forward and inverse modelling of post seismic deformation we consider a new approach to both the forward and inverse problems in post seismic deformation we present a method for forward modelling post seismic deformation in a self gravitating heterogeneous and compressible earth with a variety of linear and nonlinear rheologies we further demonstrate how the adjoint method can be applied to the inverse problem both to invert for rheological structure and to calculate the sensitivity of a given surface measurement to changes in rheology or time dependence of the source both the forward and inverse aspects are illustrated with several numerical examples implemented in a spherically symmetric earth model

generalized maxwell bodies and estimates of mantle viscosity summary three generalized maxwell bodies gmbs are discussed whose transient creep properties make them interesting alternatives to the classical maxwell body cmb in interpretations of post glacial uplift the first gmb is defined by its relaxation spectrum the second by its quality factor and the third by its stress strain relation in each case the gmb is characterized by three parameters only the unrelaxed modulus the steady state viscosity or a time constant and a dimensionless parameter representative of the transient creep properties this is one parameter less than needed for the burgers body which has mainly been used so far to model transient creep the gmb models are described in some detail and numerical results for their viscoelastic moduli relaxation spectra and relaxation and creep functions are given for practical applications it is most important that the viscoelastic moduli can be calculated at arbitrary frequencies in the second part of the paper post glacial uplift data from canada and fenno scandia are interpreted with the gmb rheologies described otherwise the earth model used and the assumptions on the form and history of the ice load are very simple effective viscosities of the successful models i e inverse creep rates at times representative of mantle convection fall in the range 1022 1024p 1021 1023pas and are in part considerably larger than mantle viscosities derived under the assumption of a cmb rheology hence post glacial uplift data constrain mantle viscosity less than has been believed until recently

a complete formalism for the inversion of post glacial rebound data resolving power analysis summary a formal inverse theory for mantle viscosity based upon the data of glacial isostatic adjustment is here formulated and applied to synthetic data in this theory full account has been taken of the normal mode nature of the forward problem for realistic viscoelastic maxwell models of the planet since it is impossible to accurately infer the excitation and decay constants of the individual normal modes from the observations the formalism is cast in terms of the observed gross earth data in the time domain in this analysis expressions are required for the first order perturbations in both the modal amplitudes and relaxation times that are induced by an arbitrary radial perturbation to the starting viscosity profile a numerical technique is developed which enables us to accurately determine differential kernels for the modal amplitudes the analogous kernels for the modal decay times are derived analytically peltier 1976 and the complete set of kernels is shown to satisfy the physical constraint imposed by the uniqueness of the state of isostatic equilibrium for the viscously incompressible maxwell models that we employ when the problem is parametrized in terms of the logarithm of viscosity the kernels are capable of accurately predicting shifts in the normal mode characteristics for at least an order of magnitude variation in mantle viscosity using bayesian statistics a formal inversion is applied to a set of synthetically generated data these data chosen to reproduce the space time coverage of the actual observables include a subset related to the global gravity field and a large sequence of idealized relative sea level rsl curves it is found that even very weak a priori constraints can provide a stable and accurate inversion a resolving power analysis indicates a spatial resolution of approximately 1200km near the core mantle boundary cmb with a gradual improvement to better than 350km in the middle of the upper mantle subsets of the synthetic data are inverted in order to examine conditions on stability and accuracy and to determine their relative contributions to the spatial resolution data from progressively older beaches are shown to contribute most to the spatial resolution at all depths though the improvement in lower mantle resolution converges for data obtained from beaches formed within the last 5000 yr furthermore the rsl curves in the vicinity of the peripheral bulge of the ancient laurentide ice sheet are significant contributors to lower mantle resolution as demonstrated in previous analyses of the forward problem the inversion of this subset of the data also appears to be encouragingly stable

on the resolution of shallow mantle viscosity structure using postearthquake relaxation data application to the 1999 hector mine california earthquake 1 most models of lower crust mantle viscosity inferred from postearthquake relaxation assume one or two uniform viscosity layers a few existing models possess apparently significant radially variable viscosity structure in the shallow mantle e g the upper 200 km but the resolution of such variations is not clear we use a geophysical inverse procedure to address the resolving power of inferred shallow mantle viscosity structure using postearthquake relaxation data we apply this methodology to 9 years of gps constrained crustal motions after the 16 october 1999 m 7 1 hector mine earthquake after application of a differencing method to isolate the postearthquake signal from the background crustal velocity field we find that surface velocities diminish from 20 mm yr in the first few months to 2 mm yr after 2 years viscoelastic relaxation of the mantle with a time dependent effective viscosity prescribed by a burgers body provides a good explanation for the postseismic crustal deformation capturing both the spatial and temporal pattern in the context of the burgers body model which involves a transient viscosity and steady state viscosity a resolution analysis based on the singular value decomposition reveals that at most two constraints on depth dependent steady state mantle viscosity are provided by the present data set uppermost mantle viscosity depth 60 km is moderately resolved but deeper viscosity structure is poorly resolved the simplest model that explains the data better than that of uniform steady state mantle viscosity involves a linear gradient in logarithmic viscosity with depth with a small increase from the moho to 220 km depth however the viscosity increase is not statistically significant this suggests that the depth dependent steady state viscosity is not resolvably different from uniformity in the uppermost mantle

global postseismic deformation deep earthquakes we study the global viscoelastic deformations associated with a shear dislocation on a fault embedded in a viscoelastic mantle to address this problem we extend the theory of the quasi static deformations of a maxwell spherical n layer incompressible earth previously limited to the modeling of the effects of earthquakes occurring only within the elastic lithosphere permanent postseismic deformations which can be generated by lithospheric sources cannot be sustained if the source region is viscoelastic however the transient response of the mantle strongly depends on the viscosity of the source region we use the technique developed here to investigate thoroughly the quasi static surface deformations induced by seismic events of variable depth we show that owing to the combined effect of sphericity and viscoelastic mantle relaxation the surface displacements do not systematically decrease as the depth of the source increases instead with increasing source depth we predict an increasing efficiency of mantle relaxation in triggering postseismic deformations of large size most of our results are based on a simple four layer model with a 100 km thick lithosphere upper and lower mantle separated by the 670 km discontinuity and a fluid core the last part of this work is devoted to a study of the effects of a low viscosity asthenosphere on the rates of deformation detected at the earth s surface the findings reported here may be useful for the interpretation of the transient motions of the earth s surface in response to deep focus earthquakes

convection in a variable viscosity fluid newtonian versus power law rheology abstract a number of finite element calculations of convection in a variable viscosity fluid have been carried out to investigate the effects of non newtonian flow when rheology is also subject to a strong temperature and pressure influence a variety of cases has been studied in the range of effective rayleigh numbers between 104 and 106 including different modes of heating and a range of values for activation energy and activation volume power law creep with a stress exponent of 3 turns out to lead to considerably different flow pattern and heat transfer properties than newtonian rheology in general the effect is to reduce viscosity contrasts imposed by p t dependence which can lead in some circumstances to the mobilisation of otherwise stagnant regions within the cell the properties of non newtonian flow can be closely imitated by a newtonian fluid with a reduced value of the activation enthalpy bh with b 0 3 0 5 it appears possible that non newtonian rheology plays a key role in determining the convective style in a planetary mantle

global post seismic deformation summary we quantify the effects of post seismic deformation on the radial and horizontal components of the displacement in the near and far field of strike and dip slip point dislocations these sources are embedded in the elastic top layer of a spherical self gravitating stratified viscoelastic earth within the scheme of the normal mode technique we derive the explicit analytical expression of the fundamental matrix for the toroidal component of the field equations this component is propagated together with its spheroidal counterpart from the core mantle boundary to the earth s surface viscosity stratification at 670km depth influences the radial and horizontal deformation accompanying viscoelastic relaxation in the mantle over time scales of 103 104 yr both in the near field ranging from 100 to 500 km and in the far field from 103 to 5 x 103 km if the upper mantle is differentiated into a low viscosity zone beneath the lithosphere and a normal upper mantle faster relaxation is obtained for an asthenospheric viscosity of 1020 pa s we obtain for a strike slip dislocation and a seismic moment of 1022 n m characteristic of an average large earthquake horizontal rates of 1 4 mm yr 1 in the near field and 0 05 0 4 mm yr 1 in the far field these values are maintained over time scales of 10 103 yr larger rates with shorter duration are obtained if the viscosity is reduced in the low viscosity channel as expected strike slip dislocations are the most effective in driving horizontal deformation in the far field in comparison with dip slip ones it is noteworthy that horizontal velocities are maintained longer in the far field in comparison with radial ones which is not surprising since momentum is propagated in far regions essentially in the horizontal direction radial deformation is generally lower in the far field vlbi techniques with a precision of a few parts per billion over distances of 103 km can detect global post seismic deformation induced by large earthquakes our results affect the interpretation of the transfer of stress and seismic activity among different plate boundaries

mantle circulation models with variational data assimilation inferring past mantle flow and structure from plate motion histories and seismic tomography summary mantle convection models require an initial condition some time in the past because this initial condition is unknown for earth we cannot infer the geological evolution of mantle flow from forward mantle convection calculations even for the most recent mesozoic and cenozoic geological history of our planet here we introduce a fluid dynamic inverse problem to constrain unknown mantle flow back in time from seismic tomographic observations of the mantle and reconstructions of past plate motions using variational data assimilation we derive the generalized inverse of mantle convection and explore the initial condition problem in high resolution 3 d spherical mantle circulation models for a time period of 100 myr roughly comparable to half a mantle overturn we present a synthetic modelling experiment to demonstrate that mid cretaceous mantle structure can be inferred accurately from fluid dynamic inverse modelling assuming present day mantle structure is well known even if an initial first guess assumption about the mid cretaceous mantle involved only a simple 1 d radial temperature profile we also demonstrate that convecting present day mantle structure back in time by reversing the time stepping of the energy equation is insufficient to model the mantle structure of the past the difficulty arises because such backward convection calculations ignore thermal diffusion effects and therefore cannot account for the generation of thermal buoyancy in boundary layers as we go back in time inverse mantle convection modelling should make it possible to infer a number of flow parameters from observational constraints of the mantle

glacial isostatic adjustment on 3 d earth models a finite volume formulation summary we describe and present results from a finite volume fv parallel computer code for forward modelling the maxwell viscoelastic response of a 3 d self gravitating elastically compressible earth to an arbitrary surface load we implement a conservative control volume discretization of the governing equations using a tetrahedral grid in cartesian geometry and a low order linear interpolation the basic starting grid honours all major radial discontinuities in the preliminary reference earth model prem and the models are permitted arbitrary spatial variations in viscosity and elastic parameters these variations may be either continuous or discontinuous at a set of grid nodes forming a 3 d surface within the regional or global modelling domain in the second part of the paper we adopt the fv methodology and a spherically symmetric earth model to generate a suite of predictions sampling a broad class of glacial isostatic adjustment gia data types 3 d crustal motions long wavelength gravity anomalies these calculations based on either a simple disc load history or a global late pleistocene ice load reconstruction ice 3g are benchmarked against predictions generated using the traditional normal mode approach to gia the detailed comparison provides a guide for future analyses e g what grid resolution is required to obtain a specific accuracy and it indicates that discrepancies in predictions of 3 d crustal velocities less than 0 1 mm yr 1 are generally obtainable for global grids with 3 106 nodes however grids of higher resolution are required to predict large amplitude 1 cm yr 1 radial velocities in zones of peak post glacial uplift e g james bay to the same level of absolute accuracy we conclude the paper with a first application of the new formulation to a 3 d problem specifically we consider the impact of mantle viscosity heterogeneity on predictions of present day 3 d crustal motions in north america in these tests the lateral viscosity variation is constructed with suitable scaling from tomographic images of seismic s wave heterogeneity and it is characterized by approximately 2 orders of magnitude peak to peak lateral variations within the lower mantle and 1 order of magnitude variations in the bulk of the upper mantle below the asthenosphere we find that the introduction of 3 d viscosity structure has a profound impact on horizontal velocities indeed the magnitude of the perturbation of order 1 mm yr 1 is as large as the prediction generated from the underlying 1 d radial reference model and it far exceeds observational uncertainties currently being obtained from space geodetic surveying the relative impact of lateral viscosity variations on predicted radial motions is significantly smaller

mantle flow beneath a continental strike slip fault postseismic deformation after the 1999 hector mine earthquake two recent large earthquakes in the mojave desert california the magnitude 7 3 1992 landers and magnitude 7 1 1999 hector mine earthquakes have each been followed by elevated crustal strain rates over periods of months and years geodetic data collected after the hector mine earthquake exhibit a temporally decaying horizontal velocity field and a quadrant uplift pattern opposite to that expected for localized shear beneath the earthquake rupture we interpret the origin of this accelerated crustal deformation to be vigorous flow in the upper mantle in response to the stress changes generated by the earthquake our results suggest that transient flow in the upper mantle is a fundamental component of the earthquake cycle and that the lower crust is a coherent stress guide coupling the upper crust with the upper mantle

postearthquake relaxation evidence for laterally variable viscoelastic structure and water content in the southern california mantle i reexamine the lower crust and mantle relaxation following two large events in the mojave desert the 1992 m7 3 landers and 1999 m7 1 hector mine california earthquakes time series from continuous gps sites out to 300 km from the ruptures are used to constrain models of postseismic relaxation crustal motions in the mojave desert region are elevated above background for several years following each event to account for broadscale relaxation of the lower crust and mantle the burgers body model is employed involving transient and steady state viscosities joint afterslip postseismic relaxation modeling of the gps time series up to one decade following the hector mine earthquake reveals a significant rheological contrast between a northwest trending southwest domain that envelopes the san andreas fault system and western mojave desert and an adjacent northeast domain that envelopes the landers and hector mine rupture areas in the central mojave desert the steady state viscosity of the northeast domain mantle asthenosphere is inferred to be 4 times greater than that of the southwest domain this pattern is counter to that expected for regional heat flow which is higher in the northeast domain but it is explicable by means of a nonlinear rheology that includes dependence on both strain rate and water concentration i infer that the southwest domain mantle has a relatively low steady state viscosity because of its high strain rate and water content the relatively low mantle water content of the northeast domain is interpreted to result from the continual extraction of water through igneous and volcanic activity over the past 20 myr

on post glacial sea level i general theory summary modern analyses of sea level changes due to glacial isostatic adjustment gia are based on the classic sea level equation derived by farrell clark 1976 geophys j r astr soc 46 647 667 the connection between global sea level variations and changes to ocean height that is assumed within this equation breaks down in the presence of a time varying shoreline geometry we present a generalized sea level equation that overcomes this difficulty we also derive analytic expressions for and present schematic illustrations of the error in the ocean height change over finite time intervals introduced in published efforts to incorporate shoreline evolution into the theory of gia induced sea level change this comparison includes studies of shoreline migration due to either local sea level changes or the growth and ablation of marine based ice we conclude that the theories applied by johnston 1993 geophys j int 114 615 634 milne 1998 phd thesis university of toronto toronto and co workers are more accurate than the procedure advocated by peltier 1994 science 265 195 201 1998a geophys res lett 25 3955 3958 1998b rev geophys 114 615 634 although an improvement in the latter has recently been reported peltier drummond 2002 geophys res lett 29 10 1029 2001gl014273 our generalized theory is valid for any earth model in a companion paper we derive the equations necessary to treat the special case of a spherically symmetric linear viscoelastic and rotating earth and we quantify errors associated with previous work

source conveyance and fate of suspended sediments following hurricane irene new england usa hurricane irene passed directly over the connecticut river valley in late august 2011 intense precipitation and high antecedent soil moisture resulted in record flooding mass wasting and fluvial erosion allowing for observations of how these rare but significant extreme events affect a landscape still responding to pleistocene glaciation and associated sediment emplacement clays and silts from upland glacial deposits once suspended in the stream network were routed directly to the mouth of the connecticut river resulting in record breaking sediment loads fifteen times greater than predicted from the pre existing rating curve denudation was particularly extensive in mountainous areas we calculate that sediment yield during the event from the deerfield river a steep tributary comprising 5 of the entire connecticut river watershed exceeded at minimum 10 40 years of routine sediment discharge and accounted for approximately 40 of the total event sediment discharge from the connecticut river a seriesof surface sediment corestaken in floodplainponds adjacent to the tidalsec

gravitational viscoelastic postseismic relaxation on a layered spherical earth viscoelastic relaxation of a ductile asthenosphere underlying a purely elastic plate is a strong candidate process for explaining anomalous rates of crustal deformation observed following large earthquakes the nongravitational treatment of pollitz 1992 which is valid on a global scale and includes the effects of compressibility is here extended to permit the calculation of gravitational viscoelastic relaxation for a specified spherically layered viscoelastic rheology following an earthquake in an elastic layer the simple approximations we adopt make the resulting treatment particularly suitable for near field calculations for an asthenosphere with a maxwell rheology the effect of gravitational coupling is manifested only many relaxation times after the earthquake as obtained by previous investigators its effect is generally to speed up the long wavelength component of the relaxation process and attenuate the overall vertical displacement pattern several subtle features common to the relaxation behavior from several different fault types thrust rift and strike slip are identified the effect of gravitational coupling on horizontal displacements is consistent with flexure of the upper elastic plate driven by the corresponding effects on the vertical displacement stress diffusion away from the source region generally exhibits pulse like behavior which is dispersive in both space and time if the asthenosphere is confined to a relatively narrow channel then the dispersion branches governing relaxation are radically altered and stress diffusion effects far from the coseismic rupture zone exhibit a complicated time dependence reflecting the competing tendencies of toroidal and spheroidal mode relaxation

transient rheology of the uppermost mantle beneath the mojave desert california abstract geodetic data indicate that the m7 1 hector mine california earthquake was followed by a brief period a few weeks of rapid deformation preceding a prolonged phase of slower deformation we find that the signal contained in continuous and campaign global positioning system data for 2 5 years after the earthquake may be explained with a transient rheology quantitative modeling of these data with allowance for transient linear biviscous rheology in the lower crust and upper mantle demonstrates that transient rheology in the upper mantle is dominant its material properties being typified by two characteristic relaxation times 0 07 and 2 years the inferred mantle rheology is a jeffreys solid in which the transient and steady state shear moduli are equal consideration of a simpler viscoelastic model with a linear univiscous rheology 2 fewer parameters than a biviscous model shows that it consistently underpredicts the amplitude of the first 3 months signal and allowance for a biviscous rheology is significant at the 99 0 confidence level another alternative model deep postseismic afterslip beneath the coseismic rupture predicts a vertical velocity pattern opposite to the observed pattern at all time periods considered despite its plausibility the advocated biviscous rheology model is non unique and should be regarded as a viable alternative to the non linear mantle rheology model for governing postseismic flow beneath the mojave desert

post seismic relaxation following the great 2004 sumatra andaman earthquake on a compressible self gravitating earth summary the mw 9 0 2004 december 26 sumatra andaman and mw 8 7 2005 march 28 nias earthquakes which collectively ruptured approximately 1800 km of the andaman and sunda subduction zones are expected to be followed by vigorous viscoelastic relaxation involving both the upper and lower mantle because of these large spatial dimensions it is desirable to fully account for gravitational coupling effects in the relaxation process we present a stable method of computing relaxation of a spherically stratified compressible and self gravitating viscoelastic earth following an impulsive moment release event the solution is cast in terms of a spherical harmonic expansion of viscoelastic normal modes for simple layered viscoelastic models which include a low viscosity oceanic asthenosphere we predict substantial post seismic effects over a region several 100s of km wide surrounding the eastern indian ocean we compare observed gps time series from ten regional sites mostly in thailand and indonesia beginning in 2004 december with synthetic time series that include the coseismic and post seismic effects of the 2004 december 26 and 2005 march 28 earthquakes a viscosity structure involving a biviscous burgers body rheology in the asthenosphere explains the pattern and amplitude of post seismic offsets remarkably well

the viscoelastic relaxation of a realistically stratified earth and a further analysis of postglacial rebound summary we discuss the isostatic adjustment of the earth in response to pleistocene deglaciation and derive constraints on the earth s viscosity profile we model the earth as viscoelastic self gravitating realistically stratified and spherically symmetric it is shown that the earth s relaxation eigenspectrum besides possessing a small set of well known discrete modes also includes infinitely dense sets of modes a continuous spectrum which separate into classes that depend on the principal restoring force of the mode methods for estimating the continuous spectrum are discussed and the effects on the predicted uplift are assessed we find that the effects of the continuous spectrum are far less important than those of the discrete modes though the presence of the continuous spectrum can make the discrete modes more difficult to identify the continuous spectrum does become important when computing the rebound due to more recent loading such as might be associated with present day changes in polar ice we use our model to refine predictions of the earth s response to pleistocene glacial loading with particular emphasis on implications for mantle viscosity we have used time histories that are consistent with the recently recalibrated 14c time scale a special effort is made to match the geologic records of sea level at sites in the vicinity of hudson bay we find consistent with the results of other authors that the curvatures of the records are sensitive indicators of lower mantle viscosity but that there are inconsistencies between the records at different sites however we find we are also able to change the predicted curvatures by modifying the temporal and spatial distribution of the ice loads if we assume a factor of 50 increase in viscosity between the upper and lower mantles and if we assume that the ice on the eastern side of hudson bay melted a few thousand years later than the ice on the western side a result consistent with recent geologic evidence we are able to match the observed sea level data about as well as we are able to if we use instead a uniform mantle viscosity profile a model with a factor of 50 jump in viscosity predicts a large free air gravity anomaly over northern canada that is reasonably consistent with what is observed whereas a uniform viscosity model does not though some authors have argued that the observed north american gravity anomaly may be mostly a consequence of mantle convection and largely unrelated to postglacial rebound our objective here is not to present a preferred ice model or viscosity profile rather we conclude only that the interpretation of the data is ambiguous and that the mantle viscosity profile can not yet be uniquely inferred from them

evidence of power law flow in the mojave desert mantle studies of the earth s response to large earthquakes can be viewed as large rock deformation experiments in which sudden stress changes induce viscous flow in the lower crust and upper mantle that lead to observable postseismic surface deformation laboratory experiments suggest that viscous flow of deforming hot lithospheric rocks is characterized by a power law in which strain rate is proportional to stress raised to a power n refs 2 3 most geodynamic models of flow in the lower crust and upper mantle however resort to newtonian linear stress strain rate relations here we show that a power law model of viscous flow in the mantle with n 3 5 successfully explains the spatial and temporal evolution of transient surface deformation following the 1992 landers and 1999 hector mine earthquakes in southern california a power law rheology implies that viscosity varies spatially with stress causing localization of strain and varies temporally as stress evolves rendering newtonian models untenable our findings are consistent with laboratory derived flow law parameters for hot and wet olivine the most abundant mineral in the upper mantle and support the contention that at least beneath the mojave desert the upper mantle is weaker than the lower crust

viscous gravitational relaxation summary this paper is concerned with a detailed examination of the response of maxwell models of the planet to surface mass loads particular attention is devoted to an examination of the factors which determine the isostatic response since the understanding of this response is crucial in a number of different geodynamic problems one particular example which we discuss in detail is concerned with the prediction of free air gravity anomalies produced by large scale deglaciation events using the methods developed here we are able to provide the first direct assessment of the importance of initial isostatic disequilibrium on the observed relative sea level variations and free air gravity anomalies forced by the melting of the laurentide ice sheet we are therefore able to estimate the extent to which such initial disequilibrium might influence the inference of mantle viscosity from isostatic adjustment data our calculations establish that free air gravity data although they are sensitive to the degree of initial disequilibrium provide an extremely high quality constraint upon the viscosity of the lower mantle

determining fluid migration and isolation times in multiphase crustal domains using noble gases geochemical characteristics in subsurface fluid systems provide a wealth of information about fluid sources migration and storage conditions determining the extent of fluid interaction aquifer hydrocarbon connectivity is important for oil and gas production and waste storage applications but is not tractable using traditional seismic methods furthermore the residence time of fluids is critical in such systems and can vary from tens of thousands to billions of years our understanding of the transport length scales in multiphase systems while equally important is more limited noble gas data from the rotliegend natural gas field northern germany are used here to determine the length scale and isolation age of the combined water gas system we show that geologically bound volume estimates i e gas to water volume ratios match closed system noble gas model predictions suggesting that the rotliegend system has remained isolated as a closed system since hydrocarbon formation radiogenic helium data show that fluid isolation occurred 63 129 m y after rock and or groundwater deposition ca 300 ma which is consistent with known hydrocarbon generation from 250 to 140 ma thus corroborating long term geologic isolation it is critical that we have the ability to distinguish between fluid systems that despite phase separation have remained closed to fluid loss from those that have lost oil or gas phases these findings are the first to demonstrate that such systems remain isolated and fully gas retentive on time scales 100 m y over 10 km length scales and have broad implications for saline aquifer co2 disposal site viability and hydrocarbon resource prediction which both require an understanding of the length and time scales of crustal fluid transport pathways