Introduction
Water vapor is a vital constituent of the Earth's electrically neutral
atmosphere (neutrosphere). Although the ratio of water vapor partial to total
atmospheric pressure is typically below 4 %, it is an important constituent
in many respects. Due to the dynamic nature of the neutrosphere and the
complex energy exchange with the Earth's surface, the spatio-temporal
distribution of water vapor can be highly variable. Accurate information
about its content and tendency is the main prerequisite for the prediction of
clouds and precipitation. Water vapor is important for studies of climate and
natural disasters such as floods, droughts or glacier melting. On the other
hand, radio signals transmitted from spaceborne sensors are refracted when
traversing the Earth's neutrosphere. The neutrospheric water vapor
contributes less than 10 % of the signal path delay; however, this error
source is not easily eliminated. Accurate information about the water vapor
concentration along the signal path is required, which is not always
obtainable. Although many efforts have been made to produce accurate
information about water vapor using ground-based, space-based or numerical
methods, the available information is often limited in the temporal
resolution, spatial resolution or accuracy . Numerical
atmospheric prediction models are increasingly used to provide simulations of
the atmospheric parameters. Various studies suggested the assimilation of
atmospheric parameters, such as water vapor, estimated from the Global
Positioning System (GPS) or Interferometric Synthetic Aperture Radar (InSAR),
into these models to improve the quality of the simulated parameters
. We want to comprehend whether the model
simulations of water vapor, in their current quality, can be used to even out
the deficits of the measurement-based estimates, particularly in regions with
no measurements. To achieve this purpose, a statistical data fusion approach
is applied. The output water vapor maps can be used in tomographic approaches
to provide 3-D water vapor grids and to adjust the parameters of numerical
atmospheric prediction models. The remainder of this section presents the
recent related research on water vapor using remote sensing data and
atmospheric models.
The amount of remote sensing data available for monitoring the Earth and its
atmosphere is growing in a rapid, continuous way. InSAR has proved its
capability for detecting surface deformation, landslides, and tectonic
movements , and for deriving digital
elevation models . The influence of water vapor in the
observations can be reduced by averaging a large number of interferograms
or by time series analysis that indicates the stable
persistent scatterers . Besides, InSAR has
recently been used to derive the phase shift caused due to the propagation in
the Earth's atmosphere from the interferograms or by time series analysis
. Global Navigation
Satellite Systems (GNSS), however, have been considered since the 1990s as an
efficient microwave-based tool for atmospheric sounding
. Since then, numerous methods have exploited the
GNSS observations to produce estimates of the integrated atmospheric water
vapor and to generate water vapor maps . InSAR and GNSS, signals are affected in a similar way by the
atmosphere . Therefore, presented
a new approach to deriving absolute, high-resolution maps of precipitable
water vapor (PWV) by combining data from InSAR and GNSS. The
SAR systems acquire the images at repeat cycles of
multiples of days. Enivsat images, which are used in this work, are available
in multiples of 35 days. The availability of the data over time can be
increased by processing data from ascending and descending modes. In
addition, new SAR missions have shorter repeat
cycles, 11 days for TerraSAR-X and 6 days for Sentinel-1. The InSAR-based PWV
estimates cannot be used to observe the variability of water vapor over a
short time, but they are important in different aspects. This geodetic-based
method produces maps of the PWV at a high spatial resolution without
additional costs. These data can be exploited, first, to model the spatial
variations of atmospheric turbulent and non-turbulent effects. Second, they
can be used to observe the variation of water content over long time periods
to detect, for example, unusual trends. Third, they can be used to
adjust/readjust the initial and boundary conditions in atmospheric prediction
models.
Atmospheric modeling systems are standard approaches to simulate
3-D distributions of
the neutrospheric water vapor at various temporal and spatial samplings.
Dynamic local area models (LAMs) are common tools for scaling down the coarse
grids of global circulation models to meso-scale applicability. Several
studies employed the Weather Research and Forecasting modeling system (WRF,
) to compare the LAM simulations of PWV with GNSS
point estimates and PWV maps from
MERIS (MEdium Resolution Imaging Spectrometer) . These
studies conclude that the medium- to long-scale (greater than 20 km) water
vapor signals can be well predicted, whereas short-scale fluctuations are
often hardly captured in a realistic way.
Despite manifold improvements over the last years, considerable uncertainties
are still connected with the parameterization of physical processes in
mesoscale-atmospheric models and biases of the driving model
. This, in addition to the configuration of the model
domains, can significantly impact the simulation output as
well as the model intrinsic water balance . Therefore, the setup of the local area model is crucial, and it
has to be proper for the study region and the research objectives.
Due to the availability of various data sources, which can be complementary
or redundant, data fusion has received increasing attention in the Earth
observation studies. The focus is put on the combination of multiple sources,
which may be spatially, temporally, or spectrally inhomogeneous, to produce
a more complete representation of a geophysical process. In this work, we use
remote sensing data and numerical atmospheric models through a data fusion
approach to provide improved information about the distribution of
atmospheric water vapor. This information is important not only for weather
forecasting and climate research, but also for better understanding how the
InSAR interferograms are affected by water vapor, and for selecting the most
appropriate method for reducing this noise. In turn, reliable local water
vapor maps can support adaptation of the WRF model configurations and, hence,
may improve the model performance.
Maps of the absolute atmospheric PWV derived by combining PSI and
GNSS data and the corresponding map from MERIS. The spatial correlation is
95 % and the rms value of the differences is 0.68 mm.
In the following, we present water vapor maps derived from microwave remote
sensing data and numerical atmospheric models. Since the available data have
different spatial levels of aggregation, it is important to discuss the
change of support problem. Then, we present the data fusion approach based on
the kriging or fixed-rank kriging techniques. We first describe the ordinary
kriging and how it can be extended for fusing multiple data sets. Then, we
present the reasons behind using the fixed-rank kriging. We use the data
fusion approach for predicting maps of the atmospheric PWV from remote
sensing data and atmospheric models.
Atmospheric water vapor
Several observation systems are commonly used to continuously monitor the
vertical and horizontal distributions of water vapor in the atmosphere. These
devices are used either from the ground, such as radiosondes and ground-based
water vapor radiometers, or from space, such as space-based water vapor
radiometers and infrared sensors. In this work, we employ microwave remote
sensing systems as well as numerical atmospheric models to provide accurate
maps of the atmospheric water vapor at a high spatial resolution.
Water vapor from remote sensing data
presented a new approach to derive absolute,
high-resolution maps of PWV by combining data from InSAR and GNSS. The data
are collected in the region of Upper Rhine Graben in Germany and France over
the period 2003–2008. Persistent scatterer InSAR (PSI) using the Stanford
Method for Persistent Scatterers (StaMPS, ) was applied
to derive PWV maps from the InSAR interferograms. These maps contain the
water vapor signal of short-scale spatial variations, while the
elevation-dependent and long wavelength water vapor components are eliminated
when forming the interferograms or by phase filtering. Therefore, GNSS-based
PWV estimates were used to reconstruct the missing components. The approach
for combining InSAR and GNSS data is presented in detail in
and .
Figure shows a map of PWV derived by
combining PSI and GNSS data and the corresponding map extracted from MERIS
observations. MERIS is a passive imaging spectrometer located on board the
Envisat platform. It measures the solar radiation reflected from the Earth's
surface or clouds. The ratio of the radiance values measured at channels 14
and 15, located respectively at 885 and 900 nm, are used to determine the
vertical PWV content in the neutrosphere . MERIS provides
maps of the PWV at a spatial resolution of 260 m × 290 m
(full-resolution mode). Under cloud weather conditions MERIS measurements are
highly underestimated since the measured PWV represents only the water vapor
existing between the sensor and cloud top; therefore, only five MERIS PWV
images were available for this study.
The PSI method produces information where stable persistent scatterers are
identified, which requires a high coherence between the SAR images. In
forests and vegetated areas, the probability of identifying persistent
scatterers is low; therefore, in these regions, only sparse points are found.
The white areas within the left figure indicate regions of low coherence and
the corresponding data from MERIS are masked out. The spatial correlation
between the maps is 95 % and the root mean square (rms)
value of the differences is 0.68 mm. We can observe that the persistent
scatterers are dense in the urban areas, while they almost disappear in the
low coherence regions. Since PWV data are spatial, their covariance function
is exploited by geostatistical techniques to reasonably infer the PWV at
regular grids. In order to improve the inferred PWV maps, especially in the
areas where the PWV estimates are sparse, we apply data fusion of the
remotely sensed PWV maps with maps produced by the WRF model.
WRF model set up with a parent domain of resolution
27 km × 27 km and two nests of 9 km × 9 km and
3 km × 3 km, respectively.
Maps of PWV content as received from MERIS and WRF, where a linear
trend is subtracted from each map. The upper data are received on 27 June
2005 (09:51 UTC), the lower data on 5 September 2005 (09:51 UTC). Gaussian
averaging is applied to scale the MERIS data at WRF resolution,
3 km × 3 km. The spatial correlation coefficient between the upper
maps is 0.8 and 0.71 for the lower maps.
Water vapor from regional atmospheric models
As depicted in Fig. , the WRF model (version 3.1.1,
) was set up with a parent domain of
27 km × 27 km resolution and two nests with 9 km × 9 km
and 3 km × 3 km, respectively. Feedback from the nests to their
parent domain was not activated. Vertically, the model is divided into 42
layers with variable distance. The resolution is increased for the lower
troposphere where most of the atmospheric vapor resides. The model top is
defined at 50 hPa. The selection of the physical modules is based on the
study of ; accordingly, the WRF single-moment (WSM) 5-class
scheme was selected for microphysics. Shortwave and longwave
radiation was computed with the community atmospheric model (CAM) scheme
. The processes in the planetary boundary layer were
represented by the Yonsai University scheme . The surface
layer was simulated with the Monin–Obukhov scheme, and the Noah land-surface
model was applied for the surface physics. Sub-grid
convective processes were included with the Kain–Fritsch parametrization
. The global dynamic boundary conditions were
ingested from
the European Center for Medium-Range Weather Forecasts (ECMWF) ERA-INTERIM
reanalysis at a 6 h interval . In ERA-INTERIM, a broad
range of different data sources is assimilated. For the atmospheric moisture
analysis, ground-based station observations, radiosonde profiles, and GPS
radio occultation are exploited. Additionally, total column water vapor
information from the Special Sensor Microwave/Imager (SSM/I) and the Advanced
Microwave Scanning Radiometer for the Earth Observing System (AMSR-E) is
assimilated . MERIS retrievals of column water vapor are not
ingested into ERA-INTERIM, and thus they depict an independent data set for
our approach.
The WRF simulations cover the period between July 2004 and September 2005,
such that the first 5 months were considered as spin-up. The PWV content was
determined at every output time step (10 min) by a vertical integration of
all moisture fields from the land surface to the model top. Two output time
slices were compared with the simultaneous MERIS observations. The long-scale
signal is modeled by a linear trend and subtracted from the maps; hence,
negative values are observed on the color bars. From the compared maps shown
in Fig. , we observe that the spatial atmospheric
patterns are not always correctly resembled by the model. On 27 June 2005
(09:51 UTC), WRF and MERIS PWV maps are strongly correlated with
a coefficient of 0.8, whereas the analysis of 5 September 2005 (09:51 UTC)
shows a lower spatial correlation (0.71). While the patterns east of the
Upper Rhine valley are reasonably resembled, an unexpected discontinuity
exists in the area around 7.7∘ E, 48.7∘ N.
At the lateral boundaries, WRF ingests the mixing ratio concentration from
the global model. Thus, for the presented simulation, the global climate
model lateral boundary conditions were applied to the first (outer) domain.
Neither gridded nor spectral nudging was activated in order to conserve the
model's internal water balance. Hence the GCM boundary fluxes and the local
area model physics solely determine the propagation of moisture through the
respective domains. For the analysis of 27 June 2005, the atmospheric
conditions were rather unexcited and varied slowly, resulting in a good
agreement between MERIS and WRF data. On 5 September, a quickly moving
frontal system with a strong west-to-east gradient and a notch in the
atmospheric vapor over the Upper Rhine Graben characterized the study region.
It is not clearly distinguishable whether the structure and dynamics of the
ERA-INTERIM boundaries or the WRF model configurations are responsible for
the discontinuity in PWV.
Change of support problem
Spatial data, for which close observations correlate more than distant ones,
can be collected at points or areal units. The former are called point-level
data or simply point data and the latter are areal-level or block data
. In geostatistics, this defines the spatial support of
the data. When both data types are available, data fusion can be applied to
infer the underlying process at any level of support. The change of support
problem is concerned with the inference of the underlying process at point
levels or block levels different from those at which the data are available.
This also includes fusing data at different support levels. Based on the
available input data and the desired output grid, there are four prediction
possibilities: points to points, points to blocks, blocks to points, or
blocks to blocks. These prediction possibilities may be collected under the
umbrella of kriging .
For block data that can be expressed as an average of point data as if it is
collected within the block, such as rainfall, temperature, surface elevation,
and atmospheric water vapor, the following model is
appropriate:
Y(Bi)=1|Bi|∫BiY(s)ds,
where Y(Bi) and Y(s) define the block and point data,
respectively (Fig. ). Bi refers to the block
over which the data are aggregated and |Bi| is the volume (or
cardinality) of the data. The block-level covariance can then be related to
the point-level covariance as follows:
C(Bi,Bj)=cov1|Bi|∫BiY(u)du,1|Bj|∫BjY(v)dv=1|Bi||Bj|∫Bi∫BjC(u,v)dudv,
where C(Bi,Bj) is the block-to-block or block covariance function and
C(u,v) is the point covariance function.
Point and block data, such that for spatial data, Y(Bi) represents the average of the point data within the block.
Spatial data fusion using kriging methods
Ordinary kriging
In geostatistics, a spatial process can be inferred over a continuous spatial
domain by exploiting the covariance function as an important source of
information. Predictions are obtained based either on single or multiple
sets. Kriging is a geostatistical interpolation technique that infers values
at new locations by considering spatial correlations . The
spatial density of the data points has to be enough to capture the covariance
structure of the process. This information is represented by a variogram or
covariance function, which is used to determine the predictions. If the
considered spatial data set is denoted by Z, then the kriging
estimator Y^(s0) at the location s0 is determined
as follows:
Y^(s0)=a′Z̃,
where the vector a contains the kriging weighting coefficients and
Z̃ is the centered data set (see Eq. ). The
best linear unbiased estimator is found by solving the following constrained
minimization problem:
minaE(Y^(s)-Y(s))2subject toE{Y^(s)}=E{Y(s)}.
The constraint is added to guarantee that the estimator is unbiased with
respect to the true process Y(s). A semivariogram function that
reflects the spatial correlations is required to solve the minimization
problem, which is determined from the detrended data in
Eq. .
The kriging method extends the spatial process using the following linear
model:
Z(s)=T(s)⋅α+ν(s)︸Y(s)+ϵ(s)︸noise,
where ϵ(s) is an independent error term, which is assumed to
be a white noise process with a mean zero and variance
σϵ2. T(s)⋅α
defines a deterministic linear trend, T has a size of N×3
and each row has the following entries: [1 longitude(s)
latitude(s)]. N is the number of observations and
α is a vector of the least squares regression
coefficient. ν(s) captures the spatial covariance structure of the
process, and it is assumed to have a mean zero and generally a non-stationary
covariance function. Before inferring the signal at a new location, it is
required to center the data by estimating and subtracting the linear trend,
i.e.,
Z̃=Z-Tα^withα^=(TT′)-1T′Z.
The detrended signal Z̃ is used to determine the predictions
in Eq. () and the deterministic signal is calculated from
T(s0)α^. The sum of the two terms
gives the total estimated value of Y(s0). In the next section,
a similar strategy is followed to solve for the best unbiased estimator using
two data sets as presented in .
Spatial statistical data fusion
Spatial statistical data fusion (SSDF) is a method that statistically
combines two data sets to optimally infer the quantity of interest and
calculate the corresponding uncertainties at any predefined grid
. This method extends the kriging technique
described above to find the optimal estimator using multiple data sets. Let
the underlying process Y(s) to be estimated at the location s
from the data in Z1 and Z2 with the sizes N1 and
N2, respectively. The estimator Y^(s) at the location
s is obtained from the two data sets as follows:
Y^(s)=a1′Z̃1+a2′Z̃2,
where a1 and a2 are the fusion weighting coefficients,
and Z̃1 and Z̃2 are detrended data sets
of Z1 and Z2, respectively. Following
Eq. () and Eq. (), the
Lagrangian function L for the minimization problem under the unbiasedness
constraint is
L=a1′Σ11a1+a2′Σ22a2+2a1′Σ12a2′-2a1′c1-2a2′c2+2m(a1′1N1+a2′1N2-1),
where Σii=cov(Z̃i),
Σij=cov(Z̃i,Z̃j), and ci=cov(Z̃i,Y(s)) are the covariance functions. 1Ni is a vector with
all entries 1 and a length Ni, and m denotes the Lagrange multiplier.
The last term of L accounts for the unbiasedness constraint. By
differentiating L with respect to a1,a2,m and assigning
the results to zero, we get, in the following system of equations,
Σ11Σ121N1Σ21Σ221N21N1′1N2′0a1a2m=c1c21
and hence
a1a2m=Σ11Σ121N1Σ21Σ221N21N1′1N2′0-1c1c21.
There are several important discussion points for the solution in
Eq. (). The covariance matrices
Σij should be determined without assuming that the
underlying process is isotropic or stationary. This is important for
atmospheric parameters, particularly the atmospheric water vapor that shows
spatial anisotropy as observed from the spatial autocorrelation function in
Fig. . The covariance function ci should account
for the change in the support between the input and the output data. For
massive data sets, the size of the covariance matrix is huge and the solution
in Eq. () is not feasible anymore. Also, the
covariance matrices should be modeled such that they would allow data
prediction to any level of aggregation. The fixed-rank kriging covariance
model suggested by provides a comprehensive solution for
these problems for single data sets and the generalized model for fusing
multiple data sets was presented by and
. In the next section, we describe the fixed-rank kriging
method and the associated covariance model. Then, we describe how the data
fusion approach is applied to our data sets.
Spatial autocorrelation function for a PWV map, with the long-wavelength component removed, computed from remote sensing data acquired on 5 September 2005, 10:51 UTC.
Fixed-rank kriging
The fixed-rank kriging (FRK) approach splits the spatial process into two or
three components depending on the spatial wavelength, i.e,
Y(s)=T(s)⋅α︸linear trend+S(s)⋅η+ζ(s)︸ν(s).
The model in Eq. () is called the spatial random effects
(SRE) model . The first component represents
a deterministic linear trend that reflects the large-scale spatial
variations. The second component S(s)⋅η
captures the relatively smooth spatial variations, which form the covariance
structure of the process. That is,
cov(S(u)⋅η,S(v)⋅η)=S(u)KS′(v),
with K the covariance function of η. This component is
modeled by a linear combination of spatial random effects at multiple spatial
scales. The vector η contains r hidden spatial random effects,
which are estimated from the data at predefined nodes. Therefore, we should
be able to estimate η regardless of the aggregation level of the
input data. When neglecting the last term in Eq. (), the
weighted sum ∑j=1rSj(s)ηj should give the detrended
value of Y at the location s.
The weights stored in the matrix S for each location s
depend on the distance between s and each node. The weighting
function S(s) has the following form:
S(s)=1-(||s-mi||/ri)22,for||s-mi||≤ri,0otherwise.
mi is the node location and ri is a predefined effective
radius. The formula in Eq. () represents a bi-square bell-shaped
function that has its maximum value at mi and decreases smoothly
until it reaches zero outside the circle. To demonstrate this, a schematic
diagram for the node setup is shown in Fig. . Within the
domain of the data, four nodes, m1,⋯,m4, are
defined with a corresponding radius. In Fig. , if s
is located within the radius of a certain node, it gets a positive weight;
otherwise, the weight is zero. Hence, S(s)=[0, 0, 0,S(s)].
The observation domain with the black dots defines the locations at
which the data are available. The black little squares indicate the nodes.
The weights for each location s are related to the distances di.
The dashed circles define the radius for each node.
The last component in Eq. () accounts for the variations
of the process that has not been captured so far . The
component ζ is assumed to be an uncorrelated Gaussian process with
a mean zero and a variance σζ2.
Obtaining predictions via the FRK method.
FRK nodes or center locations of 93 basis functions at three spatial
resolutions. The first resolution is 40 km, the second resolution is 20 km,
and the third resolution is 10 km.
(a) The experimental semivariogram and the fitted spherical
variogram model. (b) Covariance matrix used to predict the wet delay
maps in Fig. .
Based on the model in Eq. (), the FRK estimator is found
when η and ζ are determined; i.e.,
Y^(so)=Sp(so)⋅η^+ζ^(so)=Sp(so)KS′Σ-1Z̃+σζ2E(so=s)Σ-1Z̃,
where Sp(so) is the weighting matrix for the
prediction location and Σ is the covariance matrix of the
input data. The matrix E in Eq. () has a value of
one if s=so and zero elsewhere. Y^ represents the
detrended estimator. η^ and ζ^ are the optimal
a posteriori estimates of η and ζ, respectively
. In order to get the total value of Y^t, we
calculate
Y^t(so)=T(so)⋅α^+Y^(so).
The steps followed to obtain the predictions based on the FRK method are
summarized in Fig. . The methods to estimate the noise
variance σϵ2, the covariance matrix K, and the
variance of the fine-scale signal σζ2 are shown in
Appendix .
We classify the spatial variations of the atmospheric water vapor signal into
three components: long wavelength, medium to short wavelength, and
uncorrelated fine scale. Therefore, we split the water vapor signal using the
linear model in Eq. () and use the FRK method for
prediction.
Wet delay prediction map using block OK and FRK. The resolution of
the grid is 3 km × 3 km. A point-level wet delay map, on 23 May
2005 at 09:51 UTC, is used as input to the algorithms.
We applied the OK and FRK to estimate the zenith-directed wet delay derived
from remote sensing data. For the FRK, the matrix S is constructed
using the node setup shown in Fig. . The nodes or
center locations of 93 basis functions are established at three spatial
resolutions: the first resolution is 40 km, the second resolution is 20 km,
and the third resolution is 10 km. The semivariogram and the fitted
spherical variogram model are shown in
Fig. a, while the covariance matrix
determined using the FRK method is shown in
Fig. b. The predicted maps with
3 km × 3 km resolution are shown in Fig. . Due to
the lack of ground truth data that should be used to estimate the bias in the
model data, we do not add the long-wavelength component into the figures to
enable unbiased comparison. We observe similar results from both ordinary
kriging and fixed-rank kriging that agree with the original WRF map. The
spatial correlation coefficients with the corresponding WRF data are
approximately 85 and 83 % for FRK and OK, respectively. When using OK, we
assumed the signal to be spatially isotropic to ease the computations;
therefore, the OK prediction map shows results sightly different from the
FRK. The most impressive point here is the computational time reported for
both algorithms. The FRK algorithm is fast, so that it requires significantly
shorter time to produce the predictions. Most of the time is invested in the
calculations of the covariance model parameters and constructing the matrices
S and Σ. We implemented the OK algorithm such
that the predictions are found iteratively. Also, to estimate a value at
location s, we do not use the entire data, but only those that exist
within a predefined radius around the prediction location. Nevertheless, the
OK algorithm requires computational time with an order of magnitude higher
than that required by the FRK method, on the same machine.
In the next section, we describe the extension of the FRK method for
predicting the atmospheric PWV by fusing remote sensing data and the WRF
model.
Data fusion for water vapor estimation
In this section, we fuse the PWV maps derived from the remote sensing data
and WRF model. Since we classify the spatial variations of the atmospheric
water vapor signal into long wavelength, medium to short wavelength, and
uncorrelated fine-scale components, we use the following model setup for
prediction.
Model setup
PWV maps will be derived from the remote sensing data, denoted Z1,
and those from the WRF model denoted Z2 with the sizes N1 and
N2, respectively. Z1 contains the point PWV estimates from
remote sensing data and Z2 contains the block WRF data. Following
the SME model in Eq. (), the two data sets can be
expressed as
Z1Z2=T1T‾2α+S1S‾2η+ζ10+ϵ1ϵ2.
The regression coefficient α should be estimated jointly
from both data sets. However, we do not have a priori information about the
biases; therefore, we estimate α in this contribution
independently for each data set. The matrices S1 and
S‾2 contain the weights of each location for each
data set. To distinguish between point and block data, we used the notation
S‾2 for block-level data. The model components for
point and block data are given in
Table . The WRF data are available at
a resolution of 3km×3km; therefore, the highly
variable signal of water vapor is smoothed. Hence, we do not add the
component ζ for the model data.
To solve the system in Eq. (), we determine the
covariance structure associated with each SRE model in
Eq. (), i.e.,
Σ11=var(Z̃1)=S1KS1′+σζ2Vζ+σϵ12Vϵ1,Σ22=var(Z̃2)=S̃2KS̃2′+σϵ22Vϵ2,Σ12=cov(Z̃1,Z̃2)=S1KS̃2′=Σ21′,
where σζ2Vζ and σϵ2Vϵ are diagonal covariance matrices for ζ and
ϵ, respectively. Note that when computing the cross-covariance
functions Σ12 and Σ21, the only
part of the signals that is assumed correlated is η. In order to
solve Eq. (), we need not only to specify the
covariance matrices of the input data, but also to find the covariance
between the observations and the spatial process at the prediction locations.
The covariance terms are obtained from
c1=cov(Z̃1(s),Y(so))=Sp(so)KS1′(s)+σζ2E(s=so);c2=cov(Z̃2,Y(so))=Sp(so)KS̃2′.
The matrix E in Eq. () has a value of one if
s=so and zero elsewhere. By solving for a1
and a2 in Eq. () and substituting
the results in Eq. (), the estimator
Y^(so) becomes
Y^(so)=Sp(so)KS1′S̃2′+σζ2E0Σ11Σ12Σ21Σ22-1Z̃1Z̃2.
The mean squared prediction error (MSPE) corresponding to Y^
can be obtained from
MSPE=a1′Σ11a1+a2′Σ22a2+2a1′Σ12a2-2a1′c1-2a2′c2.
Model components from
point-level and areal-level data.
Point data
Block data
True process
Y(s)
Y(Bi)=1|Bi|∑s⊂BiY(s)
Trend
T(s)α
1|Bi|∑s⊂BiT(s)α
Weighting matrix
S(s)
S̃(Bi)=1|Bi|∑s⊂BiS(s)
Medium-scale signal
S(s)η
S̃(Bi)η
Fine-scale signal
ζ(s)
ζ(Bi)=1|Bi|∑s⊂Biζ(s)
Error
ϵ(s)
ϵ(Bi)
Using the FRK covariance model in Eq. () makes the matrix
inversion of Eq. () scalable. That is, the matrix
inversion can be achieved by applying a recursive block-wise inversion as
follows:
ABCD-1=O1O2O3O4,
where
O1=A-1+A-1B(D-CA-1B)-1CA-1,O2=-A-1B(D-CA-1B)-1,O3=-(D-CA-1B)-1CA-1,O4=(D-CA-1B)-1,
and A,B,C, and D are matrices of
any size, and A and D must be square. The inversion of
individual matrices in Eq. () is achieved by applying
the formula of Sherman–Morrison–Woodbury, which is made possible due to the
FRK covariance structure,
Σii-1=(Di+SiKiSi′)-1=Di-1-Di-1Si(K-1+Si′Di-1Si)-1Si′Di-1.
The computations require the inversion of the matrices K and
(K-1+Si′Di-1Si), where each of them has the size r×r with r
significantly smaller than the data size. Note that Di is
a diagonal matrix, for which the inversion is achieved by inverting the
diagonal elements. Using the FRK covariance model makes the computational
burden for the matrix inversion linear with the data size
.
Application to the data
In this section, we build PWV maps from remote sensing and WRF model data
using a spatial statistical data fusion method. The first PWV map, derived by
combining GNSS and PSI, has 169 688 data points. The WRF model provides
a block-level map of 1296 cells of the size 3 km × 3 km. The data
to be fused have different qualities, a huge size, different spatial
supports, and gaps in the remote sensing data. The output grid is defined at
3 km × 3 km (block-level support) and MERIS PWV maps are used for
evaluation.
Following the work flow in Fig. , we first estimate the
long wavelength trends and remove them from the data using
Eq. (). By comparing the PWV from the WRF model and remote
sensing data, we found it is most likely that the model data have a bias. Due
to the lack of a priori information about the bias and the absence of
accurate ground truth data to estimate it, we estimated α
independently for each data set. The centered maps are shown in
Fig. .
PWV maps from the PSI + GNSS combination and WRF on 5 September
2005, with a linear trend subtracted from each map. PSI + GNSS provide
point-level observations, while WRF generates block data with a block size of
3 km × 3 km. The predictions will be obtained within the area
indicated by the black box.
Spatial correlation coefficients (CC) and rms values when comparing
the prediction maps with MERIS PWV maps.
Method
5 September 2005
27 June 2005
Spatial CC
rms (mm)
Spatial CC
rms (mm)
WRF data
0.70
1.33
0.85
0.87
Remote sensing data
0.87
0.90
0.72
1.13
Data fusion
0.91
0.82
0.86
0.92
Second, the matrices S1 and S2 are constructed
for the first data set (remote sensing data) and the second data set (model
data). The node setup is shown in Fig. . The number
of nodes must be the same for both data sets, and they are selected such that
S does not contain columns of zeros; otherwise, the corresponding
node has to be removed. If PWV data are available at point level, a weighting
value is calculated for each point with respect to all nodes. However, the
WRF model simulates data at block level; hence, we superimpose the model grid
with a lattice of regular points such that each cell in the WRF grid contains
nine points. A weighting value is calculated for each point; these values are
averaged to get one weighing value for each WRF cell to form the matrix
S‾2. Building the matrix Sp for
the prediction locations is done in a similar way, either at point level or
block level, depending on the output grid.
PWV prediction and MSPE maps obtained by data fusion of PWV
estimates from PSI and GNSS and maps from WRF as well as predictions obtained
by applying FRK to individual data sets. The data are available on
5 September 2005 at 09:51 UTC. The output grid has a block size of
3 km × 3 km. The label A defines a region of sparse remote sensing
data and the model data in region B are highly overestimated.
PWV maps from remote sensing (PSI+GNSS) and WRF model data on
27 June 2005 at 09:51 UTC as well as prediction maps obtained by data fusion
and individual data sets. The output grid has a block size of
3 km × 3 km over the area indicated by the black box in
(a) and (b).
In the third step, the covariance parameters (K,σζ2,σϵ2) are estimated from the centered
data Z̃1 and Z̃2. The error variances for
both data sets, K and σζ2, are estimated as
described in Appendix . Note that when the two data
sets are combined to infer a single process, i.e., PWV, one K is
estimated for all data sets.
Results
So far, all components required to produce the predictions using
Eq. () have been obtained. In
Fig. , we show the prediction maps
obtained by applying FRK to individual data sets as well as the map obtained
by data fusion. The figure also shows the MSPE maps associated with each
prediction map. We compare the interpolations obtained by applying FRK to
single data sets with those obtained by SSDF, and we compare both with the
MERIS data. The results show that the map obtained by data fusion correlates
more consistently with the map predicted only from PSI + GNSS
(Table ). In the PWV map generated by WRF, shown
in Fig. , the area in the lower left corner
shows artifacts that do not reflect the correct values of PWV as observed
from the MERIS PWV map (Fig. c and d). Applying
FRK to the WRF data does not remove these artifacts from the prediction map.
However, in the map obtained by the fusion of both data sets, the artifacts
in the lower corner disappeared, but the corresponding MSPE values are large
for this region. The MSPE values corresponding to the SSDF predictions are
generally smaller, and we should note that in the regions of sparse
observations, the corresponding MSPE values tend to increase. For regions of
sparse observations in the PWV map (Fig. ),
i.e., the areas in the west of the Rhine valley or in the lower right corner,
the map from the WRF model contributes to improving the estimation of the PWV
values in the prediction map. The region in the lower right corner has
a higher topography and the wet delay values are expected to decrease, as we
observe from the map of WRF. In the prediction map obtained by applying FRK
to PWV from PSI and GNSS, the predicted values tend to increase since the
data in this area are sparse and partially biased. By applying the SSDF
approach, the data available from WRF influence the predictions such that the
PWV values in this area are more reasonable, and they decrease by moving to
the lower right corner. In a similar way, the data from WRF improve the
predictions in the region around 7.8∘ E, 49.25∘ N, where
only sparse PWV data exist. The data from the model, however, affect the
prediction in the lower left corner such that they are smaller than those
observed in the MERIS map.
In addition, we show the PWV profiles over the line drawn horizontally at the
latitude 49.37∘ N in Fig. h. It
is observed from the plots that the predictions made by data fusion are
affected more by the data from WRF in region A, where the remote sensing data
are sparse. However, in region B, the WRF data are significantly
overestimated. In the prediction map made by data fusion, these data have
a lower effect in than those received from the remote sensing data. The map
received by applying the data fusion shows the best spatial correlation with
the data from MERIS and the smallest rms value (see
Table ).
In the above example, the data from remote sensing have a more significant
influence on the output. In Fig. , we
show another example where the model highly affects the predicted map. The
predicted map based on model data shows a better spatial correlation and a
lower uncertainty value compared to the map predicted using remote sensing
data. In this case, the fusion map is more affected by the model data. The
spatial correlation coefficients and the values of uncertainty are given in
Table . In the first example
(Fig. ), the effect of the remote
sensing data on the prediction map is significant. The other examples in
Fig. and
Table show that the model has a larger effect on
the output map.
Conclusions and outlook
We presented a method to obtain the atmospheric PWV over any aggregation
level by the fusion of remote sensing data and atmospheric models. The PWV
maps derived by combining data from PSI and GNSS are available at discrete
points that are absent in regions of low coherence. On the other hand, the
WRF model provides simulations of PWV in the atmosphere on regular grids at
a coarse spatial resolution. Both the quality of the model data and the model
skills for representing mesoscale atmospheric structures should be improved.
The quality of the prediction maps should be improved by data fusion. For
data fusion, the method of spatial statistical data fusion, first presented
in , was employed. This method is based on the fixed-rank
kriging approach that attempts to solve the problems of computational
complexity of huge data sets, change of support, and bias. We inferred PWV
data on a grid of 3 km × 3 km and compared the results with PWV
maps inferred from MERIS data on the same grid. The results show a strong
correlation between data fusion maps and those maps from MERIS. The
difference between both maps shows uncertainty values of less than 1 mm,
which is lower than that obtained from inferring data based on single sets.
To further improve the results, we suggest the following. The matrix
Si has so far been constructed for each data source by defining
a set of spatial nodes. The number of the nodes is empirically adjusted such
that the covariance function computed for the data set based on the estimated
matrix K approximates the empirical covariance. In future work,
the size and the locations of nodes have to be optimized by minimizing the
difference between the empirical and the estimated covariance functions. We
should also estimate the biases for each data set (if they exist), so that
they can be accounted for in the fusion approach. The data fusion approach
can be extended such that more than two data sets are used, for example, by
including the MERIS maps in the fusion. With the increasing number of
satellite missions and improved atmospheric models, we are able to produce
complete, accurate information about the Earth's atmosphere based on data
fusion approaches. Moreover, the improved PWV maps can be iteratively
assimilated into the local area atmospheric model to generate more accurate
3-D water vapor fields. Also, testing other combinations of physical schemes
within the WRF model can further improve the resulting water vapor maps. In
this paper, we compared the prediction maps with the data from MERIS;
however, in future work, the results should be validated using bootstrapping
or jackknifing techniques.