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Computing the Area Affected by
Phosphorus Runoff in an Everglades
Wetland Using Bayesian and
Universal Kriging
Song S. Qian
-
Abstract. --Phosphorus-enriched agriculture runoff is believed to be the leading cause
of ecosystem changes in an Everglades wetland. In order to study the effects of the
added nutrients to the wetland ecosystem, it is necessary to estimate the acreage of the
affected region. In this study, Bayesian and universal kriging are used to analyze the
data collected by Reddy, et a1 (1991). The background level of the STP concentration is
used as an indicator of whether the region is affected or not through an indicator
function. The expected value of the affected area is calculated using the results from
Bayesian and universal kriging. The results show that universal kriging is sensitive to
the specification of covariance function. Universal and Bayesian kriging yield
comparable results when the covariance functions are specified in the like manner.
INTRODUCTION
The Water Conservation Area-2A (WCA2A) is part of the Everglades wetlands in
south Florida (Figure 1). It receives the agricultural runoff from the Everglades
Agriculture Area. The phosphorus enriched agriculture runoff caused some significant
changes in this phosphorus limited wetland ecosystem. The most obvious change was in
the change of the dominant vegetation species from sawgrass to cattail near the inlet of the
runoff. The significant increase of phosphorus level in the water of WCA2A is considered
a major threat to the Everglades National Park. To protect the Park, the South Florida
Water Management District proposed to use constructed wetlands, a man-made marsh like
buffer area, to remove excess phosphorus before the water enters WCA2A.
One important design parameter of a constructed wetland is the unit mass loading rate.
The unit area loading rate is usually calculated by measuring the influent mass loading rate
(in mass of phosphorus per unit time) and dividing it by the area of the receiving wetland.
Because WCA2A covers such a large region, not all parts of the region are effective in
removing phosphorus. (In other words, not all parts are affected.) If the entire WCA2A
were taken as affected and used to calculate the mass loading rate, the phosphorus
assimilating capacity of the wetland is underestimated.
The data used in this paper are from Reddy et al. (1991), who sampled soils at 74
stations on seven north -- south transacts, spaced at 2 mile intervals (Figure 2). Richardson
(personal communication, 1993) believes that the STP content should be fairly stable if
there were no agricultural runoff problem, and the background level of the STP content is
Lake Okeechobee
A
Everglades
National
Figure 1. Map of south Florida and location of the study region
Figure 2. Locations of sampling sites in WCA2A
about 500 pg of phosphorus per gram of soil ( ~ g l g )in the top 20 cm. Universal knging
(Cressie, 1991) and Bayesian kriging (Handcock and Stein, 1993) are used to model the
phosphorus concentration over the region. When the concentrations are larger than the
background level, it is believed that the corresponding region is affected by the agricultural
runoff.
METHODS
Data analysis
The mean STP content in the top 20 cm layer of the Reddy samples were used. The
STP values are log-transformed to stabilize the variance. The spatial coordinates were
converted from latitude and longitude to the Universal Transverse Mercator (UTM) grid
system
Universal kriging (UK) can be interpreted as a Gaussian random field model:
z=xp+q
(1)
where: Z is the log transformed STP content, X is the known design matrix
containing the longitude and latitude, and the distance to the nearest pollution
source, p is a vector of unknown parameters, and q is the error term which is
assumed to be normally distributed with mean 0 and covariance C.The covariance
function is cov{Z(s,), Z(s,)} = aK0(s,, s,), for any pair of spatial coordinate points
(s,, s,) of interest, where CL > 0 is a scale parameter, and 0 is a vector of structural
parameters which specify the shape of the covariance function. In the case of
kriging, we observe {Z(s,), Z(s,), ..., Z(sn)}' = Z and the prediction of Z(so) is the
objective, where sois a location with no observed data. The UK predictor is the
best linear unbiased predictor (Cressie, 1991), of the form io(so) = h(8)'Z. See
Ripley ( 1981) for details.
The covariance matrix aK0 is usually unknown, and is often estimated using
the empirical semi-variogram. Once the semi-variogram model is chosen, the
covariance function is taken as known, and the uncertainty in the selected model is
ignored. To include this uncertainty in the analysis, Handcock and Stein (1993)
introduced Bayesian kriging (BK) using the Matern class of covariance functions,
with a parameter 8 = (8,, 8,).
The posterior predictive distribution for Z(so) is derived by adapting a
noninformative prior:
pr(a, P, 0) ~ r c e m
(2)
Handcock and Stein (1993) showed that the conditional posterior distribution
of Z(s,) is:
-
where q is the number of parameters in p.
The marginal posterior distribution of 8 is:
pr(0IZ) = pr(8) IK~I-"IX'K;'XI-"&
(0)-(nnq)'2
(4)
The Bayesian predictive distribution for Z(sJ is:
pr(Z(sJZ) = pr(Z(sJ0, 2) x p r ( 0 W 0
J
Q
(5)
~b),
where: i&,) is the UK point predictor, $0) = &(z- x~)'K;'(z aV,(O) is
the UK prediction error variance, and tn-qis a student-t distribution with (n - q)
degrees of freedom. See Handcock and Stein (1993) for details.
The prior distribution of the parameters of the covariance function was chosen
as:
This prior distribution represents the belief that large values of the parameters
are less likely than small ones.
Area estimation
where I(.) is an indicator function, X is the coordinate vector, and t is the
background level.
When UK is used, the integrand in the RHS of equation (7) can be evaluated
by using the CDF of a normal density. When BK is used:
pr(Z(X) > t 10, Z ) x pr(0 I Z)d0dX
E(Ap)=
(8)
X E R ~@
The uncertainty of the estimated area using BK is evaluated using a Monte
Carlo simulation method. The procedure is based on the fact that given 0, the area
is An = ~ z ( z ( X ) > ti5 0)dX. It can be approximated by sampling Z(X) and
evaluating the integrand, and the mean of these values is approximately equal to
the proportion of the affected area over the entire WCA2A. A Monte Carlo
simulation algorithm was used to sample 0 and then Z(X).
RESULTS
The residuals of the model (1) are found to be intrinsically stationary and
isotropic through visual inspection of the semi-variograms in the north-south and
east-west directions. For UK, four different covariance functions are used. One is
the Matern covariance function using the maximum posterior estimate of 0
(UK-M). The other are: exponential (UK-exp), spherical (UK-sph), and Gaussian
(UK-Gau). Figure 3 shows the log-transformed data.
Covariance function
The covariance function used in BK is presented in Figure 4, the log
transformed posterior as a function of the parameters (0, and 8,). The posterior
Eastinf (km)
Firure 3. Log-transformeddata
The range parameter
Figure 4. Posterior of the covariance function parameters
Distance (km)
Figure 5. Covariance functions
Log transformed STP
Figure 6. Predictive distributions, site 10
Log transformed STP
Figure 7. Predictive distributions,site 25
mode is at (3.6515, -0.4697), or, the Maximum Posterior Estimates (MPE) of the
parameters are (4482,0.3391). The estimated a given the MPE of 0 is 0.032363.
The covariance functions in UK are estimated through fitting the
semi-variogram model. Figure 5 compares the covariance functions computed
from different semi-variogram models. In Figure 5, the line labeled "Matem" is the
Matern covariance function using the MPE of 0.
Predictive distributions
The predictive distributions of the log-transformed STP are estimated for the
10th sampling site, which has the largest STP content, and the 25th sampling site,
which has a below average STP content. The density functions are shown in
Figures 6 and 7. Table 1 compares the results from BK and UK using the three
semi-variogram models.
Model
BK
UK-M
UK-exp
UK-Gau
UK-sph
BK
UK-M
UK-exp
UK-Gau
Table 1. Predictive distributions
Sampling Site Predicted Mean Prob. > 500 Crg/g Measured STP
25
2.48754
0.076779
2.398536
25
2.478 177
0.052068
2.398536
25
2.4627 11
0.030408
2.398536
25
2.179858
0.000001
2.398536
25
2.453698
0.022335
2.398536
10
2.826391
0.78698 1
3.100039
10
2.850576
0.848553
3.100039
10
2.8375 1
0.823649
3.100039
10
2.814715
0.736275
3.100039
Comparing the five predictive distributions for the 10th sampling site, we note
that the predictive distributions from UK-exp, UK-sph, and UK-Mat are nearly
identical. BK yields a similar mean value but slightly larger estimation variance.
The predictive distribution from UK-Gau has the largest variance. For the 25th
sampling site, we see comparable predictive distributions from UK-exp, UK-sph,
and UK-M. BK yields a similar mean and slightly larger variance. UK-Gau
produces a much different predictive distribution from the other four. The mode is
significantly less then the modes from the other four predictive distributions. The
Gaussian covariance function is equivalent to the Matern covariance function with
0, -+ -. This means that UK-Gau describes a function which is infinitely
differentiable. This representation may not be realistic.
Computation of the affected area
Equation (6) was used for UK and Equation (7) was used for BK. The results
are listed in Table 2.
In this study, we did not try to locate the boundary of the affected region.
However, the shape of the affected region may be stable and we plot the contour
lines of the probability of STP larger than 500 pglg to indicate the possible shape.
Figure 8 is plotted using BK, and Figures 9 to 12 show the surfaces from UK. The
surface from the UK-Gau is mostly flat but has sudden jumps. The reason for this
unnaturally shaped surface is that the Gaussian covariance function describes
"super smooth" surfaces, in this case, super smooth for the residuals. A model
with very smoothed residual surface must have a jumpy mean surface.
I~odel
Table 2. Expected area
affected area (km2)1% of the total areal
1
From Table 2, we note that the estimated area using BK is very close to the
estimated area using UK-M and UK-exp. The estimated area is sensitive to the
specification of the covariance function when UK is used. It is reasonable to state
BK is more appropriate in this study, since the covariance function is unknown and
the uncertainty in estimating the covariance function is considered.
The uncertainty of the estimated area is evaluated only for BK. Figure 13
presents the histogram of the percentages of the affected area from the Monte
Carlo simulation.
DISCUSSION
In conclusion, it is seen that one significant difference is an increased
computational intensity of BK. However, the results indicate that UK is sensitive
to the specification of the covariance function. The difference between two
different covariance functions may be significant even though the semi-variogram
models of the two are fitted to the data equally well. This sensitivity to the
covariance function justifies the use of BK.
There are several studies that estimated the phosphorus affected area of
WCA2A. Craft and Richardson (1993) use the same data set as presented in this
paper. In their study, the boundary of the affected region was delineated by using
both the STP content and the phosphorus accumulation rate. The delineated
boundary is plotted onto a USGS topographic map, and the area is thus measured.
Their estimate of the area is 115 km2. Only the top 10 cm layer of the Reddy data
was used and 600 pg/g is the background level. The STP contour lines were
produced using the ordinary kriging algorithm from a commercial graphic software
(surfero). The semi-variogram model used is not reported.
Using the STP content as an indicator of the influence of the agricultural runoff
is a more appropriate way of estimating the affected area. A spatial statistical
Easting (km)
Figure 8. Bayesian
Easting (km)
Figure 9. Matern
Easting (km)
Figure 10. Exponential
Easting (krn)
Figure 11. Gaussian
Easting (km)
Figure 12. Spherical
Figure 13. dff8f@dr&ae8dimated area
approach accounts for the uncertainty in the sample data. Considering the high
level of uncertainty involved in the estimating process (Figure 13), the estimated
area using BK should be considered comparable to the result in Craft and
Richardson (1993). The discrepancy between the estimates may be caused by two
factors, without considering the error introduced by the process of measuring the
area from a map. They are their use of ordinary kriging and may be a different
covariance function, and the different background level. There is no reason to
believe that the background level is a constant. The field samples from the region
that is unaffected by the agriculture runoff show a significant variation (Craft and
Richardson, 1993). In a current study by the author, the background level is
represented by a probability distribution.
ACKNOWLEDGMENT
Drs. Richard Smith and Peter Miiller motivated the author to pursue this
project. Dr. Marcia Gumpertz's comments and suggestions significantly improved
this article. The author appreciate the encouragement and support from Drs. C.J.
Richardson and K.H. Reckhow.
REFERENCES
Craft, C.B. and C.J. Richardson. 1993. Peat accretion and phosphorus
accumulation along a eutrophication gradient in the northern Everglades.
Biogeochemistry, 22, 133-156.
Cressie, N.A.C., 1991. Statistics for Spatial Data, John Wiley & Sons, Inc. New
York.
Handcock, M.S., and M.L. Stein. 1993. A Bayesian analysis of kriging,
Technometrics, 35(4), 403 -4 10.
Howard-Williams, C. 1985. Cycling and retention of nitrogen and phosphorus in
wetlands: a theoretical and applied perspective, Freshwater Biology, 15,
391-43 1.
Reddy, K.R., W.F. DeBusk, Y. Wang, R. DeLaune, and M. Koch, 1991.
Physico-chemical properties of soils in the Water Conservation Area 2 of the
Everglades. Final report submitted to South Florida Water Management
District, West Palm Beach, FL 33416.
Ripley, B.D. 1981. Spatial Statistics, John Wiley & Sons, Inc. New York.
BIOGRAPHICAL SKETCH
Song S. Qian is a postdoctoral research associate at the Duke Wetland Center of the
Nicholas School of the Environment of Duke University. He received his Ph.D. in
environmental sciences and MS degree in statistics from Duke University. He has a
Master's degree in environmental systems engineering from Nanjing University, China.