiently large to cause mean surface velocities to be 1.2 and 1.5 times larger in spring and summer than in winter. These results were inferred with a simple inverse finite-element flow model that can successfully reproduce bulk surface velocities and tiltmeter data. Stress redistribution between the well-defined zone and the margins may also occur over much shorter time periods as a result of rapidly changing basal conditions (ice–bed decoupling or enhanced till deformation), thereby causing large variations in surface velocity and strongly influencing the glacier’s net motion during summer.

Resolving the Geometry of Hydraulic Fractures from Tilt Measurements

This paper discusses the resolution of geometrical characteristics of pressurized fractures from tiltmeter data. The quasi-static deformation and tilt field induced by such fractures can be modeled by superposition of displacement discontinuity (DD) singularities. Despite the relatively common use of such measurements to infer fracture characteristics, there is a widespread misunderstanding of what can be accurately determined, depending on the relative distance between the tiltmeter array and the fracture. We investigate in detail the resolution of the dimensions and orientation of hydraulic fractures or faults from tilt measurements. In particular, we formally prove that at a distance larger than about twice the characteristic length of the fracture, elastostatic measurements such as those measured by tiltmeters are not able to resolve independently all the dimensions of the fracture, although the fracture volume can be robustly inverted from the data. The resolution of fracture orientation is also discussed using an analysis based on a spatial Fourier Transform of the tilt field. The relative angle between the plane where the measurements are located and the fracture plane plays a major role in the accuracy of this estimation. In an illustrative field example, where the measurements are located in the far-field of the fracture deformation field, we show how a single DD singularity can be used to model tiltmeter data and efficiently obtain the fracture orientation and volume.

Propagation of the 2001–2002 silent earthquake and interplate coupling in the Oaxaca subduction zone, Mexico

Abstract The aseismic slow slip event of 2001–2002 in Guerrero, Mexico, with an equivalent magnitude MW ~ 7.5, is the largest silent earthquake (SQ) among many recently recorded by GPS in different subduction zones (i.e. Japan, Alaska, Cascadia, New Zealand). The sub-horizontal and shallow plate interface in Central Mexico is responsible for specific conditions for the ~100 km long extended transient zone where the SQs develop from ~80 to ~190 km inland from the trench. This wide transient zone and relatively large slow slips of 10 to 20 cm displacements on the subduction fault result in noticeable surface displacements of 5–6 cm during the SQs. Continuous GPS stations allow one to trace the propagation of SQs, and to estimate their arrival time, duration and geometric attenuation. These propagation parameters must be accounted in order to locate source of slow slips events and to understand the triggering effect that they have on large subduction earthquakes. We use long-baseline tiltmeter data to define new time limits (onset and duration) for the SQs and continuous records from 8 GPS stations to determine the propagation of the 2001–2002 SQ in Central Mexico. Data from the CAYA and IGUA GPS stations, separated by ~170 km and located along the profile perpendicular to the trench, are used to determine that the surface deformation from the 2001–2002 SQ started almost instantaneously. It propagated parallel to the coast at ~2 km/day with an exponential attenuation of the horizontal surface displacement and a linear decrease of its duration with distance. Campaign data obtained yearly from 2001 to 2005 at the Oaxaca GPS network have been modeled according to a propagation of the 2001–2002 SQ step-like displacement anomaly. This modeling shows that the SQ ceased gradually in the central part of the Oaxaca segment of the subduction zone (west of Puerto Angel, PUAN) and then it apparently triggered another SQ in SE Oaxaca (between PUAN and Salina Cruz, SACR). The estimated horizontal velocities for inter-event epochs at each GPS site are used to assess an average interplate coupling in the Central Oaxaca subduction zone.

Seasonal variations in ice deformation and basal motion across the tongue of Haut Glacier d’Arolla, Switzerland

AbstractRecords of surface motion, englacial tilt and repeat inclinometry are used to determine patterns of surface, internal and basal motion across the tongue of Haut Glacier d’Arolla, Switzerland, over temporal scales ranging from days to months. Findings are interpreted with reference to contemporaneous measurements of subglacial water pressures, and prior knowledge of the glacier’s subglacial drainage-system structure. Long-term inclinometry results show pronounced extrusion flow over a subglacial drainage axis, with basal velocities up to twice those measured at the glacier surface. Deformation profiles are more conventional away from the drainage axis, with basal velocities ∼60–70% of surface velocities. Comparison of long-term tilt rates from repeat inclinometry and englacial tiltmeters shows close correspondence. Englacial tiltmeter data are used to reconstruct internal velocity profiles and to split surface velocities into internal deformation and basal motion contributions over spring, summer and autumn/winter periods. Although, spatial patterns of surface movement are similar between periods, patterns of internal and basal motion are not. Results are interpreted in terms of the location of sticky and slippery spots, with temporally changing patterns of basal drag reflecting changing distributions of water pressure.

Cascade Inversion for Tilt Data

We investigate an inverse problem for mapping hydraulic fracture geometry using surface tiltmeter data. It is well known that this problem becomes a non-linear inverse problem. If the inversion algorithm can be separated into two stages as: (1) the identification of the fracture plane and (2) the estimation of the fracture aperture distribution, the computation scheme becomes robust and stable. We call this methodology a cascade inversion scheme for tilt data. In the first stage of the inversion, the fracture plane is estimated by the Nelder-Mead simplex method whereby we obtain the volume of the hydraulic fracture. The fracture aperture distribution is then determined by the successive linear inversion stage. In this second stage, the fracture plane is divided into small rectangular pieces and each piece has a different fracture aperture. The normal equation, where the fracture apertures are unknowns, can be solved by the least squared method with two constraints, which are the smoothness and non-negative values of the fracture apertures. The proposed methodology was applied to synthetic and field data. The inversion results are quite acceptable and we conclude that this cascade inversion scheme is a robust method and easy to apply to field data.

Fracturing in low-permeability soils for remediation of contaminated ground

This paper presents a field study on hydraulic fracturing for in situ remediation of contaminated ground. Sand-propped hydraulic fractures were placed from vertical and horizontal wells at a test facility. Field excavations were conducted to expose the fractures and inspect their distribution and geometry. Fractures that were mapped by field excavation were found to be near horizontal, implying that the soil formation is overconsolidated. It was also observed that the sand "proppant" was thicker at locations where the soil layers were relatively weak or contained weak fissures. Electrical resistivity tomography (ERT) was also conducted in an attempt to map the fractures. There was no indication that fractures were being mapped by this geophysical technique. Fracture mapping based on tiltmeter data analyses conformed closely with the actual fracture placement in the vertical well but did not properly predict the actual fracture placement in the horizontal well.Key words: hydraulic fracturing, field test, low-permeability soil, electrical resistivity tomography, tiltmeters, horizontal well, vertical well.

Analytic crack solutions for tilt fields around hydraulic fractures

The recent development of downhole tiltmeter arrays for monitoring hydraulic fractures has provided new information on fracture growth and geometry. These downhole arrays offer the significant advantages of being close to the fracture (large signal) and being unaffected by the free surface. As with surface tiltmeter data, analysis of these measurements requires the inversion of a crack or dislocation model. To supplement the dislocation models of Davis [1983], Okada [1992], and others, this work has extended several elastic crack solutions to provide tilt calculations. The solutions include constant‐pressure two‐dimensional (2‐D), penny‐shaped, and 3‐D‐elliptic cracks and a 2‐D‐variable‐pressure crack. Equations are developed for an arbitrary inclined fracture in an infinite elastic space. Effects of fracture height, fracture length, fracture dip, fracture azimuth, fracture width, and monitoring distance on the tilt distribution are given, as well as comparisons with the dislocation model. The results show that the tilt measurements are very sensitive to the fracture dimensions but also that it is difficult to separate the competing effects of the various parameters.

Using an analytical geometry method to improve tiltmeter data presentation

Abstract The tiltmeter is a useful tool for geologic and geotechnical applications. To obtain full benefit from the tiltmeter, easy and accurate data presentations should be used. Unfortunately, the most commonly used method for tilt data reduction now may yield inaccurate and low-resolution results. This article describes a simple, accurate, and high-resolution approach developed at the Illinois State Geological Survey for data reduction and presentation. The orientation of tiltplates is determined first by using a trigonometric relationship, followed by a matrix transformation, to obtain the true amount of rotation change of the tiltplate at any given time. The mathematical derivations used for the determination and transformation are then coded into an integrated PC application by adapting the capabilities of commercial spreadsheet, database, and graphics software. Examples of data presentation from tiltmeter applications in studies of landfill covers, characterizations of mine subsidence, and investigations of slope stability are also discussed.

Using surface deformation to image reservoir dynamics

The inversion of surface deformation data such as tilt, displacement, or strain provides a noninvasive method for monitoring subsurface volume change. Reservoir volume change is related directly to processes such as pressure variations induced by injection and withdrawal. The inversion procedure is illustrated by an application to tiltmeter data from the Hijiori test site in Japan. An inversion of surface tilt data allows us to image flow processes in a fractured granodiorite. Approximately 650 barrels of water, injected 2 km below the surface, produces a peak surface tilt of the order of 0.8 microradians. We find that the pattern of volume change in the granodiorite is very asymmetrical, elongated in a north‐northwesterly direction, and the maximum volume change is offset by more than 0.7 km to the east of the pumping well. The inversion of a suite of leveling data from the Wilmington oil field in Long Beach, California, images large‐scale reservoir volume changes in 12 one‐ to two‐year increments from 1976 to 1996. The influence of various production strategies is seen in the reservoir volume changes. In particular, a steam flood in fault block II in the northwest portion of the field produced a sudden decrease in reservoir volume.

Tropospheric corrections of SAR interferograms with strong topography. Application to Etna

The accuracy of spaceborne geodetic techniques, including SAR interferometry, is limited by the time and spatial variation and altitude dependance of the propagation delay of electomagnetic waves in the lower troposphere, particularly in mountainous areas. In this paper, we use a 1D model developed for tropospheric corrections of GPS and DORIS measurements to correct SAR data. The differential tropospheric delay is computed at each pixel of the interferogram from ground temperature, humidity and pressure using two empirical parameters calibrated from several radio‐soundings acquired in various latitude and climate conditions. It is shown that with such a model, given the 3300 meters topography of Etna, tropospheric variations can generate up to 4π phase rotations between the top and the bottom of the volcano. In 16 out of the 20 interferograms processed with images acquired between August 1992 and October 1993, correction of the tropospheric effect reduces the number of fringes associated with the 1991–93 eruption from previous estimates. The remaining deformation is consistant with a deforming source located at a depth of 14±1 km. During the second half of the eruption, the subsidence rate at the top of the volcano is roughly stable at 13±3 mm/month. These values are in good agreement with tiltmeter data collected on Etna during the same period and with the estimated volume of erupted material. No significant deformation can be observed during the last month of eruption. Inflation of the volcano seems to resume immediately after the end of the eruption at a rate of 3 mm/month.