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Geophysical inversion with a neighbourhood algorithmöI. Searching a parameter space, Geophys
 J. Int
, 1999
"... etc., are often used to explore a ¢nitedimensional parameter space. They require the solving of the forward problem many times, that is, making predictions of observables from an earth model. The resulting ensemble of earth models represents all `information ' collected in the search process. ..."
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Cited by 144 (8 self)
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etc., are often used to explore a ¢nitedimensional parameter space. They require the solving of the forward problem many times, that is, making predictions of observables from an earth model. The resulting ensemble of earth models represents all `information ' collected in the search process. Search techniques have been the subject of much study in geophysics; less attention is given to the appraisal of the ensemble. Often inferences are based on only a small subset of the ensemble, and sometimes a single member. This paper presents a new approach to the appraisal problem. To our knowledge this is the ¢rst time the general case has been addressed, that is, how to infer information from a complete ensemble, previously generated by any search method. The essence of the new approach is to use the information in the available ensemble to guide a resampling of the parameter space. This requires no further solving of the forward problem, but from the new `resampled ' ensemble we are able to obtain measures of resolution and tradeo ¡ in the model parameters, or any combinations of them. The new ensemble inference algorithm is illustrated on a highly nonlinear waveform inversion problem. It is shown how the computation time and memory requirements scale with the dimension of the parameter space and size of the ensemble. The method is highly parallel, and may easily be distributed across several computers. Since little is assumed about the initial ensemble of earth models, the technique is applicable to a wide variety of situations. For example, it may be applied to perform `error analysis ' using the ensemble generated by a genetic algorithm, or any other direct search method. Key words: numerical techniques, receiver functions, waveform inversion. 1
2011), A synoptic view of the distribution and connectivity of the midcrustal low velocity zone beneath Tibet
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Precise absolute earthquake location under SommaVesuvius volcano, Geophys
 J. Int
, 2001
"... The Somma–Vesuvius volcanic complex and surroundings are characterized by topographic relief of over 1000 m and strong 3D structural variations. This complexity has to be taken into account when monitoring the background volcano seismicity in order to obtain reliable estimates of the absolute epic ..."
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Cited by 20 (4 self)
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The Somma–Vesuvius volcanic complex and surroundings are characterized by topographic relief of over 1000 m and strong 3D structural variations. This complexity has to be taken into account when monitoring the background volcano seismicity in order to obtain reliable estimates of the absolute epicentres, depths and focal mechanisms for events beneath the volcano. We have developed a 3D Pwave velocity model for Vesuvius by interpolation of 2D velocity sections obtained from nonlinear tomographic inversion of the Tomoves 1994 and 1996 active seismic experiment data. The comparison of predicted and observed 3D traveltime data from active and passive seismic data validate the 3D interpolated model. We have relocated about 400 natural seismic events from 1989 to 1998 under Vesuvius using the new interpolated 3D model with two different VP /VS ratios and a global search, 3D location method. The solution quality, station residuals and hypocentre distribution for these 3D locations have been compared with those for a representative layered model. A relatively high VP/VS ratio of 1.90 has been obtained. The highestquality set of locations using the new 3D model falls in a depth range of about 1–3.5 km below
A transdimensional Bayesian Markov chain Monte Carlo algorithm for model assessment using frequencydomain electromagnetic data
 Geophysical Journal International
, 2011
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2002), An inquiry into the lunar interior: A nonlinear inversion of the Apollo lunar seismic data
 J. Geophys. Res
"... [1] This study discusses in detail the inversion of the Apollo lunar seismic data and the question of how to analyze the results. The wellknown problem of estimating structural parameters (seismic velocities) and other parameters crucial to an understanding of a planetary body from a set of arrival ..."
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Cited by 12 (1 self)
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[1] This study discusses in detail the inversion of the Apollo lunar seismic data and the question of how to analyze the results. The wellknown problem of estimating structural parameters (seismic velocities) and other parameters crucial to an understanding of a planetary body from a set of arrival times is strongly nonlinear. Here we consider this problem from the point of view of Bayesian statistics using a Markov chain Monte Carlo method. Generally, the results seem to indicate a somewhat thinner crust with a thickness around 45 km as well as a more detailed lunar velocity structure, especially in the middle mantle, than obtained in earlier studies. Concerning the moonquake locations, the shallow moonquakes are found in the depth range 50–220 km, and the majority of deep moonquakes are concentrated in the depth range 850–1000 km, with what seems to be an apparently rather sharp lower boundary. In wanting to further analyze the outcome of the inversion for specific features in a statistical fashion, we have used credible intervals, twodimensional marginals, and Bayesian hypothesis testing. Using this form of hypothesis testing, we are able to decide between the relative importance of any two hypotheses given data, prior information, and the physical laws that govern the relationship between model and data, such as having to decide between a thin crust of 45 km and a thick crust as implied by the generally assumed value of 60 km. We obtain a Bayes factor of 4.2, implying that a thinner crust is strongly favored. INDEX TERMS: 6250 Planetology: Solar
Towards a quantitative interpretation of global seismic tomography
 American Geophysical Union, Washington DC
, 2005
"... We review the success of seismic tomography in delineating spatial variations in the propagation speed of seismic waves on length scales from several hundreds to many thousands of kilometers. In most interpretations these wave speed variations are thought to reflect variations in temperature. Carefu ..."
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We review the success of seismic tomography in delineating spatial variations in the propagation speed of seismic waves on length scales from several hundreds to many thousands of kilometers. In most interpretations these wave speed variations are thought to reflect variations in temperature. Careful consideration of shear wave, bulk sound, and, most recently, density variations is, however, producing increasingly compelling evidence for chemical heterogeneity (that is, spatial variations in bulk major element composition) having a firstorder effect on the lateral variations in mass density and elasticity of the mantle. This has profound consequences for our understanding of mantle dynamics and the thermochemical evolution of our planet. We argue that the quantitative integration of constraints from seismology, mineral physics, and geodynamics, which underlies the inference of thermochemical parameters, requires careful uncertainty analyses and should move away from emphasizing visually pleasing images and single, nonunique solutions. 1.
and A.M.Stuart. Variational data assimilation using targetted random walks
 Int. J. Num. Meth. Fluids
"... The variational approach to data assimilation is a widely used methodology for both online prediction and for reanalysis. In either of these scenarios, it can be important to assess uncertainties in the assimilated state. Ideally, it is desirable to have complete information concerning the Bayesian ..."
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Cited by 10 (4 self)
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The variational approach to data assimilation is a widely used methodology for both online prediction and for reanalysis. In either of these scenarios, it can be important to assess uncertainties in the assimilated state. Ideally, it is desirable to have complete information concerning the Bayesian posterior distribution for unknown state given data. We show that complete computational probing of this posterior distribution is now within the reach in the offline situation. We introduce a Markov chain–Monte Carlo (MCMC) method which enables us to directly sample from the Bayesian posterior distribution on the unknown functions of interest given observations. Since we are aware that these methods are currently too computationally expensive to consider using in an online filtering scenario, we frame this in the context of offline reanalysis. Using a simple random walktype MCMC method, we are able to characterize the posterior distribution using only evaluations of the forward model of the problem, and of the model and data mismatch. No adjoint model is required for the method we use; however, more sophisticated MCMC methods are available which exploit derivative information. For simplicity of exposition, we consider the problem of assimilating data, either Eulerian or Lagrangian, into a low Reynolds number flow in a twodimensional periodic geometry. We will show that in many cases it is possible to recover the initial condition and model error (which we describe as unknown forcing to the model) from data, and that with increasing amounts of informative data, the uncertainty in our estimations reduces. Copyright! 2011 John Wiley
Reflecting Uncertainty in Inverse Problems: A Bayesian Solution using Lévy Processes
 INVERSE PROBLEMS
, 2004
"... We formulate the inverse problem of solving Fredholm integral equations of the first kind as a nonparametric Bayesian inference problem, using Lévy random fields (and their mixtures) as prior distributions. Posterior distributions for all features of interest are computed employing novel Markov chai ..."
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We formulate the inverse problem of solving Fredholm integral equations of the first kind as a nonparametric Bayesian inference problem, using Lévy random fields (and their mixtures) as prior distributions. Posterior distributions for all features of interest are computed employing novel Markov chain Monte Carlo numerical methods in infinitedimensional spaces, based on generalizations and extensions of the authors ’ Inverse Lévy Measure (ILM) algorithm. The method is also well suited for deconvolution problems, for inverting Laplace and Fourier transforms, and for other linear and nonlinear problems in which the unknown feature is high (or even infinite) dimensional and where the corresponding forward problem may be solved rapidly. The methods are illustrated for an application to an important problem in rheology: that of inferring the molecular weight distribution of polymers from conventional rheological measurements, in which we achieve not just a point estimate but a posterior probability density plot representing all uncertainty about the weight.
Linear inverse Gaussian theory and geostatistics
"... Inverse problems in geophysics require the introduction of complex a priori information and are solved using computationally expensive Monte Carlo techniques �where large portions of the model space are explored�. The geostatistical method allows for fast integration of complex a priori information ..."
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Cited by 8 (4 self)
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Inverse problems in geophysics require the introduction of complex a priori information and are solved using computationally expensive Monte Carlo techniques �where large portions of the model space are explored�. The geostatistical method allows for fast integration of complex a priori information in the form of covariance functions and training images. We combine geostatistical methods and inverse problem theory to generate realizations of the posterior probability density function of any Gaussian linear inverse problem, honoring a priori information in the form of a covariance function describing the spatial connectivity of the model space parameters. This is achieved using sequential Gaussian simulation, a wellknown, noniterative geostatistical Consider the expression