Aqeel:
Sorry for the long explanation; we helped a client here time ago with a
similar problem, and I thought this e-mail could be useful to other members
of the list.
The reflectance is the relationship between the true radiance and the energy
irradiance that reaches the soil in a given wavelenght. This is what is
normally called "true reflectance", but that one is difficult to obtain
because the radiance and irradiance are affected by the atmosphere. The
correction of the atmospheric effect for both parameters is complex, and it
involves the estimation of the composition of the atmosphere in the moment
the image is taken.
The TRUE reflectance is not easy to calculate because the atmospheric
transmisivity is difficult to estimate. A simplification of the problem is
given by the calculation of what is normally called in the literature the
APPARENT REFLECTANCE, that is, the reflectance if we do not take in account
the atmospheric effects. This simplification is useful to explain how the
calculation could be done in ER Mapper. I don't know if what you are looking
for is the true reflectance (that includes an atmospheric correction), but
the example below will explain how to implement any formula into ER Mapper;
I hope it helps.
The formula of the apparent reflectance is as follows:
K*PI*L / E cos(alpha)
where:
K = Correction factor for the distance Sun-Earth. The energy that reaches
the Earth from the Sun is modified with the variable distance between the
two bodies due to the ellipticity of the Earth's orbit. It can be calculated
with:
K = (1 + 0.0167 (sin(2*PI(julian_day - 93.5)/365))^2
PI = 3.14159etc
L = Spectral radiance (that is, the radiance that reaches the sensor in a
given spectral band, suppossed no atmospheric effects). The original values
that can be readen from a Landsat image are in the 8 bit range (0-255), and
should be transformed into radiance before trying to calculate the
reflectance. In the header of any image, you will find the coefficients used
to transform theDigital Numbers (DNs) into radiance. For Landsat 7, for
instance, there are two sets of parameters, depending if the image was
acquired in low gain or in high gain. Normally the high gain is given for
dark images or images that cover a lot of "dark bodies" like water.
Depending on when the image was acquired, the parameters can be found in
this web address (it is the IAS CPF archive, maintained by NASA):
http://edcwww.cr.usgs.gov/l7dhf/ias_folder/cpf.html
For instance, for an image from May 2000, the two sets of parameters are:
a) HIGH
TM1 a = 1.176078, b = -6.2
TM2 a = 1.205098, b = -6.4
TM3 a = 0.938824, b = -5.0
TM4 a = 0.965490, b = -5.1
TM5 a = 0.190471, b = -1.0
TM7 a = 0.066235, b = -0.35
b) LOW
TM1 a = 0.775686, b = -6.2
TM2 a = 0.795686, b = -6.4
TM3 a = 0.619216, b = -5.0
TM4 a = 0.637255, b = -5.1
TM5 a = 0.125725, b = -1.0
TM7 a = 0.043725, b = -0.35
Those measurements are given in Radiance values (L): Watts per square meter,
per stereo radian per nanometer. For any of the sets of values, the maximum
and minimum Digital Number available for the image (0-255) in each one of
the bands should be adjusted to obtain a linear function, that in reverse
will translate between the 8 bit digital numbers and the equivalent apparent
radiance on the ground for each one of the pixels. That function should
therefore use the radiance as the dependent variable for the linear
equations, and the 8-bit translated Digital Number as the independent
variable. So, for the first pair of values:
293.7 = 255a + b
-6.2 = b
The second equation is obvious, since the minimum ND is 0. This means that
if you sum both values in the table and divide them by 255 you will be able
to translate any value to radiance, simply by adjusting the coefficients a
and b. The tables are the following:
a) HIGH
TM1 a = 1.176078, b = -6.2
TM2 a = 1.205098, b = -6.4
TM3 a = 0.938824, b = -5.0
TM4 a = 0.965490, b = -5.1
TM5 a = 0.190471, b = -1.0
TM7 a = 0.066235, b = -0.35
b) LOW
TM1 a = 0.775686, b = -6.2
TM2 a = 0.795686, b = -6.4
TM3 a = 0.619216, b = -5.0
TM4 a = 0.637255, b = -5.1
TM5 a = 0.125725, b = -1.0
TM7 a = 0.043725, b = -0.35
With these values calculated, you can now calculate the radiance if you
apply this formula into ER Mapper:
I1*par1+ par2
where par1 is a and par2 is b .
E = Solar irradiance in the Top of the Atmosphere (TOA) for a given spectral
band. The TOA irradiance is the result of the integration of the energy that
reaches the Earth from the Sun in the complete range of wavelenghts that
compose a spectral band. Normally, that is given in the header of the
satellite image. For Landsat 5 and 7 bands, the irradiance in the TOA is:
TM1: L5:1957, L7:1970
TM2: L5:1829, L7:1843
TM3: L5:1557, L7:1555
TM4: L5:1047, L7:1047
TM5: L5:219.3, L7:227.1
TM7: L5:74.52, L7:80.530
(The observed differences between these values seem to be explained by the
algorithm used to calculate that spectral irradiances on top of the
atmosphere, and from the slight differences in certain bandwidths between
the two sensors).
alpha = Zenith angle of the Sun. This is the complementary angle of the Sun
elevation angle (90- sun elevation angle). The Sun elevation angle for the
centre of the scene is normally given in the image header.
Now, we are ready to include all this in ER Mapper. That should be done
using the formula editor. The correction should be done for each one of the
bands. You can create an algorithm with a layer per spectral band (so, 6
pseudo layers). In each one of them, you can write the following formula,
using the formula editor:
((I1*par1 + par2)* (POW(((1+0.0167*SIN(2*PI*(julian_day -
93.5)))/365),2))*PI)/(irradiance*COS(incidence_angle*(PI/180)))
Irradiance is the TOA solar irradiance, and the incidence_angle is a
variable for the solar Incident angle (90 - sun elevation angle). Since ER
Mapper calculates all the angles in radians, we multiply the incidence_angle
variable by PI/180; that will allow the user to enter the angle variable in
decimal degrees.
The formula will allow you to specify any of this values for each band you
add. So, this is a "general purpose" formula, that can be used with any
sensor as long as you know the pertinent parameters. Remember that this
formula calculates the APPARENT REFLECTANCE; the parameters for other
sensors can be found in the literature; the parameters vary with the sensor
because the bandwidth is variable depending on the sensor - so is the energy
reflected and irradiated - and the way to obtain the radiance could vary
depending on how the Digital Values of the image are codified on board.
You can of course modify these method to calculate the REAL reflectance; for
doing that, the formula has to be modified to account for a) the
transmittance of the atmosphere (ascendent and descendent), b) the diffuse
irradiance and c) the atmospheric radiance due to scattering. That
parameters are normally estimated using "standard atmospheres"; there are
some code available in public domains to perform atmospheric corrections
plus reflectance calculations (for instance, Lowtran, Modtran or 6S); I
believe that these programs and its code is available as source code in the
Internet, so perhaps you should give a look to them. For Lowtran, try:
http://www.ncdc.noaa.gov/pub/software/lowtran/
For S6, check the page from the University of Lille:
http://loasys.univ-lille1.fr/informatique/sixs_gb.html
I hope this can help you. If you have any question regarding the
implementation of the formulas, etc. in ER Mapper, please let me know
([EMAIL PROTECTED]).
Regards,
Manuel Alonso
Earth Resource Mapping
Blenheim House
Crabtree Office Village
Eversley Way, Egham
Surrey, TW20 8RY, UK
Telephone: +44 1784 430-691
Facsimile: +44 1784 430-692
-----Original Message-----
From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED]]On
Behalf Of Aqeel Kidwai
Sent: 19 July 2001 22:52
To: [EMAIL PROTECTED]
Subject: Reflectance
Hello all,
Does anyone know how to convert the Digital Numbers (of the image) into
REFLECTANCE using ERMapper ?
Aqeel Kidwai (Ph.D.)
Remote Sensing / GIS Manager
Spatial Mapping Ltd.,
200-484 Second Avenue
Prince George, V2L 2Z7 BC
Tel.: 250-564-1928
Fax: 250-564-0751
E-mail: [EMAIL PROTECTED]
Website: www.spatialmapping.com
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