Parameterization and Visualization of the Errors in Areal Interpolation

Areal interpolation involves the transfer of data from one zonation of a region to another, where the two zonations of space are geographically incompatible. By its very nature this process is fraught with errors. However, only recently have there been specific attempts to quantify these errors. Fisher and Langford (1995) employed Monte Carlo simulation methods, based on modifiable areal units, to compare the errors resulting from selected areal interpolation techniques. This paper builds on their work by parameterizing and visualizing the errors resulting from the areal weighting and dasymetric methods of areal interpolation. It provides the basis for further research by developing the methodology to produce predictive models of the errors in areal interpolation. Random aggregation techniques are employed to generate multiple sets of source zones and interpolation takes place from these units onto a fixed set of randomly generated target zones. Analysis takes place at the polygon, or target zone level, which enables detailed analysis of the error distributions, basic visualization of the spatial nature of the errors and predictive modeling of the errors based on parameters of the target zones. Correlation and regression analysis revealed that errors from the areal weighting technique were related to the geometric parameters of the target zones. The dasymetric errors, however, demonstrated more association with the population or attribute characteristics of the zones. The perimeter, total population, and population density of the target zones were shown to be the strongest predictive parameters.     

Multi‐Type,Multi‐Zone Facility Location

The placement of facilities according to spatial and/or geographic requirements is a popular problem within the domain of location science. Objectives that are typically considered in this class of problems include dispersion, median, center, and covering objectives—and are generally defined in terms of distance or service‐related criteria. With few exceptions, the existing models in the literature for these problems only accommodate one type of facility. Furthermore, the literature on these problems does not allow for the possibility of multiple placement zones within which facilities may be placed. Due to the unique placement requirements of different facility types—such as suitable terrain that may be considered for placement and specific placement objectives for each facility type—it is expected that different suitable placement zones for each facility type, or groups of facility types, may differ. In this article, we introduce a novel mathematical treatment for multi‐type, multi‐zone facility location problems. We derive multi‐type, multi‐zone extensions to the classical integer‐linear programming formulations involving dispersion, centering and maximal covering. The complexity of these formulations leads us to follow a heuristic solution approach, for which a novel multi‐type, multi‐zone variation of the non‐dominated sorting genetic algorithm‐II algorithm is proposed and employed to solve practical examples of multi‐type, multi‐zone facility location problems.     

Nonparametric Significance Test for Discovery of Network‐Constrained Spatial Co‐Location Patterns

Spatial co‐location patterns are useful for understanding positive spatial interactions among different geographical phenomena. Existing methods for detecting spatial co‐location patterns are mostly developed based on planar space assumption; however, geographical phenomena related to human activities are strongly constrained by road networks. Although these methods can be simply modified to consider the constraints of networks by using the network distance or network partitioning scheme, user‐specified parameters or priori assumptions for determining prevalent co‐location patterns are still subjective. As a result, some co‐location patterns may be wrongly reported or omitted. Therefore, a nonparametric significance test without priori assumptions about the distributions of the spatial features is proposed in this article. Both point‐dependent and location‐dependent network‐constrained summary statistics are first utilized to model the distribution characteristics of the spatial features. Then, by using these summary statistics, a network‐constrained pattern reconstruction method is developed to construct the null model of the test, and the prevalence degree of co‐location patterns is modeled as the significance level. The significance test is evaluated using the facility points‐of‐interest data sets. Experiments and comparisons show that the significance test can effectively detect network‐constrained spatial co‐location patterns with less priori knowledge and outperforms two state‐of‐the‐art methods in excluding spurious patterns.     

A Geostatistical Framework for Area‐to‐Point Spatial Interpolation

The spatial prediction of point values from areal data of the same attribute is addressed within the general geostatistical framework of change of support; the term support refers to the domain informed by each datum or unknown value. It is demonstrated that the proposed geostatistical framework can explicitly and consistently account for the support differences between the available areal data and the sought‐after point predictions. In particular, it is proved that appropriate modeling of all area‐to‐area and area‐to‐point covariances required by the geostatistical frame‐work yields coherent (mass‐preserving or pycnophylactic) predictions. In other words, the areal average (or areal total) of point predictions within any arbitrary area informed by an areal‐average (or areal‐total) datum is equal to that particular datum. In addition, the proposed geostatistical framework offers the unique advantage of providing a measure of the reliability (standard error) of each point prediction. It is also demonstrated that several existing approaches for area‐to‐point interpolation can be viewed within this geostatistical framework. More precisely, it is shown that (i) the choropleth map case corresponds to the geostatistical solution under the assumption of spatial independence at the point support level; (ii) several forms of kernel smoothing can be regarded as alternative (albeit sometimes incoherent) implementations of the geostatistical approach; and (iii) Tobler's smooth pycnophylactic interpolation, on a quasi‐infinite domain without non‐negativity constraints, corresponds to the geostatistical solution when the semivariogram model adopted at the point support level is identified to the free‐space Green's functions (linear in 1‐D or logarithmic in 2‐D) of Poisson's partial differential equation. In lieu of a formal case study, several 1‐D examples are given to illustrate pertinent concepts.     

The Pricing of Delivery Services*

Abstract. Due to spatial separation between points of distribution and consumption, a vital feature of retailing is the provision of delivery services by customers, retailers, or transport intermediaries. Some store retailers, many Internet retailers, and most transport intermediaries use a multizone pricing plan or a generalized N‐zone menu plan. We extend spatial pricing theory to encompass multizone plans. We prove that the optimal number of zones varies with the level of demand relative to the spatial dispersion of customers, and with the cost of implementing a zonal plan. We show that a small number of zones capture almost all the profit attainable with a spatially discriminatory plan.     

THE WEBER PROBLEM: FREQUENCY OF DIFFERENT SOLUTION TYPES AND EXTENSION TO REPULSIVE FORCES AND DYNAMIC PROCESSES*

ABSTRACT. The frequency of occurrence of the different types of solutions to the Weber problem is studied. These solutions are: a location at an attraction point due to a dominant force, to incompatible angles, or to concavity; a location at infinity; a location inside the polygon; and a location outside the polygon. Situations involving both attraction and repulsion points are examined in the triangle and in the more-than-three-sided polygon context, and methods for solving the corresponding problems are compared. A trigonometric solution is proposed for the triangle case involving one repulsion and two attraction points. The variation in the frequency of a location at an attraction point when the number of attraction points increases while the number of repulsion points remains the same is observed as well. Implications of the results are studied for the analysis of dynamic location processes.     

Distance in Spatial Analysis: Measurement,Bias, and Alternatives

Distance is an important and basic concept in geography. Many theories, methods, and applications involve distance explicitly or implicitly. While measuring the distance between two locations is a straightforward task, many geographical processes involve areal units, where the distance measurement can be complicated. This research investigates distance measurement between a location (point) and an area (polygon). We find that traditional polygon-to-point distance measurements, which involve abstracting a polygon into a central or representative point, could be problematic and may lead to biased estimates in regression analysis. To solve this issue, we propose a new polygon-to-point distance metric along with two algorithms to compute the new distance metric. Simulation analysis shows the effectiveness of the new distance metric in providing unbiased estimates in linear regression.     

Analysis of a Distribution of Point Events Using the Network‐Based Quadrat Method

This study proposes a new quadrat method that can be applied to the study of point distributions in a network space. While the conventional planar quadrat method remains one of the most fundamental spatial analytical methods on a two‐dimensional plane, its quadrats are usually identified by regular, square grids. However, assuming that they are observed along a network, points in a single quadrat are not necessarily close to each other in terms of their network distance. Using planar quadrats in such cases may distort the representation of the distribution pattern of points on a network. The network‐based units used in this article, on the other hand, consist of subsets of the actual network, providing more accurate aggregation of the data points along the network. The performance of the network‐based quadrat method is compared with that of the conventional quadrat method through a case study on a point distribution on a network. The χ² statistic and Moran's I statistic of the two quadrat types indicate that (1) the conventional planar quadrat method tends to overestimate the overall degree of dispersion and (2) the network‐based quadrat method derives a more accurate estimate on the local similarity. The article also performs sensitivity analysis on network and planar quadrats across different scales and with different spatial arrangements, in which the abovementioned statistical tendencies are also confirmed.     

Computing geochemical mass transfer and water/rock ratios in submarine hydrothermal systems: implications for estimating the vigour of convection

A method of calculating chemical water/rock ratios is presented that enables the estimation of fluid velocities in open, flow‐through hydrologic systems. The approach is based on relating the gain/loss of a chemical species per kilogram of solid phase to the loss/gain of that species in the fluid phase, integrated across a specified length of the flowpath. After examining the underlying approximations of the approach using a one‐dimensional model of seawater moving through a basalt under nonisothermal conditions, the method is applied to representative zones within a two‐dimensional hydrothermal convective system. The method requires that regions within the flow system can be identified in which the direction of flow is steady for an extended period of time. Estimates of fluid velocity are spatial and temporal averages for the length of the flowpath used in the calculation. The location within the flow system and the nature of the alteration reactions determine which species can provide reliable values of the chemical water/rock ratio and useful estimates of fluid velocities. Over the length of the flowpath considered, the calculation of water/rock ratios works best when a species is controlled by a single reaction. Accurate estimates are favoured if the concentration profile of a species along the flowpath increases or decreases monotonically. If the length of the flowpath extends over more than one reaction zone, then erroneous estimates of the water/rock ratio and fluid velocity are more likely. Model calculations suggest that the quartz/silica system should provide reliable estimates for fluid velocity under a wide range of temperature and flow conditions, in particular in those regions of a system at or near quartz equilibrium, so that the aqueous silica concentration is buffered by quartz and correlated with the temperature distribution.     

Sub‐micron spongiform porosity is the major ultra‐structural alteration occurring in archaeological bone

Total pore volume and pore size distribution are indicators of the degree of post‐mortem modification of bone. Direct measurements of pore size distribution in archaeological bones using mercury intrusion porosimetry (HgIP) and back scattered scanning electron microscopy (BSE‐SEM) reveal a common pattern in the changes seen in degraded bone as compared to modern samples. The estimates of pore size distribution from HgIP and direct measurement from the BSE‐SEM images show remarkable correspondence. The coupling of these two independent approaches has allowed the diagenetic porosity changes in human archaeological bone in the >0.01 µm range to be directly imaged, and their relationship to pre‐existing physiological pores to be explored. The increase in porosity in the archaeological bones is restricted to two discrete pore ranges. The smaller of these two ranges (0.007–0.1 µm) lies in the range of the collagen fibril (0.1 µm diameter) and is presumably formed by the loss of collagen, whereas the larger pore size distribution is evidence of direct microbial alteration of the bone. HgIP has great potential for the characterization of microbial and chemical alteration of bone. Copyright © 2002 John Wiley & Sons, Ltd.     

On the Finite Optimality Set of the Vector Assignment p‐Median Problem

The vector assignment p‐median problem (VAPMP) is one of the first discrete location problems to account for the service of a demand by multiple facilities, and has been used to model a variety of location problems in addressing issues such as system vulnerability and reliability. Specifically, it involves the location of a fixed number of facilities when the assumption is that each demand point is served a certain fraction of the time by its closest facility, a certain fraction of the time by its second closest facility, and so on. The assignment vector represents the fraction of the time a facility of a given closeness order serves a specific demand point. Weaver and Church showed that when the fractions of assignment to closer facilities are greater than more distant facilities, an optimal all‐node solution always exists. However, the general form of the VAPMP does not have this property. Hooker and Garfinkel provided a counterexample of this property for the nonmonotonic VAPMP. However, they do not conjecture as to what a finite set may be in general. The question of whether there exists a finite set of locations that contains an optimal solution has remained open to conjecture. In this article, we prove that a finite optimality set for the VAPMP consisting of “equidistant points” does exist. We also show a stronger result when the underlying network is a tree graph.     

An Area‐Based Approach for Estimating Extreme Precipitation Probability

Accurate estimates of heavy rainfall probabilities reduce loss of life, property, and infrastructure failure resulting from flooding. NOAA's Atlas‐14 provides point‐based precipitation exceedance probability estimates for a range of durations and recurrence intervals. While it has been used as an engineering reference, Atlas‐14 does not provide direct estimates of areal rainfall totals which provide a better predictor of flooding that leads to infrastructure failure, and more relevant input for storm water or hydrologic modeling. This study produces heavy precipitation exceedance probability estimates based on basin‐level precipitation totals. We adapted a Generalized Extreme Value distribution to estimate Intensity‐Duration‐Frequency curves from annual maximum totals. The method exploits a high‐resolution precipitation data set and uses a bootstrapping approach to borrow spatially across homogeneous regions, substituting space in lieu of long‐time series. We compared area‐based estimates of 1‐, 2‐, and 4‐day annual maximum total probabilities against point‐based estimates at rain gauges within watersheds impacted by five recent extraordinary precipitation and flooding events. We found considerable differences between point‐based and area‐based estimates. It suggests that caveats are needed when using pointed‐based estimates to represent areal estimates as model inputs for the purpose of storm water management and flood risk assessment.     

Geographic information system‐based edge effect correction for Ripley's K‐function under irregular boundaries

Ripley's K‐function is a test to detect geographically distributed patterns occurring across spatial scales. Initially, it assumed infinitely continuous planar space, but in reality, any geographic distribution occurs in a bounded region. Hence, the edge problem must be solved in the application of Ripley's K‐function. Traditionally, three basic edge correction methods were designed for regular study plots because of simplified geometric computation: the Ripley circumference, buffer zone, and toroidal methods. For an irregular‐shaped study region, a geographic information system (GIS) is needed to support geometric calculation of complex shapes. The Ripley circumference method was originally implemented by Haase and has been modified into a Python program in a GIS environment via Monte Carlo simulation (hereafter, the Ripley–Haase and Ripley–GIS methods). The results show that in terms of the statistical powers of clustering detection for irregular boundaries, the Ripley–GIS method is the most stable, followed by the buffer zone, toroidal, and Ripley–Haase methods. After edge effects of irregular boundaries have been eliminated, Ripley's K‐function is used to estimate the degree of spatial clustering of cities in a given territory, and in this paper, we demonstrate that by reference to the relationship between urban spatial structure and economic growth in China.     

RETURNS TO SCALE AND REGIONAL GROWTH: THE STATIC‐DYNAMIC VERDOORN LAW PARADOX REVISITED*

ABSTRACT It has long been an article of faith amongst regional economists that increasing returns to scale are necessary to explain the punctiform location of economic activity and population. However, there is no consensus in the empirical literature over whether returns to scale are constant or increasing. A notable example of this lack of agreement is provided by the static‐dynamic Verdoorn law paradox. While the dynamic Verdoorn law (specified using growth rates) yields estimates of substantial increasing returns to scale, the static Verdoorn law (specified using log‐levels) indicates only the presence of constant returns to scale. In this paper, we explain the static‐dynamic Verdoorn law paradox by showing that estimates of returns to scale obtained using the static law are subject to a spatial aggregation bias, which biases the estimates towards constant returns to scale. We illustrate our arguments by means of simulation exercises. The results obtained hold general lessons for applied economic analysis using spatial data.     

A vector of high‐temperature paleo‐fluid flow deduced from mass transfer across permeability barriers (quartz veins)

Quartz veins acted as impermeable barriers to regional fluid flow and not as fluid‐flow conduits in Mesoproterozoic rocks of the Mt Painter Block, South Australia. Systematically distributed asymmetric alteration selvedges consisting of a muscovite‐rich zone paired with a biotite‐rich zone are centered on quartz veins in quartz–muscovite–biotite schist. Geometric analysis of the orientation and facing of 126 veins at Nooldoonooldoona Waterhole reveals a single direction along which a maximum of all veins have a muscovite‐rich side, irrespective of their specific individual orientation. This direction represents a Mesoproterozoic fluid‐flow vector and the veins represent permeability barriers to the flow. The pale muscovite‐rich zones formed on the downstream side of the vein and the dark biotite‐rich zones mark the upstream side. The alteration couplets formed from mica schist at constant Zr, Ga, Sc, and involved increases in Si, Na, Al and decreases in K, Fe, Mg for pale alteration zones, and inverse alteration within dark zones. The asymmetry of the alteration couplets is best explained by the pressure dependence of mineral–fluid equilibria. These equilibria, in combination with a Darcian flow model for coupled advection and diffusion, and with permeability barriers imposed by the quartz veins, simulate the pattern of both fluid flow and differential, asymmetric metasomatism. The determined vector of fluid flow lies along the regional foliation and is consistent with the known distribution of regional alteration products. The presence of asymmetric alteration zones in rock containing abundant pre‐alteration veins suggests that vein‐rich material may have generally retarded regional fluid flow.     

Variations in Mid‐Latitude Auroral Activity During the Holocene*

Estimates of the past location of the auroral zone can aid archaeologists in the interpretation of documented light phenomena in the night sky; in particular, during periods without written records and in regions where aurorae are scarce today. Aurora occurrence largely depends on solar activity as well as the tilt and strength of the geomagnetic field. Here, we introduce a tool (AUREST) that visually combines the time‐dependent variations of these parameters to provide relative estimates of aurora occurrence in mid‐latitude regions of the Northern Hemisphere over the past 10 000 years. Tests against historical aurora observations show good compatibility with AUREST.     

THE ANALYSIS OF DYESTUFFS FROM FIRST‐ TO SECOND‐CENTURY TEXTILE ARTEFACTS FOUND IN THE MARTRES‐DE‐VEYRE (FRANCE) EXCAVATIONS*

A series of Gallo‐Roman clothes and miscellaneous textile fragments from the first and second centuries from the Martres‐de‐Veyre necropolis (Puy‐de‐Dôme, France) was investigated. The objects studied were excavated between 1851 and 1923, during successive rounds of archaeological work. Since that time, they have been conserved in the Barguin Museum at Clermont‐Ferrand. The target of our research was dye identification, as it gives very interesting and useful information regarding the ancient technology. This approach has until now not been considered to be relevant and, unfortunately, is not systematic. The analysis was carried out by high‐performance liquid chromatography with photodiode array detection (HPLC–PDA). The small samples (threads up to 0.6 mg, less than 1 cm long) of coloured textile fibres were submitted to this technique. In one case, X‐ray fluorescence (XRF) analysis was applied. According to the analysis, the dyeing sources proposed are the Rubiaceae family for red, a luteolin ‘base’ for yellow, ellagitannins for brown and indigo for blue colours. Several samples did not seem to contain any dye. We point out possible factors for the partial dye source determination: instrumental limits, sample size, low dyestuff content in analysed samples and the dyes’ decomposition during ageing. Some improvements of dye analysis in archaeological samples by separation techniques based on recent analytical instrumentation and combined approaches are proposed.     

Evaluating an Entropy‐Maximizing Solution to the Ecological Inference Problem: Split‐Ticket Voting in New Zealand, 1999

For geographers, one important aspect of the ecological inference problem relates to the analysis of spatial variations in individual behavior. Obtaining estimates of this behavior, in the absence of direct data, is often difficult, and the “solution” to the ecological inference problem propounded by G. King is not relevant in all circumstances. An alternative, using a different approach, has been used for some time in electoral studies but has lacked “real” data against which to assess the accuracy of its estimates. The availability of such data for New Zealand's 1999 general election allows such an assessment to be made—with very favourable results. The estimates are then used to test hypotheses regarding the spatial variation in split‐ticket voting, again with considerable success.     

Stress‐dependent transport properties of fractured arkosic sandstone

We measure the fluid transport properties of microfractures and macrofractures in low‐porosity polyphase sandstone and investigate the controls of in situ stress state on fluid flow conduits in fractured rock. For this study, the permeability and porosity of the Punchbowl Formation sandstone, a hydrothermally altered arkosic sandstone, were measured and mapped in stress space under intact, microfractured, and macrofractured deformation states. In contrast to crystalline and other sedimentary rocks, the distributed intragranular and grain‐boundary microfracturing that precedes macroscopic fracture formation has little effect on the fluid transport properties. The permeability and porosity of microfractured and intact sandstone depend strongly on mean stress and are relatively insensitive to differential stress and proximity to the frictional sliding envelope. Porosity variations occur by elastic pore closure with intergranular sliding and pore collapse caused by microfracturing along weakly cemented grain contacts. The macroscopic fractured samples are best described as a two‐component system consisting (i) a tabular fracture with a 0.5‐mm‐thick gouge zone bounded by 1 mm thick zones of concentrated transgranular and intragranular microfractures and (ii) damaged sandstone. Using bulk porosity and permeability measurements and finite element methods models, we show that the tabular fracture is at least two orders of magnitude more permeable than the host rock at mean stresses up to 90 MPa. Further, we show that the tabular fracture zone dilates as the stress state approaches the friction envelope resulting in up to a three order of magnitude increase in fracture permeability. These results indicate that the enhanced and stress‐sensitive permeability in fault damage zones and sedimentary basins composed of arkosic sandstones will be controlled by the distribution of macroscopic fractures rather than microfractures.     

Spatial Distribution of Shopping Malls and Analysis of their Trade Areas in Istanbul

This paper investigates the spatial distribution of shopping malls with respect to population and analyses the factors which effect the shopping mall location. According to the results, while the shopping mall space ratio is higher than population ratio in the intermediate zone, the reverse is true in the periphery. The relationships between the shopping mall space and income, population and distance to the central business district (CBD) of the locations are investigated by the use of regression analysis. The results reveal that income is the only factor affecting the location of shopping malls among those considered. In addition, three shopping malls from the inner and peripheral zones of Istanbul were analysed with respect to frequency to shopping centres and characteristics of trade areas. According to the results, the size of the catchment area of the shopping mall from the inner zone is larger then the peripheral ones due to higher accessibility with alternative transportation systems and supporting functions in its surrounding areas. The results fall within the concept of central place theory. Further research is suggested by expending the study to the other areas of the city with different characteristics to calculate their retail potential and their spatial implications.