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The spatial inhomogeneity of changes in global precipitation, which is directly related to extreme flooding or drought events, represents a significant feature of climate change. To date, limited studies quantified the spatial inhomogeneity of global precipitation and its long-term change. Aiming at solving the above challenge, this study introduces a novel but simple methodological framework that is able to (1) quantify the spatial inhomogeneity of global precipitation and its variability, (2) estimate contributions of different precipitation intensities and (3) assess contributions of different regions. In this framework, the spatial inhomogeneity is quantified by the spatial variance of gridded precipitation normalized by the global mean precipitation, namely spatial coefficient of variation (SCV). Then, the spatial probability density function (PDF) of precipitation intensity is introduced to identify contributions of different precipitation intervals that lead to changes in global precipitation inhomogeneity. Finally, the global inhomogeneity can be decomposed into intra-regional and inter-regional inhomogeneity components to estimate the contributions from different regions. The result shows that the inhomogeneity of global annual precipitation has increased consistently across multiple datasets in the satellite era (1979–2021), attributed to the increasing area of both extremely high and low precipitation. Based on the Global Precipitation Climatology Project (GPCP V2.3) dataset, the increase in inhomogeneity of global annual precipitation is primarily contributed by the intra-regional inhomogeneity component of Northern Hemispheric tropical ocean (+ 60.2%) and Southern Hemispheric tropical ocean (+ 40.3%), and is partly offset by the inter-regional inhomogeneity component of Northern Hemispheric mid-latitude ocean (− 4.5%). This study demonstrates how the spatial inhomogeneity of global precipitation can be easily estimated, which is implications for quantifying, monitoring, and understanding changes in climate extremes. Our framework offers a tool for dataset or model assessment, particularly useful for the regions susceptible to extreme precipitation events.

This work presents an improved modelling approach for wind turbine power curves (WTPCs) using fractional differential equations (FDE). Nine novel FDE-based models are presented for mathematically modelling commercial wind turbine modules’ power–velocity (P-V) characteristics. These models utilize Weibull and Gamma probability density functions to estimate the capacity factor (CF), where accuracy is measured using relative error (RE). Comparative analysis is performed for the WTPC mathematical models with a varying order of differentiation (α) from 0.5 to 1.5, utilizing the manufacturer data for 36 wind turbines with capacities ranging from 150 to 3400 kW. The shortcomings of conventional mathematical models in various meteorological scenarios can be overcome by applying the Riemann–Liouville fractional integral instead of the classical integer-order integrals. By altering the sequence of differentiation and comparing accuracy, the suggested model uses fractional derivatives to increase flexibility. By contrasting the model output with actual data obtained from the wind turbine datasheet and the historical data of a specific location, the models are validated. Their accuracy is assessed using the correlation coefficient (R) and the Mean Absolute Percentage Error (MAPE). The results demonstrate that the exponential model at α=0.9 gives the best accuracy of WTPCs, while the original linear model was the least accurate.

This paper describes a case study that converts pre-stack seismic data into meaningful rock properties by employing non-parametric probability density functions through a probabilistic modelling approach. This study used the simultaneous pre-stack inversion method to transform pre-stack seismic data into seismic attributes like compressional impedance, shear impedance, density, and V P / V S ratio. Then cross plot analysis was conducted on selected wireline log data to identify reservoir lithofacies zones based on the ranges of properties like P-impedance and V P / V S ratio. Hydrocarbon zone was identified with the range of V P / V S ratio between 1.15 and 1.82 and Z P from 3800 to 12400 ((m/s) × (g/cc)). Water bearing sand zone was separated with V P / V S ratio with 1.85–2.12 and Z P with 3500–14900 ((m/s) × (g/cc)), and 3500–14900 ((m/s) × (g/cc)) of Z P , V P / V S ratio between 2.14 and 3.1 was used to characterize the shale zone. A non-parametric kernel density estimator is used on cross-plot data points to generate a probability density function for each lithofacies. These non-parametric PDFs were incorporated with seismic attributes using a probabilistic modelling approach based on Bayes' classification to generate a lithofacies model. The application of methodology provides a better insight into predicting and discriminating lithofacies in the study area. Highlights Applied seismic inversion to obtain seismic elastic attributes such as compressional impedance ( Z P ), shear impedance ( Z S ), V P / V S ratio, and density. Shale, water-bearing zone, and hydrocarbon zone were identified using the cross plot analysis of well log data. Probability density functions (PDFs) for lithologies were generated on cross-plot space using the non-parameter statistical classification. Finally, hydrocarbon zones were identified using the Bayes' rule by combining the seismic data with PDFs.

Haralick texture features are common texture descriptors in image analysis. To compute the Haralick features, the image gray-levels are reduced, a process called quantization. The resulting features depend heavily on the quantization step, so Haralick features are not reproducible unless the same quantization is performed. The aim of this work was to develop Haralick features that are invariant to the number of quantization gray-levels. By redefining the gray-level co-occurrence matrix (GLCM) as a discretized probability density function, it becomes asymptotically invariant to the quantization. The invariant and original features were compared using logistic regression classification to separate two classes based on the texture features. Classifiers trained on the invariant features showed higher accuracies, and had similar performance when training and test images had very different quantizations. In conclusion, using the invariant Haralick features, an image pattern will give the same texture feature values independent of image quantization.

The Climate Prediction Center (CPC) morphing technique (CMORPH) satellite precipitation estimates are reprocessed and bias corrected on an 8 km × 8 km grid over the globe (60°S–60°N) and in a 30-min temporal resolution for an 18-yr period from January 1998 to the present to form a climate data record (CDR) of high-resolution global precipitation analysis. First, the purely satellite-based CMORPH precipitation estimates (raw CMORPH) are reprocessed. The integration algorithmis fixed and the input level 2 passivemicrowave (PMW) retrievals of instantaneous precipitation rates are fromidentical versions throughout the entire data period. Bias correction is then performed for the raw CMORPH through probability density function (PDF) matching against the CPC daily gauge analysis over land and through adjustment against the Global Precipitation Climatology Project (GPCP) pentadmerged analysis of precipitation over ocean. The reprocessed, bias-corrected CMORPH exhibits improved performance in representing the magnitude, spatial distribution patterns, and temporal variations of precipitation over the global domain from 60°S to 608N. Bias in the CMORPH satellite precipitation estimates is almost completely removed over land during warm seasons (May–September), while during cold seasons (October–April) CMORPH tends to underestimate the precipitation due to the less-thandesirable performance of the current-generation PMWretrievals in detecting and quantifying snowfall and cold season rainfall. An intercomparison study indicated that the reprocessed, bias-corrected CMORPH exhibits consistently superior performance than the widely used TRMM 3B42 (TMPA) in representing both daily and 3-hourly precipitation over the contiguous United States and other global regions.

The power spectral density (PSD) of any time-dependent stochastic process Xt is a meaningful feature of its spectral content. In its text-book definition, the PSD is the Fourier transform of the covariance function of Xt over an infinitely large observation time T, that is, it is defined as an ensemble-averaged property taken in the limit T → ∞ . A legitimate question is what information on the PSD can be reliably obtained from single-trajectory experiments, if one goes beyond the standard definition and analyzes the PSD of a single trajectory recorded for a finite observation time T. In quest for this answer, for a d-dimensional Brownian motion (BM) we calculate the probability density function of a single-trajectory PSD for arbitrary frequency f, finite observation time T and arbitrary number k of projections of the trajectory on different axes. We show analytically that the scaling exponent for the frequency-dependence of the PSD specific to an ensemble of BM trajectories can be already obtained from a single trajectory, while the numerical amplitude in the relation between the ensemble-averaged and single-trajectory PSDs is a fluctuating property which varies from realization to realization. The distribution of this amplitude is calculated exactly and is discussed in detail. Our results are confirmed by numerical simulations and single-particle tracking experiments, with remarkably good agreement. In addition we consider a truncated Wiener representation of BM, and the case of a discrete-time lattice random walk. We highlight some differences in the behavior of a single-trajectory PSD for BM and for the two latter situations. The framework developed herein will allow for meaningful physical analysis of experimental stochastic trajectories.

The ground validation of satellite-based precipitation products (SPPs) is very important for their hydroclimatic application. This study evaluated the performance assessment of four soil moisture-based SPPs (SM2Rain, SM2Rain- ASCAT, SM2Rain-CCI, and GPM-SM2Rain). All data of SPPs were compared with 64 weather stations in Pakistan from January 2005 to December 2020. All SPPs estimations were evaluated on daily, monthly, seasonal, and yearly scales, over the whole spatial domain, and at point-to-pixel scale. Widely used evaluation indices (root mean square error (RMSE), correlation coefficient (CC), bias, and relative bias (rBias)) along with categorical indices (false alarm ratio (FAR), probability of detection (POD), success ratio (SR), and critical success index (CSI) were evaluated for performance analysis. The results of our study signposted that: (1) On a monthly scale, all SPPs estimations were in better agreement with gauge estimations as compared to daily scales. Moreover, SM2Rain and GPM-SM2Rain products accurately traced the spatio-temporal variability with CC >0.7 and rBIAS within the acceptable range (±10) of the whole country. (2) On a seasonal scale (spring, summer, winter, and autumn), GPM-SM2Rain performed more satisfactorily as compared to all other SPPs. (3) All SPPs performed better at capturing light precipitation events, as indicated by the Probability Density Function (PDF); however, in the summer season, all SPPs displayed considerable over/underestimates with respect to PDF (%). Moreover, GPM-SM2RAIN beat all other SPPs in terms of probability of detection. Consequently, we suggest the daily and monthly use of GPM-SM2Rain and SM2Rain for hydro climate applications in a semi-arid climate zone (Pakistan).

An Infrared (IR)-passive microwave (PMW) blended technique is developed to derive precipitation estimates over the Asia-Pacific domain through calibrating the temperature of brightness blackbody from the Japanese Himawari-8 satellite to precipitation derived from the combined PMW retrievals (currently MWCOMB2x) based on the probability density function (PDF)-matching concept. Called IRQPE, the technique is modified and fine-tuned to better represent the spatially rapidly changing cloud–precipitation relationship over the target region with PDF-matching tables established over a refined spatial resolution of 0.5° lat/lon grid. The evaluation of the IRQPE shows broadly comparable performance to that of the CMORPH2 in detecting rainfall systems of large and medium-scales at a resolution of 1.0° degree. Rainfall variations from the two datasets over El Niño-Southern Oscillation and the Madden Julian Oscillation active convective centers show well consistency of each other, suggesting usefulness of the IRQPE in climate applications. Two approaches for regional improvements are explored by establishing the PDF tables for a further refined spatial resolution and by replacing the PMW-based precipitation ‘truth’ fields with the surface gauge data to overcome the shortcoming of PMW-based retrievals in capturing orographic rainfall over the Taiwan area. The results show significant improvements. The rainfall patterns of revised the IRQPE at a resolution of 0.1° degree on above the 5-day timescale correlate well with the Taiwan official surface ground truth called the QPESUMS, which is a gridded set of gauge-corrected Radar quantitative precipitation estimations. The root mean square error of the revised IRQPE on estimating the Taiwan overall land rainfall is close to Radar-derived rainfall accumulations on a 30-day time-scale.

One of the most serious limitations of current astrochemical models with the rate equation (RE) approach is that only a single type of binding site is considered in grain surface chemistry, although laboratory and quantum chemical studies have found that surfaces contain various binding sites with different potential energy depths. When various sites exist, adsorbed species can be trapped in deep potential sites, increasing the resident time on the surface. On the other hand, adsorbed species can be populated in shallow sites, activating thermal hopping and thus two-body reactions even at low temperatures, where the thermal hopping from deeper sites is not activated. Such behavior cannot be described by the conventional RE approach. In this work, I present a framework for incorporating various binding sites (i.e., binding energy distribution) in gas-ice astrochemical models as an extension of the conventional RE approach. I propose a simple method to estimate the probability density function (pdf) for the occupation of various sites by adsorbed species, assuming a quasi-steady state. By using thermal desorption and hopping rates weighted by the pdfs, the effect of binding energy distribution is incorporated into the RE approach without increasing the number of ordinary differential equations to be solved. This method is found to be accurate and computationally efficient, and enables us to consider binding energy distribution even for a large gas-ice chemical network which contains hundreds of icy species. The impact of the binding energy distribution on interstellar ice composition is discussed quantitatively for the first time.

Accurate determination of binary fractions (fb) and mass ratio (q) distributions is crucial for understanding the dynamical evolution of open clusters. We present an improved multiband fitting technique to enhance the analysis of binary properties. This approach enables an accurate photometric determination of fb and q distribution in a cluster. The detectable mass ratio can be down to the qlim , limited by the minimum stellar mass in theoretical models. First, we derived an empirical model for magnitudes of Gaia DR3 and 2MASS bands that match the photometry of single stars in the Pleiades. We then performed a multiband fitting for each cluster member, deriving the probability density function (PDF) of its primary mass ( M1 ) and q in the Bayesian framework. 1154 main-sequence (MS) single stars or unresolved MS+MS binaries are identified as members of the Pleiades. By stacking their PDFs, we conducted a detailed analysis of binary properties of the cluster. We found the fb of this sample is 0.34 ± 0.02. The q distribution exhibits a three-segment power-law profile: an initial increase, followed by a decrease, and then another increase. This distribution can be interpreted as a fiducial power-law profile with an exponent of −1.0 that is determined in the range of 0.3 < q < 0.8, but with a deficiency of binaries at lower q and an excess at higher q. The variations of fb and q with M1 reveal a complex binary distribution within the Pleiades, which might be attributed to a combination of primordial binary formation mechanisms, dynamical interactions, and the observational limit of photometric binaries imposed by qlim(M1) .