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Scale is a fundamental geographic concept, and a substantial literature exists discussing the various roles that scale plays in different geographical contexts. Relatively little work exists, though, that provides a means of measuring the geographic scale over which different processes operate. Here we demonstrate how geographically weighted regression (GWR) can be adapted to provide such measures. GWR explores the potential spatial nonstationarity of relationships and provides a measure of the spatial scale at which processes operate through the determination of an optimal bandwidth. Classical GWR assumes that all of the processes being modeled operate at the same spatial scale, however. The work here relaxes this assumption by allowing different processes to operate at different spatial scales. This is achieved by deriving an optimal bandwidth vector in which each element indicates the spatial scale at which a particular process takes place. This new version of GWR is termed multiscale geographically weighted regression (MGWR), which is similar in intent to Bayesian nonseparable spatially varying coefficients (SVC) models, although potentially providing a more flexible and scalable framework in which to examine multiscale processes. Model calibration and bandwidth vector selection in MGWR are conducted using a back-fitting algorithm. We compare the performance of GWR and MGWR by applying both frameworks to two simulated data sets with known properties and to an empirical data set on Irish famine. Results indicate that MGWR not only is superior in replicating parameter surfaces with different levels of spatial heterogeneity but provides valuable information on the scale at which different processes operate.

Geographically weighted regression (GWR) is a spatial statistical technique that recognizes that traditional ‘global’ regression models may be limited when spatial processes vary with spatial context. GWR captures process spatial heterogeneity by allowing effects to vary over space. To do this, GWR calibrates an ensemble of local linear models at any number of locations using ‘borrowed’ nearby data. This provides a surface of location-specific parameter estimates for each relationship in the model that is allowed to vary spatially, as well as a single bandwidth parameter that provides intuition about the geographic scale of the processes. A recent extension to this framework allows each relationship to vary according to a distinct spatial scale parameter, and is therefore known as multiscale (M)GWR. This paper introduces mgwr, a Python-based implementation of MGWR that explicitly focuses on the multiscale analysis of spatial heterogeneity. It provides novel functionality for inference and exploratory analysis of local spatial processes, new diagnostics unique to multi-scale local models, and drastic improvements to efficiency in estimation routines. We provide two case studies using mgwr, in addition to reviewing core concepts of local models. We present this in a literate programming style, providing an overview of the primary software functionality and demonstrations of suggested usage alongside the discussion of primary concepts and demonstration of the improvements made in mgwr.

Humans and climate affect ecosystems and their services, which may involve continuous and discontinuous transitions from one stable state to another. Discontinuous transitions are abrupt, irreversible and among the most catastrophic changes of ecosystems identified. For terrestrial ecosystems, it has been hypothesized that vegetation patchiness could be used as a signature of imminent transitions. Here, we analyse how vegetation patchiness changes in arid ecosystems with different grazing pressures, using both field data and a modelling approach. In the modelling approach, we extrapolated our analysis to even higher grazing pressures to investigate the vegetation patchiness when desertification is imminent. In three arid Mediterranean ecosystems in Spain, Greece and Morocco, we found that the patch-size distribution of the vegetation follows a power law. Using a stochastic cellular automaton model, we show that local positive interactions among plants can explain such power-law distributions. Furthermore, with increasing grazing pressure, the field data revealed consistent deviations from power laws. Increased grazing pressure leads to similar deviations in the model. When grazing was further increased in the model, we found that these deviations always and only occurred close to transition to desert, independent of the type of transition, and regardless of the vegetation cover. Therefore, we propose that patch-size distributions may be a warning signal for the onset of desertification.

The enhanced warming trend and precipitation decline in the Mediterranean region make it a climate change hotspot. We compare projections of multiple Coupled Model Intercomparison Project Phase 5 (CMIP5) and Phase 6 (CMIP6) historical and future scenario simulations to quantify the impacts of the already changing climate in the region. In particular, we investigate changes in temperature and precipitation during the 21st century following scenarios RCP2.6, RCP4.5 and RCP8.5 for CMIP5 and SSP1-2.6, SSP2-4.5 and SSP5-8.5 from CMIP6, as well as for the HighResMIP high-resolution experiments. A model weighting scheme is applied to obtain constrained estimates of projected changes, which accounts for historical model performance and inter-independence in the multi-model ensembles, using an observational ensemble as reference. Results indicate a robust and significant warming over the Mediterranean region during the 21st century over all seasons, ensembles and experiments. The temperature changes vary between CMIPs, CMIP6 being the ensemble that projects a stronger warming. The Mediterranean amplified warming with respect to the global mean is mainly found during summer. The projected Mediterranean warming during the summer season can span from 1.83 to 8.49 ∘C in CMIP6 and 1.22 to 6.63 ∘C in CMIP5 considering three different scenarios and the 50 % of inter-model spread by the end of the century. Contrarily to temperature projections, precipitation changes show greater uncertainties and spatial heterogeneity. However, a robust and significant precipitation decline is projected over large parts of the region during summer by the end of the century and for the high emission scenario (−49 % to −16 % in CMIP6 and −47 % to −22 % in CMIP5). While there is less disagreement in projected precipitation than in temperature between CMIP5 and CMIP6, the latter shows larger precipitation declines in some regions. Results obtained from the model weighting scheme indicate larger warming trends in CMIP5 and a weaker warming trend in CMIP6, thereby reducing the difference between the multi-model ensemble means from 1.32 ∘C before weighting to 0.68 ∘C after weighting.

Despite showing great success of applications in many commercial fields, machine learning and data science models generally show limited success in many scientific fields, including hydrology (Karpatne et al., 2017). The approach is often criticized for its lack of interpretability and physical consistency. This has led to the emergence of new modelling paradigms, such as theory-guided data science (TGDS) and physics-informed machine learning. The motivation behind such approaches is to improve the physical meaningfulness of machine learning models by blending existing scientific knowledge with learning algorithms. Following the same principles in our prior work (Chadalawada et al., 2020), a new model induction framework was founded on genetic programming (GP), namely the Machine Learning Rainfall–Runoff Model Induction (ML-RR-MI) toolkit. ML-RR-MI is capable of developing fully fledged lumped conceptual rainfall–runoff models for a watershed of interest using the building blocks of two flexible rainfall–runoff modelling frameworks. In this study, we extend ML-RR-MI towards inducing semi-distributed rainfall–runoff models. The meaningfulness and reliability of hydrological inferences gained from lumped models may tend to deteriorate within large catchments where the spatial heterogeneity of forcing variables and watershed properties is significant. This was the motivation behind developing our machine learning approach for distributed rainfall–runoff modelling titled Machine Induction Knowledge Augmented – System Hydrologique Asiatique (MIKA-SHA). MIKA-SHA captures spatial variabilities and automatically induces rainfall–runoff models for the catchment of interest without any explicit user selections. Currently, MIKA-SHA learns models utilizing the model building components of two flexible modelling frameworks. However, the proposed framework can be coupled with any internally coherent collection of building blocks. MIKA-SHA's model induction capabilities have been tested on the Rappahannock River basin near Fredericksburg, Virginia, USA. MIKA-SHA builds and tests many model configurations using the model building components of the two flexible modelling frameworks and quantitatively identifies the optimal model for the watershed of concern. In this study, MIKA-SHA is utilized to identify two optimal models (one from each flexible modelling framework) to capture the runoff dynamics of the Rappahannock River basin. Both optimal models achieve high-efficiency values in hydrograph predictions (both at catchment and subcatchment outlets) and good visual matches with the observed runoff response of the catchment. Furthermore, the resulting model architectures are compatible with previously reported research findings and fieldwork insights of the watershed and are readily interpretable by hydrologists. MIKA-SHA-induced semi-distributed model performances were compared against existing lumped model performances for the same basin. MIKA-SHA-induced optimal models outperform the lumped models used in this study in terms of efficiency values while benefitting hydrologists with more meaningful hydrological inferences about the runoff dynamics of the Rappahannock River basin.

Environmental heterogeneity is ubiquitous, but environmental systems are often analyzed as if they were homogeneous instead, resulting in aggregation errors that are rarely explored and almost never quantified. Here I use simple benchmark tests to explore this general problem in one specific context: the use of seasonal cycles in chemical or isotopic tracers (such as Cl−, δ18O, or δ2H) to estimate timescales of storage in catchments. Timescales of catchment storage are typically quantified by the mean transit time, meaning the average time that elapses between parcels of water entering as precipitation and leaving again as streamflow. Longer mean transit times imply greater damping of seasonal tracer cycles. Thus, the amplitudes of tracer cycles in precipitation and streamflow are commonly used to calculate catchment mean transit times. Here I show that these calculations will typically be wrong by several hundred percent, when applied to catchments with realistic degrees of spatial heterogeneity. This aggregation bias arises from the strong nonlinearity in the relationship between tracer cycle amplitude and mean travel time. I propose an alternative storage metric, the young water fraction in streamflow, defined as the fraction of runoff with transit times of less than roughly 0.2 years. I show that this young water fraction (not to be confused with event-based \"new water\" in hydrograph separations) is accurately predicted by seasonal tracer cycles within a precision of a few percent, across the entire range of mean transit times from almost zero to almost infinity. Importantly, this relationship is also virtually free from aggregation error. That is, seasonal tracer cycles also accurately predict the young water fraction in runoff from highly heterogeneous mixtures of subcatchments with strongly contrasting transit-time distributions. Thus, although tracer cycle amplitudes yield biased and unreliable estimates of catchment mean travel times in heterogeneous catchments, they can be used to reliably estimate the fraction of young water in runoff.

China is experiencing increasingly serious ambient ozone pollution, including the economically developed Pearl River Delta (PRD) region. However, the underlying reasons for the ozone increase remain largely unclear, leading to perplexity regarding formulating effective ozone control strategies. In this study, we quantitatively examine the impacts of meteorology and precursor emissions from within and outside of the PRD on the evolution of ozone during the past decade by developing a statistical analysis framework combining meteorological adjustment and source apportionment. We found that meteorological conditions mitigated ozone increase, and that their variation can account for a maximum of 15 % of the annual ozone concentration in the PRD. Precursor emissions from outside the PRD (“nonlocal”) have the largest contribution to ambient ozone in the region and show a consistently increasing trend, whereas emissions from within the PRD (“local”) show a significant spatial heterogeneity and play a more important role during ozone episodes over the southwest of the region. Under general conditions, the impact on the northeastern PRD is positive but decreasing, and in the southwest it is negative but increasing. During ozone episodes, the impact on the northeastern PRD is negative and decreasing, whereas in the southwestern PRD it is positive but decreasing. The central and western PRD are the only areas with an increasing local ozone contribution. The spatial heterogeneity in both the local ozone contribution and its trend under general conditions and during ozone episodes is well interpreted by a conceptual diagram that collectively takes ozone precursor emissions and their changing trends, ozone formation regimes, and the monsoonal and microscale synoptic conditions over different subregions of the PRD into consideration. In particular, we conclude that an inappropriate NOx∕VOC control ratio within the PRD over the past few years is most likely responsible for the ozone increase over southwest of this region, both under general conditions and during ozone episodes. By investigating the ozone evolution influenced by emission changes within and outside of the PRD during the past decade, this study highlights the importance of establishing a dichotomous ozone control strategy to tackle general conditions and pollution events separately. NOx emission control should be further strengthened to alleviate the peak ozone level during episodes. Detailed investigation is needed to retrieve appropriate NOx∕VOC ratios for different emission and meteorological conditions, so as to maximize the ozone reduction efficiency in the PRD.

While soil respiration is known to be controlled by a range of biotic and abiotic factors, its temperature sensitivity in global models is largely related to climate parameters. Here, we show that temperature sensitivity of soil respiration is primarily controlled by interacting soil properties and only secondarily by vegetation traits and plant growth conditions. Temperature was not identified as a primary driver for the response of soil respiration to warming. In contrast, the nonlinearity and large spatial variability of identified controls stress the importance of the interplay among soil, vegetation and climate parameters in controlling warming responses. Global models might predict current soil respiration but not future rates because they neglect the controls exerted by soil development. To accurately predict the response of soil respiration to warming at the global scale, more observational studies across pedogenetically diverse soils are needed rather than focusing on the isolated effect of warming alone.Understanding the temperature sensitivity of soil respiration is critical to determining soil carbon dynamics under climate change. Spatial heterogeneity in controls highlights the importance of interactions between vegetation, soil and climate in driving the response of respiration to warming.

Westerly driven, long-range transportation of dust particles in elevated aerosol layers (EALs) is a persistent phenomenon during spring and summer over the Indian subcontinent. During the snow accumulation season, EALs transport substantial amounts of dust to the snow-covered slopes of high-mountain Asia (HMA). Here we use multiple satellite-based estimates to demonstrate a robust physical association between the EALs and dust-induced snow darkening over HMA. Results from a fully coupled atmosphere–chemistry–snow model support these observations, revealing across HMA a signature of increasing dust-induced snow darkening with surface elevation that peaks near 4,500 m. Moreover, the influence of dust on snow darkening is greater than that of black carbon above 4,000 m. Our findings suggest a discernible role of dust in the observed spatial heterogeneity of snowmelt and snowline trends over HMA and highlight an increasing contribution of dust to snowmelt as the snowline rises with warming.Dust deposition in high-mountain Asia lowers snow albedo and hastens melt. Satellite data and models show that dust arrives via transport in elevated aerosol layers and outweighs black carbon impacts at high altitudes, suggesting a growing importance of dust on snowmelt as snowlines rise with warming.

A theoretically based relationship for the Darcy–Weisbach friction factor$f$for rough-bed open-channel flows is derived and discussed. The derivation procedure is based on the double averaging (in time and space) of the Navier–Stokes equation followed by repeated integration across the flow. The obtained relationship explicitly shows that the friction factor can be split into at least five additive components, due to: (i) viscous stress; (ii) turbulent stress; (iii) dispersive stress (which in turn can be subdivided into two parts, due to bed roughness and secondary currents); (iv) flow unsteadiness and non-uniformity; and (v) spatial heterogeneity of fluid stresses in a bed-parallel plane. These constitutive components account for the roughness geometry effect and highlight the significance of the turbulent and dispersive stresses in the near-bed region where their values are largest. To explore the potential of the proposed relationship, an extensive data set has been assembled by employing specially designed large-eddy simulations and laboratory experiments for a wide range of Reynolds numbers. Flows over self-affine rough boundaries, which are representative of natural and man-made surfaces, are considered. The data analysis focuses on the effects of roughness geometry (i.e. spectral slope in the bed elevation spectra), relative submergence of roughness elements and flow and roughness Reynolds numbers, all of which are found to be substantial. It is revealed that at sufficiently high Reynolds numbers the roughness-induced and secondary-currents-induced dispersive stresses may play significant roles in generating bed friction, complementing the dominant turbulent stress contribution.