The Importance of Saltmarshes

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sampled and measured in various places in California, United States (Figure 1). .... She has just obtained her MSc degree in Spatial Information Technology at ...
Integrating GIS with Fuzzy Logic and Geostatistics: Predicting Air Pollutant PM10 for California, Using Fuzzy Kriging Danni Guo, Renkuan Guo, Christien Thiart Department of Statistical Sciences University of Cape Town, Private Bag , Rhodes Gift, Rondebosch 7707, Cape Town, South Africa [email protected] [email protected] [email protected]

1. Introduction PM10 is one of the seven air pollutants the Environmental Protection Agency regulates, exposure to high outdoor PM10 concentrations causes increased disease and death. PM10 concentrations are sampled and measured in various places in California, United States (Figure 1). However, it is too costly in terms of time, finance, and manpower to sample the entire state. Facing the problem of the lack of samples and lack of analysis for actual decision-making, what is the solution? As air pollution is spatial in nature and the measurement or the degree of pollution is fuzzy (the boundaries between negligible, moderate and severe is not clear cut), Geographic information systems (GIS) and the fuzzy approach to geostatistics can help in environmental assessment and decision-making. The fuzzy approach presents a way to process vague or imprecise environmental data, and geostatistics can be used where information is fragmentary, and there is a need to predict the values of properties at unsampled locations.

Figure 1. PM10 Sample Locations in California In this project, we integrate GIS with fuzzy logic and geostatistics, to predict PM10 concentrations for the entire California State, based on existing sample data. Based on the indicator vaiogram and kriging developments in the literature, a methodology of fuzzy variogram and fuzzy kriging is proposed. Furthermore, unlike previous studies that used assumed membership functions, the sample membership functions are extracted from the data itself. The predicted membership grades are also transformed back into PM10 concentrations by using inverse functions. Using the converted prediction maps, we could clearly identify areas that are higher than the threshold, which are dangerous to human health.

2. Fuzzy variogram The experimental indicator semi-variogram is defined as:

ˆ (h) 

(1)

1 n (h) 2   (s  h)    (s)  n (h) i 1

Correspondingly, the fuzzy semi-vairogram, for fuzzy event A is defined as:

 A (h)

  

(2)

2

A

(s  h)   A (s)  dP

Rd

Therefore the experimental fuzzy variogram is defined as:

 A (h) 

2 1 n (h)   A (s j  h)   A (s j )   n(h) j 1

(3)

3. Computations and Analysis The data preparations for the fuzzy variogram and kriging is very essential to the fuzzy geostatistical modelling. Therefore, similar to the indicator kriging, the fuzzy kriging can be done by simply replacing the spatial z(sj), j=1,…,m observations by the corresponding membership grades  A (s j  h) , j=1,…,m respectively. The public’s interest is in whether living in California is safe or not, in terms of air pollution. Therefore fuzzy event A ={High hazard level of PM10 in CA} is worthwhile to try to investigate. It is meaningful to argue that the higher of the PM10 content in the air, the higher the degree of belongingness to fuzzy set A should be. It is reasonable to assume that the linear sample membership function of the fuzzy event A is linear. It has the feature of PM10 level at x=9.0 being assigned a 0 membership, and at x=90.3 being assigned a 1 membership. That is:

 0  x  9 if 0  x  9.0  ˆ A ( x)   if 9.0  x  90.3  81.3 if x  90.3  1 The linear sample membership function  Ar ( x) (Figure 2) is in general simple and mathematically easily manipulated.

(4)

Figure 2. Linear Sample Membership Function of PM10 Using the linear sample membership function, a set of new membership values is calculated and then the Geostatistical Analyst Extension of ArcGIS is used to perform ordinary kriging on the new fuzzy data set. The rational quadratic model is used to fit the variogram model (Figure 3).

Figure 3. Fuzzy Variogram of Linear Sample Membership Function The QQ plot (Figure 4) shows how well the model fits. A good fit is when the sample quantile distribution is very similar to the normal quantile line. In this plot, the middle sample values (dots), are very similar to the dash straight line. However, the extreme upper and lower values do not fit well.

Figure 4. Standardized Error Plot of Linear Sample Membership Function The prediction map (Figure 5) shows the predicted membership grades. The dark brown colours represent higher membership grades, near 1, which are of high PM10 concentrations. The light yellow areas represent low membership grades, near 0, which are of low PM10 concentrations. Therefore, the

three dark brown areas on the lower half of California, show high PM10 concentrations. The upper areas of California are light yellow, and it shows areas of low PM10 concentrations.

Figure 5. Prediction Map of Linear Sample Membership Function The predicted error map (Figure 6) shows the standard error (uncertainty of the predicted value). The light yellow areas are areas of low error, and the dark brown areas are areas of high error. When one compares this map to the map of sample locations, one can understand the errors much better. Since most of the sample locations are near the coast, it is obvious that the errors are less in the sampled locations, and more errors in the unsampled locations.

Figure 6. Prediction Standard Error Map of Linear Sample Membership Function.

The fuzzy membership is defined in terms of PM10 hazard levels. The form of membership is monotone-increasing continuous function, and so the inverse function exists. The linear sample membership function takes the form:

 ( x) 

xa ,a  x  b ba

(5)

For any given membership value 0   0,1 :

xa ba

(6)

xlinear (0 )  0 (b  a)  a

(7)

0  Then:

Conversion based on the membership prediction map class interval limits. In the prediction map, it is typically classified into ten classes, say, class 0, class 1, … class 9. Assume the class interval limits







are  l0 , u0 ,  l1 , u1 ,...,  l9 , u9 . The class limits for the PM10 predicted value with linear sample membership function is:

 xl , xu    xlinear (l ), xlinear (u )  This makes it easy to convert the linear sample membership function back to the original units of measurement, PM10 concentration μg/m3. These values are given in Table 1 and the map using the concentration values as the legend is given in Figure 7. Class Linear Membership PM10 (μg/m3) 0 0.001230 - 0.065054 9.1 - 14.3 1 0.065054 - 0.099254 14.3 - 17.1 2 0.099254 - 0.117579 17.1 - 18.6 3 0.117579 - 0.127399 18.6 - 19.4 4 0.127399 - 0.145724 19.3 - 20.8 5 0.145724 - 0.179923 20.8 - 23.6 6 0.179923 - 0.243748 23.6 - 28.8 7 0.243748 - 0.362860 28.8 - 38.5 8 0.362860 - 0.585151 38.5 - 56.6 9 0.585151 - 1.000000 56.4 - 90.3 Table 1. Conversion from Linear Membership to PM10 Concentrations The current EPA standard for PM10 has set an annual allowable arithmetic average of PM10 not to exceed 50 μg/m3. However, we defined the hazard level differently in this case. Once PM10 concentrations are over 38 μg/m3, it is classified a hazard area.

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Figure 7. PM10 Concentrations Predicted by Linear Membership Function In figure 7, the last two classes, very dark brown colours, show areas of high levels of PM10 concentrations. The level of safeness regarding PM10 concentrations is shown clearly here, by the class divisions of concentration level. The upper area of California and parts of the coastline are shown to have very low PM10 concentrations, and are the safest areas. The linear membership show much finer divisions of areas of low PM10 concentrations.

4. Conclusion In this project, a theoretical integration of fuzzy logic and geostatistics is explored, named fuzzy variogram and fuzzy kriging. The fuzzy kriging is developed as a natural extension of indicator kriging, and then ordinary kriging is performed on the membership results. This way fuzzy kriging would have the advantages of both indicator and ordinary kriging. A new set of methodology is developed for this project. The fuzzy methodology developed here is different from previous work on GIS and fuzzy. Unlike previous studies that used assumed membership functions, in this project, a sample membership function is extracted from the data itself. Therefore, the fuzzy methodology used in this project is more solid and objective. In this project, we successfully developed a set of methodology for fuzzy logic and geostatistics to be integrated together in GIS.

Bibliography Burrough, P.A., McDonnell, R.A. (1998), Principles of Geographical Information Systems (New York: Oxford University Press). Chen, S.Y. (1998) Engineering Fuzzy Set Theory and Application (Beijing: Defense Industry Publishing House). Cressie, N.A.C. (1993) Statistics for Spatial Data (New York: John Wiley & Sons). Guo, R., Love, E. (2003) Reliability Modelling with Fuzzy Covariates. International Journal of Reliability and Safety Engineering, 10, pp.131-157. Journel, A.G. (1983) Nonparametric Estimation of Spatial Distributions. Journal of the International Association for Mathematical Geology, 15, pp. 445-468.

Journel, A.G., Huijbregts, C.J. (1978) Mining Geostatistics (London: Academic Press). Vega, N.A. (2002) Potential Applicability of Fuzzy Logic in Geostatistics, Department of Mining and Minerals Engineering, VirginiaTech, http://filebox.vt.edu/users/anieto/web/fuzzy/fuzzy.htm Zadeh, A. (1965) Fuzzy Sets. Information and Control, 8, pp. 338-353. Biography Danni Guo completed her undergraduate degree and her honours at the University of Cape Town, South Africa. She has just obtained her MSc degree in Spatial Information Technology at the University of Durham, United Kingdom. She is now studying for her PhD at the University of Cape Town, South Africa.