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This work is devoted to the study of rates of convergence of the empirical measures \\mu_{n} = \\frac {1}{n} \\sum_{k=1}^n \\delta_{X_k}, n \\geq 1, over a sample (X_{k})_{k \\geq 1} of independent identically distributed real-valued random variables towards the common distribution \\mu in Kantorovich transport distances W_p. The focus is on finite range bounds on the expected Kantorovich distances \\mathbb{E}(W_{p}(\\mu_{n},\\mu )) or \\big [ \\mathbb{E}(W_{p}^p(\\mu_{n},\\mu )) \\big ]^1/p in terms of moments and analytic conditions on the measure \\mu and its distribution function. The study describes a variety of rates, from the standard one \\frac {1}{\\sqrt n} to slower rates, and both lower and upper-bounds on \\mathbb{E}(W_{p}(\\mu_{n},\\mu )) for fixed n in various instances. Order statistics, reduction to uniform samples and analysis of beta distributions, inverse distribution functions, log-concavity are main tools in the investigation. Two detailed appendices collect classical and some new facts on inverse distribution functions and beta distributions and their densities necessary to the investigation.

We analyse the impact of the North Atlantic Oscillation (NAO) on the climate of East Asia at subseasonal time scales during both winter and summer. These teleconections have mainly been investigated at seasonal and longer time scales, while higher-frequency links are largely unexplored. The NAO is defined using extended empirical orthogonal functions on pentad-mean observations, which allows to elucidate the oscillation’s spatial and temporal evolution and clearly separate the development and decay phases. The downstream dynamical imprint and associated temperature and precipitation anomalies are quantified by means of a linear regression analysis. It is shown that the NAO generates a significant climate response over East Asia during both the dry and wet seasons, whose spatial pattern is highly dependent on the phase of the NAO’s life cycle. Temperature and precipitation anomalies develop concurrently with the NAO mature phase, and reach maximum amplitude 5–10 days later. These are shown to be systematically related to mid and high-latitude teleconnections across the Eurasian continent via eastward-propagating quasi-stationary Rossby waves instigated over the Atlantic and terminating in the northeastern Pacific. These findings underscore the importance of rapidly evolving dynamical processes in governing the NAO’s downstream impacts and teleconnections with East Asia.

The summer spatial structure and sub-monthly temporal evolution of one of the key dynamical features of Central American climate, the Caribbean Low-Level Jet (CLLJ), is investigated by means of extended empirical orthogonal functions (EEOFs). The Caribbean 925-hPa zonal wind from the CFSR reanalysis for the period 1979 – 2010 is used for the analysis. This approach reveals new insights into the dynamical processes and spatio-temporal evolution of the CLLJ summer intensification, and through lead and lag linear regressions, significant climate links in the broader Caribbean region are identified. The results show that the CLLJ generates significant precipitation and temperature responses with a distinct temporal evolution over the Caribbean-Atlantic domain to that over the tropical Pacific, which hints at different underlying controlling mechanisms over these two large-scale regions. These anomalies are linked with the mid and upper tropospheric circulation, where a vertical cell over the Caribbean (ascending at the jet exit and subsiding at its entrance) varies in phase with large-scale divergence over the Pacific Ocean. Extratropical hemispheric-wide waves and the weakening of a thermal low in northeast Mexico-central US are identified as potential triggering factors for the CLLJ summer intensification. Two leading modes of tropical variability, El Niño Southern Oscillation and the Madden-Julian Oscillation, are found to modulate the CLLJ by intensifying it and prolonging its life cycle. Details of the underlying mechanisms are provided. These results help to advance the understanding of the processes that modulate local climate variations, which is an important issue in view of the rapid climate change the region is undergoing.

Under the background of climate warming, drought disasters occur frequently in China, especially in the Central China. In this study, drought disaster grade sequences from 14 representative stations were chosen from 268 years of drought disaster historical data for the Hunan Province of China collected during the Qing Dynasty (1644–1911). The empirical orthogonal function (EOF) and rotated empirical orthogonal function (REOF) methods were used to conduct a spatial characteristics analysis of these drought disasters. The results are as follows. (1) There was an inconsistency between the frequency and intensity of the drought disasters. (2) The spatial distribution of the first four EOF loads for the drought disasters showed regional consistency; however, there was a difference in the anti-phase changes in the east–west, south–north, and central directions. (3) The features of the drought disaster distribution in the Hunan Province during the Qing Dynasty are evident, and according to the high values (absolute value ≥ 0.6) for the first six EOF rotational loads, the study area can be divided into the following sections: northeast Xiang (Section I), west Xiang (Section II), southeast Xiang (Section III), middle Xiang (Section IV), north Xiang (Section V), and southwest Xiang (Section VI). The study of drought disasters in Qing Dynasty is of great significance and reference value for disaster prevention, disaster reduction and climate prediction.

Validation of ICON model configurations optimized over a limited domain is essential before accepting new semi-empirical parameters that influence the behavior of subgrid-scale schemes. Because such parameters can modify the dynamics of a numerical weather prediction (NWP) model in highly nonlinear ways, we analyze one season of forecasts (December 2022, January and February 2023) generated with the NWP ICON-LAM through the lens of machine learning–based diagnostics as a complement to traditional evaluation metrics. The goal is to extract physically interpretable information on the model behavior induced by the optimized parameters. This work represents the first part of a wider study exploring machine learning tools for model validation, focusing on two specific approaches: Empirical Orthogonal Functions (EOFs), which are widely used in meteorology and climate science, and autoencoders, which are increasingly adopted for their nonlinear feature extraction capability. In this first part, EOF analysis is used as the primary tool to decompose weather fields from observed reanalysis and forecast datasets. Hourly 2-m temperature forecasts for winter 2022–2023 from multiple regional ICON configurations are compared against downscaled ERA5 data and in situ observations from ground station. EOF analyses revealed that the optimized configurations demonstrate a high skill in predicting surface temperature. From the signal error decomposition, the fourth EOF mode is effective particularly during night-time hours, and contributes to enhancing the performance of ICON. Analyses based on autoencoders will be presented in a companion paper (Part II).

Chlorophyll- a (Chl- a ) concentration is a primary indicator for marine environmental monitoring. The spatio-temporal variations of sea surface Chl- a concentration in the Yellow Sea (YS) and the East China Sea (ECS) in 2001–2020 were investigated by reconstructing the MODIS Level 3 products with the data interpolation empirical orthogonal function (DINEOF) method. The reconstructed results by interpolating the combined MODIS daily +8-day datasets were found better than those merely by interpolating daily or 8-day data. Chl- a concentration in the YS and the ECS reached its maximum in spring, with blooms occurring, decreased in summer and autumn, and increased in late autumn and early winter. By performing empirical orthogonal function (EOF) decomposition of the reconstructed data fields and correlation analysis with several potential environmental factors, we found that the sea surface temperature (SST) plays a significant role in the seasonal variation of Chl a , especially during spring and summer. The increase of SST in spring and the upper-layer nutrients mixed up during the last winter might favor the occurrence of spring blooms. The high sea surface temperature (SST) throughout the summer would strengthen the vertical stratification and prevent nutrients supply from deep water, resulting in low surface Chl- a concentrations. The sea surface Chl a concentration in the YS was found decreased significantly from 2012 to 2020, which was possibly related to the Pacific Decadal Oscillation (PDO).

Precipitation time series exhibit complex fluctuations and statistical changes. Existing research stops short of a simple and feasible model for precipitation forecasting. In this article, the authors investigate and forecast precipitation variations in South Korea from 1973 to 2021 using cyclostationary empirical orthogonal function (CSEOF) and regression methods. First, empirical orthogonal function (EOF) and CSEOF analyses are used to examine the periodic changes in the precipitation data. Then, the autoregressive integrated moving average (ARIMA) method is applied to the principal component (PC) time series derived from the EOF and CSEOF precipitation analyses. The fifteen leading EOF and CSEOF modes and their corresponding PC time series clearly reflect the spatial distribution and temporal evolution characteristics of the precipitation data. Based on the PC forecasts of the EOF and CSEOF models, the EOF–ARIMA composite model and CSEOF–ARIMA composite model are used to obtain quantitative precipitation forecasts. The comparison results show that both composite models have good performance and similar accuracy. However, the performance of the CSEOF–ARIMA model is better than that of the EOF–ARIMA model under various measurements. Therefore, the CSEOF–ARIMA composite forecast model can be considered an efficient and feasible technology representing an analytical approach for precipitation forecasting in South Korea.

Using an empirical Green’s function (EGF) approach and data from local to regional distances we analyzed rupture propagation directivity in the three mainshocks (ML 6.0–6.1) and in six of the largest aftershocks (ML 5.0 – 5.5) of the 2017 Kerman, Iran, seismic sequence. The EGF procedure was based on data from smaller events (ML 4.0 – 4.8). Deconvolution was applied separately to P and S phases. Using the P-wave data, we calculated relative source-time functions and examined azimuthal variations in rupture duration. In the S-wave analysis, we investigated along strike rupture directivity of the mainshocks and the largest aftershocks by evaluating azimuthal variation of the amplitude spectra. Two of the mainshocks and four of the aftershocks clearly showed rupture propagation from the south-east toward the north-west. The third mainshock and one of the aftershocks suggested almost bilateral rupture propagation, and one aftershock showed rupture directivity to the southeast. It seems that the rupture propagation direction in the area is generally to the north-west and the events which have different propagation directions are located within the NW and SE ends of the faulting area. We suggest that the general rupture propagation direction in the area is steered by regional tectonic stress field regarding the faulting orientations which have been affected by stress redistribution around a restraining bend.

The time-averaged shear wave velocity of top 30 m (V S30 ) is the most commonly used parameter to classify a site and evaluate its amplification characteristics for the seismic design. The in-situ seismic tests must be performed up to a depth of 30 m for obtaining the shear wave velocity (V S ) profiles to estimate V S30 . It is intimated that, in most of the cases, the measured V S profile does not extend up to 30 m due to numerous reasons including limitation of testing techniques and unfavorable field conditions. Since, the measurements of V S30 are unavailable for the majority of Pakistan and the world, the local geology and topographic slope or its combination are used to estimate V S30 . However, there is no field-based validation of the estimated V S30 is performed in Islamabad-Rawalpindi region, the proxy-based estimation may lead to unrealistic results. To accommodate this, region specific extrapolation methods are developed. This study develops an empirical data-driven function of V S30 from shallow V S profiles by correlating V S30 with the time-averaged V S to depths less than 30 m. In this regard, 85 V S profiles are used from Rawalpindi-Islamabad region. A comparative analysis of the proposed procedure is carried out with the published methods. It is revealed that V S30 predicted by the proposed function results in close matches with the data measured in the western United States. In addition, the results indicate that the local geology and topographic slope proxies may not be acceptable for usage in the region due to their greater uncertainty. Finally, a procedure for extrapolating the V S profile from available shallow depth measurements up to 30 m is proposed.

Floods and droughts are driven, in part, by spatial patterns of extreme rainfall. Heat waves are driven by spatial patterns of extreme temperature. It is therefore of interest to design statistical methodologies that allow the rapid identification of likely patterns of extreme rain or temperature from observed historical data. The standard work-horse for the rapid identification of patterns of climate variability in historical data is Principal Component Analysis (PCA) and its variants. But PCA optimizes for variance not spatial extremes, and so there is no particular reason why the first PCA spatial pattern should identify, or even approximate, the types of patterns that may drive floods, droughts or heatwaves, even if the linear assumptions underlying PCA are correct. We present an alternative pattern identification algorithm that makes the same linear assumptions as PCA, but which can be used to explicitly optimize for spatial extremes. We call the method Directional Component Analysis (DCA), since it involves introducing a preferred direction, or metric, such as “sum of all points in the spatial field”. We compare the first PCA and DCA spatial patterns for U.S. and China winter and summer rainfall anomalies, using the sum metric for the definition of DCA in order to focus on total rainfall anomaly over the domain. In three out of four of the examples the first DCA spatial pattern is more uniform over a wide area than the first PCA spatial pattern and as a result is more obviously relevant to large-scale flooding or drought. Also, in all cases the definitions of PCA and DCA result in the first PCA spatial pattern having the larger explained variance of the two patterns, while the first DCA spatial pattern, when scaled appropriately, has a higher likelihood and greater total rainfall anomaly, and indeed is the pattern with the highest total rainfall anomaly for a given likelihood. The first DCA spatial pattern is arguably the best answer to the question: what single spatial pattern is most likely to drive large total rainfall anomalies in the future? It is also simpler to calculate than PCA. In combination PCA and DCA patterns yield more insight into rainfall variability and extremes than either pattern on its own.