A singular vector perspective of 4DVAR: The spatial structure and evolution of baroclinic weather systems

The extent to which the four-dimensional variational data assimilation (4DVAR) is able to use information about the time evolution of the atmosphere to infer the vertical spatial structure of baroclinic weather systems is investigated. The singular value decomposition (SVD) of the 4DVAR observability matrix is introduced as a novel technique to examine the spatial structure of analysis increments. Specific results are illustrated using 4DVAR analyses and SVD within an idealized 2D Eady model setting. Three different aspects are investigated. The first aspect considers correcting errors that result in normal-mode growth or decay. The results show that 4DVAR performs well at correcting growing errors but not decaying errors. Although it is possible for 4DVAR to correct decaying errors, the assimilation of observations can be detrimental to a forecast because 4DVAR is likely to add growing errors instead of correcting decaying errors. The second aspect shows that the singular values of the observability matrix are a useful tool to identify the optimal spatial and temporal locations for the observations. The results show that the ability to extract the time-evolution information can be maximized by placing the observations far apart in time. The third aspect considers correcting errors that result in nonmodal rapid growth. 4DVAR is able to use the model dynamics to infer some of the vertical structure. However, the specification of the case-dependent background error variances plays a crucial role.

Optical characterization of the growth and spatial structure of a substorm onset arc

We present a detailed case study of the characteristics of auroral forms that constitute the first ionospheric signatures of substorm expansion phase onset. Analysis of the optical frequency and along-arc (azimuthal) wave number spectra provides the strongest constraint to date on the potential mechanisms and instabilities in the near-Earth magnetosphere that accompany auroral onset and which precede poleward arc expansion and auroral breakup. We evaluate the frequency and growth rates of the auroral forms as a function of azimuthal wave number to determine whether these wave characteristics are consistent with current models of the substorm onset mechanism. We find that the frequency, spatial scales, and growth rates of the auroral forms are most consistent with the cross-field current instability or a ballooning instability, most likely triggered close to the inner edge of the ion plasma sheet. This result is supportive of a near-Earth plasma sheet initiation of the substorm expansion phase. We also present evidence that the frequency and phase characteristics of the auroral undulations may be generated via resonant processes operating along the geomagnetic field. Our observations provide the most powerful constraint to date on the ionospheric manifestation of the physical processes operating during the first few minutes around auroral substorm onset.

A theory of rational spatial agglomerations

We model the behavior of rational forward-looking agents in a spatial economy. The economic geography structure is built on Fujita et al. (1999)'s racetrack economy. Workers choose optimally what to consume at each period, as well as which spatial itinerary to follow in the geographical space. The spatial extent of the resulting agglomerations increases with the taste for variety and the expenditure share on manufactured goods, and decreases with transport costs. Because forward-looking agents anticipate the future formation of agglomerations, they are more responsive to spatial utility differentials than myopic agents. As a consequence, the emerging agglomerations are larger under perfect foresight spatial adjustments than under myopic ones.

Pollinator community responses to the spatial population structure of wild plants: A pan-European approach

Land-use changes can alter the spatial population structure of plant species, which may in turn affect the attractiveness of flower aggregations to different groups of pollinators at different spatial scales. To assess how pollinators respond to spatial heterogeneity of plant distributions and whether honeybees affect visitation by other pollinators we used an extensive data set comprising ten plant species and their flower visitors from five European countries. In particular we tested the hypothesis that the composition of the flower visitor community in terms of visitation frequencies by different pollinator groups were affected by the spatial plant population structure, viz. area and density measures, at a within-population (‘patch’) and among-population (‘population’) scale. We found that patch area and population density were the spatial variables that best explained the variation in visitation frequencies within the pollinator community. Honeybees had higher visitation frequencies in larger patches, while bumblebees and hoverflies had higher visitation frequencies in sparser populations. Solitary bees had higher visitation frequencies in sparser populations and smaller patches. We also tested the hypothesis that honeybees affect the composition of the pollinator community by altering the visitation frequencies of other groups of pollinators. There was a positive relationship between visitation frequencies of honeybees and bumblebees, while the relationship with hoverflies and solitary bees varied (positive, negative and no relationship) depending on the plant species under study. The overall conclusion is that the spatial structure of plant populations affects different groups of pollinators in contrasting ways at both the local (‘patch’) and the larger (‘population’) scales and, that honeybees affect the flower visitation by other pollinator groups in various ways, depending on the plant species under study. These contrasting responses emphasize the need to investigate the entire pollinator community when the effects of landscape change on plant–pollinator interactions are studied.

Designing a sampling scheme to reveal correlations between weeds and soil properties at multiple spatial scales

Weeds tend to aggregate in patches within fields and there is evidence that this is partly owing to variation in soil properties. Because the processes driving soil heterogeneity operate at different scales, the strength of the relationships between soil properties and weed density would also be expected to be scale-dependent. Quantifying these effects of scale on weed patch dynamics is essential to guide the design of discrete sampling protocols for mapping weed distribution. We have developed a general method that uses novel within-field nested sampling and residual maximum likelihood (REML) estimation to explore scale-dependent relationships between weeds and soil properties. We have validated the method using a case study of Alopecurus myosuroides in winter wheat. Using REML, we partitioned the variance and covariance into scale-specific components and estimated the correlations between the weed counts and soil properties at each scale. We used variograms to quantify the spatial structure in the data and to map variables by kriging. Our methodology successfully captured the effect of scale on a number of edaphic drivers of weed patchiness. The overall Pearson correlations between A. myosuroides and soil organic matter and clay content were weak and masked the stronger correlations at >50 m. Knowing how the variance was partitioned across the spatial scales we optimized the sampling design to focus sampling effort at those scales that contributed most to the total variance. The methods have the potential to guide patch spraying of weeds by identifying areas of the field that are vulnerable to weed establishment.

Spatial covariation of Azotobacter abundance and soil properties: A case study using the wavelet transform

The soil microflora is very heterogeneous in its spatial distribution. The origins of this heterogeneity and its significance for soil function are not well understood. A problem for understanding spatial variation better is the assumption of statistical stationarity that is made in most of the statistical methods used to assess it. These assumptions are made explicit in geostatistical methods that have been increasingly used by soil biologists in recent years. Geostatistical methods are powerful, particularly for local prediction, but they require the assumption that the variability of a property of interest is spatially uniform, which is not always plausible given what is known about the complexity of the soil microflora and the soil environment. We have used the wavelet transform, a relatively new innovation in mathematical analysis, to investigate the spatial variation of abundance of Azotobacter in the soil of a typical agricultural landscape. The wavelet transform entails no assumptions of stationarity and is well suited to the analysis of variables that show intermittent or transient features at different spatial scales. In this study, we computed cross-variograms of Azotobacter abundance with the pH, water content and loss on ignition of the soil. These revealed scale-dependent covariation in all cases. The wavelet transform also showed that the correlation of Azotobacter abundance with all three soil properties depended on spatial scale, the correlation generally increased with spatial scale and was only significantly different from zero at some scales. However, the wavelet analysis also allowed us to show how the correlation changed across the landscape. For example, at one scale Azotobacter abundance was strongly correlated with pH in part of the transect, and not with soil water content, but this was reversed elsewhere on the transect. The results show how scale-dependent variation of potentially limiting environmental factors can induce a complex spatial pattern of abundance in a soil organism. The geostatistical methods that we used here make assumptions that are not consistent with the spatial changes in the covariation of these properties that our wavelet analysis has shown. This suggests that the wavelet transform is a powerful tool for future investigation of the spatial structure and function of soil biota. (c) 2006 Elsevier Ltd. All rights reserved.

The Representative Soil Sampling Scheme of England and Wales: the spatial variation of topsoil nutrient status and pH between 1971 and 2001

The Representative Soil Sampling Scheme (RSSS) has monitored the soil of agricultural land in England and Wales since 1969. Here we describe the first spatial analysis of the data from these surveys using geostatistics. Four years of data (1971, 1981, 1991 and 2001) were chosen to examine the nutrient (available K, Mg and P) and pH status of the soil. At each farm, four fields were sampled; however, for the earlier years, coordinates were available for the farm only and not for each field. The averaged data for each farm were used for spatial analysis and the variograms showed spatial structure even with the smaller sample size. These variograms provide a reasonable summary of the larger scale of variation identified from the data of the more intensively sampled National Soil Inventory. Maps of kriged predictions of K generally show larger values in the central and southeastern areas (above 200 mg L-1) and an increase in values in the west over time, whereas Mg is fairly stable over time. The kriged predictions of P show a decline over time, particularly in the east, and those of pH show an increase in the east over time. Disjunctive kriging was used to examine temporal changes in available P using probabilities less than given thresholds of this element. The RSSS was not designed for spatial analysis, but the results show that the data from these surveys are suitable for this purpose. The results of the spatial analysis, together with those of the statistical analyses, provide a comprehensive view of the RSSS database as a basis for monitoring the soil. These data should be taken into account when future national soil monitoring schemes are designed.

A geostatistical analysis of the spatial variation of soil mineral nitrogen and potentially available nitrogen within an arable field.

The technology for site-specific applications of nitrogen (N) fertilizer has exposed a gap in our knowledge about the spatial variation of soil mineral N, and that which will become available during the growing season within arable fields. Spring mineral N and potentially available N were measured in an arable field together with gravimetric water content, loss on ignition, crop yield, percentages of sand, silt, and clay, and elevation to describe their spatial variation geostatistically. The areas with a larger clay content had larger values of mineral N, potentially available N, loss on ignition and gravimetric water content, and the converse was true for the areas with more sandy soil. The results suggest that the spatial relations between mineral N and loss on ignition, gravimetric water content, soil texture, elevation and crop yield, and between potentially available N and loss on ignition and silt content could be used to indicate their spatial patterns. Variable-rate nitrogen fertilizer application would be feasible in this field because of the spatial structure and the magnitude of variation of mineral N and potentially available N.

Toward a combined seasonal weather and crop productivity forecasting system: Determination of the working spatial scale

A methodology is presented for the development of a combined seasonal weather and crop productivity forecasting system. The first stage of the methodology is the determination of the spatial scale(s) on which the system could operate; this determination has been made for the case of groundnut production in India. Rainfall is a dominant climatic determinant of groundnut yield in India. The relationship between yield and rainfall has been explored using data from 1966 to 1995. On the all-India scale, seasonal rainfall explains 52% of the variance in yield. On the subdivisional scale, correlations vary between variance r(2) = 0.62 (significance level p < 10(-4)) and a negative correlation with r(2) = 0.1 (p = 0.13). The spatial structure of the relationship between rainfall and groundnut yield has been explored using empirical orthogonal function (EOF) analysis. A coherent, large-scale pattern emerges for both rainfall and yield. On the subdivisional scale (similar to 300 km), the first principal component (PC) of rainfall is correlated well with the first PC of yield (r(2) = 0.53, p < 10(-4)), demonstrating that the large-scale patterns picked out by the EOFs are related. The physical significance of this result is demonstrated. Use of larger averaging areas for the EOF analysis resulted in lower and (over time) less robust correlations. Because of this loss of detail when using larger spatial scales, the subdivisional scale is suggested as an upper limit on the spatial scale for the proposed forecasting system. Further, district-level EOFs of the yield data demonstrate the validity of upscaling these data to the subdivisional scale. Similar patterns have been produced using data on both of these scales, and the first PCs are very highly correlated (r(2) = 0.96). Hence, a working spatial scale has been identified, typical of that used in seasonal weather forecasting, that can form the basis of crop modeling work for the case of groundnut production in India. Last, the change in correlation between yield and seasonal rainfall during the study period has been examined using seasonal totals and monthly EOFs. A further link between yield and subseasonal variability is demonstrated via analysis of dynamical data.

Power properties if invariant tests for spatial autocorrelation in linear regression

This paper derives some exact power properties of tests for spatial autocorrelation in the context of a linear regression model. In particular, we characterize the circumstances in which the power vanishes as the autocorrelation increases, thus extending the work of Krämer (2005). More generally, the analysis in the paper sheds new light on how the power of tests for spatial autocorrelation is affected by the matrix of regressors and by the spatial structure. We mainly focus on the problem of residual spatial autocorrelation, in which case it is appropriate to restrict attention to the class of invariant tests, but we also consider the case when the autocorrelation is due to the presence of a spatially lagged dependent variable among the regressors. A numerical study aimed at assessing the practical relevance of the theoretical results is included

The DigiBog model of peatland development 1: rationale, conceptual model, and hydrological basis

Using a literature review, we argue that new models of peatland development are needed. Many existing models do not account for potentially important ecohydrological feedbacks, and/or ignore spatial structure and heterogeneity. Existing models, including those that simulate a near total loss of the northern peatland carbon store under a warming climate, may produce misleading results because they rely upon oversimplified representations of ecological and hydrological processes. In this, the first of a pair of papers, we present the conceptual framework for a model of peatland development, DigiBog, which considers peatlands as complex adaptive systems. DigiBog accounts for the interactions between the processes which govern litter production and peat decay, peat soil hydraulic properties, and peatland water-table behaviour, in a novel and genuinely ecohydrological manner. DigiBog consists of a number of interacting submodels, each representing a different aspect of peatland ecohydrology. Here we present in detail the mathematical and computational basis, as well as the implementation and testing, of the hydrological submodel. Remaining submodels are described and analysed in the accompanying paper. Tests of the hydrological submodel against analytical solutions for simple aquifers were highly successful: the greatest deviation between DigiBog and the analytical solutions was 2·83%. We also applied the hydrological submodel to irregularly shaped aquifers with heterogeneous hydraulic properties—situations for which no analytical solutions exist—and found the model's outputs to be plausible.

A closer look at chaotic advection in the stratosphere: part I: geometric structure

The relevance of chaotic advection to stratospheric mixing and transport is addressed in the context of (i) a numerical model of forced shallow-water flow on the sphere, and (ii) a middle-atmosphere general circulation model. It is argued that chaotic advection applies to both these models if there is suitable large-scale spatial structure in the velocity field and if the velocity field is temporally quasi-regular. This spatial structure is manifested in the form of “cat’s eyes” in the surf zone, such as are commonly seen in numerical simulations of Rossby wave critical layers; by analogy with the heteroclinic structure of a temporally aperiodic chaotic system the cat’s eyes may be thought of as an “organizing structure” for mixing and transport in the surf zone. When this organizing structure exists, Eulerian and Lagrangian autocorrelations of the velocity derivatives indicate that velocity derivatives decorrelate more rapidly along particle trajectories than at fixed spatial locations (i.e., the velocity field is temporally quasi-regular). This phenomenon is referred to as Lagrangian random strain.

Effects of spatial resolution in the simulation of daily and subdaily precipitation in the southwestern US

We evaluate the effects of spatial resolution on the ability of a regional climate model to reproduce observed extreme precipitation for a region in the Southwestern United States. A total of 73 National Climate Data Center observational sites spread throughout Arizona and New Mexico are compared with regional climate simulations at the spatial resolutions of 50 km and 10 km for a 31 year period from 1980 to 2010. We analyze mean, 3-hourly and 24-hourly extreme precipitation events using WRF regional model simulations driven by NCEP-2 reanalysis. The mean climatological spatial structure of precipitation in the Southwest is well represented by the 10 km resolution but missing in the coarse (50 km resolution) simulation. However, the fine grid has a larger positive bias in mean summer precipitation than the coarse-resolution grid. The large overestimation in the simulation is in part due to scale-dependent deficiencies in the Kain-Fritsch convective parameterization scheme that generate excessive precipitation and induce a slow eastward propagation of the moist convective summer systems in the high-resolution simulation. Despite this overestimation in the mean, the 10 km simulation captures individual extreme summer precipitation events better than the 50 km simulation. In winter, however, the two simulations appear to perform equally in simulating extremes.

Dynamic auroral structure in the vicinity of the polar cusp - multipoint observations during southward and northward IMF

The low- and high-latitude boundary layers of the earth's magnetosphere [low-latitude boundary layer (LLBL) and mantle] play important roles in transferring momentum and energy from the solar wind to the magnetosphere-ionosphere system. Particle precipitation, field-aligned current, auroral emission, ionospheric ion drift and ground magnetic perturbations are among the low-altitude parameters that show signatures of various plasma processes in the LLBL and the magnetopause current layer. Magnetic merging events, Kelvin-Helmholtz waves, and pressure pulses excited by the variable solar wind/magnetosheath plasma are examples of boundary phenomena that may be coupled to the ionosphere via field-aligned currents. Optical auroral observation, by photometry and all-sky TV cameras, is a unique technique for investigating the spatial and temporal structure of the electron precipitation associated with such phenomena. However, the distinction between the different boundary layer plasma populations cannot in general be unambiguously determined by optics alone. Additional information, such as satellite observations of particle boundaries and field-aligned currents, is needed in order to identify the plasma source(s) and the magnetosphere-ionosphere coupling mode(s). Two categories of auroral activity/structure in the vicinity of the polar cusp are discussed in this paper, based on combined ground and satellite data. In one case, the quasi-periodic sequence of auroral events at the polar cap boundary involves accelerated electrons (< 1 keV) moving poleward (< 1 km s-1) and azimuthally along the persistent cusp/cleft arc poleward boundary with velocities (< 4 km s-1), comparable to the local ionospheric ion drift during periods of southward IMF. A critical question is whether or not the optical events signify a corresponding plasma flow across the open/closed field line boundary in such cases. Near-simultaneous observations of magnetopause flux transfer events (FTEs) and such optical/ion drift events are reported. The reverse pattern of motion of discrete auroral forms is observed during positive interplanetary magnetic field (IMF) B(Z), i.e. equatorward motion into the cusp/cleft background arc from the poleward edge. Combined satellite and ground-based information for the latter cases indicate a source mechanism, poleward of the cusp at the high-latitude magnetopause or plasma mantle, giving rise to strong momentum transfer and electron precipitation structures within a approximately 200 km-wide latitudinal zone at the cusp/cleft poleward boundary. The striking similarities of auroral electrodynamics in the cleft/mantle region during northward and southward IMF indicate that a qualitatively similar solar wind-magnetosphere coupling mode is operating. It is suggested that, in both cases, the discrete auroral forms represent temporal/spatial structure of larger-scale convection over the polar magnetosphere.