Analyzing Remote Sensing Data in R: The landsat Package

Abstract

Research and development on atmospheric and topographic correction methods for multispectral satellite data such as Landsat images has far outpaced the availability of those methods in geographic information systems software. As Landsat and other data become more widely available, demand for these improved correction methods will increase. Open source R statistical software can help bridge the gap between research and implementation. Sophisticated spatial data routines are already available, and the ease of program development in R makes it straightforward to implement new correction algorithms and to assess the results. Collecting radiometric, atmospheric, and topographic correction routines into the landsat package will make them readily available for evaluation for particular applications.

Full text

JSS Journal of Statistical Software
July 2011, Volume 43, Issue 4. http://www.jstatsoft.org/
Analyzing Remote Sensing Data in R: The landsat
Package
Sarah C. Goslee
USDA-ARS Pasture Systems and Watershed Management Research Unit
Abstract
Research and development on atmospheric and topographic correction methods for
multispectral satellite data such as Landsat images has far outpaced the availability of
those methods in geographic information systems software. As Landsat and other data
become more widely available, demand for these improved correction methods will in-
crease. Open source Rstatistical software can help bridge the gap between research and
implementation. Sophisticated spatial data routines are already available, and the ease
of program development in Rmakes it straightforward to implement new correction al-
gorithms and to assess the results. Collecting radiometric, atmospheric, and topographic
correction routines into the landsat package will make them readily available for evaluation
for particular applications.
Keywords: atmospheric correction, Landsat, radiometric correction, R, remote sensing, satel-
lite, topographic correction.
1. Introduction
Satellite remote sensing data can provide a complement or even alternative to ground-based
research for large scale studies or over long periods. The Landsat platform is the premier
example: images have been collected continuously since the early 1970s. If both Landsat 5
and 7 are considered, every location within the United States is imaged every 8 days. Other
satellite platforms have become available more recently. Images from all platforms can be
used for mapping land cover, tracking land use change, and even estimating plant biomass.
Image characteristics vary from date to date. Some of this variation is due to solar elevation
angle and can be easily corrected for given date and time (radiometric calibration). Addi-
tional variation is caused by atmospheric conditions at the time of imaging: scatter at different
wavelengths due to haze. Atmospheric corrections are very complex. If one image every few

2landsat: Analyzing Remote Sensing Data in R
years is analyzed, for example to track development patterns, atmospheric corrections may
be unnecessary because of the magnitude of the change in land use during that interval. If
researchers wish to take advantage of the temporal data density offered by Landsat, careful
correction is required. Otherwise differences in apparent reflectance due to atmospheric vari-
ation can swamp differences caused by actual change on the ground. In mountainous areas,
topographic corrections are also necessary to compensate for differences in incident radiation
due to slope and aspect. Without correction, the same ground cover on opposite sides of the
mountain could return a different signal.
Many procedures to implement atmospheric and topographic corrections have been proposed,
each with strengths and weaknesses, but few are readily available in geographic information
systems (GIS) or image processing software. It is impossible to compare methods to identify
the one most suited for a given application. Indeed, it is not always clear exactly what
proprietary GIS software is doing. Roffers an alternative to GIS software for research and
testing of satellite image processing algorithms. The interpreted nature of Rmakes it possible
to implement, test and modify algorithms easily. The available graphical and statistical tools
are vastly superior to anything available in common GIS packages, making it straightforward
to compare algorithms. Tools for processing and display of spatially-referenced data are
already available in R(RDevelopment Core Team 2011). The sp (Bivand, Pebesma, and
Gomez-Rubio 2008) and rgdal (Keitt, Bivand, Pebesma, and Rowlingson 2011) packages
provide these capabilities for several commonly-used spatial formats.
The landsat package described here, and available from the Comprehensive RArchive Network
at http://CRAN.R-project.org/package=landsat, provides basic tools for working with
satellite imagery such as automated georeferencing and cloud detection. It contains functions
for radiometric normalization, and several different approaches to atmospheric correction.
Four topographic correction algorithms have been implemented. Other useful functions such
as bare soil line and tasseled cap calculations have been included. While these functions were
developed with Landsat data in mind, they are suitable for use with satellite imagery from
other platforms as long as appropriate calibration data are used.
2. Landsat platform characteristics
Landsat 5 (TM) was launched on 1984-03-01 and Landsat 7 (ETM+) on 1999-04-15. Each
revisits a location every 16 days, and the two orbits are staggered so an image is taken for
each location every 8 days. The Landsat Data Continuity Mission is planned to launch in
2011, and will record data in bands compatible with the TM and ETM+ instruments. Due
to a hardware failure on 2003-05-31, Landsat 7 scenes are now missing 22% of the pixels. The
problem is most severe near the edges of a image. Band characteristics are largely consistent,
but ETM+ added an additional band (Table 1).
The Landsat images have been converted to integer digital numbers (DN ) before distribution
to facilitate storage and display. They may require conversion to radiance or reflectance,
topographic correction or atmospheric correction. If using a single image, or images widely
separated in time to examine gross changes, minimal processing may be required. For detailed
comparison of vegetation indices from multiple images, however, careful correction is needed.
The most accurate atmospheric corrections require ground data taken during the satellite
overpass. For retrospective studies this is impossible to obtain, and less-accurate image-based

Journal of Statistical Software 3
Landsat 5
Planned wavelength Actual wavelength Resolution
Band 1 Blue 0.45–0.52 0.452–0.518 30
Band 2 Green 0.52–0.60 0.528–0.609 30
Band 3 Red 0.63–0.69 0.626–0.693 30
Band 4 Near infrared (NIR) 0.76–0.90 0.776–0.904 30
Band 5 Middle infrared (MIR) 1.55–1.75 1.567–1.784 30
Band 6 Thermal infrared (TIR) 10.40–12.50 10.45–12.42 120
Band 7 Middle infrared (SWIR) 2.08–2.35 2.097–2.349 30
Landsat 7
Planned wavelength Actual wavelength Resolution
Band 1 Blue 0.45–0.52 0.452–0.514 30
Band 2 Green 0.52–0.60 0.519–0.601 30
Band 3 Red 0.63–0.69 0.631–0.692 30
Band 4 Near infrared (NIR) 0.77–0.90 0.772–0.898 30
Band 5 Middle infrared (MIR) 1.55–1.75 1.547–1.748 30
Band 6 Thermal infrared (TIR) 10.40–12.50 10.31–12.36 60
Band 7 Middle infrared (SWIR) 2.09–2.35 2.065–2.346 30
Band 8 Panchromatic 0.52–0.90 0.515–0.896 15
Table 1: Band wavelengths (µm) and resolutions (m) for Landsat 5 Thematic Mapper (TM)
and 7 Enhanced Thematic Mapper (ETM+). Band 8 (ETM+ only) is higher-resolution visible
light data. Actual wavelengths are from Chander et al. (2009).
correction methods must be used. The following sections describe the procedures needed for
calibration of Landsat data and the Rimplementations of these algorithms in the landsat
package.
3. Sample data
The landsat package includes a 300 ×300 pixel subset of United States Geological Survey
(USGS) Landsat ETM+ images from two dates, 2002-07-20 and 2002-11-25 (US Geological
Survey 2010b). These images are in SpatialGridDataFrame format, and can be displayed
using image(). Even after clouds are removed, the July image has a higher mean DN and
greater dynamic range than the November image (Figure 1c, d). Complete metadata are
included in the help files. A USGS digital elevation model (DEM) covering the same area and
at the same resolution has been included (Figure 1f; US Geological Survey 2010a).
4. Tools
This package provides a few basic tools for working with Landsat images. The lssub()
function is an Rinterface to the image subsetting tools from the Geospatial Data Abstraction
Library (GDAL, GDAL Development Team 2011). This function is provided for convenience;
while the same effect can be obtained using the subsetting property of sp, it is considerably
faster to use the GDAL functions to remove a smaller section of a geotiff image. Landsat
images are usually distributed as geotiff files, and can be imported in SpatialGridDataFrame

4landsat: Analyzing Remote Sensing Data in R
Figure 1: Band 4 (near infrared) from Landsat images from July and November 2002. The
July image has substantial cloud cover, as identified in the cloud mask produced by clouds().
November 2002 was cloud-free, so no mask is shown.

Journal of Statistical Software 5
format using readGDAL. The lssub() function preserves the geotiff format.
Image processing requires some information not often available in the metadata. The Earth-
Sun distance for a given date can be calculated with ESdist(). A date can be conveniently
converted to decimal format with ddist().
4.1. Automated georeferencing
A simple error-minimization routine can be used to provide relative georeferencing by match-
ing one image to a reference image (georef and geoshift by means of vertical and horizontal
shifts. If the reference image has been absolutely georeferenced, then all of the subsequent
images will also be spatially referenced. This function can find local minima, so results should
always be checked visually. The matching process only needs to be done once for each date;
Band 3 or Band 4 generally provide good results, but any band may be used. The results of
this step can then be used for each band image from that date. Sufficient padding must be
added around the edges of the image to accommodate the magnitude of the shift. The larger
the image area, the more effective the matching process. The below example illustrates the
procedure, but because of the small image sizes used for the demo data the shift coefficients
are unreliable.
R> july.shift <- georef(nov3, july3, maxdist = 50)
R> july1.corr <- geoshift(july1, padx = 10, pady = 10, july.shift$shiftx,
+ july.shift$shifty)
4.2. Topographic calculations
Topographic corrections require the use of slope and aspect data calculated from a DEM of
the same resolution as the satellite data. The slopeasp() function will calculate both given a
DEM. While all GIS software will provide topographic calculations, this function was included
for the convenience of being able to do all the processing within Rand to allow exploration
of different algorithms. Most GIS software offers only a tiny subset of the methods that have
been proposed.
The most common method for slope and aspect calculations is the third-order finite difference
weighted by reciprocal of distance (Unwin 1981;Clarke and Lee 2007). This method is the
equivalent of using a 3 ×3 Sobel filter to determine the east-west slope and the north-south
slope. Given a cell Zi,j with east-west cell size EW res and north-south cell size NSres , slope
in either percent or degrees, and aspect with north as 0◦, east as 90◦and south as 180◦can
be calculated using Equation 1. A simple smoothing correction (dividing by a smoothing
parameter before taking the arctan) can reduce extreme slope values (Riano, Chuvieco, Salas,
and Aguado 2003).
EW = [(Zi+1,j+1 + 2Zi+1,j +Zi+1,j −1)−(Zi−1,j+1 + 2Zi−1,j +Zi−1,j−1)]/(8EW res)
NS = [(Zi+1,j+1 + 2Zi,j +1 +Zi−1,j+1 )−(Zi+1,j−1+ 2Zi,j−1+Zi−1,j−1)]/(8NS res )
θp= arctan pEW 2+NS 2
slope% = 100 ·pEW 2+NS2
φo= 180◦−arctan NS
EW + 90◦EW
|EW |
(1)

6landsat: Analyzing Remote Sensing Data in R
5. Radiometric calibration
Landsat images are distributed as digital numbers, integer values from 0–255. While less
crucial now, this correction was originally necessary to make it possible to store, distribute
and portray these images efficiently. Radiometric calibration is a two-step process. First
the DN values are converted to at-satellite radiance using parameters provided in the image
metadata. Data on solar intensity are used to convert the at-satellite radiance to at-satellite
reflectance. These parameters are also in the image metadata.
5.1. At-sensor radiance
The first step in processing is to convert DN to at-sensor spectral radiance L, also called
top-of-atmosphere radiance. The conversion coefficients are available in the metadata accom-
panying the images. Whenever possible the metadata values should be used, as coefficients
vary by platform and over time, but standard coefficients are given in Table 2.
Coefficients are provided in one of three band-specific formats: gain2and offset;Grescale (also
called gain) and Brescale (bias); or radiances associated with minimum and maximum DN
values (Lmax and Lmin). Any of the three can be used to convert from DN to at-sensor
radiance (Equation 2).
L=DN −offset
gain2
L=Grescale DN +Brescale
L= ( Lmax −Lmin
DN max −DN min
)·(DN −DN min ) + Lmin
(2)
It is not always clear which form of coefficients the metadata contain because “gain” has been
used to refer to both gain2and Grescale. Most recent image metadata provide Grescale and
Brescale , but these formulations are interconvertible (Equation 3). The magnitudes of Grescale
and gain2are similar for most bands, but the former is paired with Brescale , which will have
Landsat 5 Landsat 7 low gain Landsat 7 high gain
Grescale Brescale Grescale Brescale Grescale Brescale
Band 1 0.765827 −2.29 1.180709 −7.38 0.778740 −6.98
0.671339 −2.19 for TM images taken before 1991-12-31
Band 2 1.448189 −4.29 1.209843 −7.61 0.798819 −7.20
1.322205 −4.16 for TM images taken before 1991-12-31
Band 3 1.043976 −2.21 0.942520 −5.94 0.621654 −5.62
Band 4 0.876024 −2.39 0.969291 −6.07 0.639764 −5.74
Band 5 0.120354 −0.49 0.191220 −1.19 0.126220 −1.13
Band 6 0.055376 1.18 0.067087 −0.07 0.037205 3.16
Band 7 0.065551 −0.22 0.066496 −0.42 0.043898 −0.39
Band 8 NA NA 0.975597 −5.68 0.641732 −5.34
Table 2: Default gain (Grescale;W m−2sr−1µm−1DN −1) and bias (Brescale ;W m−2sr−1µm−1)
for Landsat 5 (TM) and Landsat 7 (ETM) from Chander et al. (2009). The metadata will
state whether Landsat 7 images were taken at low gain or high gain.

Journal of Statistical Software 7
Landsat 5 Landsat 7
Band 1 1983 1997
Band 2 1796 1812
Band 3 1536 1533
Band 4 1031 1039
Band 5 220.0 230.8
Band 7 83.44 84.90
Band 8 NA 1362
Table 3: Extra-solar atmospheric constants (Esun ;W m−2µm−1) for Landsat 5 (TM) and 7
(ETM+) from Chander et al. (2009). (Note: The Landsat 5 column contained errors in
previous versions of this manuscript, corrected on 2012-01-08.)
negative values for non-thermal bands, while gain 2is paired with offset , which is positive for
non-thermal bands.
Grescale =Lmax −Lmin
DN max −DN min
Grescale =1
gain2
Brescale =Lmin −Grescale DN min
Brescale =−offset
gain2
gain2=1
Grescale
offset =−Brescale
Grescale
(3)
All radiometric functions in landsat accept either gain and offset or Grescale and Brescale .
5.2. At-sensor reflectance
The at-sensor radiance values calculated using Equation 2must be corrected for solar vari-
ability caused by annual changes in the Earth-Sun distance d, producing unitless at-sensor
(or top-of-atmosphere) reflectance ρAS (Equation 4).
ρAS =πd2L
Esun cos θz
(4)
Esun is the band-specific exoatmospheric solar constant (Table 3;W m−2µm−1). The solar
zenith angle θzcan be derived from the image metadata, where the solar elevation angle θsis
usually included; θz= 90◦−θs. At-sensor reflectance can be calculated using the radiocorr()
function with method = "apparentreflectance".
R> july4.ar <- radiocorr(july4, Grescale = 0.63725, Brescale = -5.1,
+ sunelev = 61.4, edist = ESdist("2002-07-20"), Esun = 1039,
+ method = "apparentreflectance")

8landsat: Analyzing Remote Sensing Data in R
5.3. Thermal bands
Band 6 contains thermal infrared data. Landsat 7 offers two thermal bands, while Land-
sat 5 provides one. Instead of calculating top-of-atmosphere reflectance, these data can be
converted to temperature (K◦) using thermalband(). This function provides default coeffi-
cients, and requires only the DN data and the band number (6 for Landsat 5; 61 or 62 for
Landsat 7).
R> july61.thermal <- thermalband(july61, band = 61)
6. Cloud identification
Clouds are reflective (high) in Band 1 and cold (low) in Band 6, so the ratio of the two
bands is high over clouds (Martinuzzi, Gould, and Gonz´alez 2007). The absolute value of this
ratio must be adjusted for data type, whether reflectance, radiance, or DN . The clouds()
function will create a cloud mask (1where clouds are present; NA where they are not) given
Band 1 and Band 6. The default parameters for the ratio level (level) and for adding a buffer
around the cloud edge (buffer) were adequate for the test data once converted to at-sensor
reflectance and temperature. This function can be used with DN data if the level argument
is adjusted appropriately. The mask does not demarcate areas of cloud shadow, but only the
clouds themselves (Figure 1e).
R> july.cloud <- clouds(july1.ar, july61.thermal)
R> nov.cloud <- clouds(nov1.ar, nov61.thermal)
7. Atmospheric correction
For most applications, ground reflectance is of greater interest than at-sensor reflectance so
atmospheric correction is required. Variation in atmospheric conditions at the time of over-
pass can overwhelm any changes in surface reflectance, so it is crucial to correct for these
differences. If measured atmospheric data such as optical depth are available an accurate
correction can be applied, but these are rarely available for retrospective studies. Most com-
monly, an image-based method is used instead. Two categories of corrections are available.
Relative normalization methods match the spectral characteristics of each image to a refer-
ence image in such a way that each transformed image appears to have been taken using the
same sensor and with the same atmospheric conditions as the reference image. Functions are
available for relative atmospheric correction methods using the entire image or unchanging
subsets.
Absolute atmospheric correction methods rely on a mechanistic understanding of atmospheric
effects to adjust each image individually. Instead of correcting to a reference image, informa-
tion contained in part of an image, for instance the darkest areas, is extracted and used to
correct the rest of the image. This extracted information substitutes for measured parameters.
Three such methods have been included here.
7.1. Relative atmospheric correction using the entire image
Statistical correction methods are entirely empirical and do not consider physical principles

Journal of Statistical Software 9
or atmospheric conditions in any way. These algorithms force the distribution of values in
one image to match that in another. Unless otherwise indicated, these methods can operate
on DN or reflectance.
Relative normalization
The most aggressive correction method is to regress all of the pixels in the image to be
corrected onto the corresponding pixels in the reference image. This method requires georef-
erenced images covering the same area and at the same resolution. Since both variables are
random, model II regression method such as Major Axis regression implemented in lmodel2()
from the lmodel2 package is recommended (Legendre 2008). The relnorm() function imple-
ments this method for spatial data, and returns both a corrected image and the coefficients
used. In the example given here, Band 4 of the July image is corrected to match the November
image.
R> july4.rncoef <- relnorm(nov4, july4, mask = july.cloud, nperm = 0)
R> july4.rn <- july4.rncoef[["newimage"]]
Histogram matching
Histogram matching forces the distribution of the DN values in one image to match another.
This algorithm is commonly used in other areas of image processing, and the images do not
need to cover the same area, or to match in any way. The histmatch() function should be
used only with the integer DN values. In the example given here, Band 4 of the July image
is corrected to match the November image.
R> july4.hmcoef <- histmatch(nov4, july4, mask = july.cloud)
R> july4.hm <- july4.hmcoef[["newimage"]]
For this image pair, histogram matching performs better than relative normalization (Fig-
ure 2). The former method forces the histogram for July to acquire the shape of the histogram
for November, while keeping the range of values (compare Figure 1c to Figure 2c). Relative
normalization created a similar histogram profile, but greatly compressed the dynamic range,
thereby losing much of the information contained in the data. This method also inverted the
values of the original image: the slope of the regression line was negative. This inversion
is a drawback of relative normalization methods, especially for image pairs where seasonal
vegetation differences are pronounced.
7.2. Relative atmospheric correction using a subset of the image
Using the entire image for statistical correction includes areas of the image that are likely
to change between dates, particularly vegetation, so the correction factors incorporate non-
atmospheric effects. As shown above this may have unwanted side effects. These corrections
may thus conceal actual changes between dates. Identifying pixels that cover developed areas
or other land uses that would be expected to possess constant reflectance properties from
date to date could produce more accurate statistical corrections. Two methods for doing so
are included in landsat: pseudo-invariant features and radiometric control sets.

10 landsat: Analyzing Remote Sensing Data in R
Figure 2: Whole-image relative atmospheric corrections for Band 4 of the July data.
Pseudo-invariant features
The reflectance of a specific developed area such as a large rooftop or parking lot should
not change seasonally, so differences in apparent reflectance in these areas between dates are
assumed to be due to atmospheric differences. Such areas, termed pseudo-invariant features
(PIF) by Schott, Salvaggio, and Volchok (1988), can be identified using Band 7 and the ratio
of Band 4 to Band 3. The Band 4 to Band 3 ratio is low where there is no vegetation,
including water and developed areas. Band 7 is low over water areas, so it can be used to
distinguish between the unvegetated areas identified using the Band 4 to Band 3 ratio. As
implemented here, the images must cover the same area at the same resolution, but that is not
a requirement of the method. The PIF() function can be used to identify invariant features
within an image using the above criteria. Visual inspection may be needed to ensure that
the threshold value is correct for the particular images being analyzed. Having an insufficient
number of PIF points is a concern with this method, especially with the small images in these
examples.

Journal of Statistical Software 11
Using the data included in the landsat package, November is the clearer image, as would be
expected because of the higher humidity in the summer months, so it is used to identify PIF
pixels. These PIFs are then used in major axis regression to correct the July image (only
Band 4 is shown).
R> nov.PIF <- PIF(nov3, nov4, nov7, level = 0.9)
R> july4.pifcorr <- lmodel2(nov4@data[nov.PIF@data[, 1] == 1, 1] ~
+ july4@data[nov.PIF@data[, 1] == 1, 1])
R> july4.pifcorr <- unlist(july4.pifcorr[["regression.results"]][2, 2:3])
The final steps in the correction are to use the original image as a template for putting
the corrected data into the SpatialGridDataFrame format, and to block out the previously-
identified cloud areas.
R> july4.pif <- july4
R> july4.pif@data[, 1] <- july4@data[, 1] * july4.pifcorr[2] +
+ july4.pifcorr[1]
R> july4.pif@data[!is.na(july.cloud@data[, 1]), 1] <- NA
Radiometric control sets
The radiometric control set (RCS) procedure implemented in RCS() takes a different approach
to choosing invariant features Hall, Strebel, Nickeson, and Goetz (1991). The tasseled cap
procedure (tasscap()) is used to identify the greenness and brightness components of the
image (Crist 1985;Crist and Kauth 1986;Huang, Wylie, Yang, Homer, and Zylstra 2002).
These are then used to identify dark sets and bright sets with low greenness: unvegetated
radiometric control sets. As with PIF(), the threshold value may need to be adjusted for
a particular set of images. The tasselled-cap procedure included in landsat contains default
values for at-sensor reflectance for both Landsat 5 (TM) and 7 (ETM+), although other
formulations are available in the literature. In the example given, reflectance is used for RCS
identification, but the adjustment is done on the DN values to maintain consistency with
previous correction examples.
The clearer November image is used to identify the radiometric control set, and major axis
regression is used to correct the July image based on that RCS. Finally the corrected data
are put in SpatialGridDataFrame format, and cloud areas removed.
R> nov.tasscap <- tasscap("nov", 7)
R> nov.RCS <- RCS(nov.tasscap)
R> july4.rcscorr <- lmodel2(nov4@data[nov.RCS@data[, 1] == 1, 1] ~
+ july4@data[nov.RCS@data[, 1] == 1, 1])
R> july4.rcscorr <- unlist(july4.rcscorr[["regression.results"]][2, 2:3])
R> july4.rcs <- july4
R> july4.rcs@data[, 1] <- july4@data[, 1] * july4.rcscorr[2] +
+ july4.rcscorr[1]
R> july4.rcs@data[!is.na(july.cloud@data[, 1]), 1] <- NA
The PIF correction compressed the range of DN values considerably (Figure 3c), losing much
of the detail in the original image (Figure 1a). Using only points identified by PIF() for both

12 landsat: Analyzing Remote Sensing Data in R
Figure 3: Subset-based relative atmospheric corrections for Band 4 of the July data.
dates, rather than just in the clearer November image set, did not improve the results. The
RCS correction reduced the dynamic range somewhat, and inverted the data range, as did
relative normalization (Figure 3d; Figure 2d). Of the four whole-image relative atmospheric
correction methods implemented here, only the histogram matching method produced accept-
able results.
The root mean square error (RM SE) between the target and reference image can be used
to assess the effectiveness of a relative correction. The greater the reduction in RMSE,
the more effective the transformation was at matching that image pair. The RMSE for
the untransformed Band 4 data was 0.2, and after histogram matching it was reduced to
0.067. Despite their poor overall performance in other ways, the other three methods reduced
RMSE values as much or more: 0.043 for relative normalization, 0.043 for PIF, and 0.065
for RCS. The correspondence between image pairs can increase even when other properties
such as range that are crucial for further analysis and interpretation are not preserved.

Journal of Statistical Software 13
7.3. Absolute atmospheric correction
This class of methods attempts to deduce values for atmospheric parameters from information
contained within the image itself rather than using externally-measured data. Each image is
treated on its own.
ρ=πd2(L−Lhaze )
Tv(Esun cos θzTz+Edown )(5)
The conversion of at-sensor radiance to atmospherically-corrected surface reflectance is de-
scribed in Equation 5. Absolute atmospheric correction methods use measurements or at-
mospheric simulation models to determine the parameters Tz,Tv,Edown , and Lhaze (Chavez
1989). The major difference between the relative atmospheric correction methods is the pro-
cedure for estimating these values. With default parameter choices, this simplifies to the
equation for at-sensor reflectance (Table 4; Equation 4).
Dark object subtraction
The dark object subtraction (DOS) method assumes that if there are areas in an image with
very low actual reflectance values, any apparent reflectance should be due to atmospheric
scattering effects, and this information can be used to calibrate the rest of the image (Chavez
1988,1989). The darkest pixels can be selected by examining the histogram of the DN values
in an image, or by setting a threshold such as ‘lowest DN value found in at least npixels,’ or
some other criterion appropriate for the size of image being analyzed. The chosen DN value,
the Starting Haze Value (SHV ) is then converted to radiance (e.g., Equation 2or similar
conversion). It is unlikely that most images contain entire pixels that are true black, so a
correction is applied that assumes a 1% actual reflectance of these areas.
L1% = 0.01Esun cos θz
πd2
Lhaze =SHV rad −L1% (6)
The simplest form of DOS simply converts the calculated Lhaze value to at-sensor reflectances
(Equation 4) and subtracts it from the entire image (also converted to at-sensor reflectances).
A new SHV value must be calculated for each band (Chavez 1988).
The improved method developed by Chavez (1989) uses information from a single band to
calculate Lhaze values for the remaining bands of an image. This method produces correlated
haze values, and may be more accurate if there are few shadows or dark areas. Scattering
is band-specific, and the band effects are correlated with atmospheric conditions. The DOS
method implemented here uses a realistic relative atmospheric scattering model, and maintains
the spectral relationship between bands (Chavez 1989). This may be important for vegetation
Model TzTvEdown Lhaze
Apparent reflectance 1 1 0 0
DOS 1 1 0 SH V
COSTZ cos θzcos θv0SH V
DOS4 Iterative Iterative Iterative Iterative
Table 4: Parameters values used in four different atmospheric correction methods.

14 landsat: Analyzing Remote Sensing Data in R
indices whose values depend on the ratios between bands. The DOS method works poorly
for bands 5 and 7, hugely overestimating the Lhaze value. Only small components of the total
atmospheric scattering occurs in these bands, so the DN corrections for very clear atmosphere
were used for bands 5 and 7. The DOS() function will take the SHV value for the starting
band (usually Band 1, though bands 2 or 3 may be used), and calculate the related SHV
value for each other band.
DOS correction as implemented here is a two-step process. In this example, the July image
is corrected. Band 1 is used to find the SHV , here the lowest DN with at least 1000 pixels.
R> SHV <- table(july1@data[, 1])
R> SHV <- min(as.numeric(names(SHV)[SHV > 1000]))
R> SHV
[1] 69
That SHV (69) is then used in DOS() to find the corrected SHV values for the remaining
bands (Chavez 1989).
R> july.DOS <- DOS(sat = 7, SHV = SHV, SHV.band = 1, Grescale = 0.77569,
+ Brescale = -6.2, sunelev = 61.4, edist = ESdist("2002-07-20"))
R> july.DOS <- july.DOS[["DNfinal.mean"]]
R> july.DOS
coef-4 coef-2 coef-1 coef-0.7 coef-0.5
band1 68.308152 68.30815 68.30815 68.30815 68.30815
band2 41.927435 53.23414 60.23021 62.53285 64.12406
band3 25.580966 40.56327 52.31547 56.60737 59.69712
band4 14.858007 28.34164 43.02630 49.22788 53.96081
band5 8.443139 13.20415 25.72192 33.59098 40.69352
band7 8.130282 10.87164 21.16987 28.78982 36.18461
The above Routput shows the possible SHV values calculated for five scattering coefficients,
with the smallest coefficient (-4) describing the clearest conditions and -0.5 representing
extreme haze. For this example, Band 1 was used to calculate the SHV from the actual
image, so it is not adjusted for scattering coefficient. The user must select the appropriate
set of band-specific SHV values. Chavez (1989) suggested using the original SHV as a guide:
SHV ≤55 very clear (-4); SHV 56–75 clear (-2); SHV 76–95 moderate (-1); SHV 96–115
hazy (-0.7); SHV >115 very hazy (-0.5). The SHV for Band 1 from July was 69, so the
atmosphere was clear, and the corresponding column coef-2 was selected for use in DOS().
R> july.DOS <- july.DOS[, 2]
R> july2.DOSrefl <- radiocorr(july2, Grescale = 0.79569, Brescale = -6.4,
+ sunelev = 61.4, edist = ESdist("2002-07-20"), Esun = 1812,
+ Lhaze = july.DOS[2], method = "DOS")
R> july4.DOSrefl <- radiocorr(july4, Grescale = 0.63725, Brescale = -5.1,
+ sunelev = 61.4, edist = ESdist("2002-07-20"), Esun = 1039,
+ Lhaze = july.DOS[4], method = "DOS")
R> july4.DOSrefl@data[!is.na(july.cloud@data[, 1]), 1] <- NA

Journal of Statistical Software 15
COSTZ
Chavez (1996) improved on his earlier dark object subtraction methods by adding a correction
for the multiplicative transmittance component of the atmospheric scatter (cos θz, abbreviated
as COSTZ). In this revised procedure, cos θzis used as an approximation of Tz, and cos θvis
used as an approximation of Tv. For Landsat, the latter parameter is one because the satellite
sensor has a nadir view (θv= 0◦). The value for Lhaze is determined just as for the DOS
method. This method is not appropriate for use with bands 5 or 7; the original DOS method
should be used with these data (Song, Woodcock, Seto, Lenney, and Macomber 2001).
R> july4.COSTZrefl <- radiocorr(july4, Grescale = 0.63725, Brescale = -5.1,
+ sunelev = 61.4, edist = ESdist("2002-07-20"), Esun = 1039,
+ Lhaze = july.DOS[4], method = "COSTZ")
R> july4.COSTZrefl@data[!is.na(july.cloud@data[, 1]), 1] <- NA
Modified dark object subtraction
The modified dark object subtraction method (DOS4) was developed to incorporate the effect
of atmospheric aerosols into atmospheric correction (Song et al. 2001). The value for Lhaze
calculated in the DOS method is used. Corrected values for Tzand Tvare determined through
an iterative process. Both are set to initial values of one, and Equation 7is solved for τ, the
Rayleigh atmospheric optical depth. New values are calculated for Tzand Tv(Equation 8),
and the process is repeated until τstabilizes. This generally requires fewer than ten iterations.
τ=−cos θzln 1−Grescale DN min +Brescale −0.01(Eocos θzTz+Edown )Tv/π
Eocos θz(7)
Tv= exp (−τ / cos θv)
Tz= exp (−τ / cos θz)(8)
R> july4.DOS4refl <- radiocorr(july4, Grescale = 0.63725, Brescale = -5.1,
+ sunelev = 61.4, edist = ESdist("2002-07-20"), Esun = 1039,
+ Lhaze = july.DOS[4], method = "DOS4")
R> july4.DOS4refl@data[!is.na(july.cloud@data[, 1]), 1] <- NA
When compared to at-sensor reflectance, all three absolute atmospheric correction methods
produced similar results for the July test image (Figure 4). Each reduced the dynamic range,
but only slightly, and the overall appearance was similar to the original, unlike the results of
most of the relative correction methods.
Other research has found that COSTZ was effective in the visible-light bands (1-3), but less
accurate in the near-infrared, particularly in humid conditions (Wu, Wang, and Bauer 2005).
Song et al. (2001) found that DOS4 outperformed COSTZ or DOS.
8. Topographic correction
The interaction between sun angle, surface slope and satellite position produces variations in
surface reflectance unrelated to true reflectance. The same land use type can return a different

16 landsat: Analyzing Remote Sensing Data in R
Figure 4: Absolute atmospheric corrections for Band 4 of the July data.
signal if located on opposite sides of a hill. The intent of topographic correction is to remove
this source of variation, and leave only the portion of the reflectance signal actually due to
ground cover, thus converting the reflectance from an inclined surface (ρT) to that from an
equivalent horizontal surface (ρH).
The sun angle at the time of the November image was lower, so differential shading of the
north and south faces of the ridge in the middle of the image can be clearly seen (Figure 1b,
and see also Figure 1f). A successful topographic correction would eliminate this shading
difference; corrected images should appear flat.
8.1. Illumination
Slope and aspect are used in conjunction with solar and satellite parameters to model illu-
mination (IL) conditions. Information derived from the DEM is required to compute the
incident angle (γi), defined as the angle between the normal to the ground and the sun rays.

Journal of Statistical Software 17
The IL parameter varies from −1 (minimum illumination) to 1 (maximum illumination) and
is calculated from the slope angle θp, solar zenith angle θz, solar azimuth angle φaand aspect
angle φo(Equation 9).
IL = cos γi= cos θpcos θz+ sin θpsin θzcos (φa−φo) (9)
The illumination is then used in one of several topographic correction methods. Seven such
algorithms are currently implemented in topocorr(). Many of these algorithms are described
and compared in Riano et al. (2003).
8.2. Lambertian methods
Lambertian methods assume that the reflectance for all wavelengths is constant regardless of
viewing angle. The correction factor is identical across all bands.
Cosine correction
The cosine method is a trigonometric approach that assumes that irradiance is proportional
to the cosine of the incidence angle (Teillet, Guindon, and Goodenough 1982). The cosine
correction discounts indirect illumination and saturates in dark areas.
ρH=ρTcos θz
IL (10)
R> dem.slopeasp <- slopeasp(dem)
R> nov4.cosine <- topocorr(nov4.ar, dem.slopeasp$slope, dem.slopeasp$aspect,
+ sunelev = 26.2, sunazimuth = 159.5, method = "cosine")
Improved cosine correction
The improved cosine method attempts to compensate for the overcorrection often seen in the
cosine method by including average illumination (IL) in the calculation (Civco 1989).
ρH=ρT+ρTIL −IL
IL  (11)
R> nov4.cosimp <- topocorr(nov4.ar, dem.slopeasp$slope, dem.slopeasp$aspect,
+ sunelev = 26.2, sunazimuth = 159.5, method = "improvedcosine")
Gamma correction
The Gamma method is an extension of the cosine correction that adds terms for the sensor
view angle on flat terrain (θv) and on inclined terrain (βv). This is another attempt to reduce
overcorrection in areas with low illumination (Richter, Kellenberger, and Kaufmann 2009).
ρH=ρTγ=ρTcos θz+ cos θv
IL + cos βv
βv= 90◦−(θv+θp)
(12)

18 landsat: Analyzing Remote Sensing Data in R
R> nov4.gamma <- topocorr(nov4.ar, dem.slopeasp$slope, dem.slopeasp$aspect,
+ sunelev = 26.2, sunazimuth = 159.5, method = "gamma")
Sun-canopy-sensor method
The sun-canopy-sensor method (SCS) was developed by Gu and Gillespie (1998) for use in
forested areas where the canopy geometry contributes to reflectance on a sloped surface (Gao
and Zhang 2009).
ρH=ρTcos θzcos θp
IL (13)
R> nov4.SCS <- topocorr(nov4.ar, dem.slopeasp$slope, dem.slopeasp$aspect,
+ sunelev = 26.2, sunazimuth = 159.5, method = "SCS")
8.3. Non-Lambertian methods
This group of methods recognizes that the combination of angles of incidence and observation
can affect reflectance: surface roughness matters.
Minnaert method
The Minnaert method adds a band-specific constant Kto the cosine method (Minnaert 1941,
in Riano et al. 2003; Equation 14). If K= 1, the Minnaert and cosine methods are equivalent
(Lambertian behavior is assumed).
ρH=ρTcos θz
IL K
(14)
The Kparameter is the slope of the ordinary linear regression with Kand ln ρHas regression
coefficients, and is constant across the entire image for each band (Equation 15).
ln(ρT) = ln(ρH) + Kln IL
cos θz
(15)
The topocorr() function with the argument method = "minnaert" will take care of both
steps, returning the corrected image.
R> nov4.minnaert <- topocorr(nov4.ar, dem.slopeasp$slope, dem.slopeasp$aspect,
+ sunelev = 26.2, sunazimuth = 159.5, method = "minnaert")
A further modification of the Minnaert method includes the slope explicitly as well as through
the illumination calculation (Colby 1991; Equation 16).
ρH=ρTcos θpcos θz
IL cos θpK
(16)
R> nov4.min2 <- topocorr(nov4.ar, dem.slopeasp$slope, dem.slopeasp$aspect,
+ sunelev = 26.2, sunazimuth = 159.5, method = "minslope")

Journal of Statistical Software 19
Figure 5: Topographic corrections for Band 4 of the November data.

20 landsat: Analyzing Remote Sensing Data in R
Figure 6: Topographic corrections for Band 4 of the November data (continued).
C-correction
The C-correction method is a statistical approach based on the ratio of the band-specific
regression coefficients band m, as given in Equation 17 (Teillet et al. 1982).
ρH=ρTcos θz+c
IL +c
ρT=b+mIL
c=b
m
(17)
R> nov4.ccor <- topocorr(nov4.ar, dem.slopeasp$slope, dem.slopeasp$aspect,
+ sunelev = 26.2, sunazimuth = 159.5, method = "ccorrection")
Riano et al. (2003) recommend smoothing the slope to reduce the overcorrection imposed
when the illumination is low. A smoothing option was added to slopeasp() to make this

Journal of Statistical Software 21
variant possible. They found the most improvement with a slope correction factor of 5, as
illustrated in the example.
R> dem.smooth5 <- slopeasp(dem, smoothing = 5)
R> nov4.ccorsmooth <- topocorr(nov4.ar, dem.smooth5$slope, dem.smooth5$aspect,
+ sunelev = 26.2, sunazimuth = 159.5, method = "ccorrection")
None of the topographic correction methods completely eliminated the shading effects due to
the central ridge (Figures 5,6). The gamma correction was particularly poor (Figure 5c), as
was the C-correction (Figure 6a). The cosine and Minnaert methods performed the best, but
in all the images the north-south topographic features along the main ridge can be clearly
seen.
The cosine and SCS methods also resulted in reflectances greater than one. The Minnaert
methods increased the dynamic range, though not the median, and the gamma correction
reduced both. Meyer, Itten, Kellenberger, Sandmeier, and Sandmeier (1993) found that the
Minnaert and C-correction methods gave similar results, but that the cosine method results
were very different. Other work found that Minnaert was superior under most conditions
(Richter et al. 2009).
9. Conclusions
Implementing atmospheric and topographic correction methods in Rwas straightforward,
given R’s good preexisting spatial capabilities. While large-scale production work is probably
best done in a GIS environment, the scripting features available in Rmake it a feasible solution
for small- and medium-sized tasks involving satellite remote sensing imagery.
The landsat package will be useful for researchers investigating the utility of these algorithms
for their own work, and for facilitating development of new methods. The topographic algo-
rithms especially appear to need further research. The wide range of sophisticated statistical
tools available in Rmake it suitable for research and development on these methods. The
open source nature of both Rand of the landsat package allow for examination and alteration
of the code base, facilitating further work on processing satellite imagery.
Acknowledgments
This work contributes to the Conservation Effects Assessment Project (CEAP), jointly funded,
coordinated and administered by the United States Department of Agriculture’s Natural Re-
sources Conservation Service, Agricultural Research Service, and Cooperative State Research,
Education, and Extension Service. Funding for this project was provided by the USDA-NRCS.
Forrest R. Stevens provided considerable assistance in locating a problem with an earlier ver-
sion of landsat.
Mention of trade names or commercial products in this publication is solely for the purpose
of providing specific information and does not imply recommendation or endorsement by the
US Department of Agriculture.
USDA is an equal opportunity provider and employer.

22 landsat: Analyzing Remote Sensing Data in R
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24 landsat: Analyzing Remote Sensing Data in R
A. Mathematical quantities
DN – Digital number (integer, 0–255 for Landsat).
λ– Wavelength (µm).
L– At-sensor radiance (W m−2sr−1µm−1).
ρAS – At-sensor reflectance (unitless).
ρ– Surface reflectance (unitless).
Grescale – Band-specific gain (W m−2sr−1µm−1DN −1).
Brescale – Band-specific bias (W m−2sr−1µm−1).
gain2– Band-specific gain (alternate version).
offset – Band-specific (alternate version).
Lmax – Radiance associated with maximum DN value (W m−2sr−1µm−1).
Lmin – Radiance associated with minimum DN value (W m−2sr−1µm−1).
DN max – Maximum DN value, usually 255.
DN min – Minimum DN value, usually 0.
Esun – Band-specific exoatmospheric solar constant (W m−2µm−1).
d– Earth-Sun distance (AU).
Eo– Distance-adjusted exoatmospheric solar constant d2/Esun.
θz– Solar zenith angle (degrees); 90◦−θs.
θs– Solar elevation angle (degrees).
DOY – Day of year; Julian date (days).
Lhaze – Path radiance, upwelling atmospheric scattering (W m−2sr−1µm−1).
Tv– Atmospheric transmittance from surface to the sensor (W m−2µm−1).
Tz– Atmospheric transmittance from the Sun to the surface (W m−2µm−1).
Edown – Downwelling irradiance due to atmospheric scattering (W m−2µm−1).
SHV – Starting haze value (DN ).
Z– Elevation at pixel (i, j).
EW – East-west component of slope.
NS – North-south component of slope.
EW res – East-west pixel resolution.
NSres – North-south pixel resolution.
θp– Slope angle (degrees).
θz– Solar zenith angle (degrees).
θv– Sensor zenith angle (degrees; 0◦for Landsat).
βv– Sensor view angle on inclined terrain (degrees).
φa– Solar azimuth angle (degrees)

Journal of Statistical Software 25
φo– Aspect angle (degrees, with north = 0◦and moving clockwise).
ρH– Reflectance of a horizontal surface (unitless).
ρT– Reflectance of an inclined surface (unitless).
IL – Illumination (−1 to +1).
IL – Average illumination across the image (−1 to +1).
γi– Incident angle between the normal to the ground and the sun rays (degrees).
τ– Rayleigh atmospheric optical depth.
Affiliation:
Sarah C. Goslee
USDA-ARS Pasture Systems and Watershed Management Research Unit
Bldg. 3702 Curtin Rd.
University Park PA, 16802, United States of America
E-mail: Sarah.Goslee@ars.usda.gov
Journal of Statistical Software http://www.jstatsoft.org/
published by the American Statistical Association http://www.amstat.org/
Volume 43, Issue 4 Submitted: 2010-09-29
July 2011 Accepted: 2011-06-16