Showing papers in "Nonlinear Processes in Geophysics in 2019"

Data assimilation as a learning tool to infer ordinary differential equation representations of dynamical models

Abstract: . Recent progress in machine learning has shown how to forecast and, to some extent, learn the dynamics of a model from its output, resorting in particular to neural networks and deep learning techniques. We will show how the same goal can be directly achieved using data assimilation techniques without leveraging on machine learning software libraries, with a view to high-dimensional models. The dynamics of a model are learned from its observation and an ordinary differential equation (ODE) representation of this model is inferred using a recursive nonlinear regression. Because the method is embedded in a Bayesian data assimilation framework, it can learn from partial and noisy observations of a state trajectory of the physical model. Moreover, a space-wise local representation of the ODE system is introduced and is key to coping with high-dimensional models. It has recently been suggested that neural network architectures could be interpreted as dynamical systems. Reciprocally, we show that our ODE representations are reminiscent of deep learning architectures. Furthermore, numerical analysis considerations of stability shed light on the assets and limitations of the method. The method is illustrated on several chaotic discrete and continuous models of various dimensions, with or without noisy observations, with the goal of identifying or improving the model dynamics, building a surrogate or reduced model, or producing forecasts solely from observations of the physical model.

Unravelling the spatial diversity of Indian precipitation teleconnections via a non-linear multi-scale approach

Abstract: . A better understanding of precipitation dynamics in the Indian subcontinent is required since India's society depends heavily on reliable monsoon forecasts. We introduce a non-linear, multiscale approach, based on wavelets and event synchronization, for unravelling teleconnection influences on precipitation. We consider those climate patterns with the highest relevance for Indian precipitation. Our results suggest significant influences which are not well captured by only the wavelet coherence analysis, the state-of-the-art method in understanding linkages at multiple timescales. We find substantial variation across India and across timescales. In particular, El Nino–Southern Oscillation (ENSO) and the Indian Ocean Dipole (IOD) mainly influence precipitation in the south-east at interannual and decadal scales, respectively, whereas the North Atlantic Oscillation (NAO) has a strong connection to precipitation, particularly in the northern regions. The effect of the Pacific Decadal Oscillation (PDO) stretches across the whole country, whereas the Atlantic Multidecadal Oscillation (AMO) influences precipitation particularly in the central arid and semi-arid regions. The proposed method provides a powerful approach for capturing the dynamics of precipitation and, hence, helps improve precipitation forecasting.

Generalization properties of feed-forward neural networks trained on Lorenz systems

Abstract: . Neural networks are able to approximate chaotic dynamical systems when provided with training data that cover all relevant regions of the system's phase space. However, many practical applications diverge from this idealized scenario. Here, we investigate the ability of feed-forward neural networks to (1) learn the behavior of dynamical systems from incomplete training data and (2) learn the influence of an external forcing on the dynamics. Climate science is a real-world example where these questions may be relevant: it is concerned with a non-stationary chaotic system subject to external forcing and whose behavior is known only through comparatively short data series. Our analysis is performed on the Lorenz63 and Lorenz95 models. We show that for the Lorenz63 system, neural networks trained on data covering only part of the system's phase space struggle to make skillful short-term forecasts in the regions excluded from the training. Additionally, when making long series of consecutive forecasts, the networks struggle to reproduce trajectories exploring regions beyond those seen in the training data, except for cases where only small parts are left out during training. We find this is due to the neural network learning a localized mapping for each region of phase space in the training data rather than a global mapping. This manifests itself in that parts of the networks learn only particular parts of the phase space. In contrast, for the Lorenz95 system the networks succeed in generalizing to new parts of the phase space not seen in the training data. We also find that the networks are able to learn the influence of an external forcing, but only when given relatively large ranges of the forcing in the training. These results point to potential limitations of feed-forward neural networks in generalizing a system's behavior given limited initial information. Much attention must therefore be given to designing appropriate train-test splits for real-world applications.

Revising the stochastic iterative ensemble smoother

Abstract: . Ensemble randomized maximum likelihood (EnRML) is an iterative (stochastic) ensemble smoother, used for large and nonlinear inverse problems, such as history matching and data assimilation. Its current formulation is overly complicated and has issues with computational costs, noise, and covariance localization, even causing some practitioners to omit crucial prior information. This paper resolves these difficulties and streamlines the algorithm without changing its output. These simplifications are achieved through the careful treatment of the linearizations and subspaces. For example, it is shown (a) how ensemble linearizations relate to average sensitivity and (b) that the ensemble does not lose rank during updates. The paper also draws significantly on the theory of the (deterministic) iterative ensemble Kalman smoother (IEnKS). Comparative benchmarks are obtained with the Lorenz 96 model with these two smoothers and the ensemble smoother using multiple data assimilation (ES-MDA).

Lyapunov analysis of multiscale dynamics: the slow bundle of the two-scale Lorenz 96 model

Abstract: . We investigate the geometrical structure of instabilities in the two-scale Lorenz 96 model through the prism of Lyapunov analysis. Our detailed study of the full spectrum of covariant Lyapunov vectors reveals the presence of a slow bundle in tangent space, composed by a set of vectors with a significant projection onto the slow degrees of freedom; they correspond to the smallest (in absolute value) Lyapunov exponents and thereby to the longer timescales. We show that the dimension of the slow bundle is extensive in the number of both slow and fast degrees of freedom and discuss its relationship with the results of a finite-size analysis of instabilities, supporting the conjecture that the slow-variable behavior is effectively determined by a nontrivial subset of degrees of freedom. More precisely, we show that the slow bundle corresponds to the Lyapunov spectrum region where fast and slow instability rates overlap, “mixing” their evolution into a set of vectors which simultaneously carry information on both scales. We suggest that these results may pave the way for future applications to ensemble forecasting and data assimilations in weather and climate models.

Inverting Rayleigh surface wave velocities for crustal thickness in eastern Tibet and the western Yangtze craton based on deep learning neural networks

Abstract: . Crustal thickness is an important factor affecting lithospheric structure and deep geodynamics. In this paper, a deep learning neural network based on a stacked sparse auto-encoder is proposed for the inversion of crustal thickness in eastern Tibet and the western Yangtze craton. First, with the phase velocity of the Rayleigh surface wave as input and the theoretical crustal thickness as output, 12 deep-sSAE neural networks are constructed, which are trained by 380 000 and tested by 120 000 theoretical models. We then invert the observed phase velocities through these 12 neural networks. According to the test error and misfit of other crustal thickness models, the optimal crustal thickness model is selected as the crustal thickness of the study area. Compared with other ways to detect crustal thickness such as seismic wave reflection and receiver function, we adopt a new way for inversion of earth model parameters, and realize that a deep learning neural network based on data driven with the highly non-linear mapping ability can be widely used by geophysicists, and our result has good agreement with high-resolution crustal thickness models. Compared with other methods, our experimental results based on a deep learning neural network and a new Rayleigh wave phase velocity model reveal some details: there is a northward-dipping Moho gradient zone in the Qiangtang block and a relatively shallow north-west–south-east oriented crust at the Songpan–Ganzi block. Crustal thickness around Xi'an and the Ordos basin is shallow, about 35 km. The change in crustal thickness in the Sichuan–Yunnan block is sharp, where crustal thickness is 60 km north-west and 35 km south-east. We conclude that the deep learning neural network is a promising, efficient, and believable geophysical inversion tool.

Denoising stacked autoencoders for transient electromagnetic signal denoising

Abstract: . The transient electromagnetic method (TEM) is extremely important in geophysics. However, the secondary field signal (SFS) in the TEM received by coil is easily disturbed by random noise, sensor noise and man-made noise, which results in the difficulty in detecting deep geological information. To reduce the noise interference and detect deep geological information, we apply autoencoders, which make up an unsupervised learning model in deep learning, on the basis of the analysis of the characteristics of the SFS to denoise the SFS. We introduce the SFSDSA (secondary field signal denoising stacked autoencoders) model based on deep neural networks of feature extraction and denoising. SFSDSA maps the signal points of the noise interference to the high-probability points with a clean signal as reference according to the deep characteristics of the signal, so as to realize the signal denoising and reduce noise interference. The method is validated by the measured data comparison, and the comparison results show that the noise reduction method can (i) effectively reduce the noise of the SFS in contrast with the Kalman, principal component analysis (PCA) and wavelet transform methods and (ii) strongly support the speculation of deeper underground features.

A comprehensive model for the kyr and Myr timescales of Earth's axial magnetic dipole field

Abstract: . We consider a stochastic differential equation model for Earth's axial magnetic dipole field. Our goal is to estimate the model's parameters using diverse and independent data sources that had previously been treated separately, so that the model is a valid representation of an expanded paleomagnetic record on kyr to Myr timescales. We formulate the estimation problem within the Bayesian framework and define a feature-based posterior distribution that describes probabilities of model parameters given a set of features derived from the data. Numerically, we use Markov chain Monte Carlo (MCMC) to obtain a sample-based representation of the posterior distribution. The Bayesian problem formulation and its MCMC solution allow us to study the model's limitations and remaining posterior uncertainties. Another important aspect of our overall approach is that it reveals inconsistencies between model and data or within the various data sets. Identifying these shortcomings is a first and necessary step towards building more sophisticated models or towards resolving inconsistencies within the data. The stochastic model we derive represents selected aspects of the long-term behavior of the geomagnetic dipole field with limitations and errors that are well defined. We believe that such a model is useful (besides its limitations) for hypothesis testing and give a few examples of how the model can be used in this context.

A Bayesian approach to multivariate adaptive localization in ensemble-based data assimilation with time-dependent extensions

Abstract: Ever since its inception, the ensemble Kalman filter (EnKF) has elicited many heuristic approaches that sought to improve it One such method is covariance localization, which alleviates spurious correlations due to finite ensemble sizes by using relevant spatial correlation information Adaptive localization techniques account for how correlations change in time and space, in order to obtain improved covariance estimates This work develops a Bayesian approach to adaptive Schur-product localization for the deterministic ensemble Kalman filter (DEnKF) and extends it to support multiple radii of influence We test the proposed adaptive localization using the toy Lorenz'96 problem and a more realistic 15-layer quasi-geostrophic model Results with the toy problem show that the multivariate approach informs us that strongly observed variables can tolerate larger localization radii The univariate approach leads to markedly improved filter performance for the realistic geophysical model, with a reduction in error by as much as 33 %

Compacting the description of a time-dependent multivariable system and its multivariable driver by reducing the state vectors to aggregate scalars: the Earth's solar-wind-driven magnetosphere

Abstract: . Using the solar-wind-driven magnetosphere–ionosphere–thermosphere system, a methodology is developed to reduce a state-vector description of a time-dependent driven system to a composite scalar picture of the activity in the system. The technique uses canonical correlation analysis to reduce the time-dependent system and driver state vectors to time-dependent system and driver scalars, with the scalars describing the response in the system that is most-closely related to the driver. This reduced description has advantages: low noise, high prediction efficiency, linearity in the described system response to the driver, and compactness. The methodology identifies independent modes of reaction of a system to its driver. The analysis of the magnetospheric system is demonstrated. Using autocorrelation analysis, Jensen–Shannon complexity analysis, and permutation-entropy analysis the properties of the derived aggregate scalars are assessed and a new mode of reaction of the magnetosphere to the solar wind is found. This state-vector-reduction technique may be useful for other multivariable systems driven by multiple inputs.

A fast approximation for 1-D inversion of transient electromagnetic data by using a back propagation neural network and improved particle swarm optimization

Abstract: . As one of the most active nonlinear inversion methods in transient electromagnetic (TEM) inversion, the back propagation (BP) neural network has high efficiency because the complicated forward model calculation is unnecessary in iteration. The global optimization ability of the particle swarm optimization (PSO) is adopted for amending the BP's sensitivity to its initial parameters, which avoids it falling into a local optimum. A chaotic-oscillation inertia weight PSO (COPSO) is proposed for accelerating convergence. The COPSO-BP algorithm performance is validated by two typical testing functions, two geoelectric models inversions and a field example. The results show that the COPSO-BP method is more accurate, stable and needs relatively less training time. The proposed algorithm has a higher fitting degree for the data inversion, and it is feasible to use it in geophysical inverse applications.

Data assimilation using adaptive, non-conservative, moving mesh models

Abstract: . Numerical models solved on adaptive moving meshes have become increasingly prevalent in recent years. Motivating problems include the study of fluids in a Lagrangian frame and the presence of highly localized structures such as shock waves or interfaces. In the former case, Lagrangian solvers move the nodes of the mesh with the dynamical flow; in the latter, mesh resolution is increased in the proximity of the localized structure. Mesh adaptation can include remeshing, a procedure that adds or removes mesh nodes according to specific rules reflecting constraints in the numerical solver. In this case, the number of mesh nodes will change during the integration and, as a result, the dimension of the model's state vector will not be conserved. This work presents a novel approach to the formulation of ensemble data assimilation (DA) for models with this underlying computational structure. The challenge lies in the fact that remeshing entails a different state space dimension across members of the ensemble, thus impeding the usual computation of consistent ensemble-based statistics. Our methodology adds one forward and one backward mapping step before and after the ensemble Kalman filter (EnKF) analysis, respectively. This mapping takes all the ensemble members onto a fixed, uniform reference mesh where the EnKF analysis can be performed. We consider a high-resolution (HR) and a low-resolution (LR) fixed uniform reference mesh, whose resolutions are determined by the remeshing tolerances. This way the reference meshes embed the model numerical constraints and are also upper and lower uniform meshes bounding the resolutions of the individual ensemble meshes. Numerical experiments are carried out using 1-D prototypical models: Burgers and Kuramoto–Sivashinsky equations and both Eulerian and Lagrangian synthetic observations. While the HR strategy generally outperforms that of LR, their skill difference can be reduced substantially by an optimal tuning of the data assimilation parameters. The LR case is appealing in high dimensions because of its lower computational burden. Lagrangian observations are shown to be very effective in that fewer of them are able to keep the analysis error at a level comparable to the more numerous observers for the Eulerian case. This study is motivated by the development of suitable EnKF strategies for 2-D models of the sea ice that are numerically solved on a Lagrangian mesh with remeshing.

Statistical hypothesis testing in wavelet analysis: theoretical developments and applications to Indian rainfall

Abstract: . Statistical hypothesis tests in wavelet analysis are methods that assess the degree to which a wavelet quantity (e.g., power and coherence) exceeds background noise. Commonly, a point-wise approach is adopted in which a wavelet quantity at every point in a wavelet spectrum is individually compared to the critical level of the point-wise test. However, because adjacent wavelet coefficients are correlated and wavelet spectra often contain many wavelet quantities, the point-wise test can produce many false positive results that occur in clusters or patches. To circumvent the point-wise test drawbacks, it is necessary to implement the recently developed area-wise, geometric, cumulative area-wise, and topological significance tests, which are reviewed and developed in this paper. To improve the computational efficiency of the cumulative area-wise test, a simplified version of the testing procedure is created based on the idea that its output is the mean of individual estimates of statistical significance calculated from the geometric test applied at a set of point-wise significance levels. Ideal examples are used to show that the geometric and cumulative area-wise tests are unable to differentiate wavelet spectral features arising from singularity-like structures from those associated with periodicities. A cumulative arc-wise test is therefore developed to strictly test for periodicities by using normalized arclength, which is defined as the number of points composing a cross section of a patch divided by the wavelet scale in question. A previously proposed topological significance test is formalized using persistent homology profiles (PHPs) measuring the number of patches and holes corresponding to the set of all point-wise significance values. Ideal examples show that the PHPs can be used to distinguish time series containing signal components from those that are purely noise. To demonstrate the practical uses of the existing and newly developed statistical methodologies, a first comprehensive wavelet analysis of Indian rainfall is also provided. An R software package has been written by the author to implement the various testing procedures.

A prototype stochastic parameterization of regime behaviour in the stably stratified atmospheric boundary layer

Abstract: . Recent research has demonstrated that hidden Markov model (HMM) analysis is an effective tool to classify atmospheric observations of the stably stratified nocturnal boundary layer (SBL) into weakly stable (wSBL) and very stable (vSBL) regimes. Here we consider the development of explicitly stochastic representations of SBL regime dynamics. First, we analyze whether HMM-based SBL regime statistics (the occurrence of regime transitions, subsequent transitions after the first, and very persistent nights) can be accurately represented by “freely running” stationary Markov chains (FSMCs). Our results show that despite the HMM-estimated regime statistics being relatively insensitive to the HMM transition probabilities, these statistics cannot all simultaneously be captured by a FSMC. Furthermore, by construction a FSMC cannot capture the observed non-Markov regime duration distributions. Using the HMM classification of data into wSBL and vSBL regimes, state-dependent transition probabilities conditioned on the bulk Richardson number (RiB) or the stratification are investigated. We find that conditioning on stratification produces more robust results than conditioning on RiB . A prototype explicitly stochastic parameterization is developed based on stratification-dependent transition probabilities, in which turbulence pulses (representing intermittent turbulence events) are added during vSBL conditions. Experiments using an idealized single-column model demonstrate that such an approach can simulate realistic-looking SBL regime dynamics.

Non-Gaussian statistics in global atmospheric dynamics: a study with a 10 240-member ensemble Kalman filter using an intermediate atmospheric general circulation model

Abstract: . We previously performed local ensemble transform Kalman filter (LETKF) experiments with up to 10 240 ensemble members using an intermediate atmospheric general circulation model (AGCM). While the previous study focused on the impact of localization on the analysis accuracy, the present study focuses on the probability density functions (PDFs) represented by the 10 240-member ensemble. The 10 240-member ensemble can resolve the detailed structures of the PDFs and indicates that non-Gaussianity is caused in those PDFs by multimodality and outliers. The results show that the spatial patterns of the analysis errors are similar to those of non-Gaussianity. While the outliers appear randomly, large multimodality corresponds well with large analysis error, mainly in the tropical regions and storm track regions where highly nonlinear processes appear frequently. Therefore, we further investigate the life cycle of multimodal PDFs, and show that they are mainly generated by the on–off switch of convective parameterization in the tropical regions and by the instability associated with advection in the storm track regions. Sensitivity to the ensemble size suggests that approximately 1000 ensemble members are necessary in the intermediate AGCM-LETKF system to represent the detailed structures of non-Gaussian PDFs such as skewness and kurtosis; the higher-order non-Gaussian statistics are more vulnerable to the sampling errors due to a smaller ensemble size.

Statistical post-processing of ensemble forecasts of the height of new snow

Abstract: . Forecasting the height of new snow (HN) is crucial for avalanche hazard forecasting, road viability, ski resort management and tourism attractiveness. Meteo-France operates the PEARP-S2M probabilistic forecasting system, including 35 members of the PEARP Numerical Weather Prediction system, where the SAFRAN downscaling tool refines the elevation resolution and the Crocus snowpack model represents the main physical processes in the snowpack. It provides better HN forecasts than direct NWP diagnostics but exhibits significant biases and underdispersion. We applied a statistical post-processing to these ensemble forecasts, based on non-homogeneous regression with a censored shifted Gamma distribution. Observations come from manual measurements of 24 h HN in the French Alps and Pyrenees. The calibration is tested at the station scale and the massif scale (i.e. aggregating different stations over areas of 1000 km 2 ). Compared to the raw forecasts, similar improvements are obtained for both spatial scales. Therefore, the post-processing can be applied at any point of the massifs. Two training datasets are tested: (1) a 22-year homogeneous reforecast for which the NWP model resolution and physical options are identical to the operational system but without the same initial perturbations; (2) 3-year real-time forecasts with a heterogeneous model configuration but the same perturbation methods. The impact of the training dataset depends on lead time and on the evaluation criteria. The long-term reforecast improves the reliability of severe snowfall but leads to overdispersion due to the discrepancy in real-time perturbations. Thus, the development of reliable automatic forecasting products of HN needs long reforecasts as homogeneous as possible with the operational systems.

Characterization of the South Atlantic Anomaly

Abstract: . This research intends to characterize the South Atlantic Anomaly (SAA) by applying the power spectrum analysis approach. The motivation to study the SAA region is due to its nature. A comparison was made between the stations in the SAA region and outside the SAA region during the geomagnetic storm occurrence (active period) and the normal period where no geomagnetic storm occurred. The horizontal component of the data of the Earth's magnetic field for the occurrence of the active period was taken on 11 March 2011 while for the normal period it was taken on 3 February 2011. The data sample rate used is 1 min. The outcome of the research revealed that the SAA region had a tendency to be persistent during both periods. It can be said that the region experiences these characteristics because of the Earth's magnetic field strength. Through the research, it is found that as the Earth's magnetic field increases, it is likely to show an antipersistent value. This is found in the high-latitude region. The lower the Earth's magnetic field, the more it shows the persistent value as in the middle latitude region. In the region where the Earth's magnetic field is very low like the SAA region it shows a tendency to be persistent.

Numerical bifurcation methods applied to climate models: analysis beyond simulation

Abstract: . In this special issue contribution, I provide a personal view on the role of bifurcation analysis of climate models in the development of a theory of climate system variability. The state of the art of the methodology is shortly outlined, and the main part of the paper deals with examples of what has been done and what has been learned. In addressing these issues, I will discuss the role of a hierarchy of climate models, concentrate on results for spatially extended (stochastic) models (having many degrees of freedom) and evaluate the importance of these results for a theory of climate system variability.

Fluctuations of finite-time Lyapunov exponents in an intermediate-complexity atmospheric model: a multivariate and large-deviation perspective

Abstract: . The stability properties as characterized by the fluctuations of finite-time Lyapunov exponents around their mean values are investigated in a three-level quasi-geostrophic atmospheric model with realistic mean state and variability. Firstly, the covariance structure of the fluctuation field is examined. In order to identify dominant patterns of collective excitation, an empirical orthogonal function (EOF) analysis of the fluctuation field of all of the finite-time Lyapunov exponents is performed. The three leading modes are patterns where the most unstable Lyapunov exponents fluctuate in phase. These modes are virtually independent of the integration time of the finite-time Lyapunov exponents. Secondly, large-deviation rate functions are estimated from time series of finite-time Lyapunov exponents based on the probability density functions and using the Legendre transform method. Serial correlation in the time series is properly accounted for. A large-deviation principle can be established for all of the Lyapunov exponents. Convergence is rather slow for the most unstable exponent, becomes faster when going further down in the Lyapunov spectrum, is very fast for the near-neutral and weakly dissipative modes, and becomes slow again for the strongly dissipative modes at the end of the Lyapunov spectrum. The curvature of the rate functions at the minimum is linked to the corresponding elements of the diffusion matrix. Also, the joint large-deviation rate function for the first and the second Lyapunov exponent is estimated.

The adaptive particle swarm optimization technique for solving microseismic source location parameters

Abstract: . An intelligent method is presented for locating a microseismic source based on the particle swarm optimization (PSO) concept. It eliminates microseismic source locating errors caused by the inaccurate velocity model of the earth medium. The method uses, as the target of PSO, a global minimum of the sum of squared discrepancies between differences of modeled arrival times and differences of measured arrival times. The discrepancies are calculated for all pairs of detectors of a seismic monitoring system. Then, the adaptive PSO algorithm is applied to locate the microseismic source and obtain optimal value of the P-wave velocity. The PSO algorithm adjusts inertia weight, accelerating constants, the maximum flight velocity of particles, and other parameters to avoid the PSO algorithm trapping by local optima during the solution process. The origin time of the microseismic event is estimated by minimizing the sum of squared discrepancies between the modeled arrival times and the measured arrival times. This sum is calculated using the obtained estimates of the microseismic source coordinates and P-wave velocity. The effectiveness of the PSO algorithm was verified through inversion of a theoretical model and two analyses of actual data from mine blasts in different locations. Compared with the classic least squares method (LSM), the PSO algorithm displays faster convergence and higher accuracy of microseismic source location. Moreover, there is no need to measure the microseismic wave velocity in advance: the PSO algorithm eliminates the adverse effects caused by error in the P-wave velocity when locating a microseismic source using traditional methods.

Application of fractal models to delineate mineralized zones in the Pulang porphyry copper deposit, Yunnan, southwestern China

Abstract: . The aim of this study is to delineate and identify various mineralized zones and barren host rocks based on surface and subsurface lithogeochemical data from the Pulang porphyry copper deposit, southwestern China, utilizing the number–size (N-S), concentration–volume (C-V) and power-spectrum–volume (S-V) fractal models. The N-S model reveals three mineralized zones characterized by Cu thresholds of 0.28 % and 1.45 %: % Cu represents weakly mineralized zones and barren host rocks, 0.28 %–1.45 % Cu represents moderately mineralized zones, and > 1.45 % Cu represents highly mineralized zones. The results obtained by the C-V model depict four geochemical zones defined by Cu thresholds of 0.25 %, 1.48 % and 1.88 %, representing nonmineralized wall rocks ( Cu %), weakly mineralized zones (0.25 %–1.48 %), moderately mineralized zones (1.48 %–1.88 %) and highly mineralized zones ( Cu>1.88 %). The S-V model is used by performing a 3-D fast Fourier transformation of assay data in the frequency domain. The S-V model reveals three mineralized zones characterized by Cu thresholds of 0.23 % and 1.33 %: % Cu represents leached zones and barren host rocks, 0.23 %–1.33 % Cu represents hypogene zones, and >1.33 % Cu represents supergene enrichment zones. All the multifractal models indicate that high-grade mineralization occurs in the central and southern parts of the ore deposit. Their results are compared with the alteration and mineralogical models resulting from the 3-D geological model using a log-ratio matrix. The results show that the S-V model is best at identifying highly mineralized zones in the deposit. However, the results of the C-V model for moderately and weakly mineralized zones are also more accurate than those obtained from the N-S and S-V models.

Competition between chaotic advection and diffusion: stirring and mixing in a 3-D eddy model

Abstract: . The importance of chaotic advection relative to turbulent diffusion is investigated in an idealized model of a 3-D swirling and overturning ocean eddy. Various measures of stirring and mixing are examined in order to determine when and where chaotic advection is relevant. Turbulent diffusion is alternatively represented by (1) an explicit, observation-based, scale-dependent diffusivity, (2) stochastic noise, added to a deterministic velocity field, or (3) explicit and implicit diffusion in a spectral numerical model of the Navier–Stokes equations. Lagrangian chaos in our model occurs only within distinct regions of the eddy, including a large chaotic “sea” that fills much of the volume near the perimeter and central axis of the eddy and much smaller “resonant” bands. The size and distribution of these regions depend on factors such as the degree of axial asymmetry of the eddy and the Ekman number. The relative importance of chaotic advection and turbulent diffusion within the chaotic regions is quantified using three measures: the Lagrangian Batchelor scale, the rate of dispersal of closely spaced fluid parcels, and the Nakamura effective diffusivity. The role of chaotic advection in the stirring of a passive tracer is generally found to be most important within the larger chaotic seas, at intermediate times, with small diffusivities, and for eddies with strong asymmetry. In contrast, in thin chaotic regions, turbulent diffusion at oceanographically relevant rates is at least as important as chaotic advection. Future work should address anisotropic and spatially varying representations of turbulent diffusion for more realistic models.

Joint state-parameter estimation of a nonlinear stochastic energy balance model from sparse noisy data

Abstract: . While nonlinear stochastic partial differential equations arise naturally in spatiotemporal modeling, inference for such systems often faces two major challenges: sparse noisy data and ill-posedness of the inverse problem of parameter estimation. To overcome the challenges, we introduce a strongly regularized posterior by normalizing the likelihood and by imposing physical constraints through priors of the parameters and states. We investigate joint parameter-state estimation by the regularized posterior in a physically motivated nonlinear stochastic energy balance model (SEBM) for paleoclimate reconstruction. The high-dimensional posterior is sampled by a particle Gibbs sampler that combines a Markov chain Monte Carlo (MCMC) method with an optimal particle filter exploiting the structure of the SEBM. In tests using either Gaussian or uniform priors based on the physical range of parameters, the regularized posteriors overcome the ill-posedness and lead to samples within physical ranges, quantifying the uncertainty in estimation. Due to the ill-posedness and the regularization, the posterior of parameters presents a relatively large uncertainty, and consequently, the maximum of the posterior, which is the minimizer in a variational approach, can have a large variation. In contrast, the posterior of states generally concentrates near the truth, substantially filtering out observation noise and reducing uncertainty in the unconstrained SEBM.

Exploring the sensitivity of Northern Hemisphere atmospheric circulation to different surface temperature forcing using a statistical-dynamical atmospheric model

Abstract: . Climate and weather conditions in the mid-latitudes are strongly driven by the large-scale atmosphere circulation. Observational data indicate that important components of the large-scale circulation have changed in recent decades, including the strength and the width of the Hadley cell, jets, storm tracks and planetary waves. Here, we use a new statistical–dynamical atmosphere model (SDAM) to test the individual sensitivities of the large-scale atmospheric circulation to changes in the zonal temperature gradient, meridional temperature gradient and global-mean temperature. We analyze the Northern Hemisphere Hadley circulation, jet streams, storm tracks and planetary waves by systematically altering the zonal temperature asymmetry, the meridional temperature gradient and the global-mean temperature. Our results show that the strength of the Hadley cell, storm tracks and jet streams depend, in terms of relative changes, almost linearly on both the global-mean temperature and the meridional temperature gradient, whereas the zonal temperature asymmetry has little or no influence. The magnitude of planetary waves is affected by all three temperature components, as expected from theoretical dynamical considerations. The width of the Hadley cell behaves nonlinearly with respect to all three temperature components in the SDAM. Moreover, some of these observed large-scale atmospheric changes are expected from dynamical equations and are therefore an important part of model validation.

On fluctuating momentum exchange in idealised models of air–sea interaction

Abstract: . The dynamics of three local models, for momentum transfer at the air–sea interface, is compared. The models differ by whether or not the ocean velocity is included in the shear calculation applied to the ocean and the atmosphere. All three cases are employed in climate or ocean simulations. Analytic calculations for the models with deterministic and random forcing (white and coloured) are presented. The short-term behaviour is similar in all models, with only small quantitative differences, while the long-term behaviour differs qualitatively between the models. The fluctuation–dissipation relation, which connects the fast atmospheric motion to the slow oceanic dynamics, is established for all models with random forcing. The fluctuation–dissipation theorem, which compares the response to an external forcing to internal fluctuations, is established for a white-noise forcing and a coloured forcing when the phase space is augmented by the forcing variable. Using results from numerical integrations of stochastic differential equations, we show that the fluctuation theorem, which compares the probability of positive to negative fluxes of the same magnitude, averaged over time intervals of varying lengths, holds for the energy gained by the ocean from the atmosphere.

Mahalanobis distance-based recognition of changes in the dynamics of a seismic process

Abstract: . In the present work, we aim to analyse the regularity of a seismic process based on its spatial, temporal, and energetic characteristics. Increments of cumulative times, increments of cumulative distances, and increments of cumulative seismic energies are calculated from an earthquake catalogue for southern California from 1975 to 2017. As the method of analysis, we use the multivariate Mahalanobis distance calculation, combined with a surrogate data testing procedure that is often used for the testing of non-linear structures in complex data sets. Before analysing the dynamical features of the seismic process, we tested the used approach for two different 3-D models in which the dynamical features were changed from more regular to more randomised conditions by adding a certain degree of noise. An analysis of the variability in the extent of regularity of the seismic process was carried out for different completeness magnitude thresholds. The results of our analysis show that in about a third of all the 50-data windows the original seismic process was indistinguishable from a random process based on its features of temporal, spatial, and energetic variability. It was shown that prior to the occurrence of strong earthquakes, mostly in periods of generation of relatively small earthquakes, the percentage of windows in which the seismic process is indistinguishable from a random process increases (to 60 %–80 %). During periods of aftershock activity, the process of small earthquake generation became regular in all of the windows considered, and thus was markedly different from the randomised catalogues. In some periods within the catalogue, the seismic process appeared to be closer to randomness, while in other cases it became closer to a regular behaviour. More specifically, in periods of relatively decreased earthquake generation activity (with low energy release), the seismic process appears to be random, while during periods of occurrence of strong events, followed by series of aftershocks, significant deviation from randomness is shown, i.e. the extent of regularity markedly increases. The period for which such deviation from random behaviour lasts depends on the amount of seismic energy released by the strong earthquake.

Negentropy anomaly analysis of the borehole strain associated with the M s 8.0 Wenchuan earthquake

Abstract: . A large earthquake of 8.0 magnitude occurred on 12 May 2008, 14:28 UTC, with the epicentre in Wenchuan. To investigate the pre-earthquake anomalous strain changes, negentropy is introduced to borehole strain data for three locations, approximated by skewness and kurtosis, revealing the non-Gaussianity of recorded fluctuations. We separate the negentropy anomalies from the background by Otsu's method and accumulate the anomaly frequency on different scales. The results show that the long-term cumulative frequency of negentropy anomalies follows a sigmoid behaviour, while the inflection point of the fitting curve is close to the occurrence of the earthquake. For the short-term analysis before the earthquake, there are two cumulative acceleration phases. To further verify the correlation with the earthquake, we compare our findings for different time periods and stations and rule out the possible influence of meteorological factors. We consider the negentropy analysis to exhibit potential for studying pre-earthquake anomalies.

Explosive instability due to flow over a rippled bottom

Abstract: . In this paper, we study Bragg resonance, i.e., the triad interaction between surface and/or interfacial waves with a bottom ripple, in the presence of background velocity. We show that when one of the constituent waves of the triad has negative energy, the amplitudes of all the waves grow exponentially. This is very different from classic Bragg resonance in which one wave decays to cause the growth of the other. The instabilities we observe are “explosive” and are different from normal mode shear instabilities since our velocity profiles are linearly stable. Our work may explain the existence of large-amplitude internal waves over periodic bottom ripples in the presence of tidal flow observed in oceans and estuaries.

Particle clustering and subclustering as a proxy for mixing in geophysical flows

Abstract: . The Eulerian point of view is the traditional theoretical and numerical tool to describe fluid mechanics. Some modern computational fluid dynamics codes allow for the efficient simulation of particles, in turn facilitating a Lagrangian description of the flow. The existence and persistence of Lagrangian coherent structures in fluid flow has been a topic of considerable study. Here we focus on the ability of Lagrangian methods to characterize mixing in geophysical flows. We study the instability of a strongly non-linear double-jet flow, initially in geostrophic balance, which forms quasi-coherent vortices when subjected to ageostrophic perturbations. Particle clustering techniques are applied to study the behavior of the particles in the vicinity of coherent vortices. Changes in inter-particle distance play a key role in establishing the patterns in particle trajectories. This paper exploits graph theory in finding particle clusters and regions of dense interactions (also known as subclusters). The methods discussed and results presented in this paper can be used to identify mixing in a flow and extract information about particle behavior in coherent structures from a Lagrangian point of view.