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Restoration of the missing pixel
information caused by contrails in
multispectral remotely sensed
imagery
Daxiang Zhang
Chuanrong Zhang
Weidong Li
Robert Cromley
Dean Hanink
Daniel Civco
David Travis
Restoration of the missing pixel information caused by
contrails in multispectral remotely sensed imagery
Daxiang Zhang,aChuanrong Zhang,aWeidong Li,aRobert Cromley,a
Dean Hanink,aDaniel Civco,band David Travisc
aUniversity of Connecticut, Center for Environmental Sciences and Engineering, Department
of Geography, Storrs, Connecticut 06269
chuanrong.zhang@uconn.edu
bUniversity of Connecticut, Department of Natural Resources and the Environment, Storrs,
Connecticut 06269
cUniversity of Wisconsin-Whitewater, Department of Geography and Geology, Whitewater,
Wisconsin 53190
Abstract. Although removing the pixels covered by contrails and their shadows and restoring
the missing information at the locations in remotely sensed imagery are important to understand
contrails’effects on climate change, there are no such studies in the current literature. This study
investigates the restoration of the missing information of the pixels caused by contrails in multi-
spectral remotely sensed Landsat 5 TM imagery using a cokriging approach. Interpolation
results and several validation methods show that it is practical to use the cokriging approach to
restore the contrail-covered pixels in the multispectral remotely sensed imagery. Compared to
ordinary kriging, the results are improved by taking advantage of both the spatial information in
the original imagery and information from the secondary imagery. ©2014 Society of Photo-Optical
Instrumentation Engineers (SPIE) [DOI: 10.1117/1.JRS.8.083698]
Keywords: restoration; contrail; remotely sensed images; geostatistics; cokriging.
Paper 13030 received Jan. 30, 2013; revised manuscript received Nov. 29, 2013; accepted for
publication Dec. 4, 2013; published online Jan. 6, 2014.
1 Introduction
Aircraft-generated contrails are the visible trails of condensed water vapor made by the exhaust
of aircraft engines in sufficiently cold air. The aircraft-generated contrails and their resulting
cirrus clouds have the positive (warming) component from the contrails’greenhouse effect
[longwave contrail radiative forcing (CRF)] and the negative (cooling) component from the
backscattering of solar irradiance (shortwave CRF). The net radiative forcing, that is, the differ-
ence between radiant energy received by the earth and energy radiated back to space, depends on
the longwave/shortwave cancellation. The aircraft-generated contrails and their resulting cirrus
clouds may exacerbate climate warming in subregions characterized by frequent commercial
flights and may have a potential influence on climate change.1Similar to that of an aircraft’s
carbon dioxide (CO2) emissions, the aircraft-generated contrails contribute to the greenhouse
effect by trapping heat in the Earth’s atmosphere. In addition, a significant amount of high-
level thin cloudiness can be added over high-traffic areas because of the aircraft-generated con-
trails,2and the addition of high thin cloudiness may trap energy in the atmosphere and lead to
warmer surface temperatures.3According to the results of a recent study conducted by Burkhardt
and Karcher4of the Institute for Atmospheric Physics at the German Aerospace Centre, aircraft-
generated contrails are contributing more to global warming than all the CO2that has been
caused by the entire 108 years of airplane flight.
Further, aircraft-generated contrails occurring in otherwise mostly clear skies can signifi-
cantly modify radiation and energy budgets, and help to reduce the diurnal temperature
range (DTR) at the Earth’s surface. The results of studies on the effects of contrails on climate
0091-3286/2014/$25.00 © 2014 SPIE
Journal of Applied Remote Sensing 083698-1 Vol. 8, 2014
indicated that the difference between day and night temperatures was about 1°C higher in the
absence of contrails after the September 11, 2001, event, when all aircraft in the United States
were grounded for 3 days.5The influence of aircraft-generated contrails on climate has been
explored by climate scientists.6–8Both airline companies and national governments have
expressed a desire to reduce aviation impacts on the environment and climate, including those
related to aircraft-generated contrails.
However, studying aircraft-generated contrails has always been difficult because they are at a
high altitude and of short duration, sometimes only minutes. Recently, with the development of
space technologies, remotely sensed satellite imageries have become good data sources for
studying these spreading aircraft-generated contrails. For example, the Advanced Very High
Resolution Radiometer (AVHRR) satellite images and the Moderate Resolution Imaging
Spectroradiometer (MODIS) satellite images have been employed to study the effects of the
aircraft-generated contrails on climate.9,10 To estimate contrail impacts on global climate, radi-
ative forcing due to linear-shaped jet contrails was calculated over the Northern Hemisphere for
four seasonal months using 2006 Aqua MODIS satellite images in a radiative transfer model by
Spangenberg et al.11
Simulated removal of contrails from remotely sensed satellite images may further help study-
ing the effects of the aircraft-generated contrails on climate change because such removal will
permit estimation of the change in albedo or radiation due to their presence (versus absence) over
a particular area, as has been shown in similar cloud and aerosol radiative forcing studies.12 This
information can be used in physical or climate model analysis of the impacts on radiation and
energy budgets to verify the differences in temperature or DTRs identified in weather station
observations.
Although removing the contrails and their shadows in the remotely sensed images and restor-
ing the missing information are important to understanding their effects on climate change, to the
best of our knowledge there are no such studies in the current literature. Most contrail studies in
the literature are mainly focused on detection of the contrails.13–16 For example, Zhang et al.17
recently introduced an object-based method for automatically detecting contrails in AVHRR
satellite images by utilizing not only the spectral characteristics but also the spatial information
in the images.
Although in the atmospheric literature, CRF or contrail optical properties, were estimated
from the satellite imagery to study contrail impacts on global climate,11,18–20 the models pro-
duced a wide range of CRF values. The most recent assessment of aircraft climate effects has
an uncertainty of >100% in the net CRF of linear contrails in terms of the global mean, because
the actual contrail properties and their background radiation fields are poorly known.11
Simulated removal of contrails from remotely sensed satellite images may help in understanding
the background radiation fields and reduce uncertainties of the estimated CRF, thus improving
the studies on the effects of aircraft-generated contrails on climate change.
Compared to Zhang et al.,17 this paper goes a further step: we not only identify the aircraft-
generated contrails in the multispectral Landsat 5 TM imagery but also restore the missing pixel
information [i.e., the expected digital number (DN) values of pixels without contrail contami-
nation] in the imagery based on spatial information in the original image and a secondary image.
We anticipate that with both the original satellite imagery with contrails and the simulated
imagery without contrails climate scientists will be able to better study the atmospheric effects
of contrails for potential mitigation of their climatic impacts. Note that we do not study the CRF
or contrail optical properties from the satellite imagery in this research. This study is only
focused on estimating the missing pixel information in the areas covered by contrails in satellite
imagery. It is hoped that the processed contrail-free imagery could be useful for studying CRF or
contrail optical properties by atmospheric scientists or climatologists, as has been done in a
similar cloud and aerosol radiative forcing study by Allan.21
We use a cokriging approach to restore contrail pixels in the multispectral remotely sensed
Landsat 5 TM imagery in this study. In fact, similar to the restoration of the missing information
in clouded multispectral remotely sensed imagery, many different methods may be used to re-
cover the information of contrail pixels in the multispectral remotely sensed imagery. For exam-
ple, the simple replacement method or histogram matching,22,23 regression trees, and image
fusion techniques23–27 may also be used for the restoration of contrail pixels in the multispectral
Zhang et al.: Restoration of the missing pixel information caused by contrails. . .
Journal of Applied Remote Sensing 083698-2 Vol. 8, 2014
remotely sensed imagery. However, these methods do not take full advantage of the spatial infor-
mation in the contrail imagery. Spatial independence rarely occurs in image scenes, rather there
is spatial dependence in images that is reflected in the variability of DN values.28,29 Adjacent
pixels tend to be spatially autocorrelated and it is expected that two adjacent pixels will generally
be more similar in their DN values than would be two pixels separated by a greater distance.30
Without taking full consideration of the spatially autocorrelated information in the contrail
imagery, the aforementioned methods may yield poor quality results if the contrail imagery
and the secondary imagery do not satisfy certain criteria such as a minimum date separation
and a low temporal variability.
By taking full consideration of the spatially autocorrelated information in the contrail
imagery and incorporating information from secondary imagery without contrail coverage at
the same locations, a cokriging approach may restore well the contrail pixels in the multispectral
remotely sensed Landsat 5 TM imagery. The cokriging approach was found to be a relatively
more effective method for restoring the missing information caused by clouds.31 In this study,
this approach is investigated for restoring the missing information in contrail-covered multispec-
tral remotely sensed imagery. Such a study represents a new application area for cokriging. The
method should be especially helpful for “restoring”contrail pixels because of the linear char-
acteristic of contrail shapes. In this study, we apply the cokriging method to the relatively weak
correlation situations for bands 2 and 3 to assess whether the method works well or not for
restoration of contrail pixels. Specifically, we (1) identify the aircraft-generated contrails in
the multispectral remotely sensed Landsat 5 TM imagery using the object-based method pro-
posed in Zhang et al.;17 (2) model the spatial dependence between the original contrail imagery
and the secondary imagery using a linear model of coregionalization; (3) interpolate the missing
information in contrail-covered areas in the original imagery using cokriging; (4) compare the
cokriging method with the ordinary kriging to identify the advantages of the cokriging method;
and (5) verify the restoration accuracy of the cokriging method using a manually cutting-off
linear-shaped polygon as a simulated contrail.
2 Methods
2.1 Object-Based Contrail Detection
The object-based image method for the identification of contrails in remotely sensed imagery17
includes two steps: first, the whole image is segmented into meaningful pixel groups or image
objects; second, knowledge-based classification is used to define these image objects into
desired contrail or noncontrail classes. In using these two steps, both the spectral information
and spatial information are considered.
In the first step, the multiresolution segmentation extracts image objects at different hierar-
chical segmentation levels. The objects are grouped into a larger object based on a single param-
eter called heterogeneity, which is calculated based on the spectral similarity, level of contrast
with the neighboring objects, and shape characteristics of the resulting object.
After segmentation, a nearest neighbor classification method is used to classify the primary
image into two classes—contrail and noncontrail objects. A set of training samples are used to
assign contrail and noncontrail membership values. The classification results are tested based on
the training image object samples. The whole image object domain is then classified based on the
nearest sample neighbors.
The nearest neighbor classification returns a contrail or noncontrail fuzzy membership value to
each of the segmented image objects based on its feature space distance to its nearest neighbors.
The fuzzy logic values of the feature space distance are calculated based on the spectral and shape
features. The brightness, the shape index, and the width are applied in this case study to compute
the fuzzy logic values. Because shape features and color features may not be represented on a
common scale and may have different value ranges, the fuzzy logic values are standardized
by dividing the standard deviation of both the shape and color feature values. Thus, the final
obtained fuzzy logic values range from 0 to 1.0. A value of 0 means the object does not belong
to the contrail class, while a value of 1.0 means the object most likely belongs to the contrail class.
Zhang et al.: Restoration of the missing pixel information caused by contrails. . .
Journal of Applied Remote Sensing 083698-3 Vol. 8, 2014
The Definiens Developer 8.0 software program was used to perform the segmentation and
object-based image analysis.
2.2 Cokriging
Cokriging is the extension of simple kriging to incorporate data from the related variables to
increase the accuracy of estimates by taking simultaneously into account the autocorrelation in
each variable and the cross correlation between the variables.32 There are different types of cok-
riging methods. The distinctions arise from the way in which constraints are imposed. Three
common types of cokriging are simple cokriging, traditional ordinary cokriging, and standard-
ized ordinary cokriging.33–35 Simple cokriging assumes that the trend component in kriging sys-
tem has a constant and known mean, which normally is not the case in reality. Thus, simple
cokriging usually generates poorer results than found by the traditional ordinary cokriging
and standardized ordinary cokriging. In this study, we use the traditional ordinary cokriging
for restoration of contrail pixels in the multispectral remotely sensed imagery. It was selected
because it generates results similar to those found by the standardized ordinary cokriging in
restoration of clouded pixels in the multispectral remotely sensed imagery,31 but is more widely
applied.
The linear model of coregionalization is the input parameter for cokriging. The linear model
of coregionalization is a set of direct and cross-variogram models that describes spatial depend-
ence of two or more sets of variables and their interdependences simultaneously. The direct
variogram models represent the spatial dependence in individual regionalized variables and
the cross-variogram models express the interdependence or joint spatial dependence between
the regionalized variables. A direct variogram is calculated by
γðhÞ¼ 1
2NðhÞX
NðhÞ
α¼1
½zðxαÞ−zðxαþhÞ2;(1)
where zðxαÞ;is the DN values of the pixels and NðhÞis the number of pairs of pixels for lag h.As
γðhÞincreases, the pixels become less similar. Besides the individual variograms computing the
spatial dependence within the images, the joint spatial dependence is calculated using a cross-
variogram:
γðhÞ¼ 1
2NðhÞX
NðhÞ
α¼1
½ziðxαÞ−ziðxαþhÞ½zjðxαÞ−zjðxαþhÞ;(2)
where ziðxαÞand zjðxαÞare the DN values of the pixels in two random variables Z1ðxÞ
and Z2ðxÞ.
In order to use the variogram and cross-variogram in cokriging, a linear model of coregion-
alization needs to be constructed. The linear model of coregionalization is defined as a set of
direct and cross-variogram models γijðhÞ, such that
γijðhÞ¼X
L
l¼o
bl
ijglðhÞ;(3)
where γijðhÞrepresents the variogram models between the two variables, lrepresents each of the
Lvariables, and glðhÞis a permissible variogram model.
In the cases where the two variables exist, three inequalities must be satisfied to ensure that
the coregionalization matrix Blis positive semidefinite:
bl
11 ≥0;(4)
bl
22 ≥0;(5)
bl
11bl
22 −bl
12bl
12 ≥0:(6)
Zhang et al.: Restoration of the missing pixel information caused by contrails. . .
Journal of Applied Remote Sensing 083698-4 Vol. 8, 2014
It should be noted that in ArcGIS, the cross-variograms used in cokriging are expressed by
cross-covariance functions (called cross-covariograms) through the quantitative relationship
γijðhÞ¼Cij ð0Þ−½CijðhÞþCijð−hÞ∕2, where hand −hrepresent the opposite directions
and are the same for the isotropic case.34
Cokriging is a multivariate image fusing technique. It integrates the information carried out
by a secondary variable related to the primary variable being estimated. Nonexhaustive secon-
dary information can be incorporated using the cokriging approach that explicitly accounts for
the spatial cross correlation between primary and secondary variables,34 thus the interpolation
accuracy could be improved.
For a single secondary variable case, the ordinary cokriging estimate is a linear combination
of both the primary and secondary data values and is given by
Z
1ðuÞ¼X
n1ðuÞ
α1¼1
λα1ðuÞZ1ðuα1ÞþX
n2ðuÞ
α2¼1
λα2ðuÞZ2ðuα2Þ(7)
with the following two constraints making the estimator unbiased
X
n1ðuÞ
α1¼1
λα1ðuÞ¼1;(8)
X
n2ðuÞ
α2¼1
λα2ðuÞ¼0;(9)
where n1ðuÞis the known DN value in the primary image Z1,n2ðuÞis the known DN value in the
secondary image Z2,λα1ðuÞare the weights assigned to the primary image Z1, and λα2ðuÞare the
weights applied to the secondary image Z2.
The cokriging system also aims at minimizing the variance of the errors by solving the fol-
lowing cokriging equations:
8
>
>
>
<
>
>
>
:
Pn1ðμÞ
β1¼1λβ1ðuÞC11ðuα1−uβ1ÞþPn2ðμÞ
β2¼1λβ2ðuÞC12ðuα1−uβ2Þþμ1ðuÞ¼C11 ðuα1−uÞ;α1¼1;2;:::n1ðuÞ
Pn1ðμÞ
β1¼1λβ1ðuÞC21ðuα2−uβ1ÞþPn2ðμÞ
β2¼1λβ2ðuÞC22ðuα2−uβ2Þþμ2ðuÞ¼C21 ðuα1−uÞ;α2¼1;2;:::n1ðuÞ
Pn1ðμÞ
β1¼1λβ1ðuÞ¼1&Pn2ðμÞ
β2¼1λβ2ðuÞ¼0
(10)
where C11 and C22 are the autocovariances of the primary and secondary images, C12 and C21 are
the cross-covariances between the primary and secondary images, and μ1ðuÞand μ2ðuÞare the
two Lagrange parameters accounting for the two constraints that make the estimator unbiased.
2.3 Verification Methods
In order to assess the accuracy of the restoration and confirm that it is feasible and reliable to use
the cokriging method to restore contrail pixels in Landsat 5 TM imagery, a narrow and long
polygon is cut off manually from the original image and the polygon is assumed to represent
a contrail in the primary image. Since the same linear coregionalization model is used for cok-
riging parameters, the verification can be performed on this polygon by comparing the restored
DN values with their original DN values. The root-mean-square-error (RMSE), standardized
RMSE, as well as quantile–quantile (Q─Q) plots are employed to measure the overall accuracy
of the restored pixels and error distribution maps are used to assess the spatial distribution of the
DN value errors of the restored pixels.
Zhang et al.: Restoration of the missing pixel information caused by contrails. . .
Journal of Applied Remote Sensing 083698-5 Vol. 8, 2014
2.3.1 RMSE and standardized RMSE
RMSE is determined by
RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
nX
n
i¼1
ðzi;act −zi;estÞ2
s;(11)
where nis the number of pixels, zi;act is the known value of pixel iin the primary image, and zi;est
is the restored value of pixel i.
The standardized RMSE is used to make band comparisons by normalizing the standard
deviation of the corresponding bands, and is determined by
Standardized RMS E¼RMSE
s¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
nX
n
i¼1
ðzi;act −zi;estÞ2
s2
s;(12)
where s2is the variance and sis the standard deviation of the corresponding band.
2.3.2 Quantile–quantile plot
A general Q─Qplot is employed to check whether the original imagery and the restored imagery
have the same distribution. Quantiles of the restored DN values of the test polygon are plotted
against quantiles of the original DN values of the test polygon. If the original imagery and the
restored imagery have the same distribution, then the points of the general Q─Qplot should fall
along a 45 deg reference line. The departure from the reference line means that the original
imagery and the restored imagery have a dissimilar distribution. A Q─Qplot can provide insight
into the nature of the difference in many distributional aspects, such as shifts in location, shifts in
scale, changes in symmetry, and the presence of outliers.31
2.3.3 Error distribution maps
Although RMSE, the standardized RMSE, and Q─Qplots can help to evaluate the overall per-
formance of cokriging interpolation, these global statistics cannot spatially assess the effective-
ness of cokriging as the performance of cokriging may be location-dependent. The
corresponding error (true DN value minus restored DN value) distribution maps for different
bands for the test polygon area are used in this study to spatially assess cokriging’s effectiveness.
The error distribution maps can demonstrate the spatial distribution of interpolation errors.
3 Background and Data
Two Landsat 5 TM images are chosen to restore contrail-covered pixels in the multispectral
remotely sensed imagery using the cokriging method. The images were downloaded from
the United States Geological Survey (USGS) website (http://earthexplorer.usgs.gov/). They
are level 1 systematically corrected products, which provide systematic radiometric and geomet-
ric accuracies. The study area is located in the eastern part of the state of Connecticut. Figure 1
shows the relative location of the study area. Figure 2(a) shows the primary image used in this
study, which is a subset of a Landsat 5 TM image from Path12–Row31 (World Reference System
2) collected on November 20, 2005. It has 277 rows and 253 columns. It can be seen from
Fig. 2(a) that this area has a complex heterogeneous landscape, and different types of land covers
are mingled and scattered in the primary image. There are obvious spatial variations. Obvious
contrails and contrail shadows can be seen in the image. Figure 2(b) shows the secondary
Landsat 5 TM image, which covers the same area but was acquired on February 8, 2006.
Because of the different periods within a year, we can see that the DN values between the
original image [Fig. 2(a)] and the secondary image [Fig. 2(b)] are quite different. The correlation
coefficients between the primary image and the secondary image for two different TM bands—
band 2 and band 3—are 0.572 and 0.611, respectively. From these values, we can see that the
Zhang et al.: Restoration of the missing pixel information caused by contrails. . .
Journal of Applied Remote Sensing 083698-6 Vol. 8, 2014
correlation coefficient values for band 2 and band 3 are low, which implies that there is a weak/
moderate cross-linear correlation between the original image and the secondary image for both
of these two individual bands. For bands with a strong cross correlation between the original
image and the secondary image, many image fusion methods, such as simple replacement, histo-
gram matching, or regression tree methods could work well for restoration of contrail pixels in
the multispectral remotely sensed imagery. In this study, wewill use the cokriging method for the
Fig. 1 Relative location of the study area (eastern Connecticut).
Fig. 2 Images used for cokriging interpolation and analysis. (a) Original image with contrails and
contrail shadows (an RGB composite image of bands 4, 3, and 2). (b) Secondary image for use in
cokriging (an RGB composite image of bands 4, 3, and 2). (c) Original band 2 image. (d) Original
band 3 image.
Zhang et al.: Restoration of the missing pixel information caused by contrails. . .
Journal of Applied Remote Sensing 083698-7 Vol. 8, 2014
relatively weak/moderate cross correlation situations for bands 2 and 3 to assess whether the
method works well or not for restoration of contrail pixels. Figures 2(c) and 2(d) show the
original images of band 2 and band 3. The differences of mean DN values between the original
image and the secondary image for band 2 and band 3 are −2.60 and −4.50, respectively. It can
be seen that the mean DN values for the secondary image are all lower than those for the primary
image. The mean values of these bands are not very close, which suggests that other interpolation
methods, such as maximum value composite, minimum value composite, and simple replace-
ment, may yield poor results because of the large difference between the DN values of these
individual bands. Therefore, we use the cokriging method, which considers spatial correlation
structure in the original image, to restore contrail pixels in the multispectral remotely sensed
imagery.
4 Results
4.1 Object-Based Classification Results
The individual band images are divided into various objects using the multiresolution segmen-
tation method. In the image segmentation step, several parameters are established. The first
parameter is scale. A segmentation scale value of 15 is chosen through visual inspection
and this scale makes the segmentation results depict contrails reasonably well, which means
that the segmentation results avoid both having too many image objects and being too coarse
for the detection of the contrail pixels.
Both the spectral and spatial heterogeneity criteria are applied in the segmentation process to
avoid generating the branched and fractured objects. The three parameters that represent the
spectral and spatial heterogeneity criteria are color, smoothness, and compactness. The latter
two parameters are shape criteria. A color parameter of 0.9 is set for all the scenarios to empha-
size the brightness information. The shape parameter is set to 0.1. Compactness and smoothness
each comprise 50% of the shape criterion because a certain degree of shape homogeneity often
can improve the quality of object extraction.
Evenly distributed contrail and noncontrail samples are selected for the object-based clas-
sification. The brightness, shape index, and width are used to optimize the three-dimensional
feature space and to reduce the computational requirements. Figure 3shows the object-based
classification results for band 2 and band 3. It can be seen from this figure that the linear shape of
the contrails is identified and the linear contrail features are captured reasonably well.
Fig. 3 Object-based classification results of contrail pixels. (a) Classification results for band 2
(white indicates contrail pixels and black indicates noncontrail pixels). (b) Classification results
for band 3 (white indicates contrail pixels and black indicates noncontrail pixels).
Zhang et al.: Restoration of the missing pixel information caused by contrails. . .
Journal of Applied Remote Sensing 083698-8 Vol. 8, 2014
4.2 Modeling the Spatial Dependence Between the Primary and the Secondary
Images
The construction of the variogram and cross-covariogram models is mainly based on the ana-
lyst’s judgment and decision. The choice of the lag tolerance width determines the number of
those lags in any transect. 36 We chose 30 m as an appropriate lag tolerance width for this case
study because the variograms and cross-covariograms estimated with this lag tolerance are reli-
able. This distance is consistent with the spatial resolution of the Landsat 5 TM data and it is a
proper distance to capture the detailed spatial information within the data set. In order to model
the spatial dependence between the primary image and the secondary image, we fit models of
both direct variograms and cross-covariograms simultaneously to satisfy the positive definite
constraint in solving the cokriging system. To fit a linear model of coregionalization for
each band, all models (two direct variogram models and one cross-covariogram model for
each band) must have the same common model structure and range. However, the two direct
variogram models and one cross-covariogram model for each band may have different partial
sills and nuggets. In this study, each fitted variogram or cross-covariogram model is composed of
two structures: a nugget structure and a spherical structure
YðhÞ¼c0þc13h
2a1
−1
2h
a13;(13)
where c0is the nugget, c1is the sill of the spherical structure, and a1is the range parameter of the
spherical structure. Here, sill is defined as the total magnitude of spatial variability. Range is the
distance over which data are correlated. Nugget represents the extent of random variation within
the data. Usually, variance at the distance of 0 is theoretically equal to 0. However, measurement
errors and small-scale variability may often cause DN values within a small separation to be quite
dissimilar. Thus, a discontinuity at the origin of a variogram may occur and this phenomenon is
referred to as the nugget effect. The nugget effect appears in the variogram model in this case
study. Table 1shows the parameters of the linear model of coregionalization. The permissible
coregionalization model (including models for direct variogram of contrail image, direct vario-
gram of secondary image, and cross-covariogram) for band 2 uses the spherical basic functions
as the common basic structure and 268.5 m as the common range. The permissible coregion-
alization model for band 3 also uses the spherical basic functions as the common basic structure
and 292.2 m as the common range. Figure 4illustrates the direct variogram models of the origi-
nal and secondary images and the cross-covariogram models between the original and secondary
images for bands 2 and 3.
4.3 Estimating DN Values of Contrail Pixels Using Cokriging
As noted above, individual band 2 and band 3 are used to explore the cokriging method for
restoration of the contrail pixels under a weak correlation circumstance. Ordinary kriging is
also used to interpolate the contrail pixels in the primary image in order to test whether the
Table 1 Parameters of the fitted variogram and cross-covariogram models for use in cokriging
(optical band 2 and band 3).
Band Variogram/covariogram Model type Nugget Sill Range
Band 2 Direct variogram of contrail image Spherical 0.69 2.47 268.46
Direct variogram of secondary image Spherical 0.63 5.42 268.46
Cross-covariogram Spherical 0.0 2.91 268.46
Band 3 Direct variogram of contrail image Spherical 0.47 5.28 292.23
Direct variogram of secondary image Spherical 2.15 14.67 292.23
Cross-covariogram Spherical 0.0 6.86 292.23
Zhang et al.: Restoration of the missing pixel information caused by contrails. . .
Journal of Applied Remote Sensing 083698-9 Vol. 8, 2014
introduction of the secondary variable improves the restoration results in this contrail case study.
Figure 5provides a comparison of the results of the cokriging and ordinary kriging methods.
Figures 5(a) and 5(d) show the original images for band 2 and band 3, respectively. Figures 5(b)
and 5(e) show the restored results using the cokriging for band 2 and band 3, respectively.
Through visual inspection, it can be seen that by using cokriging, the contrail effect has
been reduced substantially and most traces of the contrail have been removed. Although
there are still some smoothing effects, the restoration results appear to be good and most
parts of the edges of the contrail are matched spatially. Overall, the smoothing effect is not
significant and the detailed spatial variation can still be found in the restored contrail area.
Figures 5(c) and 5(f) show the ordinary kriging restoration results for individual band 2 and
band 3. Compared with Figs. 5(b) and 5(e), it can be seen that there is an obvious smoothing
effect in the area restored by ordinary kriging and the basic spatial structure cannot be captured
using only the information in the primary image. The restored contrail pixels do not match the
surrounding noncontrail pixels spatially. There are obvious interpolation-related artifacts and
the edges of the contrails can be identified easily. This occurs due to the fact that the shapes
of the contrail in the original image are wide and the surrounding noncontrail pixels are too far
Fig. 4 The direct variogram models of the primary and secondary images, and the cross-covario-
gram models between the primary and secondary images for band 2 and band 3. (a) The direct
variogram model of the primary image for band 2. (b) The direct variogram model of the secondary
image for band 2. (c) The cross-covariogram model between the primary and secondary images
for band 2. (d) The direct variogram model of the primary image for band 3. (e) The direct vario-
gram model of the secondary image for band 3. (f) The cross-covariogram model between the
primary and secondary images for band 3.
Zhang et al.: Restoration of the missing pixel information caused by contrails. . .
Journal of Applied Remote Sensing 083698-10 Vol. 8, 2014
away, thus they may have dissimilar DN values to the original pixels. When performing the
restoration, only using the primary image information cannot capture the details of the spatial
structure of the contrail pixels well. This verifies that by incorporating the information from the
secondary image, cokriging can produce better restoration results than the ordinary kriging.
4.4 Validation Results
Figure 6shows the location of the test polygon area that is cut off manually for validation. The
test polygon is used to simulate a contrail with a linear irregular shape. The yellow line surrounds
the test polygon. Figure 7shows the zoomed-in original and restored polygon areas. It can be
Fig. 5 Comparison of cokriging and ordinary kriging for restoring contrail-covered pixels.
(a) Original image for band 2. (b) Cokriging results for band 2. (c) Ordinary kriging results for
band 2. (d) Original image for band 3. (e) Cokriging results for band 3. (f) Ordinary kriging results
for band 3.
Fig. 6 The location of the test polygon for validation.
Zhang et al.: Restoration of the missing pixel information caused by contrails. . .
Journal of Applied Remote Sensing 083698-11 Vol. 8, 2014
seen that there are only slight differences between the original image pixels and the restored
contrail pixels. The spatial pattern of these pixels is well preserved as is the spatial continuity
of the majority of the scene in the restored pixel region. Therefore, the performance of the cok-
riging model is shown to be reliable for restoration of the contrail pixels in the multispectral
remotely sensed imagery.
The RMSE for band 2 and band 3 is 1.32 and 1.96, and the standardized RMSE for band 2
and band 3 is 0.24 and 0.09, respectively. These global statistics values indicate that overall the
cokriging performs well.
Figure 8shows the Q─Qplots of the estimated DN values versus the true DN values in the
test polygon for band 2 and band 3. From the Q─Qplots, it can be seen that most points are close
to the 45 deg line, which means that the restored pixel DN values are well correlated with the true
pixel DN values. The general Q─Qplots indicate that the restored pixel DN values in the test
polygon are similar to the corresponding true pixel DN values. However, there are still some
outliers. At the upper tails of the Q─Qplots, large deviations appear in both bands and most
points are below the 45 deg reference line, which indicates that some pixels with extremely high
values are underestimated. While it is still difficult to restore the contrail pixels with extreme DN
values, the Q─Qplots show the feasibility of using cokriging to estimate the contrail values in
the remotely sensed imagery.
Figure 9shows the error distribution maps for band 2 and band 3. The error distribution map
is useful in analyzing the reliability of the DN value of each pixel in the test polygon. The yellow
color denotes the accurately estimated pixels, whereas the red color indicates positive errors, and
the green color indicates negative errors, respectively. For band 2, the error ranges from −2.37 to
þ9.13 (unit: DN value), while for band 3, the error ranges from −3.4 to þ18.2 (unit: DN value).
From the error distribution maps, it can be seen that most pixels have small deviations from the
original pixel DN values by showing yellow, light red, and light green colors. Only a small
Fig. 7 Zoomed-in images for the purpose of validation. (a) Zoomed-in original image (bands 4, 3, 2
composite). (b) Zoomed-in restored images (bands 4, 3, 2 composite).
Fig. 8 Q─Q plots of the distributions between the true DN values and the estimated DN values in
the test polygon (assumed ing as a contrail) for band 2 and band 3.
Zhang et al.: Restoration of the missing pixel information caused by contrails. . .
Journal of Applied Remote Sensing 083698-12 Vol. 8, 2014
percentage of pixels have large deviations with a deep red color. Hence, the error distribution
maps also show that it is feasible to restore the contrail cells using the cokriging method.
Spatial autocorrelation can also be seen in the error distribution maps. Red cells and green
cells are grouped separately because the positive error cells tend to group together, while the
negative error cells have the same trend. Generally, if the values of a group of cells are quite
different from surrounding cells or if they have extremely high or low values, the errors will
Fig. 9 Spatial error distribution maps for band 2 and band 3 (the true DN values minus the cor-
responding estimated DN values).
Fig. 10 Comparison images before radiometric correction and after radiometric correction,
(a) Original November 2005 image with contrails (bands 4, 3, 2 composite). (b) November
2005 image with contrails image after radiometric correction (bands 4, 3, 2 composite).
(c) Original February 2006 image without contrails (bands 4, 3, 2 composite). (d) February
2006 image without contrails after radiometric correction (bands 4, 3, 2 composite).
Zhang et al.: Restoration of the missing pixel information caused by contrails. . .
Journal of Applied Remote Sensing 083698-13 Vol. 8, 2014
Fig. 11 The object-based classification results of contrail pixels for band 2 and band 3 after radio-
metric correction. (a) Object-classification results for band 2. (b) Object-classification results for
band 3.
Fig. 12 The results of the cokriging restoration based on the radiometric correction images.
(a) Image with radiometric correction for band 2. (b) Cokriging results for band 2. (c) Image
with radiometric correction for band 3. (d) Cokriging results for band 3.
Zhang et al.: Restoration of the missing pixel information caused by contrails. . .
Journal of Applied Remote Sensing 083698-14 Vol. 8, 2014
become large. From the error distribution maps, it can also be noted that there is a small area of
extremely high deviation at the lower left corner. This occurs because in the original image the
corresponding land cover at this place is very different from the surrounding land cover.
Therefore, it is difficult to accurately restore the high-variation pixels based on the DN values
of their neighboring pixels.
5 Discussion and Conclusions
Although studies in the literature indicate that aircraft-generated contrails may have a large
impact on climate, the extent of contrail coverage, and the amount of warming effects still remain
quite uncertain. It is important to identify and restore the contrail-covered areas in the remotely
sensed imagery for contrail effect research and climate change studies. This paper introduces a
cokriging approach for the restoration of contrail pixels in the multispectral remotely sensed
imagery. The case study shows that it is very useful to use the cokriging approach in this resto-
ration. The cokriging method takes advantage of both the spatial information in the original
imagery and information from the secondary imagery.
Initially, we thought that the restoration of contrails would be fairly simple because the
shapes of contrails are relatively narrow and the information in the surrounding pixels within
a short distance may be enough to capture the spatial structure of the contrail pixels. Thus, ordi-
nary kriging might be used for the restoration of contrail pixels. However, the case study con-
ducted in this research shows that this is not the case. Through the comparison of results between
ordinary kriging and cokriging, it can be seen that the performance of the ordinary kriging in
restoring the contrail pixels is poor, which indicates that using only the primary band information
is not enough even for restoration of the linear-shaped contrail pixels. Therefore, to restore more
accurately the contrail pixels in the multispectral images, we have to incorporate the secondary
image information to acquire detailed patterns of contrail pixels in the multispectral remotely
sensed imagery. Although the study area in this application has a complex heterogeneous land-
scape, the cokriging method still produced good results.
Because we cannot obtain the ground survey true data values at the contrail pixel locations,
we manually cut off a contrail-free polygon from the original image and assumed it to be a
contrail. The validation methods demonstrate that cokriging performed well in the restoration
of the “manual”contrail. Most of the restored contrail pixels have similar values to the corre-
sponding values in the original image, which indicates the accuracy of the cokriging method is
relatively good. Some pixels still have high deviations, mainly because the corresponding pixels
in the original image have extremely high or low values.
Although cokriging is reliable in this case study, there are still some limitations to consider. In
this study, we only restored contrail pixels with linear shapes in the multispectral remotely
sensed imagery. However, contrails are not stable spatial features and they may change their
linear shapes from their formation stage into the final dissolution of the aviation-induced cirrus
cloud, making it difficult to identify the contrail-covered pixels and restore them well no matter
what methods are used. For multispectral remotely sensed images, the cokriging method may
work well for restoration of the young contrails with linear shapes, but the method may not work
so well for restoring a cirrus contrail with a spread shape. The cokriging method works well for
linear contrails because it more effectively uses the information from neighboring pixels of the
original imagery—the linear shape is narrow and the values of the neighboring pixels are nor-
mally more similar to the expected values of the contrail-covered pixels in a linear area than in
the other shapes (such as circles). The spreading of contrails not only makes it harder to identify
contrail pixels, but also causes the smoothing effect problem for the cokriging method. The
smoothing effect is caused by the far distances between the central pixels of the contrail
areas (with nonlinear shapes) and the surrounding nearest pixels in noncontrail areas, and
these two groups of pixels tend to have very different DN values. Many interpolation methods
including cokriging have this limitation.
It should be noted that when using the cokriging method, it does not matter whether DN
values or reflectance values are used as long as the variables involved are regionalized variables
which have spatial correlations at some scales. This study was focused on introducing a method
Zhang et al.: Restoration of the missing pixel information caused by contrails. . .
Journal of Applied Remote Sensing 083698-15 Vol. 8, 2014
to restore the DN values of contrail pixels in the multispectral remotely sensed Landsat 5 TM
imagery. However, the DN values recorded by the sensor do not have corrections of object illu-
mination effects (e.g., caused by terrain slope and orientation) and intervening atmospheric
effects (e.g., caused by atmospheric and solar conditions).37 Thus, the DN values for the
same spectral class over different places or at different times over the same area may be different
to some extent. This may decrease the performance of the introduced cokriging method for resto-
ration of the missing information. Converting the DN values into the surface reflectance values
through atmospheric and topographic correction models might improve the restoration results.
To answer the question whether the radiometric correction would lead to more accurate resto-
ration results in this study, we performed the radiometric correction processing on the data set
that we used using the Geomatic 2013 software. Figures 10(a) and 10(b) show the original
November 2005 image with contrails (bands 4, 3, 2 composite) and the image with radiometric
correction (bands 4, 3, 2 composite), respectively. Figures 10(c) and 10(d) display the original
February 2006 image without contrails (bands 4, 3, 2 composite) and the image with radiometric
correction (bands 4, 3, 2 composite), respectively. The correlation coefficients between the cor-
rected and uncorrected November 2005 images for band 2 and band 3 are 0.980 and 0.985, and
those for February 2006 images are 0.988 and 0.992, respectively.
Because the training data and the image to be classified as contrails are at the same relative
scale (corrected or uncorrected), atmospheric and topographic corrections may have little effect
on identifying and removing contrails from the original image.38 We verified this point in this
study. Figure 11 shows the object-based classification results of contrail pixels for band 2 and
band 3 after radiometric correction. Comparing with Fig. 3, it can be seen that there is not much
difference on identifying and removing contrails using the radiometric correction images.
Nevertheless, the restoration results derived from different dates of images using the cokrig-
ing method or other fusion methods may be better with the atmospheric and topographic cor-
rections.39,40 To check if the cokiging method can work better for fusing the images with different
reflectance values, we performed cokriging interpolation for restoration of the missing pixel
information caused by contrails using the images after radiometric correction. Figure 12
shows the results of the cokriging restoration on the images with the radiometric correction
for band 2 and band 3.
Compared with the results on the images without the radiometric correction (Fig. 5), it can be
seen that there is no significant difference for this study. However, for application studies, we still
recommend the conversion of DN values into reflectance values before performing the restoration.
Finally, the cokriging method requires reasonably strong correlations, either direct/autocorrelations
or cross correlations among the primary and secondary images.31,34 With relatively weak corre-
lations (both the direct/autocorrelations in the primary and secondary images and the cross corre-
lations between the primary and secondary images), contrail restoration could become a tough task
even for the cokriging method. Although there are weak cross correlations between the original
images and the secondary images for bands 2 and 3 in the case studies of this research, there are
strong direct/autocorrelations from the neighboring pixels in both of the original images and the
secondary images. This may be why the cokriging method worked well. Whenever possible, one
should choose a secondary image with a short separation date from the original image to perform
the restoration of the missing pixel information caused by contrails. Further exploration for poten-
tially better methods to more effectively restore contrail-covered pixels in the multispectral
remotely sensed imagery should be conducted in the future.
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Daxiang Zhang is a PhD student in Department of Geography at the University of Connecticut.
He got his master’s degree in geography from the University of Connecticut. His research inter-
ests include geostatistics, remote sensing applications to natural environment, land use, and land
cover change study.
Chuanrong Zhang is an associate professor at the University of Connecticut and holds a joint
appointment with the Department of Geography and the Center for Environmental Sciences and
Engineering (CESE). She got her PhD degree in GIS from University of Wisconsin, Milwaukee
in 2004. She has several years of working experience in computer companies. Her current
research interests include Internet GIS, geostatistics, and applications of these techniques in
natural resource and environmental management.
Weidong Li is a research professor in the Department of Geography at University of
Connecticut. He got a master’s degree in computer science from Marquette University and
Zhang et al.: Restoration of the missing pixel information caused by contrails. . .
Journal of Applied Remote Sensing 083698-18 Vol. 8, 2014
obtained a PhD degree in soil and water science in 1995 from China Agricultural University. His
research interest is focused on geostatistics and geoinformatics for natural resource management.
Robert Cromley has a PhD from the Ohio State University and is a professor of geography at
the University of Connecticut. His research interests include different forms of spatial analysis
and geoprocessing procedures.
Dean M. Hanink is a professor of geography at the University of Connecticut, Storrs, CT. His
research interests are in regional economic–environmental change, especially urban land-use
transitions, and the economic geography of China.
Dan Civco is a professor of geomatics in the Department of Natural Resources Management and
Engineering at the University of Connecticut. He is also director of the Center for Land Use
Education and Research (CLEAR). His research interests are in remote sensing applications
to the natural environment, digital image processing algorithm development, neural processing
and expert systems methods in remote sensing image processing, geographic information sys-
tems (GIS), land cover mapping and other earth resources applications of geoprocessing.
David Travis received his PhD in geography from Indiana University in 1994 with an emphasis
on atmospheric science and climatology. His research has primarily focused on human impacts
on climate with an emphasis on aviation impacts. His work on the climatic effects of jet con-
densation trails has received international attention. He was hired onto the faculty at the
University of Wisconsin-Whitewater in 1994. In recent years, he has moved into progressively
higher administrative roles at that institution and most recently was named dean of the College of
Letters & Sciences. He received his bachelor’s and master’s degrees in geography from the
University of Georgia in 1987 and 1989, respectively.
Zhang et al.: Restoration of the missing pixel information caused by contrails. . .
Journal of Applied Remote Sensing 083698-19 Vol. 8, 2014
