A probabilistic seismic model for the European Arctic

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116,, doi: /2010jb007889, 2011 A probabilistic seismic model for the European Arctic Juerg Hauser, 1 Kathleen M. Dyer, 2 Michael E. Pasyanos, 2 Hilmar Bungum, 1 Jan I. Faleide, 3 Stephen A. Clark, 3 and Johannes Schweitzer 1 Received 27 July 2010; revised 29 September 2010; accepted 11 October 2010; published 13 January [1] The development of three dimensional seismic models for the crust and upper mantle has traditionally focused on finding one model that provides the best fit to the data while observing some regularization constraints. In contrast to this, the inversion employed here fits the data in a probabilistic sense and thus provides a quantitative measure of model uncertainty. Our probabilistic model is based on two sources of information: (1) prior information, which is independent from the data, and (2) different geophysical data sets, including thickness constraints, velocity profiles, gravity data, surface wave group velocities, and regional body wave traveltimes. We use a Markov chain Monte Carlo (MCMC) algorithm to sample models from the prior distribution, the set of plausible models, and test them against the data to generate the posterior distribution, the ensemble of models that fit the data with assigned uncertainties. While being computationally more expensive, such a probabilistic inversion provides a more complete picture of solution space and allows us to combine various data sets. The complex geology of the European Arctic, encompassing oceanic crust, continental shelf regions, rift basins and old cratonic crust, as well as the nonuniform coverage of the region by data with varying degrees of uncertainty, makes it a challenging setting for any imaging technique and, therefore, an ideal environment for demonstrating the practical advantages of a probabilistic approach. Maps of depth to basement and depth to Moho derived from the posterior distribution are in good agreement with previously published maps and interpretations of the regional tectonic setting. The predicted uncertainties, which are as important as the absolute values, correlate well with the variations in data coverage and quality in the region. A practical advantage of our probabilistic model is that it can provide estimates for the uncertainties of observables due to model uncertainties. We will demonstrate how this can be used for the formulation of earthquake location algorithms that take model uncertainties into account when estimating location uncertainties. Citation: Hauser, J., K. M. Dyer, M. E. Pasyanos, H. Bungum, J. I. Faleide, S. A. Clark, and J. Schweitzer (2011), A probabilistic seismic model for the European Arctic, J. Geophys. Res., 116,, doi: /2010jb Introduction [2] When deriving a seismic model one generally seeks to find the minimum of a misfit function between the predicted and the observed data. Misfit functions usually have a large number of secondary minima in addition to a global minimum. These minima represent other possible solutions and a local search for the global minimum could get trapped in a local minimum. Backus and Gilbert [1967] established nonuniqueness as one of the fundamental properties of geophysical inverse problems; if a model can be found that satisfies the data, then it is likely that there are an infinite number of alternative models that will fit the same data, in 1 NORSAR, Kjeller, Norway. 2 Lawrence Livermore National Laboratory, Livermore, California, USA. 3 Department of Geosciences, University of Oslo, Oslo, Norway. Copyright 2011 by the American Geophysical Union /11/2010JB particular when one tries to fit more than one data set. In a probabilistic framework the global minimum is the most likely solution and local minima are less likely solutions. Bayesian inference [Bayes, 1763] forms the basis for the probabilistic formulation of an inverse problem. In this formulation all information is represented in probabilistic terms. It combines the prior information known about the model with observed data to generate the posterior probability density function on the model parameters. Mosegaard and Tarantola [1995] used a Markov chain Monte Carlo (MCMC) technique to randomly sample the model space at a rate proportional to the posterior probabilities, a process known as importance sampling, thereby empirically reconstructing the unknown posterior probability density function. Monte Carlo methods sample the space of possible models completely at random, whereas MCMC algorithms try to move through model space according to the posterior probabilities for these models, thereby performing a guided search and focusing on regions that better fit the prior information and the data. 1of17

2 [3] The region of interest for this study is the crust and upper mantle beneath the European Arctic and in particular the Barents Sea. The target area exhibits a wide variety of tectonic features for a relatively small region, such as a midoceanic ridge, a continental margin, an ancient shield and a continental shelf region with prominent sedimentary basins (Figure 1). The development of different regional seismic models to predict traveltimes has been motivated by an interest to locate seismic events in the region, which includes the former Soviet nuclear test site at Novaya Zemlya. Various regional 1 D velocity models have been proposed and used for seismic event location [e.g., Kremenetskaya et al., 2001; Hicks et al., 2004]. These models differ slightly from each other mainly with respect to their velocities for the upper mantle. Schweitzer and Kennett [2007] compared location procedures using an event in the Kara Sea and showed that the resulting traveltime residuals are significantly lower for a regional one dimensional velocity model when compared to a global reference model, for example ak135 [Kennett et al., 1995]. Nevertheless a onedimensional velocity model cannot capture lateral changes in structure. It can only represent the regional average and will therefore always have its limitations with respect to describing a regional velocity structure, in particular if the tectonic setting is as diverse as in the European Arctic. [4] The potential for oil and gas resources in this region means that it has been targeted by various crustal seismic surveys over the last 25 years [e.g., Gudlaugsson et al., 1987; Guggisberg et al., 1991; Breivik et al., 2005]. Ritzmann et al. [2007] compiled a database of available velocity data for the Barents Sea and used it to derive BARENTS50, a threedimensional model of the crust in the Barents Sea region. The continuous curvature gridding algorithm of Smith and Wessel [1990] was used to interpolate structure for regions where no seismic lines were available. They further constrained their model layers by using regional thickness relationships. For each of the 27 geological provinces in their model they derived a linear relationship between the thickness of the sedimentary units and the crystalline crust and used it to adjust the model, where no seismic lines were available. Given the uneven distribution of seismic surveys over the target region one would expect model uncertainties to vary spatially. The interpolation methodology employed by Ritzmann et al. [2007] makes it difficult to reliably map out model uncertainties resulting from the interpolation and quality of the seismic data. We will argue later that their choice of two layers with constant seismic parameters to describe the sedimentary units is a poor approximation of the seismic parameters as a function of depth in deep sedimentary basins when compared to using two linear transitions. [5] Surface wave data, in particular Rayleigh wave dispersion curves, have been used in the European Arctic since the late 1970s to invert for an average crust and upper mantle structure using layers with constant velocity [e.g., McCowan et al., 1978; Calcagnile and Panza, 1978]. Levshin and Berteussen [1979] used surface wave observation at the NORSAR array from presumed nuclear explosions on Novaya Zemlya and in northwestern Siberia to constrain structure for the eastern Barents Sea. Their choice of parameterization of the average crust and upper mantle structure is noteworthy in that it differs from the aforementioned studies by using linear transitions in the two uppermost crustal layers to describe the seismic velocities. This allows for a more natural description of velocity changes associated with the consolidation of sedimentary rocks. Zeng et al. [1989] tried to resolve lateral changes in structure by dividing the Arctic region into 24 zones and inverting for shear wave velocity as a function of depth for each zone. More recently Levshin et al. [2007] inverted surface wave group velocities for a 3 D tomographic shear wave velocity model. They achieved a higher path density in the European Arctic than previous studies [e.g., Ritzwoller et al., 2002] by systematically searching for additional broadband waveform data. Nevertheless, due to the small number of short period observations the resolution for the surface wave tomography in the crust was limited. To alleviate this, Levshin et al. [2007] used the crustal model of Ritzmann et al. [2007] as a constraint in the inversion. In the final stage of the inversion a Monte Carlo technique is used to construct an ensemble of acceptable shear wave velocity models and thereby estimate model uncertainties. The disadvantage with using this type of model to locate seismic events is that surface waves are sensitive to S wave velocities, but predicting accurate P wave traveltimes is more important to locate seismic events. [6] In this study we develop a probabilistic seismic model for the European Arctic that is in agreement with thickness constraints, velocity profiles, gravity data, surface wave group velocities and regional body wave traveltimes. It differs from earlier studies in the region in that we seek to find the posterior distribution of models that is best able to fit all the data. Using more than one data set to constrain structure is primarily motivated by the fact that they are sensitive to different parts of the structure. Even if one were only interested in a model to predict body wave traveltimes, using more than one data set is still advantageous, as it allows one to reduce model uncertainties. We will see later that the coverage of the eastern part of the target region with body waves is sparse. Using body wave traveltimes by themselves to constrain structure would result here in high model uncertainties and therefore high location uncertainties for seismic events. Clearly in the absence of sufficient body wave traveltimes other data sets provide valuable constrains for a model that will be used to predict body wave traveltimes. [7] The inversion methodology employed here is based on that of Pasyanos et al. [2006] for the Yellow Sea and Korean Peninsula region. Important improvements are the use of linear functions to describe the seismic parameters within each layer instead of constant values, and the use of an arbitrary mesh. We begin this study with a description of the model parameterization and the MCMC algorithm. After a discussion of the data sets and the forward solvers, we focus on characterizing the posterior distribution, i.e., the ensemble of models that fit the data. We also demonstrate how a probabilistic model can be used to estimate location uncertainties that result from model uncertainties. 2. Methodology [8] Our MCMC algorithm (Figure 2) is a derivative of the Metropolis Hastings Algorithm [Metropolis et al., 1953] as described by Mosegaard and Tarantola [1995]. There are two major components to the MCMC algorithm. The first component is a set of rules that define the prior distribution, 2of17

3 Figure 1. Map of the most relevant tectonic features in the region after Ritzmann et al. [2007] and Bird [2003]. Beige areas represent the major sedimentary basins in the region, and the continent ocean boundary is given by the dashed blue line. The plate boundary is indicated by the brown line, and the boundary of Cretaceous Volcanic Province near the northern margin is given by the dashed violet line. Two cross sections (A A ) and (B B ), along which we will examine our probabilistic model in Figure 9, are outlined in red. the ensemble of plausible models. Given a model, the base sampler proposes new models that are in agreement with these rules by randomly perturbing the current model. The second component is a Metropolis Hastings algorithm that generates samples according to the unknown posterior distribution, the ensemble of models that are in agreement with the data. It does this using a randomized decision rule to accept or reject proposed models according to their consistency with the data. [9] Given a model m and data d, Bayes theorem formally relates the prior r(m) and posterior P(m d) distributions as follows: Pðmjd Þ ¼ ðmþlðmþ: ð1þ The likelihood function L(m) makes models with a better fit to the data more likely and models with a bad fit to the data less likely and is given as LðmÞ ¼ exp 1 X N dm ð Þ pred;i d 2! obs;i ; ð2þ N i¼1 i rel where N is the number of observations and d(m) pred,i is the predicted data for a given model. The corresponding observation is given by d obs,i and the estimated uncertainty as s i. The parameter s rel is the overall uncertainty we assign to the data. The larger s rel the smaller the influence of the fit to the data and the more similar the posterior distribution will become to the prior distribution. If models are tested against more than one data set, then s rel becomes a parameter used to express the overall confidence we have in a given data set. [10] The MCMC algorithm recovers the unknown posterior distribution using a cleverly constructed Markov chain, a perturbative sequence of random changes to the starting model. Constructing the Markov chain in such a way that it preferably contains models that satisfy the data allows one to reconstruct the posterior distribution, given sufficient time to sample the model space. Given the likelihood for a proposed model L(m j ) and the current model L(m i ), a decision has to be made whether or not the proposed model m j should be accepted into the set of samples of the posterior distribution. If we would accept all models, we would simply recover the prior distribution, the ensemble of models based on prior information contained in the base sampler and starting model and not constrained by the data. For L(m j ) L(m i ) we accept the proposed model and add it to our set of samples of the posterior distribution. For L(m j )<L(m i ), the proposed model is accepted if the probability L(m j )/L(m i )is larger than a random number drawn from a uniform distribution between 0 and 1. Otherwise, we reject the proposed model and add a copy of the current model to the set of samples of the posterior distribution. This means that the worse the fit to the data the more likely it is that a proposed model will be rejected. The randomized rule to accept or reject models with a worse fit to the data than the current model with a certain probability theoretically ensures that the chain will never get trapped in a local minimum. Metropolis et al. [1953] proved that samples generated for the posterior distribution through this process known as importance sampling have a limiting distribution proportional to the desired posterior distribution. [11] This study uses different types of data with independent uncertainties. The total likelihood for a model can therefore be expressed as the product of the individual likelihoods for each data type. Applying the MCMC algorithm directly to the total likelihood would require one to predict all data types before being able to apply the randomized decision rule. Solving the forward problem for the various data sets is the most time consuming part of a probabilistic inversion. In addition there are significant differences in computation time necessary to predict the various data sets. It is therefore more efficient to organize the various data sets in a cascade [Mosegaard and Tarantola, 1995], sorted according to computation time required to solve the forward problem. The prediction of body wave traveltimes is the most expensive stage in this study and therefore the last stage of the cascade (Figure 2). Once a model is proposed by the base sampler the forward problem is performed for the first data set. The proposed model may then be accepted or rejected. If it is rejected, there is no need to solve computationally more expensive forward problems for the data sets placed at later stages in the cascade and the base sampler proposes another model. Only when a model has been accepted in all stages is it added to the set of samples of the posterior distribution. If it is rejected by any of the stages, then a copy of the last model that was accepted into the set of samples of the posterior distribution gets added instead. [12] Arranging the data sets in a cascade does not mean that a model with a worse fit to one of the data sets than the last accepted one is always automatically rejected. The use of a randomized decision rule in the MCMC algorithm allows for such a model to reach the next stage with a certain prob- 3of17

4 Figure 2. Flowchart showing the multistage MCMC inversion methodology used in this study. ability. It makes it possible for a model with a slightly worse fit to all the data sets to nevertheless be accepted in all the stages with a certain probability. For a sufficiently long chain the ordering of the stages should have no statistically significant influence on the result. The ordering of the stages is however important to improve the computational efficiency Model Parameterization [13] We cover the region of interest with a triangular mesh with a distance of 83 km between neighboring nodes (Figure 3a). The spatial resolution was chosen based on the computational resources available and the expected resolution of the data. The boundaries for the target region were chosen so that the seismicity associated with the midoceanic ridge to the north and west of the Barents Sea is inside the model. Each nonboundary node of the 592 nodes has 6 neighbors. Two linear functions in the sediment, three in the crystalline crust, and three in the mantle are used to describe the seismic parameters (i.e., Vp, Vs and density) as a function of depth (Figure 3b) at the nodes of the mesh. The parameterization allows for a jump in seismic parameters at the Moho and at the basement. We also define a water layer that is kept constant during the inversion. The model extends down to 250 km depth, where it sits over the ak135 model [Kennett et al., 1995]. [14] The use of layers that allow for linear changes of the seismic parameters is an improvement, when trying to image the true Earth, over layers with constant seismic parameters commonly used to describe the crust and upper mantle [e.g., Bassin et al., 2000; Pasyanos et al., 2006; Ritzmann et al., 2007]. Density versus depth functions for sedimentary rocks of the region [Ebbing et al., 2007] show a logarithmic behavior, with the rate of increase of Vp, Vs and density decreasing with increasing depth. Two layers with a linear increase of Vp, Vs and density with depth approximate this more closely than two layers with constant velocity and density. For example, two layers with constant seismic parameters to describe the sedimentary units would introduce an artificial discontinuity that can, for deep sedimentary basins, be as large or even larger than the discontinuity at the Moho. We noted earlier that Levshin and Berteussen [1979] also used linear transitions to describe seismic velocities for the two uppermost layers of their average crustal model for surface wave paths crossing the East Barents Sea Basin. Regional phases like Pn and Sn provide valuable information about the upper mantle structure; they correspond to rays that either have traveled as a head waves (or whispering gallery waves) along the Moho or have turned in the upper mantle due to a velocity gradient. The velocity gradient in the upper mantle is approximated by a linear transition in our parameterization. [15] If a model is developed for the sole purpose of predicting observables and their uncertainties, using layers that allow for a linear increase of the seismic parameters instead of homogeneous layers is unlikely to result in a better model. However, in this study the focus is also on imaging the structure. Using layers that allow for a linear increase of the seismic parameters results in a parameterization that is closer to what the true Earth might look like when compared to using the same number of homogeneous layers. [16] Common to the various forward solvers is that they rely on an interpolation routine to obtain seismic parameters at an arbitrary location inside the target region. Given the relatively low spatial resolution, the choice of interpolation algorithm is as important as the values at the supporting points. We use a linear natural neighbor interpolation [Sibson, 1981], a local interpolation method based on a Voronoi tessellation. It has a number of useful properties, including, localized influence of data points, continuity (C 0 everywhere and C 1 everywhere except at data points), and allowance for an arbitrary distribution of supporting points Base Sampler [17] The success of an MCMC algorithm relies on an appropriate mechanism (base sampler) for generating model realizations from the prior distribution. Defining the rules for the prior distribution and which models are accessible from a given model are essential ingredients of the method. Only models that are part of the prior distribution are proposed and tested against the data. If the prior distribution does not include the true Earth, models that fit the data might never be found by the MCMC algorithm. New models are proposed 4of17

5 Figure 3. (a) Map of the target region; blue and green diamonds show the distribution of the 1 D models used to describe structure. Green diamonds are used for nodes that are in the region covered by the crustal model of Ritzmann et al. [2007] and the mantle model of Levshin et al. [2007]. The blue diamonds mark the nodes where the starting model was initialized using CRUST 2.0 [Bassin et al., 2000]. Earthquakes ( ) according to the Global CMT Database ( are plotted using red stars, and stations used in this study are given by the orange triangles. Two nodes, for which the model profile distribution is shown in Figure 6, are marked with a violet circle. (b) Diagram showing the parameterization used to describe the seismic parameters as a function of depth. The seismic parameters in each layer are defined by the values on the layer boundaries and within the layers and the use of a linear interpolation to retrieve values at an arbitrary depth within a specific layer. Note that we allow for discontinuities at the basement and the Moho. by randomly perturbing the last model that was accepted. The definition of which models are accessible from a given model (i.e., the amount of perturbation to the previous model) when constructing the Markov chain is critical for the success of a probabilistic inversion. In practice this means that if the size of the step in model space taken by the base sampler is too large, then the MCMC algorithm can no longer perform a guided search through model space and it becomes more of a random sampling of model space. On the other hand, if the step size is too small it might take a very long time to explore the model space sufficiently. [18] We randomly swap the profiles for 10% of the nodes in the starting model to introduce some variation in the initial model for the Markov chain and to reduce dependence on the starting model. The base sampler changes the model by randomly perturbing the model at a number of nodes. At each node we allow the seismic parameters to move a random amount of up to 10% (as characterized by a normal distribution) from the current state. Table 1 lists the lower and upper boundaries for the seismic parameters and the layer thicknesses for the individual units. When proposing a new model it is important to keep in mind that the seismic parameters are not completely independent from each other. Given a proposed value for Vp the range of plausible values for Vs is limited by the plausible values for the Vp/Vs ratio; the same applies to the density. We allow for the Vp/Vs ratio to vary between 1.5 and 2.1. The range of plausible values for the Vp/density ratio as a function of Vp was obtained by digitizing Figure 18 of Breivik et al. [2002] and is listed in Table 2. We set up our base sampler so that every proposed model observes these limitations for the ratios between the seismic parameters where they apply. For the top of the sedimentary units, we do not constrain the Vp/Vs ratios directly, which have been measured at up to 13 for water saturated sediments [Hamilton, 1979]. We also do not impose any constraints on the Vp/density ratio if Vp is less than 3.0 km/s. [19] In joint inversions of gravity and traveltime data, density and velocities are often combined using functional relationships [e.g., Christensen and Mooney, 1995]. Laboratory measurements show however that for a given velocity thereisarangeofpossibledensityvalues.barton [1986] questions the usefulness of simple relationships between seismic velocity and density to constrain the continental crust, obtaining a good fit to the gravity along a seismic profile by assuming a uniform density, i.e., one that is independent of Vp. Clearly, using a functional relationship between seismic parameters would lead to a prior distribution that only contains a subset of the plausible models. In such a situation the true Earth might not be part of the prior distribution and, if so, models with a good fit to the data would never be proposed by the base sampler. In the upper mantle, chemical composition and mineralogy greatly influence the nature of the correlations between seismic velocities and density. Seismic and density modeling along the QUARTZ geotraverse by Romanyuk [1996] indicated that there is not even a weak correlation between Vp and density in the uppermost mantle under the Baltic shield. This is further supported 5of17

6 Table 1. Ranges for Thickness, Vp, Vs, and Density for the Individual Parameters That Define the Structure at the Nodes of the Underlying Mesh a Unit/Layer Node Thickness (km) Vp (km/s) Vs (km/s) Density (g/cm 3 ) Sediments top middle bottom Crystalline crust top upper lower bottom Mantle top upper lower bottom a Note that the water layer is not shown as it is kept constant during the inversion. by the findings of Ishii and Tromp [2004], which indicate an anticorrelation between density and Vp in the transition zone. We therefore do not constrain the Vp/density ratio in the upper mantle in models proposed by the base sampler. [20] None of the data used in this study are sensitive to the thickness of sedimentary units covering Novaya Zemlya. Stratigraphic studies for Novaya Zemlya indicate that the upper limit for the thickness of rocks with seismic properties typical for sedimentary rocks is about 1 km [Korago et al., 2004], which is included as prior information. Additionally the Precambrian sedimentary rocks covering Fennoscandia have undergone metamorphism and their seismic parameters are characteristic for crystalline lithologies. In our model parameterization the Precambrian sedimentary units in Fennoscandia are considered to be a part of the crystalline crust. Therefore we condition the base sampler so that it does not propose models with a sediment layer covering Fennoscandia. We also constrain the thickness of the crystalline crust in the oceanic part of our model. Since true oceanic crystalline crust will seldom be thicker than 10 km [e.g., White et al., 1992], models with oceanic crystalline crust thicker than 10 km are not proposed. The final condition on our base sampler is to impose loose lateral smoothness constraints between nodes of the proposed models. Each node that has been perturbed is smoothed with a randomly chosen subset of its neighbors that are in the same region of the model (i.e., oceanic crust, continental crust, continental crust with limited sediment coverage). This avoids testing models with unreasonably large changes in the seismic parameters between neighboring nodes in the same region Seismic Event Location Using a Probabilistic Model [21] The nonlinear problem of seismic event location using body wave traveltimes is often characterized as only weakly nonlinear and solved using iterative nonlinear approaches [e.g., Buland, 1976; Schweitzer, 2001]. A poor station distribution and a complex 3 D velocity structure can contribute to the nonlinearity of the location problem and create potential instabilities. The potential failure of linearization has lead to the formulation of numerous probabilistic approaches [e.g., Kennett and Sambridge, 1992; Billings, 1994; Lomax et al., 2000]. These approaches also provide more comprehensive location uncertainty information in the form of a probability density function. However, the estimates of location uncertainty provided by these methods do not take into account the uncertainties in the model used to predict the traveltimes. A probabilistic model, on the other hand, allows one to predict observables and their uncertainties. The distribution of an observable (i.e., its predicted value and uncertainty) given a probabilistic model consists of predicting its value for every model belonging to the set of samples that approximate the posterior distribution. Similarly, an estimate for the location uncertainty of a seismic event due to model uncertainty is obtained by locating the event for all the models that form the probabilistic model. We locate the event on an individual model using an MCMC approach to recover the posterior distribution for the location and origin time. The maximum of the posterior distribution is then used to define the location and origin time of the event for the individual model. The traveltimes are computed using the technique described in section Data [22] To ensure that the Markov chain has had adequate time to sample the model space several thousand iterations are required. The particular techniques used here to predict the data sets have therefore been chosen because they offer a good trade off between accuracy and computational efficiency. The base sampler only perturbs a part of the model; as a consequence only data that has been influenced by this part of the model has to be updated. Together with Table 2. Empirical Relationship Between Vp and Ranges for the Density Used in the Base Sampler When Perturbing the Density a Vp (km/s) Density (g/cm 3 ) a This data was obtained by digitizing Figure 18 of Breivik et al. [2002]. 6of17

7 precomputing interpolation weights, this reduces the computation time to calculate model predictions, in particular for body and surface waves. Nevertheless, the development of a probabilistic model still requires a significant amount of computational resources, in the range of several tens of cores for several days. The data sets that we include in our study are: thickness constraints, 1 D velocity profiles, gravity data, surface wave group velocities, and body wave traveltimes. Figure 4 shows which model properties we expect to be constrained by the different data sets. [23] The percentage of accepted models is an important measure when tuning an MCMC algorithm. We generally aim to tune the algorithm so that overall between 20 and 30% of the models are accepted. To achieve this we have to tune each stage so that it has an acceptance rate of about 76%. In practice the acceptance rate depends on the amount of perturbation when proposing a new model and the parameter s rel in the likelihood functions. Once we have decided on the model perturbation strategy we adjust the influence of the individual data sets by adjusting s rel. The larger s rel for a data set is, the less important the fit to this data set becomes. It is important to keep in mind that the data sets are not equally sensitive. A body wave traveltime for a given great circle path is much more sensitive to small changes in a model than a surface wave group velocity observation along the same great circle path. The body wave stage should therefore for the same s rel reject a higher percentage of models than for example the surface wave stage. On the other hand we do want a final model with an improved fit to all the data sets and no stage should dominate the inversion. Through trial and error we have chosen a set of values for s rel so that first the fit to all data sets is improved when compared to the starting model, and second that each stage has an acceptance rate between 50 and 90% with an overall acceptance rate between 20 and 30% Thickness Constraints [24] Ritzmann et al. [2007] derived for each of their geological provinces linear relationships between the thickness of sedimentary rocks and underlying crystalline rocks. The fundamental assumption behind these relationships is that the development of a sedimentary basin coincides with thinning of the underlying crystalline crust. These relationships were used in the development of BARENTS50 to adjust the thickness of the crystalline crust given the thickness of the sedimentary units where no other constraints were available. For our probabilistic approach such thickness relationships are interesting, as they provide a fast stage that allows us to exclude models which are very likely to be rejected by one of the later stages, where solving the forward problem would be computationally much more expensive. Given the proposed model, we compute a thickness for the crystalline crust based on the proposed thickness of the sedimentary units and the thickness relationships for the geological province. The proposed and computed thicknesses of the crystalline crust are then compared. The uncertainties for the thickness relationships are given by the standard deviation of the residuals of the linear regressions published by Ritzmann et al. [2007]. The second type of data used in this stage are thicknesses for the sediment layer and the crystalline crust along profiles resulting from gravity and seismic modeling taken from Minakov et al. [2009] and Matveev et al. [2007], some of which are the result of analyzing newly collected data along the earlier seismic lines used by Ritzmann et al. [2007]. The uncertainties for the thicknesses were determined by taking into account the type of data they are based on and also by assessing the quality of the velocity models. The assigned uncertainties are on average 5 km Velocity Profiles [25] The primary data for the crustal model by Ritzmann et al. [2007] was a database of 1 D velocity profiles. The velocity profiles were obtained by sampling 2 D seismic velocity transects every 25 km. They sorted the velocity profiles into four data classes from A (best) to D (least good) using 10 quality control parameters. The more quality control criteria a velocity profile fulfilled, the better the final class assigned to it. We assigned to each data class an uncertainty in Vp and classified new profiles used in this study using the same quality control parameters. The velocity transects are the results of traveltime tomography of reflection and refraction data, and they are ambiguous as they are results of their own inversion [e.g., Zelt and Smith, 1992]. Nevertheless, the velocity profiles are the best data available locally for the crust to constrain P wave velocities and interfaces. We chose a subset of the crustal scale velocity profile database of Ritzmann et al. [2007] to optimize regional coverage and data quality. We also added, where possible, new not yet published profiles [e.g., Czuba et al., 2010]. Figure 5a shows the distribution of the velocity profiles and the velocity uncertainties for the seismic lines used in this study. Given a proposed model we obtain the proposed velocity profile data using natural neighbor interpolation. The proposed and observed velocities are then compared. In the previous thickness stage we use thickness constraints from density and seismic modeling across the northern continent ocean boundary. Figure 5a shows the importance of these constraints in that they cover a part of the target region not covered by the seismic lines Gravity Data [26] Deep sedimentary basins like the East Barents Sea Basin are generally characterized by negative gravity anomalies. However, neither maps of the free air gravity nor the Bouguer gravity (Figure 5b) show significant negative gravity anomalies for the East Barents Sea or Kara Sea basins. This indicates that the mass deficiency due to sedimentary rocks having a lower density than crystalline rocks is compensated by denser material underlying these basins in the lower crust and/or upper mantle [e.g., Ritzmann and Faleide, 2009].Given free air gravity data and elevation data, Bouguer gravity data can be computed by applying a Bouguer plane and a terrain correction. Bouguer gravity data is preferable to free air gravity data as it only shows the effects of density variations in the subsurface. We used the computer code and methodology described by Fullea et al. [2008] to compile a Bouguer gravity data set covering the region of interest. North of the 68th parallel, free air gravity, bathymetry and topography data were taken from the Arctic Gravity Project ( info.nga. mil/gandg/wgs84/agp/index.html). South of the 68th parallel, ETOPO2 (V12.1) [Sandwell and Smith, 1997] wasusedfor the elevation and bathymetry data, while the free air gravity data was taken from the global satellite altimeter data compilation (V18.1) [Sandwell and Smith, 2009]. The accuracy of 7of17

8 Figure 4. Diagram showing the model properties that are constrained by each of the 5 data sets used to derive the model. For each of the data sets the model properties it is sensitive to (i.e., constrains) are highlighted in red. the satellite derived gravity of is about 4 7 mgal for random ship tracks [Sandwell and Smith, 1997]. The accuracy improves to 3 mgal when the ship track follows a satellite line that has been measured multiple times. Using this information and taking into account that in the Arctic Gravity Project various data were combined into one data set, we have set the uncertainty (i.e., standard deviation) for this data set to 5 mgal. We predict for each node of our model the Bouguer gravity by subdividing its vicinity into a set of prisms and then computing for the individual prisms their contribution to the gravity anomaly at the node using the equations given by Plouff [1976]. For this computation the water layer is filled with sediments having the density of the top of the sediment layer Surface Wave Group Velocities [27] Surface wave group velocities are mostly sensitive to the average S wave velocity structure. Surface waves with shorter periods are sensitive to shallow structure, like the extent of sedimentary basins, while longer period surface waves are more sensitive to upper mantle structure. The major drawback of surface waves is that they are not very sensitive to details of the topography of the Moho and basement. The probabilistic engine uses surface wave group velocities in two ways: first as group velocity observations along paths that are fully contained in the region (and have poor path coverage), and second as surface wave group velocity maps that cover the whole region. We use both observed paths and maps of Rayleigh waves at 18 s and every 5 s from s. Observed paths and maps of Love waves are used at the same periods and in addition also at a period of 16 s. [28] Given a proposed model, group velocities are calculated for different periods using the mineos package ( The code is known to have its limitations for periods of a few seconds. We therefore only use group velocity data for periods above 15 s. In addition the dispersion curves computed using the mineos package are checked for features that would indicate a failure of the methodology, i.e., unreasonable large or small group velocities. If the computation of a group velocity for a certain period at a given node has failed a value is obtained by interpolation from the neighboring nodes. [29] Group velocity observations for the surface wave paths used in this study are computed assuming that energy travels along the great circle paths. The surface wave paths are taken from the data sets of Levshin et al. [2007] and Pasyanos [2005]. In these data sets a quality mark according to the data quality is assigned to each observation. The three categories and our assigned traveltime uncertainties are: best (0.05 km/s), good (0.1 km/s), poor (0.2 km/s); ones with no quality estimate were estimated as good (0.1 km/s). [30] Both of the aforementioned studies cover a wider area and have a higher path density for the European Arctic. Figure 5c shows the coverage of the region with Love and Rayleigh wave paths for a period of 25 s. Using paths with a source and/or receiver outside the region of interest together with a reference model would lead to a higher path density but one would risk mapping traveltime anomalies from outside the region of interest into the target region. We therefore limit ourselves to paths that are entirely within the region of interest. The limited amount of surface wave group velocity observations for the eastern part of the region would result here in a mantle velocity model with uncertainties that are as large as the velocity anomalies observed by Levshin et al. [2007]. [31] Reducing uncertainties for the mantle model is only possible if one can make use of additional constraints, i.e., the group velocity maps from Levshin et al. [2007]. For each node the group velocity for the proposed model is compared with the one in the group velocity model of Levshin et al. [2007]. For the Rayleigh wave group velocity maps we assume an uncertainty of km/s (0.1 km/s at periods larger than 75 s or smaller than 20 s). For Love wave group velocity maps we assume an uncertainty of 0.1 km/s (0.125 km/s at periods larger than 75 s or smaller than 20 s) Body Wave Traveltimes [32] The final data set used for testing proposed models are traveltimes of regional phases, namely Pn (a P wave bottoming in the upper mantle), Pg (a P wave bottoming in the crust), Sn (an S wave bottoming in the upper mantle), and Sg (an S wave bottoming in the crust). The body wave traveltimes are limited to ground truth (GT) events [Bondár et al., 2004]. [33] Due to the use of linear transitions to represent the seismic parameters within layers, Pn and Sn can correspond 8of17

9 Figure 5. (a) Velocity profiles and thickness constraints used in this study. The velocity profiles are colored according to the uncertainty assigned to the profiles. (b) Bouguer gravity data. The dashed black line shows the continent ocean boundary and the dashed white lines the outlines of the East Barents Sea and Kara Sea basins. (c) Path coverage for Rayleigh and Love waves with a period of 25 s. Note the difference in raypath coverage between the east and west in the region of interest. (d) Path coverage for the regional P and S wave phases. 9 of 17

10 to one of two paths; either the head wave traveling along the underside of the Moho, or the ray bottoming in the upper mantle. The traveltime of the first arriving ray is used as the predicted Pn and Sn traveltimes. A total 12,635 phase readings of regional traveltimes from ground truth events along with their uncertainties have been collected at Lawrence Livermore National Laboratory. Of 1704 the events in the database 0.53% are GT1 events, 61.09% GT2 events, 0.06% GT3 events, 0.35% GT5 events, 18.08% GT15 events and 19.89% GT25 events. [34] The data set was complemented with additional data from NORSAR s published [Hicks et al., 2004; Schweitzer and Kennett, 2007; Pirli et al., 2010] and unpublished database ( We also improved the path coverage by adding 565 P observations from seismic experiments recorded at the permanent stations in the region, collected by NORSAR during the past decades [e.g., Schweitzer, 2000], and at OBS stations, temporarily deployed as part of the International Polar Year [Schweitzer, 2008]. Given the GT level and the phase for an observation an estimate for the traveltime uncertainty due to the location uncertainty is given by d/(2v), where d is the GT level in km and v the average velocity for the observed phase. The result is then added to the estimated arrival time uncertainty to obtain an overall traveltime uncertainty. We predict body wave traveltimes by sampling our model along great circle paths and performing a two dimensional ray tracing on the cross section using the computer code of Zelt and Ellis [1988]. Like for the surface waves, there are more paths in the western part than in the eastern part of the model (Figure 5d). 4. Results and Discussion [35] The aim of our MCMC algorithm is to sample the posterior distribution. We randomize the starting model to ensure that model space is explored sufficiently. The randomized starting model and the models accepted at the beginning of the MCMC run are models that are unlikely to fit the data and thus not representative for the models in the posterior distribution. Only after the MCMC run has had time to move through the model space are we actually sampling the posterior distribution. This initial phase of moving through the model space toward models that fit the data is known as burn in. Once the burn in phase is finished we have reached equilibrium and the MCMC run is sampling the posterior distribution. Based on tests, we have discarded the first 8,000 iterations to ensure that the result has reached equilibrium. To obtain our probabilistic model we have run two chains with different seeds for the random number generator. This means that the randomized starting model will look different and the base sampler will perturb the model in a different order. Once however convergence is reached the two chains look statistically similar Probabilistic Model [36] In this work a probabilistic model is a set of samples of the posterior distribution that describes the ensemble of models that fit the data. Having run two chains means that we have two independent sets of samples of the posterior distribution to define our probabilistic model. We could base our model on just one chain by using the last 1/3 of the chain, i.e., the last 4000 models. By mixing the two chains we can however be much more confident that model space has been explored sufficiently and easily verify the convergence by inspecting the similarity of the two chains, which can then be combined. Our probabilistic model is therefore given by the last 4000 models of one chain with every second model being replaced by a model from the second chain. Separately we have also performed two runs where we automatically accept all the models, i.e., sample the prior distributions. This allows us to compare prior and posterior distribution for individual nodes in our model. Figure 6 shows two Vp profiles: one in a part of the model we expect to be well constrained by the data (node 1) and one in a part of the model we expect to only be weakly constrained by the data (node 2). For both nodes the chains are well mixed, indicating that we have reached equilibrium and that the chains have become independent of their starting conditions. The range of models forming the posterior distribution for node 2 is as wide as the prior distribution. On the other hand, the range of the posterior distribution for node 1 is smaller than the prior distribution. [37] The results of this study are compared to other studies of the same region in Figures 7 and 8. While this is a probabilistic model and the overall distribution is important (as we shall see for locations), for the sake of comparison, we determine an average model. It is important to keep in mind that the various models do have different spatial resolutions and have been resampled for the comparison. Figure 7 shows the depth to Moho in this study, CRUST 2.0 [Bassin et al., 2000], BARENTS50 [Ritzmann et al., 2007] and an isostatic solution computed by Ebbing et al. [2007]. In our study we have hatched the map where the standard deviation on the depth to Moho exceeds 3 km, indicating where this parameter is less well constrained. While we find general agreement between the models, we also find intriguing differences. For example, most models see more complexity within the major tectonic provinces (oceanic crust, continental, continental shelf) than the relatively simple CRUST 2.0 model (provided for reference). The isostatic modeling of Ebbing et al. [2007] suggests, as expected, a shallower and smoother Moho than the other, seismically based models that do not assume an isostatic compensation of the crust. The models differ the most around Novaya Zemlya and in the Kara Sea. This is also where the uncertainties in the depth to Moho are generally larger than 3 km in our study. [38] Figure 8 shows a similar comparison among models for the depth to basement. Here we compare our study with CRUST 2.0 [Bassin et al., 2000], BARENTS50 [Ritzmann et al., 2007] and a compilation of depth to basement by Smelror et al. [2009] derived from magnetic and seismic data based on the work by Skilbrei [1991, 1995] and Gramberg et al. [2001]. For our study we hatched areas in the map where the standard deviation in depth to basement exceeds 2.0 km, indicating regions with poor constraints. In contrast to the depth to Moho comparison, we find significant variations among the studies. While there is some similarity between our study and BARENTS50, there are differences in the overall depth and shape of the major basins. The most likely reason for this mismatch is that the seismic parameters for sedimentary rocks at the bottom of a deep sedimentary basin like the East Barents Sea Basin are comparable to those of the underlying basement. As a consequence, the data sets 10 of 17

11 Figure 6. Model profile distribution for a node in a part of the model with (a) good constraints and (b) poor constraints. The locations of the two nodes are shown in Figure 3. Note the narrower distribution of models forming the posterior distribution for the well constrained node. used in this study are not very sensitive to the position of the interface. This could also explain the relatively large uncertainties for the southern end of the East Barents Sea Basin. In the southwestern Barents Sea the depth to basement based on magnetic and seismic data (Figure 8d) has been resolved at a 5 km spacing. Our model, on the other hand, has a node spacing of 83 km. As a consequence, our model cannot recover the basins that are only tens of kilometers wide. Interestingly, the model based mainly on magnetic data combined with seismic data (Figure 8d) shows the most pronounced East Barents Sea Basin. The CRUST 2.0 depth to basement map (Figure 8b) is significantly different from all the others. [39] In Figure 9, we examine two cross sections through the model, indicated by red lines in Figure 1. The first (A A ; Figure 9a) traverses the model from south to north. The top panel shows the mean model along the path while the bottom one shows the standard deviation of the layer velocities and layer depths. The most obvious feature is the change in crustal thickness from the Baltic Shield ( 40 km) to the continental shelf ( 30 km) and in the oceanic crust ( 10 km) in the North. Uncertainties for velocity ranges tend to be higher in the oceanic crust while depth to Moho has a higher uncertainty near the continent ocean boundary and the outer continental slope. The higher uncertainty in depth to Moho on the continental shelf side of the transition between the Baltic Shield and the continental shelf and the higher uncertainties in Vp for the crystalline crust could be related to the fact that the body wave paths are here located in the mantle. Further north we have P wave observations from the seismic experiments in the region that provide additional constraints for Vp in the crystalline crust. [40] The next cross section (B B ; Figure 9b) traverses our region from west to east. Unlike cross sections of the region further north (near Svalbard), we find a relatively rapid transition in crustal thickness across the continent ocean boundary. Within the Barents Sea region we find variations in crustal thickness of about 5 km. We see an increase in crustal thickness associated with Novaya Zemlya. The relatively high uncertainty in depth to Moho in the Kara Sea correlates with the fact that in this region the Moho is only constrained by gravity data and a velocity profile with a relatively high uncertainty, and that there are no body waves sampling the Moho. Velocity uncertainties in the crystalline crust seem to vary little along cross section B B. The expected lower uncertainties for the western side are visible further north along cross section A A. Cross section B B is in the west near the continental shelf side of the transition between the Baltic Shield and the continental shelf where we found high uncertainties on cross section A A. [41] For cross section B B we have also computed the Vp/Vs ratio (Figure 9c) to highlight the differences between Vp and Vs of the mean model. For the sediments with lower Vp velocities near the seafloor we obtain Vp/Vs ratios between 2.0 and 3.7. This is typical for water saturated and unconsolidated sediments. For the crystalline crust we observe the highest Vp/Vs ratio in the oceanic part of the cross section. Oceanic crust is mafic and has a higher Vp/Vs ratio than continental crust that tends to be felsic. Within the continental shelf part of the crystalline crust the higher Vp/Vs ratios to the east of the continent ocean transition are indicative of the magmatic activity associated with thinning of the continental crust and rifting in this region. To the east of this feature we recover a generally more felsic upper crust and a more mafic lower crust. This is typical for continental crust. Overall the Vp/Vs ratios recovered here are what one would expect for oceanic and continental crust and emphasize the importance of using a range for the Vp/Vs ratio instead of a fixed value. [42] On both cross sections the sedimentary units covering the oceanic crust have lower velocities and higher Vp/Vs ratios than the ones covering the continental shelf. This feature can be linked to uplift in the Neogene and repeated phases of glaciation in the Barents Sea during the late Pliocene and Pleistocene [Smelror et al., 2009]. Uplift and glaciation led to erosion of the Barents Shelf and the deposition of large amounts of young sediments into major submarine depositional fans along the western and northern margins. These relatively young sedimentary rocks are less consolidated and have lower seismic velocities when compared to the older sedimentary rocks covering the Barents Shelf. Interestingly the upper sedimentary units in the Kara Sea Basin show slightly reduced velocities when compared to the ones on the Barents Shelf. A possible explanation is that only during the maximum extent of glaciation in the late Pleistocene did the ice sheet reach into the Kara Sea. Therefore sedimentary units in the Kara Sea have experienced less erosion, and less compacted sedimentary units are exposed at the seafloor, possibly together with deposits from other periods of glaciation. [43] The mantle is parameterized in this study using three linear transitions. Such a parameterization is sufficient with respect to deriving a model that fits traveltimes for regional phases that sample the upper mantle. Our mean model recovers the differences in velocity between the subcontinental and suboceanic mantle, with the former exhibiting faster velocities. The surface wave velocity model of Levshin et al. [2007] showed for the upper mantle an eastward dipping high velocity structure below the Barents Sea and Kara Sea region. Possible explanations for its existence have been 11 of 17

12 Figure 7. Depth to Moho: (a) mean model obtained in this study, (b) CRUST 2.0 after Bassin et al. [2000], (c) BARENTS50 after Ritzmann et al. [2007], and (d) isostatic Moho of Ebbing et al. [2007]. discussed by Ritzmann and Faleide [2009]. We do see signs for such a high velocity structure in our probabilistic model. Nevertheless, the choice of model parameterization for the mantle in this work is far from ideal for recovering such a feature. Improving the mantle model would require using a parameterization similar to the approach of Ritzwoller et al. [2002], where the mantle is described by four cubic B spline perturbations to an average mantle model Probabilistic Location [44] While we have been examining the average properties of our model, the real power of probabilistic models is in using the whole posterior distribution to predict observables. Here we determine the predicted traveltimes from an event in the western Barents Sea to nearby stations. We then can propagate model uncertainties into location uncertainties. The station distribution is shown in Figure 10a and the distribution of the mean path velocities between the event and two selected stations is shown in Figure 10b. As we would expect, the mean velocities along longer paths, which reside primarily in the mantle, are less influenced by model uncertainties than along shorter paths that stay above the Moho. 12 of 17

13 Figure 8. Depth to basement: (a) mean model obtained in this study, (b) CRUST 2.0 after Bassin et al. [2000], (c) BARENTS50 after Ritzmann et al. [2007], and (d) compilation by the Geological Survey of Norway after Skilbrei [1991, 1995] for the western part and Gramberg et al. [2001] for the eastern part. [45] We have located the earthquake using each of our models in the posterior distribution. This provides us with 4000 locations: one for each model forming our probabilistic model. This allows us to estimate the location uncertainty that results from model uncertainties; we do not aim to verify our model by locating this event. A 1 D location for the event ( N E) was obtained previously by M. Pirli et al. (personal communication, 2010) using a 1 D velocity model, BAREY [Hicks et al., 2004], to predict traveltimes. Figure 10c shows the location distribution thereby providing an estimate for the location uncertainty from model uncertainty alone. All the stations lie to the west of the event resulting for the 1 D velocity model solution in an error ellipse with the major half axis in the east west direction. The point cloud of locations projected on the Earth s surface is also elongated in the east west direction with a narrower end in the west when compared with the eastern end. There is a linear trend between late deep event locations to the southwest and early shallow locations to the northeast. Bondár et al. [2004] showed that for an excellent station distribution, depth and origin time are more sensitive to the velocity model than the epicenter location. For an 13 of 17

14 Figure 9. (a) South north (A A ) and (b) west east (B B ) cross section through the probabilistic model. For each cross section the top panel shows Vp and the bottom panel shows the uncertainty in Vp. In the bottom panels the interfaces are colored according to the uncertainty in depth. (c) The Vp/Vs ratio for the west east cross section B B. Note that the same color is used to map Vp/Vs ratios equal or larger than of 17

15 Figure 10. Probabilistic location example taking model uncertainties into account: (a) station distribution, (b) distribution of average path velocities for regional phases between the event and two stations used in the location example, and (c) hypocenter and origin time of the earthquake computed for each of the 4000 models forming our probabilistic model. The black circle marks the location of the event computed using a 1 D velocity model, BAREY [Hicks et al., 2004], and a fixed depth of 0 km. The error ellipse is given by the gray shaded area. The black diamond marks the mean of our set of locations. The points are colored according to the deviation from the mean origin time of our set of locations. uneven station distribution, as shown here, the epicenter location seems to be equally sensitive to the velocity model as origin time and depth. 5. Conclusions [46] The probabilistic approach employed here for the development of a data driven regional seismic model is a promising technique that provides a number of advantages compared to traditional approaches. Among them are (1) the ability to easily reconcile different types of geophysical data and (2) robust estimates for the uncertainty of individual model parameters. In addition to providing images of the subsurface together with estimates of uncertainties, a probabilistic model allows prediction of observables and their uncertainties. We have shown that this can be used to derive seismic event location uncertainties from model uncertainties. This can in the future be used for the formulation of location algorithms that take model uncertainties in addition to uncertainties in onset time into account. [47] When comparing the mean model of our posterior distribution with other models that cover the region we find that our model has captured the features that can be resolved with a node spacing of 83 km. The data quality and density varies in the region to a degree that would for future models justify the use of a self adapting model parameterization, where we would as part of the inversion try to find the spatial distributions of nodes that allow to fit the data better than others. Part of optimizing the model parameterization is finding the optimum number of 1 D models needed to constrain the structure. Transdimensional MCMC algorithms [e.g., Green, 1995] provide the framework for situations where the optimum number of model parameters is unknown. [48] While the crustal structure in the European Arctic is relatively well known, the nature of the upper mantle, in particular of the high velocity anomaly under the Barents Sea and Kara Sea, is still the subject of ongoing research. Addressing this question would require the development of a probabilistic model for composition and temperature instead of velocity and density. This would then allow defining the prior distribution in such a way that it only contains models that are plausible from a geodynamical and compositional point of view. [49] Acknowledgments. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE AC52 07NA This is LLNL contribution LLNL JRNL Stephen Myers (LLNL) contributed the ground truth data. We thank The Norwegian Metacenter for Computational Science (NOTUR) for providing the necessary computational resources on the Titan III high performance computing facilities. Figures were generated using the Generic Mapping Tools (GMT) software [Wessel and Smith, 1998]. The Geological Survey of Norway (NGU) is thanked for providing the depth to Moho and depth to basement data shown in Figures 7d and 8d. We thank Anatoli Levshin and Christian Weidle for providing us the phase readings and group velocity maps published by Levshin et al. [2007]. 15 of 17