The problem of the optimal combination of rain-gauge measurements and radar precipitation estimates has been investigated. A method that attempts to generalize wellestablished geostatistical techniques, such as kriging with external drift, is presented. The new method, besides allowing spatial information to be incorporated into the modelling and estimation process, also allows temporal information to be incorporated. This technique employs temporal data as secondary co-kriged variables. The approach can be considered both straightforward and practical as far as design and programming aspects are concerned. Co-kriging with external drift leads to significant improvements in the results compared with typical radar estimates. It also seems to be more advantageous than kriging with external drift in modelling stability terms. Evidence is provided showing the advantages of co-kriging with external drift modelling over kriging with external drift. The difference becomes particularly pronounced for non-robust input data. The theoretical background and mathematical structure of the method is demonstrated. The method has been applied to four events, three during summer and one during winter, that took place over the complex Swiss orography. It is shown that cross-validation skill scores improve when the aggregation period of the input data increases from ten minutes to one hour. This improvement can be attributed to the increasing robustness of the input data with the period of aggregation. Moreover, a straightforward disaggregation scheme, which starts from hourly precipitation maps, produced by means of the aforementioned geostatistical technique and generating precipitation estimates at a temporal resolution of five minutes, is proposed.
Machine learning algorithms are trained on a 10-yr archive of composite weather radar images in the Swiss Alps to nowcast precipitation growth and decay in the next few hours in moving coordinates (Lagrangian frame). The hypothesis of this study is that growth and decay is more predictable in mountainous regions, which represent a potential source of practical predictability by machine learning methods. In this paper, artificial neural networks (ANN) are employed to learn the complex nonlinear dependence relating the growth and decay to the input predictors, which are geographical location, mesoscale motion vectors, freezing level height, and time of the day. The average long-term growth and decay patterns are effectively reproduced by the ANN, which allows exploring their climatology for any combination of predictors. Due to the low intrinsic predictability of growth and decay, its prediction in real time is more challenging, but is substantially improved when adding persistence information to the predictors, more precisely the growth and decay and precipitation intensity in the immediate past. The improvement is considerable in mountainous regions, where, depending on flow direction, the root-mean-square error of ANN predictions can be 20%–30% lower compared with persistence. Because large uncertainty is associated with precipitation forecasting, deterministic machine learning predictions should be coupled with a model for the predictive uncertainty. Therefore, we consider a probabilistic perspective by estimating prediction intervals based on a combination of quantile decision trees and ANNs. The probabilistic framework is an attempt to address the problem of conditional bias, which often characterizes deterministic machine learning predictions obtained by error minimization.
This paper explores how orbits in a galactic potential can be impacted by large amplitude time-dependences of the form that one might associate with galaxy or halo formation or strong encounters between pairs of galaxies. A period of time-dependence with a strong, possibly damped, oscillatory component can give rise to large amounts of transient chaos, and it is argued that chaotic phase mixing associated with this transient chaos could play a major role in accounting for the speed and efficiency of violent relaxation. Analysis of simple toy models involving time-dependent perturbations of an integrable Plummer potential indicates that this chaos results from a broad, possibly generic, resonance between the frequencies of the orbits and harmonics thereof and the frequencies of the time-dependent perturbation. Numerical computations of orbits in potentials exhibiting damped oscillations suggest that, within a period of 10 dynamical times t_D or so, one could achieve simultaneously both `near-complete' chaotic phase mixing and a nearly time-independent, integrable end state.Comment: 11 pages and 12 figures: an extended version of the original manuscript, containing a modified title, one new figure, and approximately one page of additional text, to appear in Monthly Notices of the Royal Astronomical Societ
This paper summarizes a numerical investigation of the statistical properties of orbits evolved in "frozen," time-independent N-body realizations of smooth, time-independent density distributions corresponding to integrable potentials, allowing for 10(2.5) < or = N < or = 10(5.5). Two principal conclusions were reached: (1) In agreement with recent work by Valluri and Merritt, one finds that, in the limit of a nearly "unsoftened" two-body kernel, i.e., V(r) approximately equals (r(2) + epsilon(2))(-1/2) for epsilon --> 0, the value of the largest Lyapunov exponent chi does not decrease systematically with increasing N, so that, viewed in terms of the sensitivity of individual orbits to small changes in initial conditions, there is no sense in which chaos "turns off" for large N. However, it is clear that, for any finite epsilon, chi will tend to zero for sufficiently large N. (2) Even though chi does not decrease for an unsoftened kernel, there is a clear, quantifiable sense in which, as N increases, chaotic orbits in the frozen-N systems remain "close to" integrable characteristics in the smooth potential for progressively longer times. When viewed in configuration or velocity space, or as probed by collisionless invariants like angular momentum, frozen-N orbits typically diverge from smooth potential characteristics as a power law in time, rather than exponentially, on a time scale approximately equals N(p)t(D), with p approximately 1/2 and t(D) a characteristic dynamical, or crossing, time. For the case of angular momentum, the divergence is well approximated by a t(1/2) dependence, so that, when viewed in terms of collisionless invariants, discreteness effects act as a diffusion process that, presumably, can be modeled by nearly white Gaussian noise in the context of a Langevin or Fokker-Planck description. For position and velocity, the divergence is more rapid, characterized by a nearly linear power-law growth, t(q) with q approximately 1, a result that likely reflects the effects of linear phase mixing. The inference that, pointwise, individual N-body orbits can be reasonably approximated by orbits in a smooth potential only for times < N(1/2)t(D) has potential implications for various resonance phenomena that can act in real self-gravitating systems.
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