simultaneous multifrequency inversion of full waveform seismic data we present a simultaneous multifrequency inversion approach for seismic data interpretation this algorithm inverts all frequency data components simultaneously a data weighting scheme balances the contributions from different frequency data components so the inversion process does not become dominated by high frequency data components which produce a velocity image with many artifacts a gauss newton minimization approach achieves a high convergence rate and an accurate reconstructed velocity image by introducing a modified adjoint formulation we can calculate the jacobian matrix efficiently allowing the material properties in the perfectly matched layers pmls to be updated automatically during the inversion process this feature ensures the correct behavior of the inversion and implies that the algorithm is appropriate for realistic applications where a priori information of the background medium is unavailable two different regularization schemes an l2 norm and a weighted l2 norm function a

stochastic inversion for electromagnetic geophysics practical challenges and improving convergence efficiency traditional deterministic geophysical inversion algorithms are not designed to provide a robust evaluation of uncertainty that reflects the limitations of the geophysical technique stochastic inversions which do provide a sampling based measure of uncertainty are computationally expensive and not straightforward to implement for nonexperts nonstatisticians our results include stochastic inversion for magnetotelluric and controlled source electromagnetic data two markov chain sampling algorithms metropolis hastings and slice sampler can significantly decrease the computational expense compared to using either sampler alone the statistics of the stochastic inversion allow for 1 variances that better reveal the measurement sensitivities of the two different electromagnetic techniques than traditional techniques and 2 models defined by the median and modes of parameter probability density functions which produce amplitude and phase data that are consistent with the observed data in general pa

stabilization effect of magnetic fields on two dimensional compressible current vortex sheets we analyze the linear stability of rectilinear compressible current vortex sheets in two dimensional isentropic magnetohydrodynamics which is a free boundary problem with the boundary being characteristic in the case when the magnitude of the magnetic field has no jump on the current vortex sheets we find a necessary and sufficient condition of linear stability for the rectilinear current vortex sheets showing that magnetic fields exert a stabilization effect on compressible vortex sheets in addition a loss of regularity with respect to the source terms both in the interior domain and on the boundary occurs in a priori estimates of solutions to the linearized problem for a rectilinear current vortex sheet as the kreiss lopatinskii determinant associated with this linearized boundary value problem has roots on the boundary of frequency spaces in this study the construction of symmetrizers for a reduced differential system which has poles at which the kreiss lopatinskii condition may fail simultaneously plays a crucial role in the a priori estimates

three dimensional earth conductivity inversion abstract we have constructed two inverse algorithms using tikhonov regularization one follows the well known method of solving the linearized data model relation subject to some regularizing operator which forces the solution to satisfy some desirable features we refer to the other more novel approach as the inverse scattering currents method this procedure uses the linear relation between data and scattering currents induced in an inhomogeneous structure and the relation between the scattering currents and the unknown conductivity the inverse problem is solved iteratively using these two relations

the effect of local distortions on time domain electromagnetic measurements based on the time domain integral equation we derive expressions for the effect of an anomalous body close to the receiver or close to the transmitter on transient electromagnetic measurements similar to magnetotellurics the distortion of electric fields at late times can be described by a constant distortion tensor relating the secondary electric field to the primary field components that would be obtained in the absence of the body the distortion of a single electric field transient is a static shift only for particular configurations over a layered half space in the general case the perturbation is time dependent because the direction of the total electric field vector varies with time the theory nicely explains spatial variations in electric field transients measured during a high redundancy long offset transient electromagnetics lotem survey over an underground gas storage site an inversion example with synthetic data illustrates how distortion can be corrected the elements of the distortion tensor are determined simultaneously with the model parameters ambiguity is reduced by a regularization of the distortion parameters in the example the background model is recovered well even for the difficult case where only one transmitter is used the distortion of the magnetic field time derivatives caused by bodies close to the receiver is proportional to the time derivative of the primary electric step response the distortion is generally not limited to early times and cannot be neglected in general transmitter overprint effects resulting in static shifts of vertical magnetic field time derivatives may also be understood from the theory

imaging low frequency and dc electromagnetic fields using a simple linear approximation we consider that all types of electromagnetic measurements represent weighted averages of the subsurface electrical conductivity distribution and that to each type of measurement there corresponds a different weighting function we use this concept for the quantitative interpretation of dc resistivity magnetometric resistivity and low frequency electric and magnetic measurements at low induction numbers in all three cases the corresponding inverse problems are nonlinear because the weighting functions depend on the unknown conductivity distribution we use linear approximations that adapt to the data and do not require reference resistivity values the problem is formulated numerically as a solution of a system of linear equations the unknown conductivity values are obtained by minimizing an objective function that includes the quadratic norm of the residuals as well as the spatial derivatives of the unknowns we also apply constraints through the use of quadratic programming the final product is the flattest model that is compatible with the data under the assumption of the given weighting functions this approximate inversion or imaging technique produces reasonably good results for low and moderate conductivity contrasts we present the results of inverting jointly and individually different data sets using synthetic and field data

mean field critical behaviour for percolation in high dimensions the triangle condition for percolation states that sum limits x y tau 0 x tau 0 y cdot tau y 0 is finite at the critical point where x y is the probability that the sitesx andy are connected we use an expansion related to the lace expansion for a self avoiding walk to prove that the triangle condition is satisfied in two situations i for nearest neighbour independent bond percolation on thed dimensional hypercubic lattice ifd is sufficiently large and ii in more than six dimensions for a class of spread out models of independent bond percolation which are believed to be in the same universality class as the nearest neighbour model the class of models in ii includes the case where the bond occupation probability is constant for bonds of length less than some large number and is zero otherwise in the course of the proof an infrared bound is obtained the triangle condition is known to imply that various critical exponents take their mean field bethe lattice values gamma beta 1 delta delta t 2 t underset raise0 3em hbox smash scriptscriptstyle geqslant 2 and that the percolation density is continuous at the critical point we also prove thatv2 in i and ii wherev2 is the critical exponent for the correlation length

resinvm3d a 3d resistivity inversion package we have developed an open source 3d matlab based resistivity inversion package the forward solution to the governing partial differential equation is efficiently computed using a second order finite volume discretization coupled with a preconditioned biconjugate stabilized gradient algorithm using the analytical solution to a potential field in a homogeneous half space we evaluate the accuracy of our numerical forward solution and subsequently develop a source correction factor that reduces forward modeling errors associated with boundary effects and source electrode singularities for the inversion algorithm we have implemented an inexact gauss newton solver with the model update being calculated using a preconditioned conjugate gradient algorithm the inversion uses a combination of zero and first order tikhonov regularization two synthetic examples demonstrate the usefulness of this code the first example considers a surface resistivity survey with 3813 measurements the discretized model s

the existence of current vortex sheets in ideal compressible magnetohydrodynamics we prove the local in time existence of solutions with a surface of current vortex sheet tangential discontinuity of the equations of ideal compressible magnetohydrodynamics in three space dimensions provided that a stability condition is satisfied at each point of the initial discontinuity this paper is a natural completion of our previous analysis trakhinin in arch ration mech anal 177 331 366 2005 where a sufficient condition for the weak stability of planar current vortex sheets was found and a basic a priori estimate was proved for the linearized variable coefficients problem for nonplanar discontinuities the original nonlinear problem is a free boundary hyperbolic problem since the free boundary is characteristic the functional setting is provided by the anisotropic weighted sobolev spaces h m the fact that the kreiss lopatinski condition is satisfied only in a weak sense yields losses of derivatives in a priori estimates therefore we prove our existence theorem by a suitable nash moser type iteration scheme

a priori estimates for 3d incompressible current vortex sheets we consider the free boundary problem for current vortex sheets in ideal incompressible magneto hydrodynamics it is known that current vortex sheets may be at most weakly neutrally stable due to the existence of surface waves solutions to the linearized equations the existence of such waves may yield a loss of derivatives in the energy estimate of the solution with respect to the source terms however under a suitable stability condition satisfied at each point of the initial discontinuity and a flatness condition on the initial front we prove an a priori estimate in sobolev spaces for smooth solutions with no loss of derivatives the result of this paper gives some hope for proving the local existence of smooth current vortex sheets without resorting to a nash moser iteration such result would be a rigorous confirmation of the stabilizing effect of the magnetic field on kelvin helmholtz instabilities which is well known in astrophysics

generalized covariance functions associated with the laplace equation and their use in interpolation and inverse problems this work examines which generalized covariance function when used in the stochastic approach produces the flattest possible estimate of an unknown function that is consistent with the data such an estimate is the plainest possible continuous function thus in a sense eliminating details that are irrelevant or unsupported by data the answer is found from the solution of the following variational problem determine the function that reproduces the data has the smallest gradient in the square norm sense and has a gradient that vanishes at large distances from the observations the generalized covariance functions are shown to be the green s functions for the free space laplace equation the linear distance in one dimension the logarithmic distance in two dimensions and the inverse distance in three dimensions it is demonstrated that they are appropriate covariance functions for intrinsic random fields a modification is proposed to facilitate numerical implementation and a couple of examples are presented to illustrate the applicability of the methodology

joint analysis of refractions with surface waves an inverse solution to the refraction traveltime problem we describe a possible solution to the inverse refraction traveltime problem irtp that reduces the range of possible solutions nonuniqueness this approach uses a reference model derived from surface wave shear wave velocity estimates as a constraint the application of the joint analysis of refractions with surface waves jars method provided a more realistic solution than the conventional refraction tomography methods which did not benefit from a reference model derived from real data this confirmed our conclusion that the proposed method is an advancement in the irtp analysis the unique basic principles of the jars method might be applicable to other inverse geophysical problems

applying compactness constraints to differential traveltime tomography tomographic imaging problems are typically ill posed and often require the use of regularization techniques to guarantee a stable solution minimization of a weighted norm of model length is one commonly used secondary constraint tikhonov methods exploit low order differential operators to select for solutions that are small flat or smooth in one or more dimensions this class of regularizing functionals may not always be appropriate particularly in cases where the anomaly being imaged is generated by a nonsmooth spatial process time lapse imaging of flow induced velocity anomalies is one such case flow features are often characterized by spatial compactness or connectivity by performing inversions on differenced arrival time data the properties of the time lapse feature can be directly constrained we develop a differential traveltime tomography algorithm whichselects for compact solutions i e models with a minimum area of support through application of model space iteratively reweighted leas

existence and stability of compressible current vortex sheets in three dimensional magnetohydrodynamics compressible vortex sheets are fundamental waves along with shocks and rarefaction waves in entropy solutions to multidimensional hyperbolic systems of conservation laws understanding the behavior of compressible vortex sheets is an important step towards our full understanding of fluid motions and the behavior of entropy solutions for the euler equations in two dimensional gas dynamics the classical linearized stability analysis on compressible vortex sheets predicts stability when the mach number m sqrt 2 and instability when m sqrt 2 for the euler equations in three dimensions every compressible vortex sheet is violently unstable and this instability is the analogue of the kelvin helmholtz instability for incompressible fluids the purpose of this paper is to understand whether compressible vortex sheets in three dimensions which are unstable in the regime of pure gas dynamics become stable under the magnetic effect in three dimensional magnetohydrodynamics mhd one of the main features is that the stability problem is equivalent to a free boundary problem whose free boundary is a characteristic surface which is more delicate than noncharacteristic free boundary problems another feature is that the linearized problem for current vortex sheets in mhd does not meet the uniform kreiss lopatinskii condition these features cause additional analytical difficulties and especially prevent a direct use of the standard picard iteration to the nonlinear problem in this paper we develop a nonlinear approach to deal with these difficulties in three dimensional mhd we first carefully formulate the linearized problem for the current vortex sheets to show rigorously that the magnetic effect makes the problem weakly stable and establish energy estimates especially high order energy estimates in terms of the nonhomogeneous terms and variable coefficients then we exploit these results to develop a suitable iteration scheme of the nash moser hormander type to deal with the loss of the order of derivative in the nonlinear level and establish its convergence which leads to the existence and stability of compressible current vortex sheets locally in time in three dimensional mhd

well posedness for the motion of an incompressible liquid with free surface boundary we study the motion of an incompressible perfect liquid body in vacuum this can be thought of as a model for the motion of the ocean or a star the free surface moves with the velocity of the liquid and the pressure vanishes on the free surface this leads to a free boundary problem for euler s equations where the regularity of the boundary enters to highest order we prove local existence in sobolev spaces assuming a physical condition related to the fact that the pressure of a fluid has to be positive

intersections of random walks in four dimensions ii letf n be the probability that the paths of two simple random walks of lengthn starting at the origin in 4 have no intersection it has previously been shown thatf n c logn 1 2 here it is proved that for allr 1 2 mathop lim limits n to infty log n r f n infty

self avoiding walk in 5 or more dimensions using an expansion based on the renormalization group philosophy we prove that for a t step weakly self avoiding random walk in five or more dimensions the variance of the endpoint is of order t and the scaling limit is gaussian as t

reconstruction of 1 d conductivity from dual loop em data inversions of electromagnetic data from different coil configurations provide independent information about geological structures we develop a 1 d inversion algorithm that can invert data from the horizontal coplanar hc vertical coplanar coaxial ca and perpendicular coil configurations separately or jointly the inverse problem is solved by minimizing a model objective function subject to data constraints tests using synthetic data from 1 d models indicate that if data are collected at a sufficient number of frequencies then the recovered models from individual inversions of different coil systems can be quite similar however if only a limited number of frequencies are available then joint inversion of data from different coils produces a better model than the individual inversions tests on 3 d synthetic data sets indicate that 1 d inversions can be used as a fast and approximate tool to locate anomalies in the subsurface also for the test example presented here the joint inversion of hc a

existence of compressible current vortex sheets variable coefficients linear analysis we study the initial boundary value problem resulting from the linearization of the equations of ideal compressible magnetohydrodynamics and the rankine hugoniot relations about an unsteady piecewise smooth solution this solution is supposed to be a classical solution of the system of magnetohydrodynamics on either side of a surface of tangential discontinuity current vortex sheet under some assumptions on the unperturbed flow we prove an energy a priori estimate for the linearized problem since the tangential discontinuity is characteristic the functional setting is provided by the anisotropic weighted sobolev space w21 despite the fact that the constant coefficients linearized problem does not meet the uniform kreiss lopatinskii condition the estimate we obtain is without loss of smoothness even for the variable coefficients problem and nonplanar current vortex sheets the result of this paper is a necessary step in proving the local in time existence of current vortex sheet solutions of the nonlinear equations of magnetohydrodynamics

a priori estimates for fluid interface problems we consider the regularity of an interface between two incompressible and inviscid fluid flows in the presence of surface tension we obtain local in time estimates on the interface in h 3 2 k 1 and the velocity fields in h 3 2 k these estimates are obtained using geometric considerations which show that the kelvin helmholtz instabilities are a consequence of a curvature calculation 2007 wiley periodicals inc

on the water wave equations with surface tension the purpose of this article is to clarify the cauchy theory of the water waves equations as well in terms of regularity indexes for the initial conditions as for the smoothness of the bottom of the domain namely no regularity assumption is assumed on the bottom our main result is that following the approach developped by t alazard and g metivier in 1 after suitable paralinearizations the system can be arranged into an explicit symmetric system of schrodinger type we then show that the smoothing effect for the one dimensional surface tension water waves proved by h christianson v m hur and g staffilani in 9 is in fact a rather direct consequence of this reduction which allows also to lower the regularity indexes of the initial data and to obtain the natural weights in the estimates

on the cauchy problem for gravity water waves we are interested in the system of gravity water waves equations without surface tension our purpose is to study the optimal regularity thresholds for the initial conditions in terms of sobolev embeddings the initial surfaces we consider turn out to be only of c 3 2 epsilon class for some epsilon 0 and consequently have unbounded curvature while the initial velocities are only lipschitz we reduce the system using a paradifferential approach

well posedness of the free boundary problem in incompressible elastodynamics under the mixed type stability condition abstract we consider the free boundary problem for the flow of an incompressible inviscid elastic fluid the columns of the deformation gradient are tangent and the pressure vanishes along the free interface we prove the local existence of a unique smooth solution to the free boundary problem under the mixed type stability condition provided that the rayleigh taylor sign condition is satisfied at all the points of the initial interface where the non collinearity condition for the deformation gradient among three columns of the deformation gradient there are two non collinear vectors fails in particular we solve an open problem proposed by y trakhinin in the paper 29 under the incompressible setting

intersection equivalence of brownian paths and certain branching processes we show that sample paths of brownian motion and other stable processes intersect the same sets as certain random cantor sets constructed by a branching process with this approach the classical result that two independent brownian paths in four dimensions do not intersect reduces to the dying out of a critical branching process and estimates due to lawler 1982 for the long range intersection probability of several random walk paths reduce to kolmogrov s 1938 law for the lifetime of a critical branching process extensions to random walks with long jumps and applications to hausdorff dimension are also derived

enhancing the resolving power of least squares inversion with active constraint balancing most geophysical inverse problems are solved using least squares inversion schemes with damping or smoothness constraints to improve stability and convergence rate since the lagrangian multiplier controls resolution and stability of the inverse problem we always want to use the optimum multiplier which is not easy to get and is usually obtained by experience or a time consuming optimization process we present a new regularization approach in which the lagrangian multiplier is set as a spatial variable at each parameterized block and automatically determined via the parameter resolution matrix and spread function analysis for highly resolvable parameters a small value of the lagrangian multiplier is assigned and vice versa this approach named active constraint balancing acb tries to balance the constraints of the least squares inversion according to sensitivity for a given problem so that it enhances the resolution as well as the stability of the inversion process we demonstrate the performa

automatic 1 d inversion of magnetotelluric data finding the simplest possible model that fits the data we demonstrate that automatic layered inversion of plane wave electromagnetic data can be carried out by modifying standard least squares inversion schemes the modifications include a logarithmic reparameterization of the unknown model parameters whereby all layer parameters are forced to remain within given bounds however the most important modification to help the optimization process find the best model and to avoid local minima is to split the data into several subbands starting from the highest frequencies by this stripping procedure the shallower part of the model becomes well estimated first as more data are introduced more layers may be required to improve the data fit the new inversion procedure has been applied to many sets of theoretical data representing increasingly complicated layering that is often found in near surface studies the main result of these simulations is that there is a very strong coupling between the resolution power of plane wave data and their random errors part

the probability of intersection of independent random walks in four dimensions lets1 ands2 be independent simple random walks of lengthn inz4 starting at 0 andx0 respectively if x0 2 n it is shown that the probability that the paths intersect is of order logn 1 ifx0 0 it is shown that the probability of no intersection of the paths decays no faster than logn 1 and no slower than logn 1 2 it is conjectured that logn 1 2 is the actual decay rate

stability of incompressible current vortex sheets abstract we revisit the study in y trakhinin on the existence of incompressible current vortex sheets study of a linearized free boundary value problem math methods appl sci 28 2005 917 945 where an energy a priori estimate for the linearized free boundary value problem for planar current vortex sheets in ideal incompressible magnetohydrodynamics was proved for a part of the whole stability domain found a long time ago in s i syrovatskij the stability of tangential discontinuities in a magnetohydrodynamic medium zh eksper teor fiz 24 1953 622 629 in russian w i axford note on a problem of magnetohydrodynamic stability canad j phys 40 1962 654 655 in this paper we derive an a priori estimate in the whole stability domain the crucial point in deriving this estimate is the construction of a symbolic symmetrizer for a nonstandard elliptic problem for the small perturbation of total pressure this symmetrizer is an analogue of kreiss type symmetrizers as in hyperbolic theory the failure of the uniform lopatinski condition i e the fact that current vortex sheets are only weakly neutrally stable yields loss of derivatives in the energy estimate the result of this paper is a necessary step to prove the local in time existence of stable nonplanar incompressible current vortex sheets by a suitable nash moser type iteration scheme

imaging of 3d electromagnetic data at low induction numbers we expressed electromagnetic measurements at low induction numbers as spatial averages of the subsurface electrical conductivity distribution and developed an algorithm for the recovery of the latter in terms of the former the basis of our approach is an integral equation whose averaging kernel is independent of the conductivity distribution that is the recovery of conductivity from the measurements leads to a linear inverse problem previous work in one and two dimensions demonstrated that using a kernel independent of conductivity leads to reasonably good results in quantitative interpretations this study extended the approach to 3d models and to data taken along several profiles over a given area the algorithm handles vertical and horizontal magnetic dipoles with multiple separations for appropriate depth discrimination the approximation also handles issues like negative conductivity measurements which commonly appear when crossing near surface conductors this happens particularly when using vertical magnetic dipoles whose averaging kernel has significant negative weights in the space between the dipoles something that does not happen for the horizontal dipoles in general the more complex the kernel the more complicated the signature of any given anomaly this makes qualitative interpretations of pseudosections somewhat difficult when dealing with more than one conductive or resistive body the algorithm was validated using synthetic data for imaging data from horizontal or vertical coils or from a combination of them imaging of field data from a mine tailings site recovered a shallow 3d conductive anomaly associated with the tailings

intersection properties of simple random walks a renormalization group approach we study estimates for the intersection probability g m of two simple random walks on lattices of dimensiond 4 4 e as a problem in euclidean field theory we rigorously establish a renormalization group flow equation forg m and bounds on the function which show that ind 4 g m tends to zero logarithmically as the killing rate mass m tends to zero and that the fixed point g ind 4 e is bounded by const e g conste our methods also yield estimates on the intersection probability of three random walks ind 3 3 e for e 0 these results were first obtained by lawler 1

a hybrid regularization scheme for the inversion of magnetotelluric data from natural and controlled sources to layer and distortion parameters magnetotelluric mt radiomagnetotelluric rmt and in particular controlled source audiomagnetotelluric csamt data are often heavily distorted by near surface inhomogeneities we developed a

direct estimates on intersection probabilities of random walks a study is made of intersection properties of independent random walks in d dimensional lattice space z sup d a simple method is developed which makes it possible to estimate intersection probabilities of two random walks with killing rate m directly it is expected that the method can be generalized and extended to other issues

nonlinear conjugate gradients algorithm for 2 d magnetotelluric inversion we investigate a new algorithm for computing regularized solutions of the 2 d magnetotelluric inverse problem the algorithm employs a nonlinear conjugate gradients nlcg scheme to minimize an objective function that penalizes data residuals and second spatial derivatives of resistivity we compare this algorithm theoretically and numerically to two previous algorithms for constructing such minimum structure models the gauss newton method which solves a sequence of linearized inverse problems and has been the standard approach to nonlinear inversion in geophysics and an algorithm due to mackie and madden which solves a sequence of linearized inverse problems incompletely using a linear conjugate gradients technique numerical experiments involving synthetic and field data indicate that the two algorithms based on conjugate gradients nlcg and mackie madden are more efficient than the gauss newton algorithm in terms of both computer memory requirements and cpu time needed to find accurate solutio

an automatic technique for presentation of coincident loop impulse response transient electromagnetic data coincident loop tem sounding data are often presented by plotting the half space apparent conductivity as a function of delay time a new algorithm generates an improved presentation that plots the apparent conductivity as a function of depth the resulting data may be further processed to sharpen or spike the smoothly varying apparent conductivity depth curves in an attempt to better represent the rapid changes in conductivity that often exist in the earth the algorithm described involves an approximation but is simple easy to use and computationally efficient a layered conductivity structure is assumed so the algorithm is best for areas where the geology is approximately horizontal however the algorithm can also be used to identify anomalous features that are not infinite horizontal layers the spiked conductivity models derived from synthetic data are consistent with the original layered earth models and show a greater resolution than the apparent conductivity depth curves and sometimes ampl

review on airborne electromagnetic inverse theory and applications airborne electromagnetic aem data processing and inversion has progressed from the early conductivity depth imaging to 1d inversion for a layered earth model and most recently to 2d 3d inversions the new processes have used methods for searching for solutions applying constraints and focusing images such as occam s laterally constrained and holistic inversions for airborne data imaging and 1d inversion some algorithms are fast whereas others have control over data misfit and the vertical or lateral smoothness beyond 1d algorithms 2d 3d inversions are based on currently used methods such as isolated conductor models which are good for highly conductive orebodies in resistive background and techniques that discretized regions that are used for more complex structures and backgrounds in the latter case the computations are slow so research is focusing on time efficient computer algorithms for solving the equations such as gauss newton quasi newton and nonlinear conjugate gradient algorithms for the electromagnetic em problem the solution can be obtained in a more practical time frame if material that has minimal impact is ignored two approaches can be used to speedup the em inversion process the moving footprint and direct solver methods we hope our work will to some extent help stimulate and focus the research in aem inversion

nonlinear stability of the current vortex sheet to the incompressible mhd equations in this paper we solve a long standing open problem nonlinear stability of the current vortex sheet in the ideal incompressible magnetohydrodynamics under the syrovatskij stability condition this result gives the first rigorous confirmation of the stabilizing effect of the magnetic field on kelvin helmholtz instability 2017 wiley periodicals inc

the zero surface tension limit of two dimensional water waves we consider two dimensional water waves of infinite depth periodic in the horizontal direction it has been proven by wu in the slightly different nonperiodic setting that solutions to this initial value problem exist in the absence of surface tension recently ambrose has proven that solutions exist when surface tension is taken into account in this paper we provide a shorter more elementary proof of existence of solutions to the water wave initial value problem both with and without surface tension our proof requires estimating the growth of geometric quantities using a renormalized arc length parametrization of the free surface and using physical quantities related to the tangential velocity of the free surface using this formulation we find that as surface tension goes to 0 the water wave without surface tension is the limit of the water wave with surface tension far from being a simple adaptation of previous works our method requires a very original choice of variables these variables turn out to be physical and well adapted to both cases c

multitrace impedance inversion with lateral constraints abstractwe have developed a lateral constraint to the inversion of 1d seismic impedance models to suppress the effect of data noise and improve the fidelity of formation boundaries in 2d models for situations with dips of less than 20 typical inversion frameworks rely on a 1d forward model with each 1d trace being inverted independently adjacent inversion models are combined together to form a 2d impedance model adding a lateral constraint improves the fidelity of the 2d impedance models while retaining much of the advantage of the low computational cost associated with typical 1d inversion schemes solving the 1d lateral constraint inversion 1d lci problem involves the simultaneous inversion of multiple 1d traces producing layered sections with lateral smoothed transition in addition to enforcing lateral continuity in the inversion model this algorithm allows for the inclusion of a priori knowledge from boreholes we determined the effectiveness of this algorithm on two synthetic models as well

magnetotelluric inversion for minimum structure structure can be measured in terms of a norm of the derivative of a model with respect to a function of depth f z where the model m z is either the conductivity or log an iterative linearized algorithm can find models that minimize norms of this form for chosen levels of chi squared misfit the models found may very well be global minima of these norms since they are not observed to depend on the starting model overfitting data causes extraneous structure some choices of the depth function result in systematic overfitting of high frequencies a blue fit and extraneous shallow structure others result in systematic overfitting of low frequencies a red fit and extraneous deep structure a robust statistic is used to test for whiteness the fit can be made acceptably white by varying the depth function f z which defines the norm an optimum norm produces an inversion which does not introduce false structure and which approaches the true structure in a reasonable way as data errors decrease

transforming a time domain electromagnetic signal to a frequency domain electromagnetic response using regularization inversion abstractwide applications of time domain electromagnetic tem data require 3d inversion a possible strategy is to use the developed 3d inversion algorithms in frequency domain fd electromagnetic em methods thus the key of the strategy is how to transform the time domain t d em signal into the fd an inversion algorithm has been developed to transform the t d signal into a corresponding fd response in this method a step off current is presumed under this assumption the fourier transform relating the em fd response to the t d signal becomes a sine or cosine transformation using the polynomial approximation method the transformation turns into a linear equation from a set of t d signals fd responses could be obtained by solving these linear equations in the least squares sense to reduce the nonuniqueness of the solution and enhance the solution stability an additional smoothness constraint on the fd response is imposed thus converting the minimization problem into a regularization inver

straightforward inversion of vertical electrical sounding data a straightforward inversion scheme sis has been developed to interpret vertical electrical sounding data this scheme does not require quasi linearization of the inverse resistivity problem and thereby dispenses with the iterative process and the necessity of guessing the number of layers and their resistivities and thicknesses the entire solution domain is divided into uniform thickness layers whose scale must be judiciously selected for the desired resolution the apparent resistivity formula can now be posed as an underdetermined matrix equation whose minimum norm solution is downward continued to obtain the reflection coefficients which in turn yield the vertical resistivity distribution a recurrence relation has been developed especially for this purpose in general when data are expected to be noisy a regressed minimum norm solution is used exhaustive tests of the algorithm have established its numerical efficiency results of six typical synthetic models representing diverse geological conditions as well as results of two field examples are included to demonstrate this claim

the random walk representation of classical spin systems and correlation inequalities ii the skeleton inequalities on utilise la representation marche aleatoire pour prouver les premieres d une nouvelle famille d inegalites de correlation pour des modeles reticules ferromagnetiques 4 ces inegalites etablissent que les sommes partielles finies du developpement en perturbation resomme du propagateur pour la fonction a 4 points forment un ensemble alterne de limites superieures et inferieures rigoureuses pour la fonction a 4 points exactes

conductivity depth imaging of airborne electromagnetic step response data an adaptation of the macnae lamontagne method allows transform of airborne step response electromagnetic em data to a conductivity depth image the algorithm is based on a nonlinear transformation of the amplitude of the measured response at each delay time to an apparent mirror image depth using matrix algebra the set of mirror image depth delay time data pairs can then be used to derive a conductivity section data can be efficiently processed on a personal computer at rates faster or comparable to the rate required for collection stable conductivity fitting as a function of depth is obtained by damping the matrix inversion by specification of the first and second derivative smoothness weights of the fitted conductivity depth sounding damping parameters may be either fixed or varied along the profile their choice can be constrained by geologic control stability of the process is enhanced by accounting for the transmitter and receiver tilts the mirror image depth delay time data can also be use

the intersection of brownian paths as a case study of a renormalization group method for quantum field theory a new approach is presented for the study of the probability that the random paths generated by two independent brownian motions in d intersect or more generally are within a short distance a of each other the well known behavior of that function of a above below and at the critical dimension d 4 as well as further corrections are derived here by means of a single renormalization group equation the equation s derivation is expected to shed some light on the function of the o d 4 quantum field theory

a homotopy method for nonlinear inverse problems we develop a homotopy method for nonlinear inverse problems where the forward problems are governed by some forms of differential equations a tikhonov style regularization approach yields an optimization problem ordinary iterative methods may fail to solve this problem due to their locally convergent properties then the fixed point homotopy method is introduced to solving the normal equation of the optimization problem and a new and globally convergent algorithm is constructed which is highly effective in the aspects of speed of computation ability of noise suppression and wide region of convergence as a practical application the method is used to solve the inverse problem of 2 d acoustic wave equation we demonstrate the merits and effectiveness of our algorithm on two realistic model problems

image guided regularization of marine electromagnetic inversion abstractmarine electromagnetic em inverse methods have recently been rapidly developed for offshore exploration however inverted resistivity results with low resolution are provided by the em method to improve this quality of the results we have developed an image guided regularization method for inversion of the marine em data the method incorporates seismic constraints into em inversion information is extracted from seismic geologic images and consists of the metric tensor field and sampling on the geologic structure in addition to the regularization geologic horizons picked from the seismic images and samplings on the structure can be used to generate an irregular sparse mesh compared with an unstructured regular dense mesh a coherence based irregular sparse mesh can reduce computational costs furthermore image guided regularization represents an improvement compared with traditional regularization that are structurally based on seismic images by following geologic features more closely a

the random walk representation of classical spin systems and correlation inequalities ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion by using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities as an application of these inequalities we obtain a simple proof of the mass gap for the 4 2 quantum field model we also establish new upper bounds on critical temperatures

inversion of magnetotelluric data using a practical inverse scattering formulation we present a flexible one dimensional magnetotelluric mt inversion algorithm based on inverse scattering theory the algorithm easily generates different classes of conductivity depth profiles so the interpreter may choose models that satisfy any external geologic or geophysical constraints the two stage process is based on the work of weidelt the first stage uses the mt frequency domain data to construct an impulse response analogous to a deconvolved seismogram with or without a free surface assumption since this is a linear problem a laplace transform numerous impulse responses may be generated by linear inverse techniques which handle data errors robustly we minimize four norms of the impulse response in order to construct varied classes of limited structure earth models we choose such models to prevent overinterpreting the limited number of inaccurate mt observations the second stage of the algorithm maps the impulse response to the conductivity model using any of four fredholm integral eq

a priori estimates for free boundary problem of incompressible inviscid magnetohydrodynamic flows in the present paper we prove the a priori estimates of sobolev norms for a free boundary problem of the incompressible inviscid magnetohydrodynamics equations in all physical spatial dimensions n 2 and 3 by adopting a geometrical point of view used in christodoulou and lindblad commun pure appl math 53 1536 1602 2000 and estimating quantities such as the second fundamental form and the velocity of the free surface we identify the well posedness condition thattheouternormalderivativeofthetotalpressureincludingthefluidandmagnetic pressuresisnegativeonthefreeboundary whichissimilartothephysicalcondition taylor sign condition for the incompressible euler equations of fluids

crosshole seismic inversion with normalized full waveform amplitude data we investigate a simple scheme for full waveform amplitude spectrum inversion of crosshole seismic data with an unknown source wavelet the method is based on our 2d 2 5d finite element method of acoustic wave modeling the normalized amplitude data defined as the spectral ratio of the original trace amplitude to the average amplitude for the entire common shot gather are used for full waveform inversion in the frequency domain in essence the normalization of amplitudes removes the source wavelet and is easily carried out in the time domain or frequency domain from crosshole seismic surveying two synthetic models simulating respectively a dipping fracture model and a complicated sedimentary structure are examined with the inversion scheme the numerical results show that clear images of the targets can be obtained by using the single and multiple frequency data for comparison two other amplitude inversion schemes in which the source signatures are known and estimated with an iterative procedure were carried out for the same models the results show that the iteratively source estimated procedure also produces satisfactory images of the velocity structure and yields an approximate amplitude of the source wavelet with the multifrequency data three such schemes yield very competitive results

smooth inversion of induction logs for conductivity models with mud filtrate invasion a robust efficient method of inversion of induction logging data for smooth 2 d models appropriate to an environment in which mud filtrate invades flat lying layers is described an infinite number of solutions exist to the problem of determining a conductivity structure from a finite number of imprecise induction data therefore the inverse problem is regularized such that the smoothest model is sought subject to the condition that the resulting computed log agrees with the field log to a given preset level at each iteration the jacobian sensitivities are approximated using the distorted born approximation in most cases the algorithm converges in 3 to 4 iterations the resulting maximally smooth models reflect the resolution power of the induction data and are unlikely to result in overinterpretation of the data inversion of both synthetic and field data indicates that layer boundaries are well resolved but radial boundaries are poorly resolved by conventional induction logging data

the inverse problem of resistivity sounding the electric potential due to a single point electrode at the surface of a layered conducting medium is calculated by means of a linear combination of the potentials associated with a set of two layer systems this new representation is called the bilayer expansion for the green s function it enables the forward problem of resistivity sounding to be solved very efficiently even for complicated profiles also the bilayer expansion facilitates the solution of the resistivity inverse problem the coefficients in the expansion are linearly related to apparent resistivity as it is measured and they are readily mapped into parameters for a model specifically i consider models comprising uniformly conducting layers of equal thickness for a given finite data set a quadratic program can be used to find the best fitting model in this class for any specified thickness as the thickness is reduced models of this kind can approximate arbitrary profiles with unlimited accuracy if there is a model that satisfies

what is the depth of investigation of a resistivity measurement abstractever since the first computation of resistivity sounding curves there has been the impression that somehow they are averages of the vertical resistivity profile this prompted the idea to represent apparent resistivity as an integral over depth and to define depth of investigation using the integrands of the integrals as elementary contributions however elementary contributions for a boundary value problem cannot be uniquely defined and are not physically meaningful many practical applications that have been derived from this approach might be at stake regarding their theoretical basis on the other hand a sensitivity function has a definite physical meaning and it is uniquely defined but it offers a different picture for a layered earth the concept of elementary contributions must then be abandoned as not real as some respected scholars have suggested or it must be put on solid ground if we are going to continue using it our claim is that any definition of elementary contributions must

two ways to quantify uncertainty in geophysical inverse problems we present two approaches to invert geophysical measurements and estimate subsurface properties and their uncertainties when little is known a priori about the size of the errors associated with the data we illustrate these approaches by inverting first arrival traveltimes of seismic waves measured in a vertical well to infer the variation of compressional slowness in depth first we describe a bayesian formulation based on probability distributions that define prior knowledge about the slowness and the data errors we use an empirical bayes approach where hyperparameters are not well known ahead of time e g the variance of the data errors and are estimated from their most probable value given the data the second approach is a non bayesian formulation that we call spectral in the sense that it uses the power spectral density of the traveltime data to constrain the inversion e g to estimate the variance of the data errors in the spectral approach we vary assumptions made about the characteristics of the slowness signal and evaluate the resulting slowness estimates and their uncertainties this approach is computationally simple and starts from a few assumptions these assumptions can be checked during the analysis on the other hand it requires evenly spaced traveltime measurements and it cannot be extended easily e g to data that have gaps in contrast the bayesian framework is based on a general theory that can be generalized immediately but it is more involved computationally despite the conceptual and practical differences we find that the two approaches give the same results when they start from the same assumptions the allegiance to a bayesian or non bayesian formulation matters less than what one is willing to assume when solving the inverse problem

well posedness of the free boundary problem in incompressible elastodynamics abstract in this paper we prove the local well posedness of the free boundary problem in incompressible elastodynamics under a natural stability condition which ensures that the evolution equation describing the free boundary is strictly hyperbolic our result gives a rigorous confirmation that the elasticity has a stabilizing effect on the rayleigh taylor instability

geometric analysis of φ4 fields and ising models parts i and ii we provide here the details of the proof announced in 1 that ind 4 dimensions the even 4 euclidean field theory with a lattice cut off is inevitably free in the continuum limit in the single phase regime the analysis is nonperturbative and is based on a representation of the field variables or spins in ising systems as source sink creation operators in a system of random currents which may be viewed as the mediators of correlations in this dual representation the onset of long range order is attributed to percolation in an ensemble of sourceless currents and the physical interaction in the 4 field and other aspects of the critical behavior in ising models are directly related to the intersection properties of long current clusters an insight into the criticality of the dimensiond 4 is derived from an analogy foreseen by k symanzik with the intersection properties of paths of brownian motion other results include the proof that in certain respect the critical behavior in ising models is in exact agreement with the mean field approximation in high dimensionsd 4 but not in the low dimensiond 2 for which we establish the universality of hyperscaling

well posedness of the water waves equations 1 1 presentation of the problem the water waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect incompressible irrotational fluid under the influence of gravity in this paper we restrict our attention to the case when the surface is a graph parameterized by a function x where t denotes the time variable and x xi xd rd the horizontal spatial variables the method developed here works equally well for any integer d 1 but the only physically relevant cases are of course d 1 and d 2 the layer of fluid is also delimited from below by a not necessarily flat bottom parameterized by a time independent function b x we denote by q t the fluid domain at time t the incompressibility of the fluids is expressed by 1 1 div v 0 in qu t 0

well posedness of the free surface incompressible euler equations with or without surface tension we provide a new method for treating free boundary problems in perfect fluids and prove local in time well posedness in sobolev spaces for the free surface incompressible 3d euler equations with or without surface tension for arbitrary initial data and without any irrotationality assumption on the fluid this is a free boundary problem for the motion of an incompressible perfect liquid in vacuum wherein the motion of the fluid interacts with the motion of the free surface at highest order

the diffusion of self avoiding random walk in high dimensions we use the brydges spencer lace expansion to prove that the mean square displacement of at step strictly self avoiding random walk in thed dimensional square lattice is asymptotically of the formdt ast approaches infinity ifd is sufficiently large the diffusion constantd is greater than one

an efficient algorithm for solving the inverse problem of locating the interfaces using the frequency sounding data we consider the inverse coefficient problem of locating the interface positions arising in frequency sounding of layered media such a problem is of particular interest in the exploration of geophysics underwater acoustics and electromagnetics optical sensing and so forth we found that a simplified algorithm can be constructed to determine the approximate positions of interfaces unlike the conventional non linear least squares this algorithm does not require the time consuming constrained optimisation instead the predictor corrector method is applied to solve numerically the auxiliary cauchy problem for a riccati equation the feasibility of this algorithm is demonstrated in computational experiments with a model problem of electromagnetic frequency sounding of layered media

well posedness in sobolev spaces of the full water wave problem in 2 d we consider the motion of the interface of 2 d irrotational incompressible inviscid water wave with air above water and surface tension zero we show that the interface is always not accelerating into the water region normal to itself as rapidly as the normal acceleration of gravity as long as the interface is nonself intersect we therefore obtain the existence and uniqueness of solutions of the full water wave problem locally in time for any initial interface which is nonself intersect including the case that the interface is of multiple heights

fast 3d multichannel deconvolution of electromagnetic induction loop loop apparent conductivity data sets acquired at low induction numbers abstractelectromagnetic induction emi sensors using sufficiently low frequency harmonic sources and sufficiently small loop separations operate in the low induction number lin domain for a relatively wide range of background conductivity these systems are used in diverse near surface investigations including applications from soil sciences hydrology and archaeology the special case of portable multiconfiguration emi sensors operating at frequencies 20 khz offers the possibility of using a fast linear deconvolution method to interpret multichannel data sets in three dimensions here we have developed a fast 3d inversion deconvolution method regularized with 3d smoothness constraints and formulated in the hybrid spectral spatial domain compared with other linear approaches the spectral spatial domain formulation significantly reduces the computational cost of the processing and opens the door for real time 3d interpretation of large data sets consisting of more than 100 000 data points first

on the topography of the cost functional in linear and nonlinear inverse problems we analyze through linear algebra the topography of the cost functional in linear and nonlinear inverse problems with the aim of illuminating general characteristics to a firstorder approximation the local data misfit function in any inverse problem is valley shaped and elongated in the directions of the null space of the jacobian and or in the directions of the smallest singular values in nonlinear inverse problems valleys persist however local minima might also coexist in the misfit space and might be related to nonlinear effects ignored by the gauss newton approximation to the hessian the regularization term designed to provide convexity to the misfit function or to noise in the data furthermore noise perturbs the size of the equivalence region making location of solutions easier but finding a global minimum harder in the case of existence understanding the behavior of the cost functional is an important step in the developing techniques to appraise inverse solutions and estimate uncertainties caused by noise incomplete sampling regularization and more fundamentally simplified physical models

local well posedness for fluid interface problems in this paper we prove the local well posedness of the fluid interface problem with surface tension where the velocity fields are not assumed to be irrotational and the fluid domains are not assumed to be simply connected viewed as a lagrangian system with the configuration space being an infinite dimensional manifold possessing many symmetries this problem is reduced to the evolution of the interface determined by its mean curvature and the evolution of the rotational part of the velocity fields determined by the symmetries this framework also applies to several other fluid surface problems which are outlined in the paper

occam s inversion a practical algorithm for generating smooth models from electromagnetic sounding data the inversion of electromagnetic sounding data does not yield a unique solution but inevitably a single model to interpret the observations is sought we recommend that this model be as simple or smooth as possible in order to reduce the temptation to overinterpret the data and to eliminate arbitrary discontinuities in simple layered models to obtain smooth models the nonlinear forward problem is linearized about a starting model in the usual way but it is then solved explicitly for the desired model rather than for a model correction by parameterizing the model in terms of its first or second derivative with depth the minimum norm solution yields the smoothest possible model rather than fitting the experimental data as well as possible which maximizes the roughness of the model the smoothest model which fits the data to within an expected tolerance is sought a practical scheme is developed which optimizes the step size at each iteration and retains the computational efficiency of layered models resulting in a stable and rapidly convergent algorithm the inversion of both magnetotelluric and schlumberger sounding field data and a joint magnetotelluric resistivity inversion demonstrate the method and show it to have practical application

well posedness in sobolev spaces of the full water wave problem in 3 d we consider the motion of the interface separating an inviscid incompressible irrotational fluid from a region of zero density in three dimensional space we assume that the fluid region is below the vacuum the fluid is under the influence of gravity and the surface tension is zero assume that the density of mass of the fluid is one the gravitational field is 0 0 1 the free interface is t at time t 0 and the fluid occupies the region t the motion of the fluid is described by vt v v 0 0 1 p on t t 0 euler s equation 1 1 div v 0 on t t 0 incompressible 1 2 curl v 0 on t t 0 irrotational 1 3 where v v1 v2 v3 is the fluid velocity p is the fluid pressure since we neglect the surface tension the pressure is zero on the interface so on the interface p 0 on t 1 4 1 v is tangent to the free surface t t 1 5 we want to find solutions of system 1 1 1 5 taking prescribed initial data such that for every fixed t 0 t approaches the xy plane at infinity and v x y z t 0 vt x y z t 0 as x y z since the fluid is assumed irrotational incompressible we can reduce the study of the entire motion to the motion of the free surface the above model is a 3 d water wave model it is generally known that when surface tension is neglected the motion of the interface between an inviscid fluid and vacuum under the influence of gravity can be subject to taylor instability 8 22 in a previous work 24 we studied the 2 dimensional water wave model we showed that for a 2 d water wave the sign condition relating to taylor instability always holds for nonself intersecting interface that is the motion of the interface is not subject to taylor instability we showed further that the 2 d full nonlinear water wave problem is uniquely solvable in sobolev spaces locally in time for any initially nonself intersecting interface earlier works on the well posedness in sobolev spaces of the 2 d water wave problem include nalimov 18 yosihara 25 and walter craig 6 where the main results concern the well posedness of the

geometry and a priori estimates for free boundary problems of the euler s equation in this paper we derive estimates to the free boundary problem for the euler equation with surface tension and without surface tension provided the rayleigh taylor sign condition holds we prove that as the surface tension tends to 0 when the rayleigh taylor condition is satisfied solutions converge to the euler flow with zero surface tension 2007 wiley periodicals inc

occam s inversion to generate smooth two dimensional models from magnetotelluric data magnetotelluric mt data are inverted for smooth 2 d models using an extension of the existing 1 d algorithm occam s inversion since an mt data set consists of a finite number of imprecise data an infinity of solutions to the inverse problem exists fitting field or synthetic electromagnetic data as closely as possible results in theoretical models with a maximum amount of roughness or structure however by relaxing the misfit criterion only a small amount models which are maximally smooth may be generated smooth models are less likely to result in overinterpretation of the data and reflect the true resolving power of the mt method the models are composed of a large number of rectangular prisms each having a constant conductivity apriori information in the form of boundary locations only or both boundary locations and conductivity may be included providing a powerful tool for improving the resolving power of the data joint inversion of te and tm synthetic data generated from known models al

efficient estimation of nonlinear posterior model covariances using maximally sparse cubature rules we estimated posterior model covariances by randomly perturbing data and start model vectors with samples drawn from prior probability distributions solving the corresponding deterministic inversion for each sample and integrating the resulting model vectors over all realizations the computational burden of this method is the number of samples required for the integration because each sample involves a solution to a nonlinear inverse problem for certain classes of smooth prior probability functions we present a new approach to the model covariance estimation problem using grids based on maximally sparse cubature integration rules the n 1 degree two and 2n degree three cubature rules exist to exactly integrate symmetric gaussian distributions in n stochastic dimensions these represent the two sparsest integration rules possible and thus provide the most efficient estimation method for model covariances using integration in contrast to monte carlo mc integration which requires hundreds of random samples to estimate covariances and has ill defined convergence sparse cubature quadrature rules have determined numbers of samples and definite convergence the trade off is that they also have limited degrees of accuracy using a simple 1d electromagnetic example with 67 stochastic variables we demonstrated that this sparse integration is three to five times more efficient than mc methods in terms of numbers of required samples

on the free boundary problem of three dimensional incompressible euler equations in this paper we consider in three dimensions the motion of a general inviscid incompressible fluid with a free interface that separates the fluid region from the vacuum we assume that the fluid region is below the vacuum and that there is no surface tension on the free surface then we prove the local well posedness of the free boundary problem in sobolev space provided that there is no self intersection point on the initial surface and under the stability assumption that with being restricted to the initial surface 2007 wiley periodicals inc

regularization in model updating this paper presents the theory of sensitivity based model updating with a special focus on the properties of the solution that result from the combination of optimization of the response prediction with a priori information about the uncertain parameters model updating together with the additional regularization criterion is an optimization with two objective functions and must be linearized to obtain the solution structured solutions are obtained based on the generalized singular value decomposition gsvd and specific features of the parameter and response paths as the regularization parameter varies are explored the four different types of spaces that arise in the solution are discussed together with the characteristics of the regularized solution families these concepts are demonstrated on a simulated discrete example and on an experimental case study copyright 2007 john wiley sons ltd

well posedness of the free boundary problem in compressible elastodynamics abstract we study the free boundary problem for the flow of a compressible isentropic inviscid elastic fluid at the free boundary moving with the velocity of the fluid particles the columns of the deformation gradient are tangent to the boundary and the pressure vanishes outside the flow domain we prove the local in time existence of a unique smooth solution of the free boundary problem provided that among three columns of the deformation gradient there are two which are non collinear vectors at each point of the initial free boundary if this non collinearity condition fails the local in time existence is proved under the classical rayleigh taylor sign condition satisfied at the first moment by constructing an hadamard type ill posedness example for the frozen coefficients linearized problem we show that the simultaneous failure of the non collinearity condition and the rayleigh taylor sign condition leads to rayleigh taylor instability

resolution and uncertainty in 1d csem inversion a bayesian approach and open source implementation we show that resolution and uncertainty in csem inversion is most naturally approached using a bayesian framework resolution can be inferred by either hierarchical models with free parameters for effective correlation lengths bayesian smoothing or model choice frameworks applied to variable resolution spatial models bayesian splitting merging we find that typical 1d csem data can be modelled with quite parsimonious models typically o 10 parameters per common midpoint efficient opt imizations for the csem problem must address the challenges of poor scaling strong nonlinearity multimodality and the necessity of bound constraints the posterior parameter uncertainties are frequently controlled by the nonlinearity and linearised approaches to uncertainty are usually very poor in markov chain monte carlo mcmc approaches the nonlinearity and poor scaling make good mixing hard to achieve a novel approximate frequentist method we call the bayesianized parametric bootstrap sometimes called randomized maximum likelihood is much more efficient than mcmc in this problem is considerably better than linearized analysis but tends to modestly overstate uncertainties the software that implements these ideas for the 1d csem problem is made available under an open source license agreement