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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 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.

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.

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.

Most bias correction algorithms used in climatology, for example quantile mapping, are applied to univariate time series. They neglect the dependence between different variables. Those that are multivariate often correct only limited measures of joint dependence, such as Pearson or Spearman rank correlation. Here, an image processing technique designed to transfer colour information from one image to another—the N-dimensional probability density function transform—is adapted for use as a multivariate bias correction algorithm (MBCn) for climate model projections/predictions of multiple climate variables. MBCn is a multivariate generalization of quantile mapping that transfers all aspects of an observed continuous multivariate distribution to the corresponding multivariate distribution of variables from a climate model. When applied to climate model projections, changes in quantiles of each variable between the historical and projection period are also preserved. The MBCn algorithm is demonstrated on three case studies. First, the method is applied to an image processing example with characteristics that mimic a climate projection problem. Second, MBCn is used to correct a suite of 3-hourly surface meteorological variables from the Canadian Centre for Climate Modelling and Analysis Regional Climate Model (CanRCM4) across a North American domain. Components of the Canadian Forest Fire Weather Index (FWI) System, a complicated set of multivariate indices that characterizes the risk of wildfire, are then calculated and verified against observed values. Third, MBCn is used to correct biases in the spatial dependence structure of CanRCM4 precipitation fields. Results are compared against a univariate quantile mapping algorithm, which neglects the dependence between variables, and two multivariate bias correction algorithms, each of which corrects a different form of inter-variable correlation structure. MBCn outperforms these alternatives, often by a large margin, particularly for annual maxima of the FWI distribution and spatiotemporal autocorrelation of precipitation fields.

Choosing a probability distribution to represent daily precipitation depths is important for precipitation frequency analysis, stochastic precipitation modeling and in climate trend assessments. Early studies identified the two-parameter gamma (G2) distribution as a suitable distribution for wet-day precipitation based on the traditional goodness-of-fit tests. Here, probability plot correlation coefficients and L-moment diagrams are used to examine distributional alternatives for the wet-day series of daily precipitation for hundreds of stations at the point and catchment scales in the United States. Importantly, both Pearson Type-III (P3) and kappa (KAP) distributions perform very well, particularly for point rainfall. Our analysis indicates that the KAP distribution best describes the distribution of wet-day precipitation at the point scale, whereas the performance of G2 and P3 distributions are comparable for wet-day precipitation at the catchment scale, with P3 generally providing the improved goodness of fit over G2. Since the G2 distribution is currently the most widely used probability density function, our findings could be considerably important, especially within the context of climate change investigations.