Introduction
The ground-based total column ozone data set is based on Dobson and
Brewer spectrophotometer and filter ozonometer observations available
from the World Ozone and UV Data Centre (WOUDC)
(http://www.woudc.org/). Large longitudinal inhomogeneities in
the global ozone distribution and limited spatial coverage of the
ground-based network make it difficult to estimate zonal and global
total ozone values from station observations directly
. The Total Column Ozone (TCO) data set is comprised
of the ozone observations from the set of ground-based stations
worldwide. Most of those stations are located on land in the Northern
Hemisphere, and relatively few stations are over the Southern
Hemisphere and oceans. Therefore the spatial distribution of
ground-based stations is highly irregular. In addition, durations of
operations for each station are different. One of the major
difficulties in assessing long-term global total ozone variations is
thus data inhomogeneity. Indeed recalibration of ground-based
instruments, or interruptions in observation records result in data
sets which may have systematic errors that change with time
.
The TCO data set and corresponding satellite measurements have also
been widely discussed in the statistics literature. Some authors have
noticed space–time asymmetry in ozone data . The other important feature of TCO data is
that the spatial dependence of ozone distributions varies strongly
with latitude and weakly with longitude, so that homogeneous models
(invariant to all rotations) are clearly unsuitable
. This is why assume that the spatial
process driving the TCO data is an axially symmetric process
whose first two moments are invariant to rotations about the Earth's
axis, and constructed space–time covariance functions on the
sphere × time that are weakly stationary with respect to
longitude and time for fixed values of latitude. further
used linear combinations of Legendre polynomials to represent the
coefficients of partial differential operators in the covariance
functions. These covariance functions produce covariance matrices that
are neither of low rank nor sparse for irregularly distributed
observations, as it is the case with ground-based stations. Hence,
likelihood calculations can thus be difficult in that situation, and
we will not follow this approach.
The aim of this article is to apply a new technique, the stochastic
partial differential equation (SPDE) approach in spatial statistics
in order to best evaluate total column ozone spatially
from ground-based stations. The SPDE approach has already been applied
to regularly spaced ozone satellite data by but not to
ground-level stations, where gains in accuracy are potentially larger
due to the gaps in coverage. Furthermore, we quantify the impact of
the improvement of these spatial estimations on the computations of
time series over various regions. Finally, the SPDE and covariance-based
approaches are also used to calculate monthly zonal mean total ozone
values and compare them with zonal means calculated from ground-based
data and available from the WOUDC .
Section 2 gives a brief introduction to the theoretical framework of
the SPDE technique, a basic description of the covariance-based kriging and
related model selection and diagnostic techniques. Section 3 describes
the spatial analysis using TCO data from WOUDC on a monthly, seasonal
and annual basis. Furthermore, the estimated results of SPDE and covariance-based
kriging are compared with the Total Ozone Mapping Spectrometer (TOMS)
satellite data to examine which method yields approximations closer
to satellite data. Finally, the long-term zonal mean trends enable us
to conduct a sensitivity analysis by removing stations at random and
by introducing long-term drifts at some ground-based stations.
Methods
Our main problem is to estimate ozone values at places where it is not
observed. Models in spatial statistics that enable this task are
usually specified through the covariance function of the latent
field. Indeed, in order to assess uncertainties in the spatial
interpolation with global coverage, we cannot build models only for
the discretely located observations, we need to build an approximation
of the entire underlying stochastic process defined on the sphere.
We consider statistical models for which the unknown functions are assumed to be realizations
of a Gaussian random spatial process. The standard fitting approach, covariance-based
kriging, spatially interpolates values as linear combinations of the
original observations, and this constitutes the spatial predictor. Not only large data sets
can be computationally demanding for a kriging predictor but covariance-based models also struggle to
take into consideration in general nonstationarity (i.e., when physical spatial
correlations are different across regions) due to the fixed
underlying covariance structure. Recently, a different computational
approach (for identical underlying spatial covariance models) was
introduced by , in which random fields are expressed as
a weak solution to an SPDE,
with explicit links between the parameters of the SPDE and the
covariance structure. This approach can deal with large spatial data
sets and naturally account for nonstationarity. We review below some
of the recent development on the covariance structure modeling on the
sphere, with a particular focus on SPDE and covariance-based kriging. Computational
implementations of SPDE and covariance-based kriging, with mathematical details, are
relegated to Appendix A.
The Matérn covariance function is an advanced covariance
structure used to model dependence of spatial data on the plane. On
the sphere, show that the kriging prediction
using the Matérn function with chordal distance outperforms many
other types of isotropic covariance functions, both in terms of accuracy and
quantification of uncertainty. The shape parameter ν, scale
parameter κ and the marginal precision τ2 parameterize
it:
C(h)=21-ν(4π)d/2Γ(ν+d/2)κ2ντ2(κ‖h‖)νKν(κ‖h‖),
where h∈Rd denotes the difference between any
two locations s and s′:
h=s-s′, and Kν is a modified
Bessel function of the second kind of order ν>0.
SPDE approach
Let X(s) be the latent field of ozone measurements Y(s) under
observation errors ϵ(s). use the fact that
a random process X(s) on Rd with a Matérn covariance
function (Eq. ) is the stationary solution to the SPDE:
(κ2-Δ)α/2τX(s)=W(s),
where W(s) is Gaussian white noise, and Δ
is the Laplace operator. The regularity (or smoothness) parameter
ν essentially determines the order of differentiability of the
fields. The link between the Matérn covariance (Eq. )
and the SPDE formulation (Eq. ) is given by α=ν+d/2, which makes explicit the relationship between dimension and
regularity for fixed α. Unlike the covariance-based model, the SPDE approach can
be easily manipulated on manifolds. On more general manifolds than
Rd, the direct Matérn representation is not easy to
implement, but the SPDE formulation provides a natural generalization,
and the ν parameter will keep its meaning as the quantitative
measure of regularity. Instead of defining Matérn fields by the
covariance function on these manifolds, used the
solution of the SPDE as a definition, and it is much easier and
flexible to do so. This definition is valid not only on Rd
but also on general smooth manifolds, such as the sphere. The SPDE
approach allows κ and τ to vary with location:
(κ2(s)-Δ)α/2τ(s)X(s)=W(s),
where κ(s) and τ(s) are estimated by expanding them in
a basis of a function space such as the spherical harmonic basis. We
estimate the SPDE approach parameters and supply uncertainty
information about the surfaces by using the integrated nested
Laplacian approximations (INLA) framework, available as an R
package (http://www.r-inla.org/). For latent Gaussian Markov
random fields used to efficiently solve SPDEs on triangulations, INLA
provides good approximations of posterior densities at a fraction of
the cost of Markov chain Monte Carlo. Note that for models with
Gaussian data, the calculated densities are for practical purposes
exact.
Covariance-based approach
For locations on spherical domain, si=(Li,li), let
Y(si) denote the ozone measured at station i. Then the
kriging representation can be assumed additive with a polynomial model
for the spatial trend (universal kriging):
Y(si)=P(si)+Z(si)+ei,
where P is a polynomial which is the fixed part of the model, Z
is a zero mean, Gaussian stochastic process with an unknown covariance
function K and ei are i.i.d. normal errors. The estimated latent
field is then the best linear unbiased estimator (BLUE) of
P(s)+Z(s) given the observed data.
Note that the Gaussian process in the spatial model, Z(s), can be
defined as a realization of the process X(s) from the previous section.
For the covariance-based approach, the hurdle that we are facing is that we have to define
a valid (but flexible enough) covariance model and, furthermore,
compared to data on the plane, we must employ a distance on the
sphere. Two distances are commonly considered. The chordal distance
between the two points (L1,l1) and (L2,l2)
on the sphere is given by
ch(L1,L2,l1-l2)=2rsin2L1-L22+cosL1cosL2sin2l1-l221/2,
where r denotes the Earth's radius. The more physically intuitive
great circle distance between the two locations is
gc(L1,L2,l1-l2)=2rarcsin{ch(L1,L2,l1-l2)}. However, pointed out
that using the great circle distance in the original Matérn
covariance function (Eq. ) would not work, as it may not
yield a valid positive definite covariance function. Therefore in this
study we use the chordal distance for covariance-based kriging. The main advantage of
using the chordal distance is that it is well defined on spherical
domains, as it restricts positive definite covariance functions on
R3 to S2 . For the ozone data, we
specify the Matérn covariance function defined in Eq. ()
in the covariance-based approach in order to compare the performance with the
SPDE approach for exactly the same covariance function, whereas the model parameters
are optimized according to different techniques. The relevant model
diagnostic and selection criteria are described in Appendix
B. Note that the smoothness parameter ν is allowed to be selected
in the covariance-based kriging, whereas we fix it at ν=1 in the SPDE approach.
Surface predicted ozone (DU) mean and SD for SPDE (strategy D) and
CBK (strategy B) from January 2000. The red points indicate the locations of
stations.
![]()
Mapping accuracy
In this section, we produce statistical estimates of monthly ozone
maps, using TCO data from WOUDC. We consider TCO data in January 2000
as an illustration, which contain 150 ground-based ozone observations
around the world. All ozone values in this article are in Dobson units
(DU). We first choose the model setups for both SPDE and covariance-based
approaches below.
With the SPDE approach, as the smoothness parameter ν is defined through
the relationship α=ν+d/2 (see Appendix A1), we only need to choose the
basis expansion order to represent κ and τ. To choose the
best maximal order of the spherical harmonic basis, we fitted models
with different maximal orders of spherical harmonics for the
expansions of κ and τ in order to estimate them thereafter
(the default formulation in the R-INLA package). The best fitted model is
for a spherical harmonics basis with maximal order 3 since it yielded
the lowest generalized cross-validation (GCV) standard deviation (SD) computed in
Eq. (), which yields a total of nine parameters to be estimated
(four parameters for the expansions of κ and τ and one
parameter for the variance). For the covariance-based approach, we need to choose the
smoothness parameter in the Matérn covariance function before
estimating the univariate quantities κ and τ2. The same
σGCV criterion is used to evaluate the model performance. We fit
a wide range of values for the smoothing parameter, and ν=20
minimizes the σGCV for these ozone data.
To compare the performance of the SPDE approach with the covariance-based
kriging, the same σGCV criterion is used as it balances well
predictive power vs. overfitting across methods. We first estimate the
optimized model parameters, i.e., κ and τ in a SPDE and ν in a
covariance-based model, by the same generalized cross-validation criterion
(with different computational methods, the estimated method for the chordal
distance covariance model is maximum likelihood estimates), then compare the
uncertainty of spatial estimation over the surface. For the prior
specifications of the SPDE approach, κ and τ follow log normal
priors by default: precision (theta.prior.prec) = 0.1, median for τ
(prior.variance.nominal) = 1 and median for κ (prior.range.nominal)
depends on the mesh. We use a regression basis of spherical harmonics for
both of these parameters; therefore by default the coefficients follow the
log normal priors. We do not change the R-INLA default prior settings
throughout the analysis.
In order to achieve a better estimation, the monthly mean “norms”
(or total ozone “climatology”) are
calculated for each station and each month of the year over the whole
period and then subtracted from the data. The norms are used as
first approximations to remove the general spatial trend. For each
station and for each month, spatial interpolations through SPDE and covariance-based
approaches were performed to these deviations.
Specifications of the different strategies in the spatial estimations.
Strategy
Descriptions
A
A covariance-based model with ν=1
B
A covariance-based model with ν=20
C
An SPDE-based model with stationary covariance
parameters (i.e., κ and τ are unknown constants)
D
An SPDE-based model with space-varying
covariance parameters
We now compare the results estimated by four strategies
described in Table .
We start an illustration to 6 years of monthly
observations. The results of the analysis of ozone data averaged from
2000 to 2005 are shown in Table . The averaged number
of available stations is denoted by n‾, and the residual sum of square (RSS) indicates
residuals sum of square, which is defined in Eq. (). The
SPDE approach (strategies C and D) provides a better fit than
the covariance-based kriging (strategies A and B) for all
months. The effective degree of freedom neff is higher in
the SPDE approach as it is more complex than covariance-based kriging. Higher effective
degrees of freedom means smaller values in the σGCV
denominator (n-neff) and thus higher values for
σGCV to account for
overparametrization. Nevertheless, σGCV values for
the SPDE approach are still all much lower than for covariance-based kriging in all
cases. This means that the RSS in the SPDE
approach is drastically smaller than the RSS for covariance-based
kriging. Thus the SPDE approach supplies a much better fit to
the true ozone observation. Note that the covariance-based model is
computing a weighted average of the neighborhood values
around the location, while the SPDE model is constructed
through a triangular mesh. The mesh can be more adaptive and flexible
to irregularly distributed observations. In addition, with spatially varying κ
and τ, the table indicates that the results could be improved further
by applying a nonstationary extension.
Comparison of the generalized cross-validation error (σGCV)
and the residual sum of square (RSS) for different strategies averaged of 2000–2005 by
month. Statistical summaries for Northern Hemisphere (NH), tropics and Southern Hemisphere (SH).
Global statistics
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Number of obs.
135.33
144.33
146.17
146.00
142.50
143.33
143.00
145.33
146.00
142.67
140.17
129.50
A
neff
16.00
20.92
26.84
17.50
28.23
22.96
11.20
31.67
53.11
60.29
30.09
19.67
σGCV
16.62
17.51
14.97
13.29
10.83
10.23
10.35
11.20
13.19
12.67
17.17
15.58
RSS
34 584
39 599
28 017
23 405
13 884
13 333
14 300
14 688
19 312
13 379
34 470
28 416
B
neff
16.89
25.69
30.51
19.57
24.14
15.46
14.39
29.63
36.93
38.24
28.86
20.67
σGCV
15.10
14.05
12.11
12.45
10.33
10.54
9.87
9.75
9.68
9.53
12.35
12.71
RSS
28 535
25 611
18 323
20 711
13 258
14 770
12 637
11 479
10 654
9645
17 550
19 129
C
neff
70.33
60.76
81.23
59.31
64.87
76.23
48.70
54.63
65.13
75.41
72.09
51.12
σGCV
11.89
10.33
8.39
9.24
6.06
6.77
7.78
7.61
6.71
6.15
9.45
8.91
RSS
12 263
10 085
5394
9600
3428
4283
6224
6351
4400
2600
6924
6897
D
neff
67.75
59.12
73.08
61.01
72.01
84.44
60.14
64.71
73.67
75.21
72.00
62.97
σGCV
8.60
9.90
7.90
9.18
6.27
6.80
7.22
6.85
6.62
5.75
8.84
7.90
RSS
6148
9431
5312
8664
3359
3865
4326
4845
4179
2285
5924
4805
RSS by region
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Number of obs.
89.50
98.50
101.67
104.00
102.50
103.83
103.00
103.00
102.67
98.50
95.50
84.67
NH
A
29 531
34 011
23 399
17 207
9981
9692
8844
6933
6118
4159
16 117
22 115
B
26 035
22 234
15 507
16 144
10 498
11 288
8435
7238
7179
6748
13 035
16 690
C
11 543
9248
4895
8002
2932
3415
4467
4519
3490
2011
5649
6082
D
4119
7240
4183
6696
2880
3328
3384
3903
3709
1914
5104
3301
Number of obs.
27.83
27.33
26.33
26.33
26.50
26.67
27.33
27.33
26.83
26.67
27.00
27.17
Tropics
A
3706
3666
3461
4082
1758
1671
2766
2253
2898
2286
4123
3008
B
1778
2182
2120
2988
1254
1632
2264
1228
957
1042
1360
1411
C
413
489
367
1095
219
280
830
489
238
192
397
454
D
1148
1438
688
1142
331
368
646
441
171
120
309
990
Number of obs.
18.00
18.50
18.17
15.67
13.50
12.83
12.67
15.00
16.50
17.50
17.67
17.67
SH
A
1347
1922
1157
2116
2144
19 670
2689
5502
10 296
6933
14 230
3293
B
722
1195
696
1580
1505
1850
1938
3012
2518
1855
3155
1028
C
306
348
162
504
277
588
927
1343
672
397
878
361
D
882
754
442
827
148
169
295
501
299
251
510
515
Surface predicted ozone (DU) mean from SPDE approach
(strategy D) by season from 2000.
![]()
Surface predicted ozone (DU) standard deviation from SPDE approach
(strategy D) by season from 2000.
![]()
Table also reports regional RSSs by dividing the
Earth into the Northern Hemisphere (>30∘ N), tropics
(30∘ S–30∘ N) and Southern Hemisphere
(<30∘ S). In general, over half of the ground-based
stations are located on the Northern Hemisphere, which gives rise to a
higher RSS with respect to other regions as the RSSs are not
normalized by the number of observations. RSSs estimated by SPDE and covariance-based
kriging show similar patterns across months and across regions. Lower
estimation errors can be found in August–October and higher
errors occur in December–February.
Figure shows
the predicted mean and SD ozone maps by strategies B and D
on January 2000. The spatial distributions of ozone means
are similar for SPDE and covariance-based kriging in the
Northern Hemisphere, but there are differences in the Southern
Hemisphere. These differences arise from the asymmetry of available
stations in the two hemispheres. The spatial distributions of the SD
present similar general patterns for the two techniques. The
uncertainties are higher where with fewer stations are available, see the
large uncertainty distribution over the South Pacific Ocean. SDs of SPDE
predictions are much smaller than the SDs of the covariance-based kriging predictions,
especially where fewer observations are available (e.g., mid-Atlantic
and South Pacific), but are larger near the North Pole; this may be due
to covariance-based kriging underestimating its own uncertainty.
Total ozone (DU) difference mapping of SPDE and covariance-based kriging (CBK)
estimated mean with respect to satellite data from January, April
and July 2000.
![]()
Total ozone (DU) difference mapping of SPDE and covariance-based kriging (CBK)
estimated mean with respect to satellite data from October
2000. Estimation in October shows worse prediction than other
months; hence it used different scale from
Fig. .
![]()
Ozone mapping from TOMS data in (a) DJF and
(b) MAM; global difference mapping of SPDE and covariance-based kriging (CBK)
predicted mean with respect to TOMS data in DJF (c and
e) and MAM (d and f) 2000.
![]()
Ozone mapping from TOMS data in (a) JJA and
(b) SON; global difference mapping of SPDE and covariance-based kriging (CBK)
predicted mean with respect to TOMS data in JJA (c and
e) and SON (d and f) 2000.
![]()
Seasonal and annual effects
Seasonal ozone data are obtained by averaging the corresponding
monthly data (but all months of every season must be available to
create such seasonal averages). Table shows the
seasonal results for different strategies over the years 2000–2005.
Their respective highest errors are in December–January–February
(DJF) for strategies A, B and C and
in June–July–August (JJA) for strategy D. The
results are hardly different in March–April–May (MAM) and
September–October–November (SON) for strategies C and D, while
a significant improvement can be achieved in DJF by applying a
nonstationary SPDE model. Moreover, the values of
σGCV and RSS from this seasonal analysis are
smaller than the corresponding analysis for each month of the
associated season, both for SPDE and covariance-based kriging,
and are also closer across the two competing techniques due to
additional averaging smoothing out the gains in accuracy.
Nevertheless the SPDE approach still provides
a better fit than covariance-based kriging in all seasons.
Figures and show
the seasonal ozone maps by strategy D of means and
SD, respectively. The maps for SD again
reveal the higher estimated error in regions without stations.
Generalized cross-validation error (σGCV) and
residual sum of square (RSS) for different strategies averaged over 2000–2005 by season.
Season
DJF
MAM
JJA
SON
Average
n‾
117.17
132.83
132.17
130.67
128.21
A
neff
13.45
21.66
13.01
52.06
25.04
σGCV
11.95
9.96
8.65
11.23
10.45
RSS
16 194
11 513
8987
10 178
11 718
B
neff
18.73
20.68
11.53
32.77
20.93
σGCV
9.47
9.20
8.39
8.19
8.81
RSS
10 274
9990
8526
6783
8893
C
neff
48.10
60.41
50.25
75.00
58.44
σGCV
7.72
5.86
6.41
6.47
6.46
RSS
4661
3120
4140
2272
3548
D
neff
52.82
64.43
58.06
77.07
63.09
σGCV
6.55
6.00
6.13
5.82
6.12
RSS
2695
3074
3609
2210
2897
Generalized cross-validation error (σGCV) and residual
sum of square (RSS) for different strategies over 2000–2005. Monthly and seasonally results
are averaged over each year, and annual results are directly estimated using annual means from each station.
Year
2000
2001
2002
2003
2004
2005
Monthly
n‾
143.25
151.58
147.50
136.17
135.25
138.42
A
neff
34.42
19.20
35.73
31.49
20.76
27.65
σGCV
12.95
15.39
12.12
12.71
14.52
14.13
RSS
20 681
33 662
17 304
17 215
25 838
23 994
B
neff
30.95
17.88
30.82
24.52
20.51
25.81
σGCV
10.85
12.60
10.34
10.98
12.75
11.72
RSS
14 473
23 946
12 693
13 930
20 112
15 998
C
neff
75.34
51.17
76.18
76.03
52.61
58.57
σGCV
7.21
10.29
7.05
7.10
9.77
8.22
RSS
4157
12 734
3888
3019
9519
5906
D
neff
81.27
47.06
80.41
78.16
55.88
70.26
σGCV
6.73
9.48
6.62
6.66
8.79
7.64
RSS
3306
10 890
3187
2692
6923
4574
Seasonally
n‾
126.00
135.75
135.50
123.75
120.50
127.75
A
neff
33.25
21.07
26.62
32.19
19.05
18.07
σGCV
10.44
11.67
9.17
9.23
9.99
12.18
RSS
9993
16 559
9306
7630
10 243
16 576
B
neff
28.05
12.48
24.69
21.11
18.43
20.82
σGCV
8.18
10.38
7.71
8.37
8.63
9.63
RSS
6533
15 281
6552
7299
7781
9912
C
neff
47.87
48.17
99.78
69.18
41.10
44.56
σGCV
5.23
8.51
5.58
5.31
6.78
7.39
RSS
2387
7546
1025
1567
4070
4695
D
neff
63.11
55.65
84.82
68.48
51.23
55.28
σGCV
5.21
7.29
4.91
5.48
6.71
7.15
RSS
1888
5051
1372
1759
3507
3804
Annually
n
83
97
101
90
87
97
A
neff
27.90
15.96
9.94
10.43
6.02
3.00
σGCV
7.01
7.28
6.61
7.16
6.83
10.03
RSS
2704
4301
3982
4080
3782
9452
B
neff
24.10
23.67
11.21
8.35
12.65
12.70
σGCV
5.05
5.34
6.29
7.23
5.95
8.67
RSS
1501
2094
3552
4271
2629
6342
C
neff
47.32
40.50
48.22
57.58
25.63
28.00
σGCV
3.59
4.56
4.45
4.54
5.04
7.65
RSS
460
1172
1045
667
1556
4035
D
neff
56.16
49.55
49.46
67.83
36.17
48.02
σGCV
3.12
4.08
4.56
4.93
5.03
7.11
RSS
261
790
1076
539
1469
3076
The annual ozone data are obtained by creating an annual average, which
also means that stations with record interruptions are not
used. Therefore fewer stations are available for this exercise. To see
the improvement of the annual-based analysis over seasonally and
monthly analyses, Table shows the annual averaged
results by month and seasonally, and the results directly obtained by
doing the analysis on the annual mean. Although there are fewer
stations in annual-based and seasonal-based data than in monthly data,
the errors are lower than for monthly data over the years for
all strategies and the results directly obtained from annual means
yield even lower RSS and σGCV than the results
averaged by seasons due to smooth variation.
Comparison with satellite data over all months and averaged over
2000–2005 RMSEs for covariance-based kriging (CBK) and SPDE predictions.
Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Average
CBK
20.16
26.30
22.00
17.64
24.49
15.88
13.30
29.39
50.15
51.92
40.01
28.35
28.30
SPDE
12.75
10.68
9.98
7.76
16.26
7.80
8.03
13.05
11.46
12.07
10.38
15.21
11.29
Percentage of
36.76
59.37
54.64
56.00
33.60
50.87
39.61
55.58
77.15
76.76
74.06
46.36
60.12
improvement
Comparison with satellite data
In this section, we assess the match between satellite observations
and spatial predictions based on ground-level stations. The TOMS data on monthly averages are
obtained from the NASA website (http://ozoneaq.gsfc.nasa.gov/),
where we collected the Earth Probe (25 July 1996–31 December 2005)
satellite data with grid 1∘×1.25∘. We
calculate the differences over all grid cells and summarize it by the
root mean square error (RMSE). Let y^i be the estimated
result from the SPDE or covariance-based kriging on grid i, and let yis
denote the satellite value on grid i, then the (normalized) RMSE is
given by
RMSE=∑i=1ny^i-yis2n,
where n=180×288 is total number of grid cells. However,
satellite data are unavailable over high latitudes in DJF and MAM
and over low latitudes in JJA and SON. Therefore we restrict the calculations
of RMSEs between 60∘ S and 60∘ N.
From this stage we only compare the results between a nonstationary
SPDE-based model and a covariance-based model with ν=20.
Monthly comparisons over 2000–2005 are shown in
Table . Ozone surfaces predicted by the SPDE
approach are closer to the satellite data than the predictions from covariance-based
kriging over all months. The highest improvement of SPDE over covariance-based kriging
is 77.15 % in September and the lowest is 33.60 % in May. Also,
in contrast with relatively unstable monthly predictions by covariance-based kriging,
SPDE shows more consistency in predictions of monthly ozone
variation.
Figures and map the
differences of surface predictions of covariance-based and SPDE methods with
respect to satellite data over 60∘ S–60∘ N on
January, April, July and October 2000. The differences in October turn
out to be much larger than in other months, and therefore a different
scale is used. These maps indicate the overestimation (red) and
underestimation (blue) with respect to satellite data. Similar
patterns in deviations are revealed for both techniques, but SPDE
displays less magnitude in the deviations than covariance-based kriging. One noticeable
feature is that the pattern of deviation from satellite data is
strongly related to the distribution of ground-based stations: for
instance, the covariance-based kriging predicted surfaces tend to underestimate the
values over the South Pacific Ocean, where few stations are
available. The surface predictions by SPDE achieve a clear improvement
in predictions compared to covariance-based kriging over areas with less stations,
especially in January and October.
The seasonal predicted total ozone is obtained by averaging the
corresponding monthly means. We excluded stations that have
interruptions in their records. Therefore fewer observations are used
to predict seasonal means. The RMSEs between predicted surface and
satellite data are presented in Table . In
general, seasonal maps should agree better with satellite-based maps
than monthly maps. However, fewer observations are used in seasonal
predictions and that may trigger high RMSEs in the covariance-based kriging estimation
in particular. In those circumstances, the SPDE approach shows
robustness against observations loss. Figures
and show TOMS maps of all seasons in 2000 in top
panels and differences with predictions from SPDE and covariance-based
kriging. Underestimation at the South Pacific are in accordance with
expectations for both techniques, but surface predictions by SPDE
achieve a better fit than covariance-based kriging, especially in SON.
Impact on long-term changes
In this section, we show how variations in time of the zonal means can
be improved by employing the more accurate SPDE-based mapping
technique instead of covariance-based kriging.
Zonal mean time series analysis
To see how the ozone zonal means change over time over the same
stations with different algorithms, we choose the stations which
supplied data for at least 25 years between 1979 and 2010. Hence
67 stations are used to construct these zonal mean time series. There
is a strong asymmetry between the Southern Hemisphere (6 stations) and
the Northern Hemisphere (48 stations); there are 13 stations at the tropics
(defined as 30∘ S–30∘ N). The zonal means were
constructed by averaging the estimations obtained from either SPDE
or covariance-based kriging over a grid of 1∘×1∘.
In order to overcome the underestimation over the South Pacific (see
Fig. ) and achieve a better estimation of long-term
global zonal means, the monthly mean norms for each station were
subtracted from observations over all the period. Then for each month,
spatial interpolation through SPDE and covariance-based kriging were performed to the
deviations. The ozone norms were added back to these deviations in
order to compare zonal means over the corresponding belts.
Comparison with satellite data over all seasons and averaged over
2000–2005 RMSEs for covariance-based kriging (CBK) and SPDE predictions.
Season
DJF
MAM
JJA
SON
Average
CBK
21.42
19.66
15.71
44.45
25.31
SPDE
18.45
13.54
8.49
9.20
12.42
Percentage of improvement
13.88
31.12
45.94
79.30
50.92
In this study, we compare the zonal mean time series estimated by SPDE
and covariance-based kriging with Solar Backscatter Ultraviolet (SBUV) satellite
instrument merged ozone data described by
(http://acd-ext.gsfc.nasa.gov/Data_services/merged/) and
a data set based on ground-based data available from the WOUDC
. The SBUV merged satellite
data sets incorporated the measurements from eight backscatter
ultraviolet instruments (BUV on Nimbus 4, SBUV on Nimbus 7 and
a series of SBUV/2 instruments on NOAA satellites) processed with the
v8.6 algorithm
. The WOUDC ground-based zonal mean data set
is based on the following technique. Firstly, monthly means for each
point of the globe were estimated from satellite TOMS data for 1978–1989. Then for each station and for
each month the deviations from these means were calculated, and the
belt's value for a particular month was estimated as a mean of these
deviations. The calculations were done for 5∘ broad latitudinal
belts. In order to take into account various densities of the network
across regions, the deviations of the stations were first averaged
over 5 by 30∘ cells, and then the belt mean was calculated by
averaging these first set of averages over the belts. Until this point
the data in the different 5∘ belts were based on different
stations (i.e., were considered independent). However, the differences
between nearby belts are small. Hence one can reduce the errors of the
belt's average estimations by using some
smoothing or approximation. So the zonal means were then approximated
by zonal spherical functions (Legendre polynomials cosine of the
latitude) to smooth out spurious variations. This
methodology shares some ideas with SPDE
in terms of taking advantage of spherical functions for spatial
interpolation over the globe, but this methodology can only be
conducted on zonal means calculation rather than global surface
prediction. The merged satellite and the WOUDC data sets were compared
again recently and demonstrated a good agreement
.
Time series of zonal means by SBUV satellite data (black)
WOUDC data set (green), covariance-based kriging (red) and SPDE (blue) from
1979 to 2010.
![]()
Time series of 30–60∘ N zonal means
by (a) covariance-based kriging (CBK) and (b) SPDE from 1990 to 2010
for four scenarios with 5, 10, 20 and 30 stations removed globally
including 3, 6, 12 and 18 stations removed in the NH.
![]()
Time series of 30–60∘ S zonal means
by (a) covariance-based kriging (CBK) and (b) SPDE from 1990 to 2010
for four scenarios with 5, 10, 20 and 30 stations removed globally
including 1, 2, 4 and 6 stations removed in the SH.
![]()
To investigate the pattern of zonal mean long-term changes in detail,
Fig. shows the monthly means from SBUV merged data
(black), WOUDC data set (green), SPDE (blue) and covariance-based
kriging (red). SPDE and covariance-based kriging
estimated means both match well satellite data and the WOUDC
data set in the Northern Hemisphere. Covariance-based kriging means in the tropics
fluctuate heavily and are unrealistic in some years, which indicates
that covariance-based kriging may perform well at some locations but can provide
distorted results at other locations; moreover the large kriging-based
fluctuations in the beginning of the period may be due to a lack of
stations in the early years of 1979–2010. SPDE estimated means are
more robust under this circumstance. Limited stations in the South
Hemisphere may trigger underestimation and deflation of the estimated
annual cycle in SPDE and covariance-based kriging. Therefore we carry out a seasonal
smoothing by averaging September to November to estimate better the
annual peak over the Southern Hemisphere (i.e., October). This
smoothing algorithm improves the match with SBUV data.
Sensitivity analysis
The final step is to conduct a sensitivity analysis for the long-term
zonal mean estimations against either randomly removed stations or
drifts in some of the ground-based observations. To see the impact of
removing stations on the long-term ozone zonal mean change, we choose
57 stations (39 stations in the Northern Hemisphere, 10 stations in
the tropics and 8 stations in the Southern Hemisphere) which provided
data over the entire period from 1990 to 2010. We randomly remove 5, 10, 20
and 30 stations out of these set of stations by taking into account
the relative weights of the respective regions and estimate the zonal
mean trends in each case. The stations removed are randomly chosen by
the design in Table .
Furthermore, to illustrate possible variations in the sensitivity
analysis, we randomly draw 5 sets of stations which need to be removed,
labeled cases 1–5. The time series for different zonal mean
trends over the latitude band 30–60∘ N and
30–60∘ S are displayed in Figs. and , respectively. The impact of randomly removing
stations in the Northern Hemisphere is small even in the case of 30 stations removed (over half of the observations). The Southern
Hemisphere is more sensitive to a loss of information because only few
stations are located in there. The more stations are removed, the more
fluctuations appears in the time series. The main finding is that the
long-term effects estimated by SPDE are again more robust than the
ones obtained by covariance-based kriging, especially for the case of 30 stations
removed (with only two stations left in the Southern Hemisphere). The covariance-based kriging estimated
trends can become chaotic for cases 2 and 4, and under this
circumstance the total ozone annual cycle become unidentifiable.
Annual zonal mean deviances from SBUV data (black), WOUDC
data set (green), using all 57 available ground-based data
(blue), random removed 5 (red), 10 (yellow), 20 (brown) and 30
(grey) stations in SPDE and covariance-based kriging (CBK) estimation over the (1)
global (60∘ N–60∘ S), (2) NH
(30–60∘ N), (3) tropics
(30∘ N–30∘ S) and (4) SH
(30–60∘ S) from 1990 to 2010.
![]()
Time series of 30–60∘ N zonal means
by (a) covariance-based kriging (CBK) and (b) SPDE from 1990 to 2010
for four scenarios with 5, 10, 20 and 30 stations drifted
globally.
![]()
We use case 1 for further illustration, where both SPDE and covariance-based approaches
estimated well with respect to other cases. Figure
shows deviations in time series in the annual mean total ozone
estimated by SBUV data, WOUDC data set, SPDE
and covariance-based kriging. We can see
that both SPDE and covariance-based kriging estimate well in the Northern
Hemisphere. Covariance-based kriging underestimates means significantly over
the tropics and the Southern Hemisphere, while SPDE estimated means
are close to SBUV trends. Note that SPDE estimated
trends using all stations are closer to SBUV observations than the
WOUDC data set at the tropics and Southern Hemisphere overall.
Time series of 30–60∘ S zonal means
by (a) covariance-based kriging (CBK) and (b) SPDE from 1990 to 2010
for four scenarios with 5, 10, 20 and 30 stations drifted
globally.
![]()
Annual zonal mean deviances from SBUV data (black), WOUDC
data set (green), using all 57 available ground-based data
(blue), adding drift to 5 (red), 10 (yellow), 20 (brown) and 30
(grey) stations in SPDE and covariance-based kriging (CBK) estimation over the (1)
global (60∘ N–60∘ S), (2) NH
(30–60∘ N), (3) tropics
(30∘ N–30∘ S) and (4) SH
(30–60∘ S) from 1990 to 2010.
![]()
Design of the sensitivity analysis: stations to be removed are
randomly selected within each region.
Number of removed
5
10
20
30
Total
NH (90–30∘ N)
3
6
12
18
39
Tropics (30∘ S–30∘ N)
1
2
4
6
10
SH (30–90∘ S)
1
2
4
6
8
For the second part of sensitivity analysis, we add random long-term
drifts into observations due to instrument-related problems. In
reality, all observations from a ground-level station are often be
biased by 5–10 DU (2–3 %) over a period of several years. For
the setting of drifts, let yij be the ozone observations
at station i and time j. We randomly select some stations i and
set
yij∗=aiyij,
where ai∼N(1,0.032) is the slope over every 5-year periods,
i.e., one slope factor for 1990–1994, then different drifts for
1995–1999, 2000–2004 and 2005–2010. This setting means that
stations were randomly selected and drift values were then randomly
generated, but the drifts are fixed for each particular station over every
5 or 6 years.
Using the same 57 stations which provided data consistently over 1990
to 2010, the zonal mean trends were estimated with these added drifts over
subsets of randomly selected 5, 10, 20 and 30 stations. We consider five
sets of random drifts as well to account for possible random
variations in the selection process. The time series in each case are
shown in Figs. and for the
Northern and Southern Hemisphere. Covariance-based kriging estimations hold in
the case of over half of stations are drifted, SPDE also displays
robustness to drift. We use case 1 as further
illustration. Figure shows the annual mean total
ozone deviations in time series for SBUV, WOUDC data set, SPDE and
covariance-based kriging estimated means when drifts are present. SPDE estimated trends
turn out to be superior to covariance-based kriging over the tropics and Southern
Hemisphere overall.
Table reports monthly, seasonal and annual
average RMSEs obtained by comparing the WOUDC, SPDE and covariance-based kriging
estimated zonal means to the SBUV zonal means over 1990–2010. We can
see that SPDE is always superior to covariance-based kriging. Furthermore, SPDE zonal
means are closer to satellite zonal means than WOUDC zonal means for
annual and seasonal averages in the Northern Hemisphere, despite using
fewer stations (39) than WOUDC that uses all available stations each
month. It shows the remarkable ability of SPDE to interpolate variations
over the globe than WOUDC. However, for monthly zonal means, SPDE
zonal means are further away from satellite zonal means than WOUDC
zonal means. Indeed, there is much less averaging over 1 month, and
the SPDE approach can suffer from the lack of stations at some
locations to describe particular monthly features that can be more
pronounced than seasonal or annual averages. SPDE zonal means over the
tropics and Southern Hemisphere in monthly and seasonal analysis
suffer greatly as only 10 stations are used in the tropics and 8 in
the Southern Hemisphere (whereas WOUDC can use up to 20–30 in the
Southern Hemisphere). We expect that for operational purposes, using
all the available stations (usually around 130–150 as seen in
Table , not 57 as done here for convenience) for each
month would allow SPDE to clearly outperform WOUDC everywhere at all
frequencies, as it does already with fewer stations in the Northern
Hemisphere for annual and seasonal averages. Such a data set would
constitute an improvement for the study of trends based on ground-level
instruments.
RMSEs of annual, seasonal and monthly total mean ozone from WOUDC data set, and SPDE and
covariance-based kriging (CBK) estimated means (using 57 stations) against SBUV data over 1990–2010.
Annual
Seasonal
Monthly
WOUDC
SPDE
CBK
WOUDC
SPDE
CBK
WOUDC
SPDE
CBK
NH
3.17
2.37
2.94
3.40
3.21
3.54
3.95
4.21
4.40
Tropics
2.07
4.01
9.59
2.42
5.22
10.47
2.56
5.79
10.66
SH
4.51
3.96
8.39
4.39
8.84
15.33
6.10
11.07
15.58
Global
2.28
2.36
6.85
2.55
3.55
8.91
2.92
4.60
9.03