ALBERT

Description: Most available verification metrics for ensemble forecasts focus on univariate quantities. That is, they assess whether the ensemble provides an adequate representation of the forecast uncertainty about the quantity of interest at a particular location and time. For spatially indexed ensemble forecasts, however, it is also important that forecast fields reproduce the spatial structure of the observed field and represent the uncertainty about spatial properties such as the size of the area for which heavy precipitation, high winds, critical fire weather conditions, etc., are expected. In this article we study the properties of the fraction of threshold exceedance (FTE) histogram, a new diagnostic tool designed for spatially indexed ensemble forecast fields. Defined as the fraction of grid points where a prescribed threshold is exceeded, the FTE is calculated for the verification field and separately for each ensemble member. It yields a projection of a – possibly high-dimensional – multivariate quantity onto a univariate quantity that can be studied with standard tools like verification rank histograms. This projection is appealing since it reflects a spatial property that is intuitive and directly relevant in applications, though it is not obvious whether the FTE is sufficiently sensitive to misrepresentation of spatial structure in the ensemble. In a comprehensive simulation study we find that departures from uniformity of the FTE histograms can indeed be related to forecast ensembles with biased spatial variability and that these histograms detect shortcomings in the spatial structure of ensemble forecast fields that are not obvious by eye. For demonstration, FTE histograms are applied in the context of spatially downscaled ensemble precipitation forecast fields from NOAA's Global Ensemble Forecast System.

Description: In this paper, the performance of three machine-learning methods for predicting short-term evolution and for reproducing the long-term statistics of a multiscale spatiotemporal Lorenz 96 system is examined. The methods are an echo state network (ESN, which is a type of reservoir computing; hereafter RC–ESN), a deep feed-forward artificial neural network (ANN), and a recurrent neural network (RNN) with long short-term memory (LSTM; hereafter RNN–LSTM). This Lorenz 96 system has three tiers of nonlinearly interacting variables representing slow/large-scale (X), intermediate (Y), and fast/small-scale (Z) processes. For training or testing, only X is available; Y and Z are never known or used. We show that RC–ESN substantially outperforms ANN and RNN–LSTM for short-term predictions, e.g., accurately forecasting the chaotic trajectories for hundreds of numerical solver's time steps equivalent to several Lyapunov timescales. The RNN–LSTM outperforms ANN, and both methods show some prediction skills too. Furthermore, even after losing the trajectory, data predicted by RC–ESN and RNN–LSTM have probability density functions (pdf's) that closely match the true pdf – even at the tails. The pdf of the data predicted using ANN, however, deviates from the true pdf. Implications, caveats, and applications to data-driven and data-assisted surrogate modeling of complex nonlinear dynamical systems, such as weather and climate, are discussed.

Description: This article presents the results of a bifurcation analysis of a simple energy balance model (EBM) for the future climate of the Earth. The main focus is on the following question: can the nonlinear processes intrinsic to atmospheric physics, including natural positive feedback mechanisms, cause a mathematical bifurcation of the climate state, as a consequence of continued anthropogenic forcing by rising greenhouse gas emissions? Our analysis shows that such a bifurcation could cause an abrupt change to a drastically different climate state in the EBM, which is warmer and more equable than any climate existing on Earth since the Pliocene epoch. In previous papers, with this EBM adapted to paleoclimate conditions, it was shown to exhibit saddle-node and cusp bifurcations, as well as hysteresis. The EBM was validated by the agreement of its predicted bifurcations with the abrupt climate changes that are known to have occurred in the paleoclimate record, in the Antarctic at the Eocene–Oligocene transition (EOT) and in the Arctic at the Pliocene–Paleocene transition (PPT). In this paper, the EBM is adapted to fit Anthropocene climate conditions, with emphasis on the Arctic and Antarctic climates. The four Representative Concentration Pathways (RCP) considered by the IPCC (Intergovernmental Panel on Climate Change) are used to model future CO2 concentrations, corresponding to different scenarios of anthropogenic activity. In addition, the EBM investigates four naturally occurring nonlinear feedback processes which magnify the warming that would be caused by anthropogenic CO2 emissions alone. These four feedback mechanisms are ice–albedo feedback, water vapour feedback, ocean heat transport feedback, and atmospheric heat transport feedback. The EBM predicts that a bifurcation resulting in a catastrophic climate change, to a pre-Pliocene-like climate state, will occur in coming centuries for an RCP with unabated anthropogenic forcing, amplified by these positive feedbacks. However, the EBM also predicts that appropriate reductions in carbon emissions may limit climate change to a more tolerable continuation of what is observed today. The globally averaged version of this EBM has an equilibrium climate sensitivity (ECS) of 4.34 K, near the high end of the likely range reported by the IPCC.

Description: Beyond multifractional Brownian motion: new stochastic models for geophysical modelling Nonlinear Processes in Geophysics, 20, 643-655, 2013 Author(s): J. Lévy Véhel Multifractional Brownian motion (mBm) has proved to be a useful tool in various areas of geophysical modelling. Although a versatile model, mBm is of course not always an adequate one. We present in this work several other stochastic processes which could potentially be useful in geophysics. The first alternative type is that of self-regulating processes : these are models where the local regularity is a function of the amplitude, in contrast to mBm where it is tuned exogenously. We demonstrate the relevance of such models for digital elevation maps and for temperature records. We also briefly describe two other types of alternative processes, which are the counterparts of mBm and of self-regulating processes when the intensity of local jumps is considered in lieu of local regularity: multistable processes allow one to prescribe the local intensity of jumps in space/time, while this intensity is governed by the amplitude for self-stabilizing processes .

Description: Recently, in 2013, we proved that certain presentations present the Dunwoody -manifold groups. Since the Dunwoody -manifolds do not have a unique Heegaard diagram, we cannot determine a unique group presentation for the Dunwoody -manifolds. It is well known that every -knots in a lens space can be represented by the set of the 4-tuples (Cattabriga and Mulazzani (2004); S. H. Kim and Y. Kim (2012, 2013)). In particular, to determine a unique Heegaard diagram of the Dunwoody -manifolds, we proved the fact that the certain subset of representing all -bridge knots of -knots is determined completely by using the dual and mirror -decompositions (S. H. Kim and Y. Kim (2011)). In this paper, we show how to obtain the dual and mirror images of all elements in as the generalization of some results by Grasselli and Mulazzani (2001); S. H. Kim and Y. Kim (2011).

Description: Non-extensivity and long-range correlations in the earthquake activity at the West Corinth rift (Greece) Nonlinear Processes in Geophysics, 20, 713-724, 2013 Author(s): G. Michas, F. Vallianatos, and P. Sammonds In the present work the statistical properties of the earthquake activity in a highly seismic region, the West Corinth rift (Central Greece), are being studied by means of generalized statistical physics. By using a dataset that covers the period 2001–2008, we investigate the earthquake energy distribution and the distribution of the time intervals (interevent times) between the successive events. As has been reported previously, these distributions exhibit complex statistical properties and fractality. By using detrended fluctuation analysis (DFA), a well-established method for detection of long-range correlations in non-stationary signals, it is shown that long-range correlations are also present in the earthquake activity. The existence of these properties motivates us to use non-extensive statistical physics (NESP) to investigate the statistical properties of the frequency-magnitude and the interevent time distributions, along with other well-known relations in seismology, such as the gamma distribution for interevent times. The results of the analysis indicate that the statistical properties of the earthquake activity can be successfully reproduced by means of NESP and that the earthquake activity at the West Corinth rift is correlated at all-time scales.

Description: The impact of initial spread calibration on the RELO ensemble and its application to Lagrangian dynamics Nonlinear Processes in Geophysics, 20, 621-641, 2013 Author(s): M. Wei, G. Jacobs, C. Rowley, C. N. Barron, P. Hogan, P. Spence, O. M. Smedstad, P. Martin, P. Muscarella, and E. Coelho A number of real-time ocean model forecasts were carried out successfully at Naval Research Laboratory (NRL) to provide modeling support and numerical guidance to the CARTHE GLAD at-sea experiment during summer 2012. Two RELO ensembles and three single models using NCOM and HYCOM with different resolutions were carried out. A calibrated ensemble system with enhanced spread and reliability was developed to better support this experiment. The calibrated ensemble is found to outperform the un-calibrated ensemble in forecasting accuracy, skill, and reliability for all the variables and observation spaces evaluated. The metrics used in this paper include RMS error, anomaly correlation, PECA, Brier score, spread reliability, and Talagrand rank histogram. It is also found that even the un-calibrated ensemble outperforms the single forecast from the model with the same resolution. The advantages of the ensembles are further extended to the Lagrangian framework. In contrast to a single model forecast, the RELO ensemble provides not only the most likely Lagrangian trajectory for a particle in the ocean, but also an uncertainty estimate that directly reflects the complicated ocean dynamics, which is valuable for decision makers. The examples show that the calibrated ensemble with more reliability can capture trajectories in different, even opposite, directions, which would be missed by the un-calibrated ensemble. The ensembles are applied to compute the repelling and attracting Lagrangian coherent structures (LCSs), and the uncertainties of the LCSs, which are hard to obtain from a single model forecast, are estimated. It is found that the spatial scales of the LCSs depend on the model resolution. The model with the highest resolution produces the finest, small-scale, LCS structures, while the model with lowest resolution generates only large-scale LCSs. The repelling and attracting LCSs are found to intersect at many locations and create complex mesoscale eddies. The fluid particles and drifters in the middle of these tangles are subject to attraction and repulsion simultaneously from these two kinds of LCSs. As a result, the movements of particles near the Deepwater Horizon (DWH) location are severely limited. This is also confirmed by the Lagrangian trajectories predicted by the ensembles.

Description: Multifractal properties of embedded convective structures in orographic precipitation: toward subgrid-scale predictability Nonlinear Processes in Geophysics, 20, 605-620, 2013 Author(s): M. Nogueira, A. P. Barros, and P. M. A. Miranda Rain and cloud fields produced by fully nonlinear idealized cloud resolving numerical simulations of orographic convective precipitation display statistical multiscaling behavior, implying that multifractal diagnostics should provide a physically robust basis for the downscaling and sub-grid scale parameterizations of moist processes. Our results show that the horizontal scaling exponent function (and respective multiscaling parameters) of the simulated rainfall and cloud fields varies with atmospheric and terrain properties, particularly small-scale terrain spectra, atmospheric stability, and advective timescale. This implies that multifractal diagnostics of moist processes for these simulations are fundamentally transient, exhibiting complex nonlinear behavior depending on atmospheric conditions and terrain forcing at each location. A particularly robust behavior found here is the transition of the multifractal parameters between stable and unstable cases, which has a clear physical correspondence to the transition from stratiform to organized (banded and cellular) convective regime. This result is reinforced by a similar behavior in the horizontal spectral exponent. Finally, our results indicate that although nonlinearly coupled fields (such as rain and clouds) have different scaling exponent functions, there are robust relationships with physical underpinnings between the scaling parameters that can be explored for hybrid dynamical-statistical downscaling.

Description: A top-down model to generate ensembles of runoff from a large number of hillslopes Nonlinear Processes in Geophysics, 20, 683-704, 2013 Author(s): P. R. Furey, V. K. Gupta, and B. M. Troutman We hypothesize that total hillslope water loss for a rainfall–runoff event is inversely related to a function of a lognormal random variable, based on basin- and point-scale observations taken from the 21 km 2 Goodwin Creek Experimental Watershed (GCEW) in Mississippi, USA. A top-down approach is used to develop a new runoff generation model both to test our physical-statistical hypothesis and to provide a method of generating ensembles of runoff from a large number of hillslopes in a basin. The model is based on the assumption that the probability distributions of a runoff/loss ratio have a space–time rescaling property. We test this assumption using streamflow and rainfall data from GCEW. For over 100 rainfall–runoff events, we find that the spatial probability distributions of a runoff/loss ratio can be rescaled to a new distribution that is common to all events. We interpret random within-event differences in runoff/loss ratios in the model to arise from soil moisture spatial variability. Observations of water loss during events in GCEW support this interpretation. Our model preserves water balance in a mean statistical sense and supports our hypothesis. As an example, we use the model to generate ensembles of runoff at a large number of hillslopes for a rainfall–runoff event in GCEW.

Description: A mechanism for catastrophic filter divergence in data assimilation for sparse observation networks Nonlinear Processes in Geophysics, 20, 705-712, 2013 Author(s): G. A. Gottwald and A. J. Majda We study catastrophic filter divergence in data assimilation procedures whereby the forecast model develops severe numerical instabilities leading to a blow-up of the solution. Catastrophic filter divergence can occur in sparse observational grids with small observational noise for intermediate observation intervals and finite ensemble sizes. Using a minimal five-dimensional model, we establish that catastrophic filter divergence is a numerical instability of the underlying forecast model caused by the filtering procedure producing analyses which are not consistent with the true dynamics, and stiffness caused by the fast attraction of the inconsistent analyses towards the attractor during the forecast step.