test

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 53, NO. 9, SEPTEMBER 2015 5021
A Robust Fixed Rank Kriging Method for Improving
the Spatial Completeness and Accuracy
of Satellite SST Products
Yuxin Zhu, Emily Lei Kang, Yanchen Bo, Qingxin Tang, Jiehai Cheng, and Yaqian He
Abstract—Sea surface temperature (SST) plays a vital role in
the Earth’s atmosphere and climate systems. Complete and accu-
rate SST observations are in great demand for forecasting tropical
cyclones and projecting climate change. Satellite remote sensing
has been used to retrieve SST globally, but missing values and
biased observations impose difficulties on practical applications
of these satellite-derived SST data. Conventional spatial statistics
methods such as kriging have been widely used to fill the gaps.
However, when such conventional methods are used to analyze
a massive satellite data set of size n, the inversion of the n × n
covariance matrix may require O(n3
) computations, which make
the computation very intensive or even infeasible. The fixed rank
kriging (FRK) performs dimension reduction through multires-
olution wavelet analysis so that it can dramatically reduce the
computation cost of various kriging methods. However, the FRK
cannot directly be used for incomplete data over spatially irregular
regions such as SSTs, and the potential bias in the satellite data
is not addressed. In this paper, we construct a data-driven bias-
correction model for the correction of the bias in satellite SSTs
and develop a robust FRK (R-FRK) method so that the dimension
reduction can be used to the satellite data in irregular regions with
missing data. We implement the bias-correction model and the
R-FRK to the level-3 mapped night Moderate Resolution Imaging
Spectroradiometer SSTs. The accuracy of the resulting predictions
is assessed using the colocated drifting buoy SST observations,
in terms of mean bias (bias), root-mean-squared error, and R
squared (R2
). The spatial completeness is assessed by the availabil-
ity of ocean pixels. The assessment results show that the spatially
Manuscript received August 23, 2014; revised February 3, 2015; accepted
March 14, 2015. Date of publication April 17, 2015; date of current version
June 8, 2015. This work was supported by the Natural Science Foundation of
China under Grant 41271347, by the National Basic Research Program (973
Program) Project of China under Grant 2013CB733403, by the Natural Science
Foundation of China under Grant 41401405, by the China Postdoctoral Science
Foundation under Grant 2014M561039, and by the Natural Science Foundation
of Shandong Province under Grant ZR2013DL002. (Corresponding author:
Yanchen Bo.)
Y. Zhu was with the State Key Laboratory of Remote Sensing Science and
School of Geography, Beijing Normal University, Beijing 100875, China. She
is now with the School of Urban and Environmental Sciences, Huaiyin Normal
University, Jiangsu 223300, China, and also with the Institute of Geographic
Sciences and Natural Resources Research, Chinese Academy of Sciences
(CAS), Beijing 100101, China (e-mail: zhuyuxin_402@163.com).
E. L. Kang is with the Department of Mathematical Sciences, University of
Cincinnati, Cincinnati, OH 45221-0025 USA (e-mail: kangel@ucmail.uc.edu).
Y. Bo and Q. Tang are with the State Key Laboratory of Remote Sensing
Science and School of Geography, Beijing Normal University, Beijing 100875,
China (e-mail: boyc@bnu.edu.cn; tangqinxin@mail.bnu.edu.cn).
J. Cheng is with the School of Surveying and Land Information En-
gineering, Henan Polytechnic University, Jiaozuo 454003, China (e-mail:
chengjiehai@gmail.com).
Y. He is with the Department of Geology and Geography, West Virginia
University, Morgantown, WV 26506 USA (e-mail: heyaqian1987@126.com).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TGRS.2015.2416351
complete SSTs with high accuracy can be obtained through the
bias-correction model and the R-FRK method developed in this
paper.
Index Terms—Basis function, data-driven bias-correction
model, Moderate Resolution Imaging Spectroradiometer
(MODIS) sea surface temperature (SST), robust fixed rank
kriging (FRK).
I. INTRODUCTION
SEA surface temperature (SST) plays a vital role in the
Earth’s atmosphere and climate systems, for coupling the
ocean and atmosphere through exchanges of heat, momentum,
moisture, and gases [1]. Therefore, spatiotemporally continu-
ous SST observations are essential in oceanographic sciences,
weather forecasts, and in investigating global and regional
climate changes [2]–[5]. The quality of SST observations is
one of the major sources of uncertainty in numerical weather
prediction and climate models [6]. Spatiotemporally complete
and accurate SST observations are in great demand. In situ
SST measurements by moored and drifting buoys, ships, and
observation platforms off-shore [7], are sparse and irregularly
distributed over space and time. Satellite-derived SST data have
a more complete spatial coverage than in situ observations do,
and have become an important data source in atmospheric and
oceanic sciences [8].
At present, satellite-derived SST data are mainly obtained
from infrared and microwave sensors, such as the Advanced
Very High Resolution Radiometer, Moderate Resolution Imag-
ing Spectroradiometer (MODIS), Advanced Along-Track Scan-
ning Radiometer, and the Advanced Microwave Scanning
Radiometer-Earth Observing System (AMSR-E). Among these
sensors, SSTs from infrared sensors have higher spatial resolu-
tion. However, there are a large number of pixels with missing
data due to cloud and water vapor contamination [9], [10].
Meanwhile, although satellite-derived SST data are attractive
due to their superior spatial coverage compared with conven-
tional in situ data, they may have larger bias [11]. Errors in
the MODIS SST are likely from atmospheric absorption and
undetected cloud or fog and others [12], [13]. We have assessed
the bias of the MODIS and AMSR-E SST products in Joining
Area of Asia and Indian-Pacific Ocean of 2003 [14]. The result
shows that the regional and seasonal biases in satellite-derived
SST data can be substantial, although the globally averaged
bias in the satellite-derived SSTs might be small [5], [15].
These incomplete and biased satellite SST products limit their
applicability and may cause large uncertainties in potential
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5022 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 53, NO. 9, SEPTEMBER 2015
applications. Hence, it is necessary to develop methods that aim
to correcting the bias and filling the gaps, so that the satellite-
derived SST data can become more operational for applications.
Many methods for filling gaps of satellite data have been
developed in oceanic sciences, such as the Optimum Inter-
polation [8], [16]–[18], the objective analysis [19], [20], the
blended analysis [21],the data interpolating empirical orthogo-
nal functions (DINEOF) [22], [23], the Kalman filtering method
[24], and the 3-D-variation assimilation [25]. The optimum
interpolation method requires the prediction error covariance
matrix and observation error covariance matrix, which are
typically lacking in prior. In addition, similar to the kriging
method, inverting a high-dimensional matrix is computationally
intensive, which makes the optimum interpolation method im-
practical or even impossible for gap filling of massive satellite
data. The objective analysis and blended analysis generally do
not take into account the spatial dependence that is inherent in
the satellite data. DINEOF and the Kalman filtering method
are usually used to reconstruct the time series data sets, but
are not directly designed for only single time point data. Al-
though these aforementioned methods could be used to make
predictions, most of them cannot provide quantitative measure
of uncertainty associated with the predictions, which provides
the reference of the data quality in practical applications. In
contrast, kriging, as a geostatistical interpolation method, is
able to provide not only spatially complete predictions but also
the associated uncertainties. In addition, the kriging predictions,
known as the spatial best linear unbiased predictions (spatial
BLUP), are statistically optimal, since they minimize the mean-
squared prediction errors.
Using kriging, we can obtain from noisy and incomplete data
the optimal prediction and the associated prediction standard
error for any location of interest, thus generating spatially
complete maps. However, since kriging requires inverting the
n × N covariance matrix, where n denotes the size of the data
set, the computational complexity is O(n3
); thus, it could be in-
feasible to implement kriging directly for massive data, such as
satellite SST data. To alleviate such computational difficulties
for kriging, the fixed rank kriging (FRK) method to compute
the kriging equations exactly for massive data is developed
[26], [27]. The FRK method is based on the Spatial Random
Effects (SRE) model, which achieves dimension reduction by
modeling the spatial dependence through a fixed number of
spatial basis functions. Compared with the localized methods
and other approximation methods (e.g., [28], [29]), the resulting
spatial covariance structure in the FRK method is not only
statistically valid (i.e., positive definite) but also highly flexible
without assuming stationarity or isotropy. The FRK method
is linearly scalable to the number of observations, i.e., O(n),
and it can be used to analyze massive spatial data efficiently
[27], [30]. To carry out FRK, the spatial basis functions must
be specified. The basis functions used in previous applications
(e.g., [27], [30]) are only suitable for the regular regions, and
cannot be applied straightly to data over spatially irregular
regions, since the resulting matrix of basis functions may not
be full-rank, and the computation may break down. Because of
the irregular shape of the ocean–land boundary, the study area
of satellite SST data is often irregular; thus, the FRK cannot
be applied straightly to the satellite SST data. In addition, we
have investigated in previous studies the satellite SST data is
biased spatiotemporally [14], and the FRK method itself cannot
directly reduce the systematic bias lying in the satellite data
due to the instrument instability, atmosphere condition, and the
retrieval algorithm.
In this paper, in order to obtain the spatial continuous SST
predictions with high precision and the associated prediction
standard error from massive satellite SST data with missing
value in space, we developed a Robust FRK (R-FRK) though
improving the basis function selection of the FRK method. The
R-FRK method can be implemented for satellite SSTs over
irregular regions. Moreover, in order to reduce the systematic
bias of MODIS SSTs and improve the prediction accuracy,
before implementing the R-FRK, we presented a data-driven
adaptive bias-correction model for the satellite SSTs. There-
fore, the methods presented in this paper is a two-stage pro-
cedure: The first stage is to build a data-driven bias-correction
model to remove the bias of satellite SSTs, in the second stage
the R-FRK method is presented to make predictions with bias-
corrected satellite SSTs in irregular regions.
The whole paper is organized as follows. The details of the
data sets used in this study and the procedures of data sets
preprocessing are provided in Section II. Section III describes
the R-FRK method and the data-driven adaptive bias-correction
model we developed in this study. Section IV demonstrates the
implementations of the methods we developed for MODIS SST
data. The last section summarizes the approach developed in
this paper and puts forward suggestions for further research.
II. DATA AND PREPROCESSING
A. Study Region and Data
The geographical area of interest in this study is the oceanic
area between longitudes 9.33
◦
E and 180
◦
E and between
latitudes 39.33
◦
S and 46
◦
N. This study region, which covers
the joining area of Asia, Africa, Australia, and Indian-Pacific
Ocean, has the big ‘warm pool’ with the largest scope and the
most warm SST in the world. This area is a region with the
strongest tropical convection and the most water vapor in
the world, which results in missing value pixels of satellite SST
data. The air–sea interaction is very strong in this area and it is
a key area for the short-term climate variation and prediction in
China and for studying hurricanes and other forms of tropical
convections, as well as for investigating future regional and
global climate change [31].
The satellite-derived SSTs used in this paper are MODIS
Aqua night level-3 mapped products, which are processed and
distributed by the Ocean Biology Processing Group at the
NASA Goddard Space Flight Center (http://gcmd.nasa.gov).
The most recent version is 5.x. The Level-3 mapped MODIS
SSTs are available at either 4- or 9-km spatial resolution and
at daily, 8-day, monthly, seasonal, and annually temporal res-
olutions. Data with coarser temporal resolutions are generally
obtained by averaging observations over the corresponding
period of time. All these data are available at http://oceandata.
sci.gsfc.nasa.gov/MODISA/Mapped/. In our study, we used the
level-3 mapped data with 4-km spatial resolution and 8-day

ZHU et al.: FRK METHOD FOR IMPROVING THE SPATIAL COMPLETENESS AND ACCURACY OF SST PRODUCTS 5023
TABLE I
MODIS SST AND DRIFTING BUOY SST
composited. The 4-km spatial resolution MODIS SST product
was chosen for this study since we would like to demonstrate
the capability of our method in processing massive spatial
data. We chose the 8-day composited data product because it
has better spatial coverage of the study region than the daily
data does, so that to ensure enough observed data to estimate
parameters of model. The weekly or 8-day composited satellite
SST products are useful to many analyses. For example, the
weekly satellite SST products were used for analyzing the SST
anomaly [32], for analyzing the relationship between South
Atlantic SST and SACZ intensity and positioning [33], for
documenting the Spatial and temporal scales of the Brazil-
Malvinas Current confluence [34], for attesting the hurricane-
induced phytoplankton blooms in an oceanic desert [35], for
analyzing the importance of wind stress curl to the upwelling
[36], and for studying the productivity induced by cyclone [37].
In order to test the long term and seasonal stabilities of
the performances of the methods we developed in this study,
the MODIS SSTs in 2003 and 2010 were chosen for the
experiments and in each year, 4 weeks of SST data in each
season were selected for study. That is to say that 16 weeks
of data covering four seasons were considered: the fourth to
seventh week, (January 25 through February1), the 15th to 18th
week (April 23 through May 24), the 27th to 30th week (July 28
through August 28), the 38th to 41th week (October 24
through November 16). In addition, in order to investigate the
consistency of the error feature of the satellite SSTs, these
aforementioned data is from 2003 and 2010, respectively.
The drifting buoy SSTs observations were used as the bench-
marks to correct the bias of satellite SSTs and to validate the
results. We obtained drifting buoy SSTs during 2003 and 2010
from the Global Drifter Program at the Marine Environmen-
tal Data Services site (http://www.meds-sdmm.dfo-mpo.gc.
ca/isdm-gdsi/drib-bder/svp-vcs/index-eng.asp).All the drifting
buoy observations were temporally interpolated and provided
at 0:00, 6:00, 12:00, and 18:00 on every day in UTC time [38].
Note that the observation time of Aqua-MODIS is at1:30AM in
local time. In order to temporally match up the MODIS SSTs’
with the drifting buoy SSTs, we chose the drifting buoy SSTs
observed at UTC 0:00 over regions 9.33E–105E in longitude
and at UTC 18:00 over regions 105E–180E in longitude so that
the drifting buoy SST observations were mostly close to the
MODIS SST observation in time and to minimize the effects of
the ocean diurnal warm layer [1]. All of the data used in this
study are shown in Table I.
B. Data Preprocessing
The MODIS SSTs are in HDF format with two different
layers, i.e., for temperature and for quality control, respectively.
Specifically, in the layer of quality control, each pixel is flagged
as 255, 0, 1, or 2 to denote different land-surface feature or
quality level of the corresponding pixel value in temperature
layer: the number 255 represents land, gross clouds, and other
errors; the number 0 represents being good, 1 represents being
questionable and 2 represents being cloud.
All the pixels in the study region are first reflagged as sea
or land: a pixel will be flagged as land if and only if it is
identified as nonsea in both MODIS and AMSR-Elevel-3 SST
data products; otherwise, we consider it as a sea pixel.
The sea pixels flagged as 1 (questionable) or 2 (cloud) in the
quality control layer of the MODIS data file are taken as pixels
with missing data. For the sea pixels flagged as 0, the Digital
Numbers (DN) in the temperature layer is converted into SST
using
SSTvalid = 0.000717185 × DN − 2 (1)
where SSTvalid represents converted temperature (
◦
C), and
the coefficient 0.000717185 and intercept −2 are ob-
tained from http://grasswiki.osgeo.org/wiki/MODIS. Noting
that the retrieved range of MODIS SST should be from
−2
◦
C to 32
◦
C (http://podaac.jpl.nasa.gov/DATA PRODUCT/
SST/modis/modis sst.html), we thus label the pixels with the
SSTvalid value less than −2
◦
C or larger than 32
◦
C as the
pixels with missing data. After the preprocessing procedure
aforementioned, we obtain the gross quality controlled MODIT
SST data. In our study region, the proportions of the missing
data in 2003 and 2010 are 29.26% and 28.70%, respectively.
The gross error quality of drifting buoy SST data was also
controlled. The observations lower than −1.8
◦
C or higher than
35
◦
C are removed [39]. To compare the drifting buoy SST
observations with the MODIS SSTs, we aggregated the drifting
buoy SST observations to the grid with the same spatial and
temporal resolutions of the MODIS SST data by averaging
the drifting buoy SST observations within the same MODIS
pixel and in the same period of 8 days the MODIS SST
was composited. Example of the 8-day averaged drifting buoy
SST data with 4-km spatial resolution in 2003 and 2010 are
presented in Fig. 1(a) and (b).
III. METHODS
We applied the R-FRK method to the incomplete MODIS
SSTs. To reduce the systematic bias, the preprocessed MODIS
SSTs are first corrected using the data-driven bias-correction
model. Then the R-FRK method is applied to the bias-corrected
MODIS SSTs to obtain spatially complete data. The general
process is shown in Fig. 2.
In the Section III-A–C, we will describe the FRK, R-FRK,
and the data-driven bias-correction model, respectively.
A. Hierarchical Statistical Modeling and FRK
A latent true SST spatial process is made up of two com-
ponents: the large scale trend and the fine-scale variation. The
relationship between the observed SST data and the SST spatial
process is flexible to be simulated in a hierarchical framework.

Fig. 1. Study region and spatial distribution of the drifting buoy. Gray and white areas represent land and ocean, respectively, and blue points indicate the
locations of drifting buoys. (a) 2003. (b) 2010.
Fig. 2. Workflow for spatial interpolation based on the R-FRK method.
1) Hierarchical Statistical Model: In the hierarchical statis-
tical analysis, the latent true SST spatial process is modeled at
two levels: the data-model level and the process-model level.
Let D denote the rectangular study region for analysis, and
let D0 denote the irregular ocean region within D. The latent
true SST, Y (S), is a “hidden” spatial process defined on this
subregion D0: Y (s) : s ∈ Do ⊂ D.
In the data-level model, we assume that MODIS SST data
is collected from an observable spatial process Z(s), s ⊂ Do.
This data process Z(·) is modeled as a “noisy” version of Y (s)
with measurement error ε(s)
Z(s) = Y (s) + ε(s) (2)
where ε(s) is a spatial white-noise Gaussian process with mean
zero and variance σ2
, and it is assumed to be independent
of Y (s). In reality, the data process Z(·) is observed only
at a finite number of spatial locations {s1, . . . , sn}, and we
define the n-dimensional vector of available data to be Z ≡
(Z(s1), . . . , Z(sn)) , where n is the number of observed pixels
of the MODIS SSTs, and it can be very large or even massive.
The “hidden” spatial process Y (s) is modeled at the process
level, and our inferences are made on this underlying process
given the observed data Z. We assume the following linear
structure for Y (s):
Y (s) = μ(s) + ν(s) (3)
where μ(s) and ν(s) represent the large-scale (trend) and the
fine-scale spatial variations, respectively. In our analysis, we
model the trend μ(s) with a linear model
μ(s) = T (s) β (4)
where T (.) = (T1(.), . . . , Tp(.)) is a p-dimensional vector of
known covariates, such as coarser scale wavelet basis, latitude,
and square latitude, β is an unknown coefficient.
The fine-scale spatial variation ν(s) is simulated by the
Spatial Random Effects (SRE) model [see (5)] [27]
ν(s) = S(s) η (5)
where S(.) = (S1(.), . . . , Sr(.)) is a r-dimensional vector de-
fined through r not-necessarily-orthogonal-known spatial basis
functions, and η = (η1, . . . , ηr) is a zero-mean Gaussian ran-
dom effect with distribution η ∼ Gau(0, K), where K is the
associated r × r covariance matrix.
Then, we can write our two-level model hierarchically as
follows:
Y (s) ∼ Gau (T (s) β, S(s) KS(s)) (6)
Z|Y ∼ Gau(Y, σ2
I). (7)
Furthermore, it can be derived from (6) straightforwardly that
the covariance structure of the spatial process Y (s) is
Cov (Y (s), Y (u)) = S(s) KS(u). (8)
Therefore, the covariance of the vector Y is C = SKS , and
thus the covariance matrix of the data vector Z is as
Σ = SKS + σ2
I. (9)
2) FRK Method: The FRK method gives the kriging predic-
tor (also known as the spatial best linear unbiased predictor,
spatial BLUP) for the aforementioned hierarchical model de-
scribed in Section III-A1. The kriging predictor of Y (s0) and

its associated kriging standard error in the FRK is shown in (10)
and (11), respectively [30]
ˆY (s0) = T (s0)ˆβ + S(s0)KS Σ−1
(Z − T ˆβ) (10)
σk(s0) = S(s0)KS(s0) − S(s0)KS Σ−1
SKS(s0)
+ (T (s0) − T Σ SKS(s0) ) (T Σ−1
T )−1
× T (s0) − T Σ−1
SKS(s0)
1
2
(11)
where the n × n matrix Σ is the covariance matrix of the data
vector Z, as defined in (9). Note that both (10) and (11) in-
versing this n × Nmatrix, which can be computed exactly and
efficiently by using the Sherman–Morrison–Woodbury formula
shown in (12) [40]
Σ−1
=(σ2
v)−1
−(σ2
v)−1
S K−1
+S (σ2
v)−1
S
−1
S (σ2
v)−1
.
(12)
Therefore, we only need to invert fixed rank r × r matrices
and n × n diagonal matrices. As pointed out in [30], the as-
sociated computational complexity of the FRK is only linear
to the number of observations, n. So it can be used to analyze
large to even massive spatial data sets, such as the MODIS SST
data in our study. As s0 varies over Do, by using (10) and (11),
we can obtain the maps of the kriging predictors of SST and
their associated kriging standard errors, respectively. Readers
are referred to [30] and [27] for more details on the FRK.
To apply the FRK method to the massive MODIS SST data,
we need to specify T (·) in (4) and S(·) in (5). The elements of
T (·) and S(·) are chosen from multiresolution W-wavelet basis
functions as suggested in [30]. Such W-wavelets have several
advantages. First, they are multiresolution, and thus are able
to capture spatial variations at varying degrees of smoothness
and scales. Second, the associated computation of the discrete
wavelet transform (DWT) can be efficient due to selection
of fixed coarser and fine-scale basis functions. In addition, it
has been shown that such W-wavelets can provide a flexible
class of nonstationary spatial covariance models, as well as
approximation of common stationary covariance models (e.g.,
in [41] and [42]). Of course, there are bi-square function and
others which can be applied in the R-FRK method due to its
general feature. Considering the characteristics of satellite SST
data sets and the advantages of the W-wavelets, in this paper we
selected the W-wavelets.
Before applying the DWT to the MODIS SSTs, the MODIS
SSTs were detrended by a linear regression model with co-
variates including latitude, and squared latitude. Based on the
detrended data, the simple and computationally efficient mean
polishing (e.g., [43]) was used to obtain the “complete” data,
and the DWT was implemented on these “complete” data to
obtain the wavelet coefficients. It should be noted that such
“complete” data are only used to accomplish basis-function
selection for T (·) in (4) and S(·) in (5) but not for our latter
estimation and mapping procedure. All the wavelets in scales
1, . . . , J0 and those with large absolute wavelet coefficients at
scale J0 + 1 (usually J0 is small) for T (·) in (4) are chosen.
The unselected wavelets at scale J0 + 1 and wavelets with large
absolute coefficients at scale J0 + 2 are chosen for S(·) in (5).
For more details on this strategy readers are referred to [30].
B. R-FRK Method
1) Basis Function Selection: The FRK method has been
successfully applied into global satellite aerosol optical depth
(AOD) data [30]. We notice one substantial difference between
the satellite AOD data and the satellite SST data in our study.
The satellite AOD data are potentially to be observed on the
entire globe, no matter land or water; thus, it is easy to define
a study area with a regular rectangle extent. However, for the
satellite SST data, we have to mask out the land where the
SST is meaningless, which results in a spatially irregular study
region with a large number of missing data. Although the
strategy in Section III-A2 alone works well in choosing basis
functions in several analyses of the satellite AOD data (e.g., in
[44]), It cannot be used straightforwardly to the satellite SST
data in an spatially irregular region with a large number of
missing value, since the basis function selection procedure of
the FRK works only for data with regular regions. We need
to develop a robust FRK method, in which the basis function
selection is available for data in both regular regions and in any
irregular regions.
We propose to select W-wavelets for T (·) and S(·) with
the following two-stage scheme. In Stage I, we implement
the selection strategy described in Section III-A2 proposed in
[30]. Among selected basis functions in Stage I, we should not
include those W-wavelets whose supports are mostly over land,
although their corresponding coefficients are large in absolute
value or they should be chosen by default (e.g., W-wavelet
in small scales are chosen for T (·) by default), since these
wavelets will not contribute to presenting the underlying spatial
process Y (·) that is only for the region of sea. In addition,
including the wavelets with support over land and support
over regions with missing data will make the matrix T (·) and
S(·) are not full rank, so that the estimation of the regression
coefficients and the matrix K will become computationally un-
stable. To tackle this problem, we proposed another screening
procedure in Stage II. Specifically, for the wavelets already
being chosen in Stage I, we compute the two ratios defined as
R1 (f(s)) = AD (f(s)) /A (f(s)) (13)
R2 (f(s)) = AO (f(s)) /AO (f(s)) (14)
where A(f(s)) denotes the number of pixels where the wavelet
f(·) is nonzero (i.e., the support of f(·)); AD(f(s)) denotes
the number of pixels over sea where f(·) is nonzero; AO(f(s))
denotes the number of pixels with observations as well as
nonzero values of f(·). In Stage II, we first set a common
threshold aT = 0.1 − 0.3 and screen out W-wavelets chosen in
Stage I whose associated R1(·) are less than aT . This allows us
to keep W-wavelet with support covering region over both land
and sea, but also ensures that we only include wavelets with
substantial proportion of their support on the region of interest,
i.e., region over sea. For those remaining W-wavelets chosen
for T (·) in Stage I, we only keep those W-wavelets with corre-
sponding R1(·) greater than or equal to aT , while screening out

Fig. 3. Scatter plot of MODIS SSTs against aggregated drifting buoy SSTs and box plot of bias in 2003. (a) Scatter plot. (b) Box plot of bias, the left panel
denotes the bias box plot for SST over the warm temperature regions and the right panel denotes the bias box plot for SST over the cool temperature regions.
the others. Similarly, for the remaining W-wavelets chosen for
S(.), we only keep those W-wavelets with corresponding R2(·)
greater than or equal to aS = 0.3, and remove the rest. We set
aT to be smaller than aS, since T (·) is to describe the large
scale spatial variation, and a more strict threshold will avoid
colinearity and stabilize the computation of the weighted least
squares estimate of the coefficient β. On the other hand, S(·)
is to describe small-scale spatial variation through the spatial
random effects, and it allows for more flexibility, although
we need certain amount observations in the support of these
wavelets for the estimation of covariance matrix K of random
effects through binning.
2) Parameters Estimation: As in classical geostatistics, in
order to implement the R-FRK method, we need to estimate
unknown covariance parameters in our model, namely K and
σ2
. We estimated ˆσ2
through the empirical semivariogram
estimation for small spatial lags proposed in [44]. We first
compute the robust semivariogram estimator defined as (15),
shown at the bottom of the page, where C(h) ≡ (i, j) : |si −
sj| = h, and (Z(si), Z(sj)) are observed data at location si and
sj; |C(h)| is the number of distinct elements of C(h), and h is
the spatial lag, defined by the pixel size in our study of the SST
data. We fitted a straight line of the estimated semivariogram
with the spatial lag h, and then the intercept of the fitted line is
an unbiased estimate of σ2
ˆσ2
= ˆγ(0+). (16)
To estimate K, we need to detrend the data using the method
suggested in [27]. We first estimate ˆβ using the ordinary least
square estimator. With an estimate of ˆβ, we calculate the
residuals of observations that reflect the spatial random effects
and the measurement errors. We partition the domain Do into
M subregions called bins and obtained an M × M positive-
definite empirical covariance matrix from the binned data by
the way that is proposed in [26]. Lastly, we estimate K by
minimizing the Frobenius norm between this empirical covari-
ance matrix and the binned-version of the theoretical covariance
matrix derived from Σ. Interested readers are referred to [30]
and [45] for more details on parameter estimation.
C. Data-Driven Bias Correction
Satellite-derived SST can be biased due to various reasons
such as instability of sensors and contamination of cloud, water
vapor, aerosols [9], [10], and these biases can be spatially
heterogeneous. The R-FRK method itself cannot reduce the
systematic bias of satellite-derived SST. In addition, the data
model in (2) assumes that the measurement error term ε(s) is a
spatial white noise with mean zero and homogeneous variance
σ2
. It would be tantalizing to generalize this data model to
one that is more flexible to describe the potential biases and
heterogeneous variance. This direct modeling approach would
require a reasonable statistical representation of the underlying
measurement error term ε(s). It would be challenging, since
the spatial characteristics of the mean and covariance structure
for ε(s) may not be simple. Therefore, instead of estimating
the ε(s) in (2), we apply a bias-correction procedure before
applying the R-FRK to the MODIS SSTs. There are many
methods for correcting bias caused by observation accuracy, or
by spatial sampling scheme [46], [47]. Considering the error
characteristics of the MODIS SSTs, we suggested a data-driven
bias-correction procedure to the original MODIS SSTs.
We first investigate the relationship between the MODIS
SSTs and the aggregated drifting buoy by scatter. Fig. 3(a) is
the scatter plot of MODIS data against aggregated drifting buoy
data in 2003 (red points are the MODIS SSTs that are equal or
2¯γ(h) ≡ 1
|C(h)| C(h) v(si)−1/2
Z(si) − v(sj)−1/2
Z(sj)
1/2
4
0.457 + 0.494 |C(h)|
(15)

Fig. 4. Scatter plot of MODIS SSTs against aggregated drifting buoy SSTs and box plot of bias in 2010. (a) Scatter plot. (b) Box plot of bias, the left panel
denotes the bias box plot for SST over the warm temperature regions, the right panel denotes the bias box plot for SST over the cool temperature regions.
TABLE II
STATISTICS OF BIAS OVER WARM AND COOL
REGIONS IN 2003 AND 2010
larger than 12.5
◦
C blue points are the opposite). Fig. 3(b) is the
box plot of the bias shown in Fig. 3(a). Fig. 4(a) is the scatter
plot of the MODIS SSTs against the aggregated drifting buoy
SSTs in 2010 (red points are the MODIS SSTs that are equal or
larger than 9
◦
C blue points are the opposite), and Fig. 4(b) is the
box plot of the bias shown in Fig. 4(a). The statistical databased
on Figs. 3 and 4 is shown in Table II.
As a data-driven bias-correction method, the critical values
are determined using an iterative procedure: first, we empiri-
cally determine a temperature as an initial critical value based
on Figs. 3 and 4, for example, 14
◦
C in 2003 and 10
◦
C in 2010,
and build the bias-correction equation based on this empirical
value. Then the mean bias of bias-corrected MODIS SSTs was
calculated. Second, the initial critical value plus or minus 0.1
◦
C
as the new critical value, and was used to build the new bias-
correction equation. The bias of the MODIS SSTs corrected by
the model built based on the new critical value was calculated.
This procedure was iteratively repeated until we got the critical
value that came up with the minimum bias of the corrected
MODIS SSTs. The critical value of cool and warm region
is 12.5
◦
C in 2003 and 9
◦
C in 2010, respectively. From
Figs. 3(a) and 4(a) we can see that the features of the bias
between the MODIS SSTs and the drifting buoy SSTs are
obviously different over cool and warm temperature regions.
There are similar features of bias in 2003 and 2010, i.e., the
MODIS SSTs over the cool regions are generally lower than the
drifting buoy SSTs, while over the warm regions MODIS SSTs
and drifting buoy SSTs are distributed more symmetrically
along the 1 : 1 line. From Table II, the mean bias over cool
regions is larger than that of warm regions; especially in 2010.
So we build the bias-correction models by (17) for the bias
correction of the MODIS SSTs in the different spatial domains
SST A
b = δ1A + δ2ASST A
s (17)
where A indicates the type of SST s such as SST s in the warm
regions and in the cool regions, δA and δA are coefficients, the
SSTb indicates drifting buoy SST s, and the SSTs indicates
satellite SST s.
In the time domain, the matchups are divided into 46 groups
based on 46 week. The bias-correction equation for the first
week MODIS SSTs is built by using the matchups from the
second to the 46th week, and that of the second week is built by
using the matchups from the third to the 46th and the first week,
and so on. For each week, we obtained two regression equations
based on the (17), and we obtained 92 regression equations in
total to correct MODIS SSTs.
IV. RESULTS
The coefficients of data-driven bias-correction model for the
selected weeks are shown in Tables III and IV.
Fig. 5 shows the scatter plot of the original MODIS SSTs,
the bias-corrected MODIS SSTs against the aggregated drifting
buoy SSTs for all weeks in 2003 and in 2010. Table V shows
the summaries of the validation of the original MODIS SSTs
and the bias-corrected MODIS SSTs against the aggregated the
drifting buoy SSTs for all weeks. The mean bias (bias), root-
mean-square error (RMSE), and R squared (R2
) is utilized
to quantitatively evaluate the accuracy of the original MODIS
SSTs and the bias-corrected MODIS SSTs.
As shown in Fig. 5 and in Table V the error of the original
MODIS SSTs has been reduced effectively, and especially
through data-driven bias-correction model, the effect of the
bias-correction model over the cool regions is much better than
that over the warm regions. In 2003, the bias and RMSE of the
bias-corrected MODIS SSTs are 1.7790e-04
◦
C and 0.6130
◦
C

TABLE III
COEFFICIENTS OF DATA-DRIVEN BIAS-CORRECTION MODEL OVER THE WARM AND THE
COOL TEMPERATURE REGIONS FOR THE SELECTED WEEKS IN 2003
TABLE IV
COEFFICIENTS OF DATA-DRIVEN BIAS-CORRECTION MODEL OVER THE WARM AND THE
COOL TEMPERATURE REGIONS FOR THE SELECTED WEEKS IN 2010
Fig. 5. Scatter plot of MODIS SSTs against the aggregated drifting buoy SSTs. (a) Scatter plot of the original MODIS SSTs against the aggregated drifting buoy
SSTs in 2003. (b) Scatter plot of the bias-corrected MODIS SSTs against the aggregated drifting buoy SSTs in 2003. (c) Scatter plot of the original MODIS SSTs
against the aggregated drifting buoy SSTs in 2010. (d) Scatter plot of the bias-corrected MODIS SSTs against the aggregated drifting buoy SSTs in 2010.

TABLE V
SUMMARIES OF THE VALIDATION OF THE ORIGINAL MODIS SSTS AND
THE BIAS-CORRECTED MODIS SSTS AGAINST AGGREGATED DRIFTING
BUOY SSTS (THE BIAS IS DEFINED AS SST_MODIS-SST_BUOY)
while those of the original MODIS SSTs are −0.1961
◦
C and
0.6452
◦
C, respectively. In 2010, the bias and RMSE of the
bias-corrected MODIS SSTs are 0.0028
◦
C and 0.6574
◦
C
while those of the original MODIS SSTs are −0.2466
◦
C and
0.7004
◦
C. Both the absolute bias of the bias-corrected MODIS
SSTs in 2003 and that in 2010 are all within 0.003
◦
C. The
effect of the data-driven bias-correction model is obviously.
The difference of the bias between the original MODIS SSTs
in 2003 and that in 2010 is 0.0505
◦
C, and the difference of the
RMSE between the original MODIS SSTs in 2003 and that in
2010 is 0.0552
◦
C. This shows that the accuracy of the original
MODIS SSTs in 2003 is similar with that in 2010.
Based on the bias-corrected MODIS SSTs, we first im-
plement the two-stage basis-selection scheme proposed in
Section III-B1. There are 4 × 8 = 32 basis functions at the first
scale, 3 × 4 × 8 = 96 basis functions at the second scale, and
3 × 8 × 16 = 384 basis functions at the third scale, and 3 ×
16 × 32 = 1536 basis functions at the fourth scale. In Stage I,
we selected all the wavelets from the first and second scales and
wavelets with large absolute coefficients from the third scale as
candidates for T (·); the remaining wavelets from the third scale
and wavelets with large absolute coefficients from the fourth
scale are considered as candidates for S(·). In Stage II, we use
aT = 0.2 ∼ 0.3, aS = 0.3, and only kept those candidates from
Stage I that lead to stable computation in the regression and
binning procedures. After the two-stage selection scheme we
select the wavelets, as well as the intercept, latitude, and latitude
square to be included in T (·), and select r wavelets for S(·).
We then applied the SRE model to the MODIS SST data
following the estimation procedure described in Sections III,
III-A, III-A1 and III-B. We estimated the covariance matrix
Kusing the binning procedure in [27]. The bin centers are based
on 16 × 16 windows, and we then compute the method-of-
moments estimate of K using the binned data. To obtain the
R-FRK prediction ˆY (s0) and the associated R-FRK standard
error σk(s0) for a given pixel s0, we substituted the estimates
ˆσ2
and ˆK into (10) and (11). With s0 ranging over all sea
pixels in the study region, the predicted SSTs and the asso-
ciated standard errors of the R-FRK-based predictions were
obtained. The bias-corrected MODIS SSTs, the fitted trend, the
R-FRK predictions and the R-FRK standard error are shown
in Figs. 6(a)–(d) and 7(a)–(d) (considering the layout, we only
list the fourth week in 2003 and 2010). The fitted trend and the
R-FRK predictions show the same features as the original data
do. The R-FRK predictions provide more small-scale spatial
variation than the fitted trend does. The R-FRK predictions are
available for pixels the value is missing in the original MODIS
SST data. As expected, the associated R-FRK standard errors
for the pixels with missing data are larger than those pixels for
which the original observations are available.
V. EVALUATION OF THE MODEL RESULTS
A. Spatial Completeness Evaluation
We evaluated the spatial completeness of the MODIS SSTs
using the proportion of the valid SST pixels over all the oceanic
regions. Fig. 8(a) and (b) shows the spatial completeness of
the original MODIS SSTs and predictions in 2003 and 2010
by histogram.
Most of the pixels with missing SST data are in the trop-
ical areas [see Figs. 6(a) and 7(a)]. Fig. 8(a) and (b) shows
that the spatial completeness of the SST predictions based on
R-FRK method is all 100%, and comparing to the original
MODISSSTs, the spatial completeness of the predictions from
the R-FRK method have been increased 28.74% in 2003 and
26.87% in 2010 on average, respectively.
B. Accuracy Assessment
The drifting buoy SSTs are taken as the reference for the
accuracy assessment of the satellite SSTs. The mean bias (bias),
RMSE, and R squared (R2) are utilized to quantitatively eval-
uate the accuracy of the R-FRK predictions, the bias-corrected
MODIS SSTs and the original MODIS SSTs. Fig. 9(a)–(j) are
the scatter plots of the original MODIS SSTs, the bias-corrected
MODIS SSTs and the R-FRK predictions against the spatially
and temporally colocated drifting buoy SSTs for selected weeks
in 2003 and 2010, respectively. Table VI presents the summary
of the validation of the original MODIS SSTs and the bias-
corrected MODIS SSTs against the aggregated the drifting buoy
SSTs for selected weeks.
As shown in Fig. 9(a)–(f) and in Table VI, in 2003 the
bias and RMSE of the original MODIS SSTs are −0.2140
◦
C
and 0.6488
◦
C, respectively, and the bias and RMSE of the
bias-corrected MODIS SSTs are −0.0198
◦
C and 0.6110
◦
C,
respectively, and the bias and RMSE of the R-FRK predictions
are −0.0975
◦
C and 0.7038
◦
C, respectively. In 2010, the bias
and RMSE of the original MODIS SSTs are −0.2431
◦
C and
0.6922
◦
C, respectively, and the bias and RMSE of the bias-
corrected MODIS SSTs are 0.0086
◦
C and 0.6479
◦
C, respec-
tively, and the bias and RMSE of the R-FRK predictions are
−0.0298
◦
C and 0.7985
◦
C, respectively. No matter in 2003 or
in 2010, the absolute bias of the R-FRK predictions is similar
with that of the bias-corrected MODIS SSTs and much smaller
than that of the original MODIS SSTs though the RMSE of the
R-FRK predictions in 2003 and in 2010 is a little larger than that
of the original MODIS SSTs and the bias-corrected MODIS
SSTs, about 0.1
◦
C, and the value is within 0.2
◦
C. This shows
that the bias correction procedure is necessary.

Fig. 6. Image of the bias-corrected MODIS SSTs, the ﬁtted trend T ˆβ, the R-FRK predictions, and the R-FRK standard error in 2003. (a) Bias-corrected MODIS
SSTs. (b) Fitted trend T ˆβ. (c) R-FRK predictions. (d) R-FRK standard error.
Fig. 7. Image of the bias-corrected MODIS SSTs, the ﬁtted trend T ˆβ, the R-FRK predictions, and the R-FRK standard error in 2010. (a) Bias-corrected MODIS
SSTs. (b) Fitted trend T ˆβ. (c) R-FRK predictions. (d) R-FRK standard error.

Fig. 8. Histogram of the availability of the original MODIS SSTs and predictions in 2003 and 2010, blue bar denotes the availability of original MODIS SST,
and the brown bar denotes the availability of R-FRK prediction. (a) 2003. (b) 2010.
Fig. 9(g)–(j) show the information from MODIS SSTs,
which have obvious effects on the accuracy of the R-FRK
predictions. In 2003, over the regions with MODIS SSTs, the
Bias and RMSE of the R-FRK predictions are −0.0422
◦
C
and 0.6902
◦
C, respectively, and over the regions without
MODIS SSTs the bias and RMSE of the R-FRK predictions
are −0.1358
◦
C and 0.7155
◦
C, respectively. In 2010, over the
regions with MODIS SSTs the bias and RMSE of the R-FRK
predictions are −0.0129
◦
C and 0.7122
◦
C, respectively, and
over the regions without MODIS SSTs the bias and RMSE
of the R-FRK predictions are −0.0239
◦
C and 0.6931
◦
C,
respectively. This shows that the valid MODIS SSTs have
obvious effects on the accuracy of the R-FRK predictions; if the
valid MODIS SSTs has larger random error the random error of
the R-FRK predictions over the regions with MODIS SSTs is
larger than that over the regions without MODIS SSTs.
Fig. 10(a)–(d) shows the weekly error feature of the original
MODIS SSTs, the bias-corrected MODIS SSTs, and the R-FRK
predictions for selected weeks in 2003 and 2010 in this study.
As shown in Fig. 10, for selected weeks in 2003 and 2010
in this study the bias of R-FRK predictions is similar to that of
the bias-corrected MODIS though the RMSE is a little higher
than that of the bias-corrected MODIS SSTs and the original
MODIS SSTs. The tendency of the error feature of this three
type data is similar, i.e., the accuracy is lower in summer, at
around 17th weeks, than that in other season and the inner-
annual fluctuation of error feature in 2010 is smaller than that
in 2003.
VI. DISCUSSIONS AND CONCLUSIONS
The satellite SSTs with spatial completeness and high accu-
racy are obtained by the data-driven bias-correction model and
the R-FRK proposed in this paper.
The data-driven bias-correction model and the R-FRK
method are implemented to reduce the systematic error and fill
in gaps of satellite SSTs, respectively. The improved spatial ba-
sis function selection is developed so that the R-FRK method is
suitable to filling in gaps of massive satellite data over irregular
regions with a large number of missing values. The error of the
original MODIS SSTs is reduced by data-driven bias-correction
model. An assessment using the colocated aggregated drifting
buoy SSTs shows that the bias and the RMSE is reduced by
0.1959
◦
C and 0.0323
◦
C in 2003, respectively and in 2010
they are 0.2438
◦
C and 0.043
◦
C in 2010, respectively. The
accuracy of the original MODIS SSTs in 2003 is similar with
that in 2010 due to the small difference of the bias and RMSE
between the original MODIS SSTs in 2003 and those in 2010.
Although the bias and RMSE of the R-FRK predictions are a
bit larger than those of the bias-corrected MODIS SSTs they
are smaller than those of the original MODIS SSTs, and the
R-FRK predictions yield spatially coherent SSTs with availabil-
ity up to 100%. Thus, the effect of data-driven bias-correction
model on the accuracy of the R-FRK predictions is obvious.
The effect of the valid MODIS SSTs on the accuracy of the
R-FRK predictions in 2003 is a little different from those in
2010. In 2003, the accuracy of the R-FRK predictions over the
regions with MODIS SSTs is higher than that over the regions
without MODIS SSTs. In 2010, this difference is not so obvious
as in 2003. The inner-annual fluctuation of the error feature
of the R-FRK predictions is consist with that of the original
MODIS SSTs no matter in 2003 or in 2010 and the inner-annual
fluctuation in 2003 is more obvious than that in 2010 and the
consistent feature in 2003 and in 2010 is that the accuracy of the
R-FRK predictions in summer, at around 17th week, is lower
than that of the other week.
In summary, the analysis result in this paper shows that
implementing the data-driven bias correction procedure before
implementing R-FRK model is effective and the R-FRK model
is superior to the FRK model due to the improved basis function
selection. It is suitable for filling gaps of massive satellite data
over irregular regions. This method is easy to be applied into
the spatiotemporal data with both spatial and temporal auto-
correlation.
When the drifting buoy SST observations are compared with
the MODIS SSTs for bias-correction and validation of MODIS
SSTs, It would be better to take account of the difference
between the skin and subskin temperature, although we chose
the night MODIS SST for analysis in this paper to minimize the
skin and subskin difference. The difference between the skin

Fig. 9. Scatter plots of the MODIS SSTs, R-FRK predictions against aggregated drifting buoy SSTs for selected weeks in 2003 and 2010. (a) Scatter plot of the
original MODIS SSTs in 2003. (b) Scatter plot of the bias-corrected MODIS SSTs in 2003. (c) Scatter plot of the R-FRK predictions in 2003. (d) Scatter plot
of the original MODIS SSTs in 2010. (e) Scatter plot of the bias-corrected MODIS SSTs in 2010. (f) Scatter plot of the R-RFK predictions in 2010. (g) Scatter
plot of the R-FRK predictions with MODIS SSTs in 2003. (h) Scatter plot of the R-FRK predictions without MODIS SSTs in 2003. (i) Scatter plot of the R-FRK
predictions with MODIS SSTs in 2010. (j) Scatter plot of the R-FRK predictions without MODIS SSTs in 2010.

TABLE VI
SUMMARY OF THE VALIDATION OF THE ORIGINAL MODIS SSTS, THE BIAS-CORRECTED
MODIS SSTS, AND THE R-FRK PREDICTIONS FOR SELECTED WEEKS
Fig. 10. Weekly error feature of the original MODIS SSTs, the bias-corrected MODIS SSTs and the R-FRK predictions in 2003 and 2010. (a) Weekly bias in
2003. (b) Weekly RMSE in 2003. (c) Weekly bias in 2010. (d) Weekly RMSE in 2010.
and subskin temperature depends on the wind speed and the
net heat flux at the sea surface and may have differences larger
than 1 K. However, the difference between skin and subskin
temperatures seems to tend toward a constant value of about
0.2 K for wind speeds larger than 5–7 m/s [7], [13] This
difference can be considered to be 0.17 ± 0.07 K for surface
wind speed values ≥ 6 m/s and also during the night [1], [7],
[13]. While there is some research on correcting the skin effect
of SST measurements [1], [48]–[50], it is still challenging to
take account for the complicating effect of diurnal stratification
in validating satellite SST products. This issue should be ad-
dressed in further study.
ACKNOWLEDGMENT
The authors would like to thank two anonymous reviewers
for their constructive comments.
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Yuxin Zhu received the Ph.D. degree in remote
sensing science and technology from Beijing Normal
University, Beijing, China, in 2013.
She is currently a Postdoctoral Fellow with the
Institute of Geographic Sciences and Natural Re-
sources Research, Chinese Academy of Sciences,
Beijing. She is currently an Associate Professor
with the School of Urban and Environmental Sci-
ences, Huaiyin Normal University, Jiangsu Province,
China. Her research interests focus on spatiotem-
poral statistics method and uncertainty analysis of
remote sensing data.

Emily Lei Kang received the B.S. degree in applied
mathematics from Tianjin University, Tianjin, China,
the B.A. degree in ﬁnance from Nankai University,
Tianjin, in 2004, and the M.S. and Ph.D. degrees in
statistics from The Ohio State University, Columbus,
OH, USA, in 2006 and 2009, respectively.
From 2009 to 2011, she was a Postdoctoral Fel-
low with the Statistical and Applied Mathematical
Sciences Institute and North Carolina State Univer-
sity, Raleigh, NC, USA. She is currently an Assis-
tant Professor with the Department of Mathematical
Sciences, University of Cincinnati, Cincinnati, OH, USA.
Yanchen Bo received the B.S. degree in geography
from Lanzhou University, Lanzhou, China, in 1996,
and the M.S. and Ph.D. degrees in GIS and remote
sensing from the Chinese Academy of Sciences,
Beijing, China, in 1999 and 2002, respectively.
He is currently a Professor of remote sensing with
the School of Geography, Beijing Normal University,
Beijing, China, and the State Key Lab of Remote
Sensing Science that is jointly sponsored by the
Beijing Normal University and the Chinese
Academy of Sciences, Beijing. His research interests
focus on the scale effect and scaling method in remote sensing data analysis,
remote sensing data products validation and uncertainty analysis, and the
spatiotemporal merging of the multisource remote sensing products.
Qingxin Tang is currently working toward the Ph.D.
degree in the School of Geography, Beijing Normal
University, Beijing, China.
His current research interests mainly include the
study of satellite remote sensing of aerosols, data
fusion from multiple satellites, and the uncertainty
of these data sets.
Jiehai Cheng received the Ph.D. degree in cartogra-
phy and geography information system from Beijing
Normal University, Beijing, China, in 2013.
He is currently an Associate Professor with
Henan Polytechnic University, Henan, China. His re-
search interests include spatial data analysis, object-
based image analysis, and remote sensing data
quality.
Yaqian He is currently working toward the Ph.D. de-
gree in the Department of Geology and Geography,
West Virginia University, Morgantown, WV, USA.
Her research interests are land–atmosphere in-
teraction. She focuses on using remote sensing and
statistical methods to investigate how land use and
land cover change inﬂuence regional climate (e.g.,
Monsoon climate in West Africa and East Asian).

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