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Particle filters contain the promise of fully nonlinear data assimilation. They have been applied in numerous science areas, including the geosciences, but their application to high-dimensional geoscience systems has been limited due to their inefficiency in high-dimensional systems in standard settings. However, huge progress has been made, and this limitation is disappearing fast due to recent developments in proposal densities, the use of ideas from (optimal) transportation, the use of localization and intelligent adaptive resampling strategies. Furthermore, powerful hybrids between particle filters and ensemble Kalman filters and variational methods have been developed. We present a state-of-the-art discussion of present efforts of developing particle filters for high-dimensional nonlinear geoscience state-estimation problems, with an emphasis on atmospheric and oceanic applications, including many new ideas, derivations and unifications, highlighting hidden connections, including pseudo-code, and generating a valuable tool and guide for the community. Initial experiments show that particle filters can be competitive with present-day methods for numerical weather prediction, suggesting that they will become mainstream soon.
Tasking machine learning to predict segments of a time series requires estimating the parameters of a ML model with input/output pairs from the time series. We borrow two techniques used in statistical data assimilation in order to accomplish this task: time-delay embedding to prepare our input data and precision annealing as a training method. The precision annealing approach identifies the global minimum of the action (-log[P]). In this way, we are able to identify the number of training pairs required to produce good generalizations (predictions) for the time series. We proceed from a scalar time series s(tn);tn=t0+n Delta t and, using methods of nonlinear time series analysis, show how to produce a DE>1-dimensional time-delay embedding space in which the time series has no false neighbors as does the observed s(tn) time series. In that DE-dimensional space, we explore the use of feedforward multilayer perceptrons as network models operating on DE-dimensional input and producing DE-dimensional outputs.
The majority of earthquakes occur unexpectedly and can trigger subsequent sequences of events that can culminate in more powerful earthquakes. This self-exciting nature of seismicity generates complex clustering of earthquakes in space and time. Therefore, the problem of constraining the magnitude of the largest expected earthquake during a future time interval is of critical importance in mitigating earthquake hazard. We address this problem by developing a methodology to compute the probabilities for such extreme earthquakes to be above certain magnitudes. We combine the Bayesian methods with the extreme value theory and assume that the occurrence of earthquakes can be described by the Epidemic Type Aftershock Sequence process. We analyze in detail the application of this methodology to the 2016 Kumamoto, Japan, earthquake sequence. We are able to estimate retrospectively the probabilities of having large subsequent earthquakes during several stages of the evolution of this sequence.
We show that elliptic complexes of (pseudo) differential operators on smooth compact manifolds with boundary can always be complemented to a Fredholm problem by boundary conditions involving global pseudodifferential projections on the boundary (similarly as the spectral boundary conditions of Atiyah, Patodi, and Singer for a single operator). We prove that boundary conditions without projections can be chosen if, and only if, the topological Atiyah-Bott obstruction vanishes. These results make use of a Fredholm theory for complexes of operators in algebras of generalized pseudodifferential operators of Toeplitz type which we also develop in the present paper.
High-precision observations of the present-day geomagnetic field by ground-based observatories and satellites provide unprecedented conditions for unveiling the dynamics of the Earth’s core. Combining geomagnetic observations with dynamo simulations in a data assimilation (DA) framework allows the reconstruction of past and present states of the internal core dynamics. The essential information that couples the internal state to the observations is provided by the statistical correlations from a numerical dynamo model in the form of a model covariance matrix. Here we test a sequential DA framework, working through a succession of forecast and analysis steps, that extracts the correlations from an ensemble of dynamo models. The primary correlations couple variables of the same azimuthal wave number, reflecting the predominant axial symmetry of the magnetic field. Synthetic tests show that the scheme becomes unstable when confronted with high-precision geomagnetic observations. Our study has identified spurious secondary correlations as the origin of the problem. Keeping only the primary correlations by localizing the covariance matrix with respect to the azimuthal wave number suffices to stabilize the assimilation. While the first analysis step is fundamental in constraining the large-scale interior state, further assimilation steps refine the smaller and more dynamical scales. This refinement turns out to be critical for long-term geomagnetic predictions. Increasing the assimilation steps from one to 18 roughly doubles the prediction horizon for the dipole from about tree to six centuries, and from 30 to about 60 yr for smaller observable scales. This improvement is also reflected on the predictability of surface intensity features such as the South Atlantic Anomaly. Intensity prediction errors are decreased roughly by a half when assimilating long observation sequences.
This paper concerns the problem of predicting the maximum expected earthquake magnitude μ in a future time interval Tf given a catalog covering a time period T in the past. Different studies show the divergence of the confidence interval of the maximum possible earthquake magnitude m_{ max } for high levels of confidence (Salamat et al. 2017). Therefore, m_{ max } should be better replaced by μ (Holschneider et al. 2011). In a previous study (Salamat et al. 2018), μ is estimated for an instrumental earthquake catalog of Iran from 1900 onwards with a constant level of completeness ( {m0 = 5.5} ). In the current study, the Bayesian methodology developed by Zöller et al. (2014, 2015) is applied for the purpose of predicting μ based on the catalog consisting of both historical and instrumental parts. The catalog is first subdivided into six subcatalogs corresponding to six seismotectonic zones, and each of those zone catalogs is subsequently subdivided according to changes in completeness level and magnitude uncertainty. For this, broad and small error distributions are considered for historical and instrumental earthquakes, respectively. We assume that earthquakes follow a Poisson process in time and Gutenberg-Richter law in the magnitude domain with a priori unknown a and b values which are first estimated by Bayes' theorem and subsequently used to estimate μ. Imposing different values of m_{ max } for different seismotectonic zones namely Alborz, Azerbaijan, Central Iran, Zagros, Kopet Dagh and Makran, the results show considerable probabilities for the occurrence of earthquakes with Mw ≥ 7.5 in short Tf , whereas for long Tf, μ is almost equal to m_{ max }
We develop a technique for the multivariate data analysis of perturbed self-sustained oscillators. The approach is based on the reconstruction of the phase dynamics model from observations and on a subsequent exploration of this model. For the system, driven by several inputs, we suggest a dynamical disentanglement procedure, allowing us to reconstruct the variability of the system's output that is due to a particular observed input, or, alternatively, to reconstruct the variability which is caused by all the inputs except for the observed one. We focus on the application of the method to the vagal component of the heart rate variability caused by a respiratory influence. We develop an algorithm that extracts purely respiratory-related variability, using a respiratory trace and times of R-peaks in the electrocardiogram. The algorithm can be applied to other systems where the observed bivariate data can be represented as a point process and a slow continuous signal, e.g. for the analysis of neuronal spiking. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.
Let (M-i, g(i))(i is an element of N) be a sequence of spin manifolds with uniform bounded curvature and diameter that converges to a lower-dimensional Riemannian manifold (B, h) in the Gromov-Hausdorff topology. Then, it happens that the spectrum of the Dirac operator converges to the spectrum of a certain first-order elliptic differential operator D-B on B. We give an explicit description of D-B and characterize the special case where D-B equals the Dirac operator on B.
Fractures serve as highly conductive preferential flow paths for fluids in rocks, which are difficult to exactly reconstruct in numerical models. Especially, in low-conductive rocks, fractures are often the only pathways for advection of solutes and heat. The presented study compares the results from hydraulic and tracer tomography applied to invert a theoretical discrete fracture network (DFN) that is based on data from synthetic cross-well testing. For hydraulic tomography, pressure pulses in various injection intervals are induced and the pressure responses in the monitoring intervals of a nearby observation well are recorded. For tracer tomography, a conservative tracer is injected in different well levels and the depth-dependent breakthrough of the tracer is monitored. A recently introduced transdimensional Bayesian inversion procedure is applied for both tomographical methods, which adjusts the fracture positions, orientations, and numbers based on given geometrical fracture statistics. The used Metropolis-Hastings-Green algorithm is refined by the simultaneous estimation of the measurement error’s variance, that is, the measurement noise. Based on the presented application to invert the two-dimensional cross-section between source and the receiver well, the hydraulic tomography reveals itself to be more suitable for reconstructing the original DFN. This is based on a probabilistic representation of the inverted results by means of fracture probabilities.