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In this work we introduce a spatio-temporal bounded noise derived by the sine-Wiener noise and by the spatially colored unbounded noise proposed by GarcASHa-Ojalvo, Sancho, and RamASHrez-Piscina (GSR noise). We characterize the behavior of the equilibrium distribution of this novel noise by showing its dependence on both the temporal and the spatial autocorrelation lengths. In particular, we show that the distribution experiences a stochastic transition from bimodality to trimodality.

Functions to calculate measures of spatial association, especially measures of spatial autocorrelation, have been made available in many software applications. Measures may be global, applying to the whole data set under consideration, or local, applying to each observation in the data set. Methods of statistical inference may also be provided, but these will, like the measures themselves, depend on the support of the observations, chosen assumptions, and the way in which spatial association is represented; spatial weights are often used as a representational technique. In addition, assumptions may be made about the underlying mean model, and about error distributions. Different software implementations may choose to expose these choices to the analyst, but the sets of choices available may vary between these implementations, as may default settings. This comparison will consider the implementations of global Moran’s I, Getis–Ord G and Geary’s C, local \\[I_i\\] and \\[G_i\\], available in a range of software including Crimestat, GeoDa, ArcGIS, PySAL and R contributed packages.

The characteristics of subdiffusion are influenced by hydrodynamic fluctuations, not the scaling of the mean square displacement (MSD) at the long time limit, but rather the transition of this property from ballistic behaviour at early times to its final scaling at large times. Additionally, since a significant portion of the normalised velocity autocorrelation function (NVAF), a widely used motion descriptor, no longer adequately describes the antipersistent character that is typical of a subdiffusive motion, it is also impacted by these fluctuations. The combined findings can lead to misleading conclusions if hydrodynamic fluctuations are not taken into account. Diffusing and vorticity time scales are crucial for the way the motion turns into the final subdiffusion dynamics, whose exponent is determined by the scaling of the friction.

Dark-field imaging probes the projected autocorrelation function at the autocorrelation length of the grating interferometer and quantitatively accesses the parameters of a microstructure model. The National Institute of Standards and Technology has developed a novel far-field grating interferometer to study hierarchical materials in various fields such as polymer science, geology, additive manufacturing under the INFER project. In this work, we detail the simulation of dark-field imaging which is one of the goals of INFER.

A number of spatial statistic measurements such as Moran’s I and Geary’s C can be used for spatial autocorrelation analysis. Spatial autocorrelation modeling proceeded from the 1-dimension autocorrelation of time series analysis, with time lag replaced by spatial weights so that the autocorrelation functions degenerated to autocorrelation coefficients. This paper develops 2-dimensional spatial autocorrelation functions based on the Moran index using the relative staircase function as a weight function to yield a spatial weight matrix with a displacement parameter. The displacement bears analogy with the time lag in time series analysis. Based on the spatial displacement parameter, two types of spatial autocorrelation functions are constructed for 2-dimensional spatial analysis. Then the partial spatial autocorrelation functions are derived by using the Yule-Walker recursive equation. The spatial autocorrelation functions are generalized to the autocorrelation functions based on Geary’s coefficient and Getis’ index. As an example, the new analytical framework was applied to the spatial autocorrelation modeling of Chinese cities. A conclusion can be reached that it is an effective method to build an autocorrelation function based on the relative step function. The spatial autocorrelation functions can be employed to reveal deep geographical information and perform spatial dynamic analysis, and lay the foundation for the scaling analysis of spatial correlation.

Spatial autocorrelation is inherent to remotely sensed data. Nearby pixels are more similar than distant ones. This property can help to improve the classification performance, by adding spatial or contextual features into the model. However, it can also lead to overestimation of generalisation capabilities, if the spatial dependence between training and test sets is ignored. In this paper, we review existing approaches that deal with spatial autocorrelation for image classification in remote sensing and demonstrate the importance of bias in accuracy metrics when spatial independence between the training and test sets is not respected. We compare three spatial and non-spatial cross-validation strategies at pixel and object levels and study how performances vary at different sample sizes. Experiments based on Sentinel-2 data for mapping two simple forest classes show that spatial leave-one-out cross-validation is the better strategy to provide unbiased estimates of predictive error. Its performance metrics are consistent with the real quality of the resulting map contrary to traditional non-spatial cross-validation that overestimates accuracy. This highlight the need to change practices in classification accuracy assessment. To encourage it we developped Museo ToolBox, an open-source python library that makes spatial cross-validation possible.

Steady flow toward a fully penetrating well takes place in a natural porous formation, where the erratic spatial variations, and the raising uncertainty, of the hydraulic conductivity K are modeled within a stochastic framework which regards the log‐conductivity, ln K, as a Gaussian, stationary, random field. The study provides second order moments of the flow variables by regarding the variance of the log‐conductivity as a perturbation parameter. Unlike similar studies on the topic, moments are expressed in a quite general (valid for any autocorrelation function of ln K) and very simple (from the computational stand point) form. It is shown that the (cross)variances, unlike the case of mean uniform flows, are not anymore stationary due to the dependence of the mean velocity upon the distance from the well. In particular, they vanish at the well because of the condition of given head along the well’s axis, whereas away from it they behave like those pertaining to a uniform flow. Then, theoretical results are applied to a couple (one serving for calibration and the other used for validation purposes) of pumping tests to illustrate how they can be used to determine the hydraulic properties of the aquifers. In particular, the concept of head‐factor is shown to be the key‐parameter to identify the statistical moments of the random field K. Plain Language Summary Flow toward a single well takes place in a porous formation, where the hydraulic conductivity is regarded as a random space function to account for its irregular spatial variability. A simple solution to this difficult problem is achieved by adopting some simplifying assumptions which apply to numerous real settings. Theoretical results are applied to a series of pumping tests in order to demonstrate their utility in the identification of aquifers' hydraulic properties. Key Points A simple, general formulation to compute second‐order moments is presented The head factor is introduced for a robust identification of aquifers' statistical parameters The application to pumping tests is illustrated and discussed

Streamflow prediction is an important matter for the water resources management and the design of hydraulic structures that can be built on rivers. Recently, it has become a widely studied research field where data obtained from stream gauge stations can be utilized for creating estimating models by resorting to different methods such as machine and deep learning techniques. In this study, we performed monthly streamflow predictions by using the following data-driven methods of machine learning: linear regression, support vector regression, random forest and deep learning (DL) models to compare the performances of ML's and DL's techniques. A general workflow that can be applied to similar regions is presented. An estimating model containing six-input combinations and time-lagged streamflow data is improved by means of the autocorrelation function (ACF) and partial autocorrelation function (PACF). Furthermore, moving average is used as a smoothing technique to make the dataset more stable and reduce the effects of noise data. A comparative evaluation has been conducted to determine the performances of the above-mentioned methods. In this study, we proposed four different DL models and compared them with existing techniques. For the comparison of the results, we used evaluation criteria such as Nash–Sutcliffe efficiency (NSE), mean square error (MSE) and percent bias (PBIAS). The experimental results indicate that our bidirectional gated recurrent units (BiGRU) model outperforms both ML algorithms and existing solutions with 0.971 NSE, 0.001 MSE and − 1.536 PBIAS scores.

The Snow Microwave Radiative Transfer (SMRT) thermal emission and backscatter model was developed to determine uncertainties in forward modeling through intercomparison of different model ingredients. The model differs from established models by the high degree of flexibility in switching between different electromagnetic theories, representations of snow microstructure, and other modules involved in various calculation steps. SMRT v1.0 includes the dense media radiative transfer theory (DMRT), the improved Born approximation (IBA), and independent Rayleigh scatterers to compute the intrinsic electromagnetic properties of a snow layer. In the case of IBA, five different formulations of the autocorrelation function to describe the snow microstructure characteristics are available, including the sticky hard sphere model, for which close equivalence between the IBA and DMRT theories has been shown here. Validation is demonstrated against established theories and models. SMRT was used to identify that several former studies conducting simulations with in situ measured snow properties are now comparable and moreover appear to be quantitatively nearly equivalent. This study also proves that a third parameter is needed in addition to density and specific surface area to characterize the microstructure. The paper provides a comprehensive description of the mathematical basis of SMRT and its numerical implementation in Python. Modularity supports model extensions foreseen in future versions comprising other media (e.g., sea ice, frozen lakes), different scattering theories, rough surface models, or new microstructure models.

Summary Non‐structural carbohydrates (NSC) play a crucial role in tree resistance and resilience to drought. Stem sapwood parenchyma is among the largest storage tissue for NSC in mature trees. However, there is a limited mechanistic understanding of how NSC reserves, stem parenchyma abundance and growth rates are interrelated, and how they respond to changing water availability. We quantified NSC, ray parenchyma abundance and ring width along four successive 5‐year radial sapwood segments of the stem of 40 mature Pinus sylvestris trees from a 10‐year irrigation experiment conducted at a xeric site in Switzerland. Percentage of ray volume (PERPAR) varied from 3·75 to 8·94% among trees, but showed low intra‐individual variability. PERPAR responded positively to irrigation with a lag of several years, but was unrelated to %NSC. %NSC was lower in wider rings. However, wider rings still contained a larger NSC pool that was positively related to next year's ring growth. Our results suggest that stem ray parenchyma does not limit NSC storage capacity, but responds to long‐term environmental drivers with years of delay. The observed carbon allocation patterns indicate a prioritization of storage over growth independent of growth conditions, likely as a mechanism to ensure long‐term survival. Furthermore, NSC pool size proved to be a determinant for the inter‐annual autocorrelation in tree‐ring growth. Our study highlights the importance of long‐term multi‐parameter studies to better understand tree responses to environmental variability at different time‐scales. A lay summary is available for this article. Lay Summary