1. Radiometric Calibration and Atmospheric Corrections
Of Multi-Spectral Satellite Images
DOS and COST Models
Pat Chavez
Adjunct Research Professor
Northern Arizona University (Retired USGS)
September 2014

2. The purpose of this presentation is to give a general overview of radiometric calibration and atmospheric corrections required for satellite image data with the focus on the DOS and COST models (Chavez 1975, 1988, 1989, and 1996).

3. imaging system is a function of the following
The Digital Number (DN) recorded by a satellite
imaging system is a function of the following
Sensor = Function (spectral bands, gains, offsets)
Sun Elevation = Function (time of year, latitude, typically NOT
time of day because most current satellite
images are collected at around the same time
of day :40am local time for Landsat)
Atmosphere = Function (conditions–scattering/absorption )
Topography = Function (slope and aspect)
Surface Cover = Function (cover types --- vegetation, soils,
water, etc.--- this is usually what is of interest)

4. General methods for radiometric calibration
and atmospheric corrections
Field spectral measurements at time of overflight
Radiative Transfer Models (e.g. MODTRAN and 6S --- FLAASH)
Image Based Correction Techniques / Models
DOS (Dark Object Subtraction)
Improved DOS
COST (COSine Theta)
TOA (Top Of Atmosphere) Radiance
AR (Apparent Reflectance)
PLUS Other models/techniques/methods

5. Radiance versus Reflectance
Radiance relates to the TOTAL AMOUNT of energy / light
received at the sensor and is a function of the spectral band,
sun elevation, surface cover type, topography/slope/aspect,
and atmospheric conditions.
Reflectance is the RATIO of the amount of radiance reflected (bounced back) to the amount of radiance coming in at a
given spectral band (basically brightness). It is a unit less value
less than 1 and is a function of the same variables as radiance.
The reflectance of the surface cover types is usually what is important to most applications. Surface reflectance values typically range from 2 to 50 percent (0.02 to 0.50) but can be higher.

6. Absolute vs Relative Calibration
Absolute --- Convert satellite image DNs to surface reflectance
(represents a physical property of the surface)
Radiative transfer modeling
Image based models
Radiometric master made by one of several methods
Relative --- Convert satellite image DNs to match DNs of another
satellite image (may or may not represent a physical
property of the surface)
Histogram matching
Use points/areas that in theory the brightness does
not change thru time
Radiometric master and match other images to it

7. Image based models for absolute calibration
1) DOS (Dark Object Subtraction)
2) Improved DOS
3) COST (Cosine Theta)
These models are used to convert satellite DNs to
surface reflectance using ONLY information extracted
directly from the image. A major advantage over more
rigorous radiative transfer models is that surface and/or
atmospheric field data collected at the time of overflight
are not needed. Therefore, entirely image based
models, such as these, can be applied to historical
images as well as images of areas that are not easily
accessible.

8. Digital Number (DN) a function of the following
Sensor = Function (spectral bands, gains, offsets)
Sun Elevation = Function (time of year, latitude, typically NOT
(Solar) time of day because most current satellite
images are collected at around the same time
of day :40am local time for Landsat)
Atmosphere = Function (conditions–scattering/absorption )
Topography = Function (slope and aspect)
Surface Cover = Function (cover types --- vegetation, soils,
water, etc.--- this is usually what is of interest)

9. SENSOR
The first step is to convert at satellite raw Digital Numbers (DNs)
to at-satellite radiance values by removing the gain and offset
effects introduced by the imaging system (think of these as
similar to exposure and f-stop settings).
Lsat = (DN - 0ffset) / Gain (1)
DN is the raw digital number at the given pixel for the given
spectral band, Offset and Gain are those used for the given
spectral band by the given sensor. Lsat is often referred to as
the Top-Of-Atmosphere (TOA) radiance.

10. SOLAR and ATMOSPHERE
The at-satellite radiance (Lsat) is converted to surface reflectance
by correcting for solar (includes corrections for sun elevation
angle, amount of solar irradiance for the individual spectral
bands, and Earth-Sun distance) and atmospheric effects. In the
equation red are solar corrections and purple are atmospheric
corrections. 
(PI * D*D * (Lsat - Lhaze))
REF = (2)
(Eo * Cos (TZ) * TAUz)
Note:
The corrections applied are ENTIRE IMAGE BASED not
pixel-by-pixel.

11. REF = Spectral reflectance (brightness) of the surface (what we want).
The definition of the various parameters are:
REF = Spectral reflectance (brightness) of the surface (what we want).
Lhaze = Atmospheric scattering for the given spectral band (i.e., path radiance)
and is determined by the dark-object minimum DN values extracted
from the image and is a CRITICAL parameter.
Eo = Solar irradiance for the given spectral band (equal to the total amount
of radiance coming in for the given band and is constant for the given
band and sensor).
D = Earth-sun distance in astronomical units (AUs) which is a function
of time of year with a range from approximately to
TZ = Solar zenith angle which is equal to 90 minus sun elevation angle ---
I referred to this as THETA.
TAUz = Atmospheric transmittance along the path from the sun to the
ground surface (atmospheric absorption). The COST model sets
this equal to Cos(TZ) and was the first image based model to not
ignore this parameter (i.e., set it equal to 1).
PI = Constant equal to

12. DOS Model (PI * D*D * (Lsat - Lhaze))
REF = (2)
(Eo * Cos (TZ) * TAUz)
TAUz = 1.0 in the DOS model

13. Lhaze represents the amount of scattering of light by the atmosphere in a given spectral band and is used to correct for atmospheric scattering which is ‘additive’.
Most DOS methods use statistical information extracted for each individual band to select the minimum dark-object DN which is used to compute the radiance value to use as that band’s Lhaze.
In the DOS method a dark-object DN is selected for
each band using it’s histogram.

14. Band 1 (blue)

15. Band 1 (blue)

16. Band 2 (green)

17. Band 3 (red)

18. Band 4 (nir)

19. Equation 1 is used to compute the DOS Lhaze radiance values
for each individual band i using the dark-object / minimum DN
values selected using the histograms:
Lhaze(i) = (Minimum DN(i) - 0ffset(i)) / Gain(i) (1)

20. Improved DOS Model (PI * D*D * (Lsat(i) – Lhaze(i)))
REF(i) = (2)
(Eo(i) * Cos (TZ) * TAUz)
TAUz = 1.0 and Lhaze(i) PREDICTED from a single band

21. Like DOS the Improved DOS model sets TAUz = 1.0 and
Lhaze is still the main focus, however, the method used
to select and identify the Lhaze values for all the bands is
different.
Instead of using the histograms of EACH individual spectral
band to select the minimum dark-object DN values like the
DOS model the Improved DOS method uses the minimum
dark-object radiance value of a SINGLE visible band
to PREDICT the Lhaze values for the remaining bands using
atmospheric scattering functions.

22. Two well-known models that represent realistic
atmospheric scattering relationships are the
Rayleigh and Mie models.

23. The Rayleigh model states that relative scattering is
inversely proportional to the fourth power of the
wavelength (λ-4 = 1/λ4) which means that shorter
wavelengths of the spectrum are scattered more than
the longer wavelengths. For example blue and green
are scattered more than red.
This type of scattering is predominantly caused by gas
molecules that are much smaller than the wavelengths
of light.

24. The Mie model states that scattering is inversely
proportional to wavelength and in general for a
moderate atmosphere the factor is (λ-1 = 1/λ1). However,
this relationship can vary from λ0 to λ-4, with λ0
representing complete scattering (e.g., complete cloud
cover).
Mie scattering is caused by particles that are approximately
the same size as the wave lengths such as smoke and dust
particles

25. Relative Atmospheric Scattering Models
Atmospheric Conditions Relative Scattering Model
Very clear λ (Rayleigh Scattering)
Clear λ-2
Moderate λ-1
Hazy λ-0.7
Very hazy λ (Completely Mie Scattering)
λ refers to wavelength

26. Values of Specific Scattering Functions for the Landsat TM Bands
Very Very
WL* Clear Clear Moderate Hazy Hazy
λ λ-4 (%) λ-2 (%) λ-1 (%) λ-0.7 (%) λ-0.5 (%) TM
1 - blue (50.5) (36.2) (27.0) (24.0) (21.9)
2 - green (28.4) (27.1) (23.4) (21.7) (20.4)
3 - red (14.7) (19.5) (19.9) (19.3) (18.8)
4 - nir ( 5.9) (12.3) (15.8) (16.5) (16.8)
5 - swir ( 0.4) ( 3.1) ( 7.9) (10.2) (11.9)
7 - swir ( 0.1) ( 1.8) ( 5.9) ( 8.3) (10.2)
* WL = Wavelength

27. Relative Scattering Models --- TM 1 = SHV
Hazy
Moderate
Clear
Very Clear --- Rayleigh Scattering
Wavelengths --- TM Bands 1 to 7

28. COST Model (1996 paper)
The COST model is an improvement of the DOS by adding a
term to correct for atmospheric transmittance which is a
multiplicative not additive effect. The next few slides show
the improvement this correction has on Landsat TM image
data. Multi-temporal Landsat TM images of an area in
Southern Arizona were used in the study with field data
collected and made available by Susan Moran.

29. For the DOS models TAUz = 1.0 and Lhaze is the main focus.
For the COST model TAUz = Cos(Theta) is added.
(PI * D*D * (Lsat - Lhaze))
REF = (2)
(Eo * Cos (TZ) * TAUz)
The COST model was the first, and may still be the only, entirely
image based method to include a correction for TAUz.

30. Aircraft / Field Measurements vs Model / Corrections
Moran et al

31. Chavez 1996

32. and other models/techniques/methods. To review some of the
Comparisons have been made between the DOS / COST models
and other models/techniques/methods. To review some of the
comparisons google ‘remote sensing Chavez COST atmospheric
corrections’.
A few questions to keep in mind while reading articles related to
this are:
Did the images used contain VALID dark objects (i.e., was there sufficient topography in the image or does the area have mostly relatively flat terrain with few, if any, totally shadowed areas)? Related to this question is ‘was a full image used or just a
sub-area that may or may not contain a valid dark object
(Chavez 1988, 1989, 1996)?

33. Were VALID SHVs selected / used --- the identification and selection of a valid SHV is affected by the presence or absence
of a valid dark object and/or a valid realistic atmospheric
scattering model (Chavez, 1989)?
If field measurements were used to access the accuracy of the models were the measured field spectral reflectance values
error free --- did the field sample size match the satellite pixel
size AND were soils and vegetation mixed at the right
percentages to duplicate what the satellite image pixel ‘saw’
(this is particularly critical for near-infrared bands because of
the potential high contrast between vegetation and
background soils)?

34. References
Chavez, P.S., Jr., Atmospheric, solar, and MTF corrections
for ERTS digital imagery, Proceedings of the American Society of Photogrammetry, Fall Technical Meeting, Phoenix, Arizona, p,69.
Chavez, P.S., Jr., An improved dark-object subtraction technique for atmospheric scattering correction of multispectral data, Remote Sensing of Environment, Vol. 24, pp
Chavez, P.S., Jr., Radiometric calibration of Landsat Thematic Mapper multispectral images, Photogrammetric Engineering and Remote Sensing, Vol. 55, No. 9, pp
Chavez, P.S., Jr., Image-based atmospheric corrections --- revisited and improved, Photogrammetric Engineering and Remote Sensing, Vol. 62, No. 9, pp

35. corrections NEEDED and when are they NOT NEEDED ?
When are radiometric calibration and atmospheric
corrections NEEDED and when are they NOT NEEDED ?
Single image?
Multi-temporal images?
Multi-sensor images?
Spectral analyses?
Spatial analyses?
Spatial comparison / Spatial extrapolation?
Classification / Enhancements?
Digital mosaicking?
Temporal change detection?
Spatial change detection?

36. Example of current on-going project
Use of remote sensing to monitor land surface
change in the Sonoran Desert and Ironwood
Forest National Monuments