Introduction
Pollution of the ambient ozone has become a serious health hazard in cities all over the world, since high level ozone is capable of damaging human respiratory systems. It has therefore proven necessary to assess ozone trends and predict its concentrations so as to understand the changing patterns of ozone. Because meteorological, topographical and emission patterns of ozone precursors can adversely affect ozone concentrations, an appropriate spatiotemporal covariance method would be heterogeneous across space and non-stationary over time.
The Kriging model
Numerical models have been widely utilized by scientists seeking to understand and predict spatiotemporal processes. While the specific aspects of the various models are different, they are all deterministic in that they mathematically approximate underlying physical and chemical processes by the use of nonlinear partial differential equations (Berrocal, Gelfand, & Holland, 2009). As direct solutions to these equations are not available, solutions are sought by discretizing both space and time. Consequently, these predictions are given as averages over grid cells. By using a large number of grid cells, the predictions can cover large spatial domains and also give very high temporal solutions. However, since they have been derived using a deterministic paradigm, they lack information regarding inherent uncertainty in their predictions. In addition, they tend to be biased with unknown calibration.
There have been various proposals for spatiotemporal models for calibrating ambient ozone data for a variety of purposes. Guttorp, Meiring and Sampson (1994) and (1998) have used independent spatial deformation models to generate data predictions for each time period in order to evaluate deterministic models, while Carrol et al. (1997) calculated exposure indices in Texas by combining ozone predictions with population data. Dynamic linear models have been used for various purposes like performing short term forecasting over a small region (Dou, LE, & Zidek, 2010) and predicting temporal summaries of ozone and examining meteorologically adjusted trends over space (Sahu, Gelfand, & Holland, 2007). Dou et al. (2010) apply and compare two models for mapping hourly ambient ozone concentration fields over small regions of the US. These two models, namely the Dynamic Linear model (DLM) which was developed by Huerta, Sansó and Stroud in 2004, and an alternative to the Kriging model called the Bayesian model. Another Bayesian based model was again developed and used by McMillan et al. (2005) to develop the spatiotemporal behavior of ozone (O3) in the Michigan region. This model incorporated the link between ozone and meteorology by identifying meteorological regimes conducive for high ozone levels and allowing ozone behaviors during these periods to vary from typical ozone behavior.
The IDW and spine interpolation method are classified as deterministic interpolation methods since they are based on the surrounding measured values or specific mathematical formulas to determine the smoothness of the resulting surface. The other class of interpolation methods includes Kriging and is based on statistical methods that involve autocorrelation, which is the statistical relationship among the measured points (Bohling, 2005). Due to this, geostatistical methods not only contain the capacity to produce predictions, but also provide some measure of the accuracy and certainty of the predictions, making them the more efficient interpolation methods.
The Kriging model is the most popular of all stochastic methods of geostatistical interpolation (U.S. ENVIRONMENTAL PROTECTION AGENCY , 2004). Most kriging methods are gradual, local and may or may not be exact; they perfectly reproduce the measured data. In addition, kriging models do not by definition constrain the set values to the range of measured values. And just like the IDW method, kriging calculates weights for the measured points in deriving the predicted values for unmeasured locations. In kriging models, however, the weights are not only based on the distance between points, but also features the variation the measured points as a function of distance. The model is thus composed of two main parts – an analysis of the spatial variation and the calculation of predicted values.
The kriging model uses variograms for spatial variation by plotting the variance of paired sample measurements as a function of the distance between these samples. After this, a parametric model is typically fitted to the empirical variogram and then utilized to calculate distance weights for interpolation. Kriging then selects weights so that the estimates are not biased and the variance is minimal. Just like regression analysis, a continuous curve will be fitted into the data points in the variogram (Bohling, 2005). From this, identification of the best model is done by running and evaluating several models using geostatistical software applications.
Once the most suitable variogram is identified, kriging creates a seamless surface for the whole area of study using the weights calculated based on the selected variogram model and the values and the location of the points measured. This gives the analyst an opportunity to adjust the distance or number or points measured to be considered while making predictions for each point. All measured points will be considered using a fixed search radius method within the specified distance of each point to be predicted, while a variable search method is used to utilize a specified amount of measured point within varying distances for each prediction (U.S. ENVIRONMENTAL PROTECTION AGENCY , 2004).
Since kriging uses a statistical model for interpolation, certain assumptions must be met. One, it is assumed that the spatial variation is uniform across the area of study and it depends solely on the distance between measures points. Other kriging methods have their own additional assumptions that must be met. For example, simple kriging assumes that there is an established constant mean, that all variations are statistical, and above all, there is no underlying trend. The difference between this and the ordinary kriging is that instead os an established constant mean, the mean is unknown and must me estimated based on the data. On the other hand, universal kriging assumes that there is a trend in the surface which explains the variation in the data, and should only be used when it is certain that the collected data has a trend.
The main advantage of the various kriging methods, and by large other stochastic interpolation methods), is the ability to calculate the uncertainty of the estimates of the model’s prediction, consider it in the analysis, and plot it along with the predicted area (Childs, 2004). This uncertainty information comes as an integral tool in the spatial decision making process.
ESRI ArGIS
GIS is all about spatial data and the tools used in the management, compilation and analysis of the data. ArcGIS spatial analyst extension provides a toolset that is utilized in analyzing and modeling of spatial data. Interpolation tools use a set of sample points to represent the changes in landscape, population or environment to visualize the continuity or variability of data that’s observed across a surface.
ESRI ArcGIS is limited in terms of its capabilities in Spatial Analyst Extension (Landgrebe, 2003). Spatial Analyst has a ML (Maximum Likelihood) classification tool which is used to derive thematic maps from a satellite image or air photo. The most common tasks in imagery analysis is classification. A remotely sensed image with several layers is used in the development of a thematic map where areas in the image are classified as per the type of land cover or other categorical schemes. If done by humans, this process is known as photointerpretation, and is very labor intensive. Research has thus gone into developing an automated classification, and studies have been done to find the most efficient, accurate and effective classification method.
While a trained human eye can interpret grayscale images, automated classification such as the ESRI ArcGIS ML requires multilayered images (Landgrebe, 2003). An example of a multilayered data would be those taken by Landsat satellites of agricultural land at different times of growing, showing different spectral signatures of red, green and blue layers, plus an additional infrared bands.
While using the ESRI ArcGIS, the Spatial Analyst Extension must be installed so as to perform classification since it provides only one kind of supervised classifier; the Maximum Likelihood Classification tool in its toolbox. This ML classifier performs a pixel-by-pixel classification of a multilayer dataset.
ML classification in ArcGIS is a two-step process, with an additional VBA code. The first step is using the Create Signature, using the image raster as input. The output of the Create Signature tool is a “.gsg” file, a signature which ESRI uses for statistical information. the second step is the Maximum Likelihood Classification, which takes image raster bands (red, green and blue) as well as the signature file as its input (U.S. ENVIRONMENTAL PROTECTION AGENCY , 2004). There is, however, no way for verification of the ML classification’s accuracy apart from the use of the focus area itself, due to lack of “ground truth” for the rest of the image, so Visual Basic coding is used as well to complete the classification.
Figure 1 Source U.S. Environmental Protection Agency (2004)
The Spatial Analyst extension employs one of several interpolation tools so as to create a surface grid in ArcGIS. Different interpolation methods will often produce different results (Childs, 2004). The main types of interpolation methods are deterministic or geostatistical. Deterministic methods such as Inverse Distance Weight (IDW) will create surfaces based on measured points and mathematical formulas based on the similarities of cells. Geostatistical methods such as Kriging are based on statistics and are often used for more advanced prediction modeling which includes measuring the certainty and accuracy of the predictions (ESRI, n.d.).
ArcGIS is an integrated collection of several GIS software, consisting of various frameworks for deploying GIs:
- ArcGIS Desktop – an integrated suite of professional GIS software applications
- ArcGIS Engine –An embeddable developer component for custom GIS applications
- Server GIS – ArcSDE, ArcIMS, and ArcGIS Server
- Mobile GIS – ArcPad, as well as ArcGIC Desktop and Engine for Tablet PC computing.
References
Berrocal, V. J., Gelfand, A. E., & Holland, D. M. (2009). A Spatio-Temporal Downscaler for Output From Numerical Models. Journal of Agricultural, Biological, and Environmental Statistics, 15(2), 176-197.
Bohling, G. (2005). Kriging. Kansas: Kansas University.
Childs, C. (2004). Interpolating Surfaces in ArcGIS Spatial Analyst. ESRI Education Services.
Dou, Y., LE, D. N., & Zidek, J. V. (2010). Modeling hourly ozone concentration fields. Annals of Applied Statistics, 4, 1183–1213.
ESRI. (n.d.). Comparing interpolation methods. Retrieved from ArcGIS Pro for Desktop: http://pro.arcgis.com/en/pro-app/tool-reference/3d-analyst/comparing-interpolation-methods.htm
Guttorp, P., Meiring, W., & Sampson, P. D. (1994). A space-time analysis of ground level ozone data. Environmetrics, 5, 241–254.
Guttorp, P., Meiring, W., & Sampson, P. D. (1998). Space-time estimation of gridcell hourly ozone levels for assessment of a deterministic model. Environmental and. Ecological Statistics, 5, 197–222.
Landgrebe, D. A. (2003). Signal theory methods in multispectral remote sensing. Hoboken, NJ: John Wiley & Sons, Inc.
Sahu, S. K., Gelfand, A. E., & Holland, D. M. (2007). High-resolution spacetime ozone modeling for assessing trends. Journal of the American Statistical Association, 102, 1221–1234.
U.S. ENVIRONMENTAL PROTECTION AGENCY . (2004). Developing Spatially Interpolated Surfaces and Estimating Uncertainty. Research Triangle Park, NC: U.S. ENVIRONMENTAL PROTECTION AGENCY.
