Practical issues and tools for modeling
temporal and spatio-temporal trends in
atmospheric pollutant monitoring data
Paul D. Sampson
Department of Statistics
University of Washington
The International Environmetric Society
Modelling Spatio-Temporal Trends Workshop
3 November 2003
1
Our experience in analysis of trends in atmospheric pollutants
Part I: Meteorological adjustment and long-term temporal trends in ozone
• Meteorological adjustment of western Washington and northwest Oregon
surface ozone observations with investigation of trends. Reynolds, Das,
Sampson, Guttorp, NRCSE TRS #15
(http://www.nrcse.washington.edu/pdf/trs15_doe.pdf)
• Meteorological adjustment of Chicago, Illinois, surface ozone observations
with investigation of trends. Reynolds, Caccia, Sampson, Guttorp, NRCSE
TRS #25 (http://www.nrcse.washington.edu/pdf/trs25_chicago.pdf)
• A review of statistical methods for the meteorological adjustment of
tropospheric ozone. Thompson, Reynolds, Cox, Guttorp, Sampson,
Atmospheric Environment 35, 617-630, 2001.
Part II: Spatial trend for health effects studies
• Spatial estimation of ambient air concentrations for ozone, 1986-94, for
chronic health effects modeling in 83 counties in the U.S. Current contract
with U.S. EPA.
• Spatio-temporal modeling and prediction of ambient PM2.5
concentrations for acute and chronic health effects modeling with the
NIH/NHLBI cohort study, MESA (Multi-Ethnic Study of
Atheroscloerosis). Current proposal and ongoing collaboration with
colleagues at the Univ of Washington’s Northwest Center for Particulate
Matter and Health.
2
Spatio-temporal modeling of ambient PM
exposure for chronic health effect studies
Paul D. Sampson
Department of Statistics
University of Washington
Northwest Center for Particulate Matter & Health
External Science Advisory Committee Meeting
12 November 2003
3
Motivation for fine(r)-scale spatial modeling of
pollutant exposure for chronic health effect
studies:
• Major North American cohort studies of PM: single communitywide exposure/monitor to characterize a metropolitan area. Fails
to address important local spatial variation of air pollutants known
to exist within regions.
• Hoek et al.: 3-component regression model to predict exposure
to air pollutants (black smoke and NO2). Incorporates (a) regional
background levels, (b) urban gradient (based on population
density) and (c) proximity to heavily-trafficked roadways and other
point sources.
• Build on this approach to combine in a spatial model average
concentration data from fixed-site ambient monitors and spatial
covariate information encoded in a GIS, including population
density, proximity to roads, and traffic density.
4
Aside: U.S. EPA currently funded Epidemiologic Research on Health
Effects of Long-Term Exposure to Ambient P.M. and Other Air Pollutants
(June 2003)
• Laden, Schwartz et al (Harvard): Chronic Exposure to Particulate
Matter and Cardiopulmonary Disease. Nurses Health Study:
Prospective cohort study of 121,700 women throughout U.S.
• Knutson, Beeson et al (Loma Linda): Relating Cardiovascular
Disease Risk to Ambient Air Pollutants using GIS and Bayesian Neural
Networks. AHSMOG study.
• Samet, Zeger, Dominici et al (Johns Hopkins): Chronic and Acute
Exposure to Ambient Fine Particulate Matter and Other Air Pollutants:
National Cohort Studies of Mortality and Morbidity. Data from Medicate
beneficiary file and National Claims History File.
• Diez-Roux, Keeler, Samson, Lin (Michigan). Long-Term Exposure to
Ambient PM and Subclincial Atherosclerosis. MESA Study.
5
EPA apparently mandated/directed that all these studies
be concerned with computing exposure estimates from
ambient monitoring data and GIS-based information on
local traffic, pop density, … .
Jon Samet (Johns Hopkins): “EPA should invest in
drawing national maps of exposure as all our research
groups are trying to do the same thing.”
6
Applications:
• “MESA Air”: NHLBI-funded Multi-Ethnic Study of Atherosclerosis:
effects of ambient PM (and other pollutants) on subclinical
cardiovascular function
– 8700 subjects, aged 50-89, from 9 communities, assessed prospectively,
longitudinally.
– Monitoring data and exposure assessment:
• Current AQS PM monitors (mostly 3-day sampling)
• Supplemental monitors, up to 5 per community (2 week integrated msmts of
key pollutants)
• Mobile gradient monitoring (2 week integrated sampling)
PLUS
• Distances to nearest major roadways with traffic volume and composition
• Distances to pollutant point source
EVALUATION on PM2.5 and co-pollutants measured at 10% of homes
• Preliminary demonstration of spatio-temporal modeling using S.
Calif ozone data.
7
Personal PM exposure for subject I at time t:
sum of non-ambient (N) and ambient (A) components:
E E E
P
it
N
it
A
it
Ambient exposure is ambient concentration times an
ambient exposure attenuation factor reflecting time
spent outside the home and particle infiltration into the
home:
A
A
Eit   it Cit
Model for ambient concentration: trend + residual
C   ( si , t )  v( si , t )
A
it
8
Smoothly varying spatio-temporal trend is further
decomposed:
 (si , t )  1 (si )  2 (si , t )
• the 1st term represents long-term mean concentration and
will derive from a Bayesian analysis of a spatial regression
model combining average concentration data from fixedsite ambient monitors and spatial covariate information
encoded in a GIS.
• the 2nd component represents mainly smooth seasonal
temporal variation.
9
C   ( si , t )  v( si , t )
A
it
The variance model for the residual term represents the
spatio-temporal variation considered primarily at the 2week time scale of the fixed sites and mobile gradient
monitors.
Estimation of this component will be based on
(extensions to) the Bayesian model for the SampsonGuttorp spatial deformation approach to nonstationary
spatial covariance as demonstrated in Damian et al.
(2001, 2003). This modeling strategy accommodates the
spatial varying effects of predominant meteorology, coast
lines and topographic features that underlie the statistical
relationship between time varying pollutant levels at
different points in space.
10
Spatial Analysis Region Definitions
11
Region 1: Northeast, all 125 sites and target counties
12
Region 6: S Calif, all 82 sites and target counties
13
Region 6: S Calif, all 94 sites, fitting and validation
Fitting (63)
Validation (31)
Los Angeles County
14
Region 6 : S. Calif
Starplot of temporal trend coefficients (LA)
061111003
061112003
060719004
060714003
060370113
060374002
15
0.4
0.2
0.0
sqrt(max 8hr O3)
60714003
01/01/1989
01/01/1990
01/01/1991
01/01/1992
01/01/1993
01/01/1994
01/01/1993
01/01/1994
01/01/1993
01/01/1994
1987-1994
0.4
0.2
0.0
sqrt(max 8hr O3)
60719004
01/01/1989
01/01/1990
01/01/1991
01/01/1992
1987-1994
0.4
0.2
0.0
sqrt(max 8hr O3)
60374002
01/01/1989
01/01/1990
01/01/1991
01/01/1992
1987-1994
16
0.4
0.2
0.0
sqrt(max 8hr O3)
60370113
01/01/1989
01/01/1990
01/01/1991
01/01/1992
01/01/1993
01/01/1994
01/01/1993
01/01/1994
01/01/1993
01/01/1994
1987-1994
0.4
0.2
0.0
sqrt(max 8hr O3)
61112003
01/01/1989
01/01/1990
01/01/1991
01/01/1992
1987-1994
0.4
0.2
0.0
sqrt(max 8hr O3)
61111003
01/01/1989
01/01/1990
01/01/1991
01/01/1992
1987-1994
17
Some Southern California PM2.5 Monitors
s9002v1
s1201v1
s1002v1
s1103v1
s1301v1
s0002v1
s2005v1
s1601v1
s8001v1
s1003v1
s4002v1
18
01/01/2001
01/01/2001
s1003v1
s1103v1
3
2
1
0
1
2
3
log PM2.5
4
dates.99.01
01/01/2000
01/01/2001
01/01/1999
01/01/2000
01/01/2001
s1201v1
s1301v1
3
2
0
1
2
3
log PM2.5
4
dates.99.01
4
dates.99.01
1
log PM2.5
01/01/2000
dates.99.01
0
01/01/1999
3
01/01/1999
0
log PM2.5
01/01/1999
2
0
01/01/2000
4
01/01/1999
1
log PM2.5
3
2
1
0
log PM2.5
4
s1002v1
4
s0002v1
01/01/2000
01/01/2001
dates.99.01
01/01/1999
01/01/2000
01/01/2001
dates.99.01
19
01/01/2001
3
01/01/1999
01/01/2000
01/01/2001
dates.99.01
s4002v1
s8001v1
3
2
1
0
1
2
3
log PM2.5
4
dates.99.01
0
log PM2.5
01/01/1999
2
0
01/01/2000
4
01/01/1999
1
log PM2.5
3
2
1
0
log PM2.5
4
s2005v1
4
s1601v1
01/01/2000
01/01/2001
dates.99.01
01/01/1999
01/01/2000
01/01/2001
dates.99.01
3
2
1
0
log PM2.5
4
s9002v1
01/01/1999
01/01/2000
01/01/2001
dates.99.01
20
2.0
01/01/2000
01/01/2001
01/01/1999
biweekly s1103v1
01/01/2000
01/01/2001
01/01/1999
01/01/2000
01/01/2001
biweekly s1201v1
biweekly s1301v1
2.0
3.0
log PM2.5
4.0
dates(row.names(pm.biweek))
4.0
dates(row.names(pm.biweek))
3.0
log PM2.5
3.0
2.0
3.0
log PM2.5
4.0
biweekly s1003v1
2.0
01/01/1999
01/01/2001
dates(row.names(pm.biweek))
2.0
log PM2.5
01/01/1999
01/01/2000
dates(row.names(pm.biweek))
4.0
01/01/1999
3.0
log PM2.5
3.0
2.0
log PM2.5
4.0
biweekly s1002v1
4.0
biweekly s0002v1
01/01/2000
01/01/2001
dates(row.names(pm.biweek))
01/01/1999
01/01/2000
01/01/2001
dates(row.names(pm.biweek))
21
2.0
01/01/2000
01/01/2001
01/01/1999
01/01/2001
biweekly s4002v1
biweekly s8001v1
3.0
2.0
3.0
log PM2.5
4.0
dates(row.names(pm.biweek))
2.0
log PM2.5
01/01/1999
01/01/2000
dates(row.names(pm.biweek))
4.0
01/01/1999
3.0
log PM2.5
3.0
2.0
log PM2.5
4.0
biweekly s2005v1
4.0
biweekly s1601v1
01/01/2000
01/01/2001
dates(row.names(pm.biweek))
01/01/1999
01/01/2000
01/01/2001
dates(row.names(pm.biweek))
3.0
2.0
log PM2.5
4.0
biweekly s9002v1
01/01/1999
01/01/2000
01/01/2001
dates(row.names(pm.biweek))
22
(1) Estimation of the long-term mean spatial field
Following Hoek and colleagues (2002 Atmos Env, 2003
Epidemiology), assume the regression model can be
written
1 (si )  0  1vi1   2vi 2 
  k vik
Where v1 , , vk represent pop density, proximity to
roads, traffic density, and possibly local topographic and
climatic wind patterns.
The Bayesian analysis incorporates prior information on
the parameters and on the spatial covariance structure
of residuals from this regression model in a manner
similar to that of our Bayesian framework for spatial
estimation of the residual component (see (3) below).
23
Mean field
Note that monitoring observations will be used directly
in the estimation, not just in the specification or
calibration of the regression model as in the work of
Hoek et al. I.e., in Hoek et al., (long-term) exposure is
estimated as:
Cˆ (si )  ˆ1 (si )  ˆ0  ˆ1vi1  ˆ2vi 2 
 ˆk vik
In our (geostatistical) approach, we will be estimating
the space-time field, Cˆ (si , t ) ; the long-term exposure
at a point includes an estimated (“kriged”) spatial
residual and can be written:
Cˆ ( si )  ˆ1 ( si )  eˆ( si , t )
 ˆ 0  ˆ1vi1  ˆ2 vi 2 
 ˆk vik  eˆ( si , t )
24
(2) Smooth, spatially varying, temporal variation.
 (si , t )  1 (si )  2 ( si , t )
The spatial index in the 2nd component allows for the
possibility that the magnitude and precise details of the
seasonal variation may vary from location to location over
the spatial scale of the regional target communities.
Preliminary analysis of PM2.5 monitoring data in the Los
Angeles county region suggests some spatial variation in
seasonality, but in some regions we expect to find that
this seasonal variation is homogeneous, permitting an
additive (separable) decomposition of the spatio-temporal
trend.
 (si , t )  1 ( si )  2 (t )
25
Trend decomposition
Characterize and estimate the seasonal structure of air
pollutant concentrations in terms of a model written as:
C ( si , t )  1 ( si )  2 ( si , t )   ( si , t )
J
2 ( si , t )    j ( si ) f j (t )
j 1
where the f j (t ) are temporal basis functions
describing possible seasonal trend patterns, and  j ( si )
represent spatially varying coefficients of these trend
patterns.
Example: O3 trend components. (What do we
expect with PM more generally?)
26
We compute trend components empirically as smoothed
versions of the temporal singular vectors of the TN data
matrix (rather than assuming parametric forms such as
trigonometric functions). Arbitrary amounts of missing data
are accommodated in an EM-like iterative calculation of the
SVD.
The Bayesian spatial regression model can incorporate the
coefficients of these trend components as spatial fields,
and thus provide the basis for estimation of  2 ( si , t ) at
target homes.
27
400
200
0
Singular value
600
800
Singular values of T=2912 x S=545 observation matrix
0
100
200
300
Index, 1:545
400
500
28
0.04
0.02
0.0
-0.04
Annual.svd$svd$u[1:1456, j]
Annual Trend Component 1
01/01/1987
10/01/1987
07/01/1988
04/01/1989
01/01/1990
10/01/1990
01/01/1994
10/01/1994
0.04
0.02
0.0
-0.02
Annual.svd$svd$u[1457:2912, j]
dates87to94[1:1456]
01/01/1991
10/01/1991
07/01/1992
04/01/1993
dates87to94[1457:2912]
29
0.06
0.02
-0.02
-0.06
Annual.svd$svd$u[1:1456, j]
Annual Trend Component 2
01/01/1987
10/01/1987
07/01/1988
04/01/1989
01/01/1990
10/01/1990
01/01/1994
10/01/1994
0.06
0.02
-0.02
-0.06
Annual.svd$svd$u[1457:2912, j]
dates87to94[1:1456]
01/01/1991
10/01/1991
07/01/1992
04/01/1993
dates87to94[1457:2912]
30
0.04
0.0
-0.04
Annual.svd$svd$u[1:1456, j]
Annual Trend Component 3
01/01/1987
10/01/1987
07/01/1988
04/01/1989
01/01/1990
10/01/1990
01/01/1994
10/01/1994 31
0.06
0.02
-0.02
-0.06
Annual.svd$svd$u[1457:2912, j]
dates87to94[1:1456]
01/01/1991
10/01/1991
07/01/1992
04/01/1993
dates87to94[1457:2912]
0.06
0.02
-0.02
-0.06
Annual.svd$svd$u[1:1456, j]
Annual Trend Component 4
01/01/1987
10/01/1987
07/01/1988
04/01/1989
01/01/1990
10/01/1990
01/01/1994
10/01/1994
0.04
0.0
-0.04
Annual.svd$svd$u[1457:2912, j]
dates87to94[1:1456]
01/01/1991
10/01/1991
07/01/1992
04/01/1993
dates87to94[1457:2912]
32
Region 6 : S. Calif
Starplot of temporal trend coefficients (LA)
061111003
061112003
060719004
060714003
060370113
060374002
33
0.0
0.5
1.0
1.5
0.0
0.5
1.0
1.5
distance
1.5
b1.b4
0.5
lin.b4
0.2
0.6
0.0
b1.b3
0.0
0.05
0.2
0.4
b1.b2
-0.2
-0.05
lin.b3
0.0
0.4
0.8
0
-0.006
0.4
-3*10^-6
0.6 0.0
0.2
0.0 -0.012
lin.b2
0.0 0.0
10^-6
-0.006
1.2
2*10^-6
2*10^-11
0.0 -8*10^-8
lin.b1
-0.5
-0.2
mu.b4
-0.3
-10^-6
mu.b3
-0.6
2*10^-6
0.005
-0.012
mu.b2
-0.2
0
-0.005
mu.b1
-0.6
-0.6
-2*10^-6
0.010
6*10^-11
-2*10^-8
0.0001
mu.lin
-1.0
-8*10^-6
0.0
semivariance
0.0004
0.0007
mu
Linear Model of Coregionalization
with Gaussian (co)-variograms, (fit.lmc=T)
lin
b1
b2
b2.b3
b3
b2.b4
b3.b4
0.0
0.5
1.0
b4
1.5
34
Ordinary kriging prediction of mu
0.26
1.5
0.25
1.0
0.24
0.5
0.23
0.0
y
0.22
0.21
-0.5
0.20
-1.0
0.19
-1.5
0.18
-2
-1
0
x
1
2
35
Ordinary kriging prediction of b2
1.5
1.0
1.0
0.5
0.5
y
0.0
-0.5
0.0
-1.0
-0.5
-1.5
-2
-1
0
x
1
2
36
0.4
0.2
0.0
sq rt Ozone
Fitted trend (solid) vs Predicted (dashed): 060371002
01/01/1989
01/01/1990
01/01/1991
01/01/1992
01/01/1993
01/01/1994
D ate
0.4
0.2
0.0
sq rt Ozone
Fitted trend (solid) vs Predicted (dashed): 060371301
01/01/1989
01/01/1990
01/01/1991
01/01/1992
01/01/1993
01/01/1994
D ate
0.4
0.2
0.0
sq rt Ozone
Fitted trend (solid) vs Predicted (dashed): 060375001
01/01/1989
01/01/1990
01/01/1991
01/01/1992
D ate
01/01/1993
01/01/1994
37
(3) Nonstationary residual spatio-temporal variation.
C   ( si , t )  v( si , t )
A
it
Final component: spatio-temporal variation at the (2-week)
time scale of the fixed sites and mobile gradient monitors.
Sampson-Guttorp spatial deformation approach (Damian et al.
2001, 2003), to model the nonstationary spatial covariance
structure.
Allows for spatially varying effects of predominant meteorology,
coast lines and topographic features that underlie the
statistical relationship between time varying pollutant levels at
different points in space.
Bayesian analysis provides a full posterior distribution for the
model parameters, and thus a ready computation of multiple
imputations of exposures for the health effects analysis. 38
63 Region 6 monitoring sites and their representation in a
deformed coordinate system reflecting spatial covariance
Thu Oct 30 00:12:36 PST 2003
55643
26
54
5 56
2530
43
20
62
26
52
32
15
7
17
58 11
47
18
6 9468223727
41
12
42
3 51
35
45 49
44
53
57
40
59
55
61
2
63
60
4
28
38
50 29
19
23
36
48
33
39
10 31 1 21
34 24
14
54
30
25
20
62
32 478
52
7 11
69 22 27
41
12
5842
18
351
46 37
44
17
45
49
15
35
57
53
2 55
38 36
61
63
50 46059
29282310 31
48 24
33
39 34
13
16
14
40
19
1 21
13
16
39
0.8
0.6
-0.2
0.0
0.2
0.4
Correlation
0.4
0.2
0.0
-0.2
Correlation
0.6
0.8
1.0
Region 6 S. Calif
1.0
Region 6 S. Calif
0
1
2
3
Geographic Distance (km)
4
0
1
2
3
4
D-plane Distance
40
Observed vs Predicted ozone at 3 validation sites.
0.4
0.3
0.0
0.1
0.2
Zpredf[ , j]
0.3
0.2
0.0
0.1
Zpredf[ , j]
0.4
0.5
060296001
0.5
060371103
0.0
0.1
0.2
0.3
0.4
0.5
Zval[, j]
0.0
0.1
0.2
0.3
0.4
0.5
Zval[, j]
0.3
0.2
0.1
0.0
Zpredf[ , j]
0.4
0.5
060831015
0.0
0.1
0.2
0.3
Zval[, j]
0.4
0.5
41
0.2
0.0
sq rt Ozone
0.4
Observ ed (points) v s Predicted (lines): 060371002
01/01/1989
04/01/1989
07/01/1989
10/01/1989
01/01/1990
04/01/1990
07/01/1990
10/01/1990
04/01/1992
07/01/1992
10/01/1992
04/01/1994
07/01/1994
10/01/1994
0.2
0.0
sq rt Ozone
0.4
D ate
01/01/1991
04/01/1991
07/01/1991
10/01/1991
01/01/1992
0.2
0.0
sq rt Ozone
0.4
D ate
01/01/1993
04/01/1993
07/01/1993
10/01/1993
01/01/1994
D ate
42
0.2
0.0
sq rt Ozone
0.4
Observ ed (points) v s Predicted (lines): 060371301
01/01/1989
04/01/1989
07/01/1989
10/01/1989
01/01/1990
04/01/1990
07/01/1990
10/01/1990
04/01/1992
07/01/1992
10/01/1992
04/01/1994
07/01/1994
10/01/1994
0.2
0.0
sq rt Ozone
0.4
D ate
01/01/1991
04/01/1991
07/01/1991
10/01/1991
01/01/1992
0.2
0.0
sq rt Ozone
0.4
D ate
01/01/1993
04/01/1993
07/01/1993
10/01/1993
01/01/1994
D ate
43
0.2
0.0
sq rt Ozone
0.4
Observ ed (points) v s Predicted (lines): 060375001
01/01/1989
04/01/1989
07/01/1989
10/01/1989
01/01/1990
04/01/1990
07/01/1990
10/01/1990
04/01/1992
07/01/1992
10/01/1992
04/01/1994
07/01/1994
10/01/1994
0.2
0.0
sq rt Ozone
0.4
D ate
01/01/1991
04/01/1991
07/01/1991
10/01/1991
01/01/1992
0.2
0.0
sq rt Ozone
0.4
D ate
01/01/1993
04/01/1993
07/01/1993
10/01/1993
01/01/1994
D ate
44
Conclusion:
• We can estimate/predict both the day-to-day
deviations from the trend, and the seasonal
shape of the trend quite well, but
• We sometimes miss the long-term mean.
=> need to incorporate extra local information to
predict the mean concentration.
45
46
Technical details
47
Details, issues, and extensions
• Gaussian assumption after transformation
• Current AQS data sampling usually every 3 days;
proposed sampling on 2-week intervals
• Conditional, hierarchical approach to estimating the
parameters of our space-time models from this
“incomplete data,” beginning with models estimated
from the longer-term AQS data and them updating
estimated model parameters with data from the new
fixed and mobile monitors.
48
• First stage of analysis: build separate models and
estimates for the three major exposures of interest,
PM2.5, NOx, and O3.
• Second stage: take advantage of the association
between PM2.5 and NOx in a multivariate (“co-kriging”)
analysis that assumes only that spatial nonstationarity
can be expressed in a common underlying deformed
coordinate system.
49
J
C ( si , t )  1 ( si )    j ( si ) f j (t )   ( si , t )
j 1
J
   ij f j (t )   it
j 0
or
C  F  
where we are writing C as an ST (space-time) matrix
of observations,  is an S(J+1) matrix of coefficients
multiplying the matrix F, (J+1) T, with columns
containing values of the basis functions evaluated at
the S observation sites (i=1,…,S).
Obvious calculation is an SVD of the concentration
matrix C.
50
C   F    UDV   j 1 d j u j v j
S
T


 

d
u
v
 j 1 j j j 
 U
J 1
( J 1)
D
( J 1)
V

d
u
v
 j  J 2 j j j
S

N
( J 1)
where the columns of the (truncated) matrix of right
singular vectors is considered to represent the matrix of
values of the J+1 temporal basis functions:
F = V ( J 1)
Issues: Smoothness of the singular vectors as components of
trend; computation with missing data.
51
N=63, S. Calif: 4 samples from the posterior distribution of deformations reflecting spatial covariance
Tue Oct 28 22:18:29 PST 2003
56
5
2643
54
30
25
20
62
327 478
52
17 15
11
41
6922 27
58
12
18
351
42
46 37
45 49
35
44
57
53
61
38
63
55
59
502 460
231036
28
29
48 2431
33
39 34
13
16
14
55643
26
40
19
121
556
2643
54
30
62
20
25
17 15
327478
52
11
6922 27
41
12
58
18
351
42
37 45 49
35
46
40
19
44
53
57
2460
61
38 36 121
55
59
63
50
231031
28
29
48 24
3933
34
13
14 16
54
30
25
20
62
17 15
327478
52
11
58
18
6
41
27
12
351
42
922
37 45 49
46
40
35
19
44
57
53
2 460
38 36 121
61
63
55
59
50
23
28
10
29
48 2431
33
39 34 13
16
14
56
5
2643
54
15
30
25
20
62
17
32
527478
11
40
58
6
41
18
12
351
42
92227 45 49
35
46
19
37
53
44
57
21
2 460
36 1
61
38
63
59
50
55
28231031
29
48 24
33
39 34 13
16
14
52
Posterior sample
Site variances
0.2
0.0
0.5
1.0
1.5
78
71
94
103
92
83
96
88
131
139
113
59
0.1
0.0
Variogram
0.3
0.4
Region 6 : S. Calif
Empirical variogram of log site variances
Circle radii proportional to (detrended) log site variances
2.0
D-plane Distance
53