comparison of three simplified algorithms for atmospheric corrections

COMPARISON OF THREE SIMPLIFIED ALGORITHMS FOR ATMOSPHERIC CORRECTIONS OF MERIS DATA OVER LAND Jürgen Telaar, Maria von Schönermark Institute of Space Systems, University of Stuttgart, Pfaffenwaldring 31, 70550 Stuttgart, Germany

ABSTRACT Today many different algorithms for atmospheric corrections are available. Most of them require detailed knowledge of atmospheric aerosols and humidity. This report describes two approaches to derive ground reflectances from satellite images without the need for detailed atmospheric observations. The first method makes use of the spectral signature of water, while the second method is based on the spectral signature of dark vegetation. The radiative transfer program MODTRAN4 is used to estimate the functional relationships between the atmospheric radiative properties and the difference between top of atmosphere (TOA) reflectance and ground reflectance for both, water and dark vegetation. The third method, based on the assumption of a very clear atmosphere, has been developed for the case of absence of water and dark vegetation in the scene. These methods have been extended and adapted for use with MERIS data. The different methods are analyzed to determine the effect of modeling errors and the most crucial parameters and uncertainties, i.e. knowledge of the ground reflectance for the first two methods. The application of these methods to the spectral channels of MERIS and to data of different geographic regions shows the advantages and disadvantages of the different methods when applied to MERIS data.

1

ATMOSPHERIC CORRECTION ALGORITHMS

1.1 Modified clear water and dark vegetation method The atmospheric correction procedure described in [6] makes use of the unique signature of water pixels in satellite images. The basic idea is to estimate the atmospheric radiative properties above water and to use these for the atmospheric correction of the entire satellite image. This requires the knowledge of water reflectance in the wave length ranges of the channels to be corrected. Here the method in [6] utilizing clear water pixels for the atmospheric correction of AVHRR data is extended to use clear water as well as dark vegetation pixels for the atmospheric correction of MERIS data. The simplified radiation transfer equation to calculate the radiation at the top of atmosphere (TOA) is given as [4] Lm = L0 +

ρ E0Tup π (1 − s ρ )

(1)

With the measured radiance Lm, the path radiance L0, the upward radiance of the atmosphere at TOA for zero surface reflectance, the surface reflectance ρ, the downward irradiance (energy flux density) at the ground E0 for zero surface reflectance, the total radiance transmittance Tup from the ground to the satellite sensor, and the spherical albedo s. All these variables are wave length dependent. If the key atmospheric radiative parameters L0, E0, Tup and s are known, the surface reflectance ρ can be calculated by rearranging Eq. 1: f 1+ s ⋅ f

(2)

π ( Lm − L0 )

(3)

ρ=

Where f =

E0Tup

In the visible wave length range scattering processes in the atmosphere lead to higher reflectance values α at the TOA compared to the surface reflectance at the ground. The difference between these two values corresponds approximately to the atmospheric reflectance δ: δ =α − ρ

(4)

For water and dark vegetation pixels the surface reflectance as function of the wavelength is assumed to be known. The TOA reflectance α can be calculated from the measured radiance Lm: α=

π ⋅ Lm S0 ⋅ cos ϑsun

(5)

The atmospheric reflectance δ depends on the observer zenith angle ϑsensor as well as on the sun zenith angle ϑsun. Eq. 6 is used to calculate the atmospheric reflectance δ0 for a nadir satellite view (ϑsensor = 0°) with the sun directly overhead (ϑsun = 0°): δ 0 = δ ⋅ ( a1 ⋅ secϑsensor + a2 ⋅ secϑsun + a3 + a4 ⋅ cosϑsensor ⋅ cosϑsun )

(6)

The key atmospheric parameters path radiance L0, transmittance T and spherical albedo s are modeled as a function of

δ0, ϑsun and ϑsensor. L0 = b1 + b2δ 0 + b3 ( sec ϑsensor − 1) + b4 ( secϑsun − 1)

(7)

Two similar equations with different sets of coefficients are employed to calculate the transmittance from the sun down to the surface Tdown and from the surface up to the satellite Tup. Tdown = c1 + c2δ 0 + c3 ( secϑsun − 1)

(8)

Tup = d1 + d 2δ 0 + d3 ( secϑsensor − 1)

(9)

The spherical albedo s is approximated as a function of the atmospheric reflectance δ0 with a second order polynomial: s = e1δ 02 + e2δ 0 + e3

(10)

To evaluate the coefficients in Eq. 6 – 10 for all MERIS channels, except 11 (O2) and 15 (H2O), simulations with the radiative transfer code MODTRAN4 have been performed, where the input variables were varied systematically. A standard set of conditions for a midlatitude summer atmosphere, rural aerosols, and optical thickness as a function of wavelength, corresponding to a surface visibility varying between 7 and 23 km has been used. The observer zenith angle ϑsensor has been varied between 0 and 40°, the sun zenith angle between 0 and 70°. The results of these simulations were used as input data for the regression analysis to determine the coefficients for the modified clear water and dark vegetation method, respectively for each channel. These can be provided by the authors upon request. The knowledge of the ground reflectance is very critical for the success of these atmospheric correction methods and has been analyzed in more detail. Mixed vegetation (= Modtran 54 Crop Mosaic = Mosart 46) shows to be quite dark and might be assumed to be available in most scenes. For clear lake water we extended the data set no. 150 in [1] up to a wave length of 900 nm assuming that the ground reflectance of clear water in the NIR is only due to Fresnel reflection, being 0.2%. Due to the high reflectance of vegetation at wave lengths above 700 nm, the modified dark vegetation method can not be applied directly to the channels 9-14. If the TOA reflectance becomes equal or larger than the ground reflectance, their difference δ becomes very small or even negative, and the modeling as a function of δ becomes very inaccurate. Therefore another solution had to be found. Fortunately there is a strong correlation between the key atmospheric parameters of channel 8 (at 680 nm) and those for the larger wavelength (channels 9-14, λ = 700-895 nm). These correlations are employed for the atmospheric correction of the channels 9-14. The key atmospheric parameters for these channels are calculated using the equation y ( channel i ) = a ⋅ y ( channel 8 ) + b

(11)

Where y represents the key atmospheric parameter (L0, Tdown, Tup or s) to be calculated. The coefficients a and b have been determined by regression analysis. 1.2 Modified Ångström Algorithm The following algorithm refers to a clear atmosphere, specifically which means an atmosphere with rural aerosols within the boundary layer and a horizontal visibility above the ground which is identical to 11 km or more. Eq. 2 and 3 can be written as ρ=

π ⋅ ( Lm − L0 ) π ⋅ s ⋅ ( Lm − L0 ) + E0 ⋅ (Tdir + Tdiff )

(12)

The contributions to the transmittance are calculated employing the approximations published in [2] and [3] for climatologic values. The impact of the aerosols is described using the Ångström turbidity formula, which assumes that the aerosol particle size distribution can be described by a power law, but usually we observe variations in regard to the power law. The path radiance L0 depends on the sun zenith angle and on the view zenith angle as well as on the difference E p 0 = Bsun − E0 . For different sun zenith angels and view zenith angles we simulated 504 datasets per channel for

visibilities from 11 to 23 km, using MODTRAN4. Here the visibility is a measure of the aerosol loading in the lower atmosphere. In order to determine the spectral path radiance we assumed the relationship  a1  a2 + + a3 + a4 ⋅ cosϑsun cos ϑview  L0 = E p 0 ⋅  cos cos ϑ ϑ view sun  

(13)

1.3 Classification of clouds, water and vegetation MERIS data contain the L1-Flag which is used to classify bright pixels, i.e. snow and clouds, as well as land and water. The NDVI < -0.1 threshold is used to identify clear water pixels. Song [6] assumes that the smallest (most negative) NDVI indicates the clearest water. This leads to the problem that a turbid lake with high reflectance values in the red channels would be considered as very clear. Therefore we use the smallest sum of reflectance values in the channels 1, 4 and 6 to identify the clearest water in the scene. Vegetation is detected using the threshold NDVI > 0.4. Among these pixels the darkest in each channel is used for the modified dark vegetation method.

2

MODELING ACCURACY

To investigate the accuracy of the modeling 1000 MODTRAN4 simulations per channel with random input data have been performed. Then the equations 6 – 10 have been used to calculate the ground reflectance as function of the simulated TOA radiance and the observer and sun zenith angle. The mean error is below 0.5% for all channels, the standard deviation is about 1% for the NIR channels and increases to 2.5% for the first channel as shown in Fig. 1. These results reflect the accuracy we can expect, if we know the ground reflectance of water exactly. These errors occur due to the simplified modeling of the radiation transport in Eq. 1 and 6 – 10 only. 0.03 mean error

0.025

reflectance error

standard deviation 0.02 0.015 0.01 0.005 0 -0.005 -0.01 400

450

500

550

600

650

700

750

800

850

900

wave length (nm)

Fig. 1: Modeling error due to use of simplified equations (Eq. 1 and 6 – 10) instead of detailed radiation transport modeling like MODTRAN4. The simplified algorithms for the atmospheric corrections presented here have been analyzed in detail to estimate the magnitude of the expected errors. Sources of error are modeling uncertainties in the key atmospheric parameters L0, Tdown, Tup, and s, as well as the assumption of known ground reflectance for clear water or dark vegetation, for these atmospheric correction methods respectively. To determine the sensitivity of the reflectance with respect to the key atmospheric parameters, the following linearization is performed, using Eq. 2 and 3: ∆ρ L 0 ≈

∂ρ π ⋅ ∆L0 ≈ − ⋅ ∆L0 ∂L0 TdownTup S

(14)

The results are shown in Fig. 2 for the MERIS channels. Here an average transmittance has been assumed. The maximum difference between the MODTRAN4 calculations and the modified clear water method values is about 20 W/(m2 sr µm) at a wavelength between 600 and 700 nm. This results in a maximum error of approximately 8%. The reason for the large deviation at these wave lengths is the assumption of the known ground reflectance of clear water. In the same manner as for the path radiance the sensitivity of the calculated reflectance with respect to the transmittance is calculated: ∆ρT

ρ

≈−

∆Tup Tup

(15)

0 412.5nm -0.02

442.5nm

error reflectance

490nm 510nm

-0.04

560nm 620nm

-0.06

665nm 681.25nm 708.75nm

-0.08

753.75nm 778.75nm 865nm

-0.1

885nm -0.12 0

5

10

15

20

error path radiance [W/(m2 sr µm)]

Fig. 2: Reflectance errors due to path radiance errors The influence of the downward transmittance can be estimated by replacing E0 by S · Tdown. Although two different sets of constants are used to calculate the upward and downward transmittance, these transmittances and their modeling errors can not be expected to be completely independent of each other. Therefore the expression E0 · Tup in Eq. 1 could be replaced by S · Tdown · Tup ≈ S · T2 where T is an average transmittance considering the different air mass numbers due to the observer and sun zenith angle. This expression leads to a factor of 2 in the equation above and represents an approximate worst case scenario, where the modeling errors of upward and downward transmittance have the same sign and the same magnitude. This means that the relative reflectance error is proportional to the relative transmittance error. Errors in the spherical albedo show the following influence on reflectance errors: ∆ρs ≈

∂ρ ⋅ ∆s ≈ − ρ 2 ⋅ ∆s ∂s

(16)

This means that the reflectance error due to errors in the modeling of the spherical albedo will be very small compared to the others and might be neglected. The determination of the influence of deviations of the assumed ground reflectance for clear water or dark vegetation requires some more derivatives. Eq. 3 can be rewritten as f =

π ⋅ ( Lm − L0 )

(17)

TdownTup S

The derivative of ρ with respect to the assumed ground reflectance ρw is ∂f ∂s − f2⋅ ∂ρ w ∂ρ w ∂ρ = 2 ∂ρ w (1 + fs )

(18)

∂δ ∂s ∂s ∂δ 0 = ⋅ = ( 2e1 ⋅ δ 0 + e2 ) ⋅ 0 ∂ρ w ∂δ 0 ∂ρ w ∂ρ w

(19)

∂δ 0 ∂δ 0 ∂δ = ⋅ = − ( a1 ⋅ secϑsensor + a2 ⋅ secϑsun + a3 + a4 ⋅ cosϑsensor ⋅ cos ϑsun ) ∂ρ w ∂δ ∂ρ w

(20)

Using the partial derivatives:

 ∂L ∂Tup  ( L − L0 )  ∂T ⋅ 0 + m ⋅  Tdown + Tup down    ∂ρ TdownTup  ∂ρ w ∂ρ w    w ∂Tup ∂Tup ∂δ 0 ∂Tdown ∂Tdown ∂δ 0 ∂δ ∂δ = ⋅ = c3 ⋅ 0 ; = ⋅ = d3 ⋅ 0 ∂ρ w ∂δ 0 ∂ρ w ∂ρ w ∂ρ w ∂δ 0 ∂ρ w ∂ρ w

∂f π =− ∂ρ w TdownTup S

∂L0 ∂L0 ∂δ 0 ∂δ = ⋅ = b3 ⋅ 0 ; ∂ρ w ∂δ 0 ∂ρ w ∂ρ w

(21) (22)

We can calculate: ∆ρ ≈

∂ρ ⋅ ∆ρ w ∂ρ w

(23)

Fig. 3 shows the result of the linearization above. The magnitude and also the sign of the expected error depend on the wave length. At about 500 nm the influence of ground reflectance uncertainties is very small. Since the reflectance of clear water in the NIR (channels 10-14) is well known and the influence of the ground reflectance on the results is small for channels 3 and 4, the atmospheric correction will be significantly more accurate for these channels than for the others. 0.04 412.5nm

error corrected reflectance

0.02

442.5nm 490nm

0

510nm 560nm

-0.02

620nm 665nm 681.25nm

-0.04

708.75nm 753.75nm

-0.06

778.75nm 865nm

-0.08

885nm -0.1 0

0.01

0.02

0.03

0.04

0.05

error clear water reflectance

Fig. 3:

3

Influence of assumed clear water reflectance on corrected ground reflectance

RESULTS

In this chapter some exemplary results of the different methods are shown to demonstrate what the advantages and disadvantages of these methods are. Fig. 4 and Fig. 5 show the TOA reflectance compared to the results of the three different methods for two different scenes. The images shown are RGB generated using channels 7, 5 and 3 corresponding to the wave lengths 665, 560 and 490 nm.

Fig. 4:

Results for southern Germany; left to right: TOA reflectance, modified clear water method, modified dark vegetation method, modified Ångström method

Fig. 5:

Results for Kansas; left to right: TOA reflectance, modified clear water method, modified dark vegetation method, modified Ångström method

These images give a qualitative impression. While the results of the modified dark vegetation method look quite realistic, the modified clear water method always seems to underestimate the atmospheric reflectance in the green,

resulting in the green touch of these pictures. This indicates that the assumption of the clear water reflectance might be too high in the green channels. But for the NIR the results of the modified clear water method can be expected to be better than those of the modified dark vegetation method, because the reflectance of water is known to be very small here. The reflectance of vegetation is very high in the NIR and an extrapolation was necessary to apply the modified dark vegetation method to the NIR channels. The modified Ångström method produces images that look quite realistic, too, but should be used carefully, because of the assumption of a very clear atmosphere. Fig. 6 shows the average reflectance for land in two different scenes corrected with various algorithms. Additionally to the methods described in this paper, the TOA reflectance as well as the results using the SMAC method [5] with an aerosol optical depth of 0.2 at λ = 550 nm are plotted for comparison. 0.35

0.35

0.3 reflectance

reflectance

0.3 0.25 0.2

angstrom method clear water method

0.2

dark vegetation method smac algorithm

0.15

0.15 0.1

0.1

0.05

0.05

0 400

TOA reflectance

0.25

500

600

700

800

900

0 400

500

wave length [nm]

Fig. 6:

4

600

700

800

900

wave length [nm]

Results for southern Germany (left) and Kansas (right)

SUMMARY AND CONCLUSIONS

The modified clear water method works well in the infrared, since the water reflectance is known to be very small. But in the visible wave length every lake has its own spectral signature, depending on turbidity, sediments, algae etc. Therefore the uncertainties remain large in the visible spectrum for this method. Since the modified dark vegetation method cannot be directly applied to the infrared channels, the correlation between the atmospheric radiative properties in the red and the infrared is used to extend the modified dark vegetation method to the infrared. The third method is restricted to very clear atmospheric conditions. It will underestimate the influence of the atmosphere for hazy conditions. This method is advantageous if applied above mountains since the ground level elevation is considered in the algorithm. To validate the results we will apply these algorithms to several MERIS data and compare the derived optical depth of the different methods to measurements performed by the observatories Lindenberg and Hohenpeißenberg of the German Weather Service. Regarding the modified clear water method, more measurements of the ground reflectance of water would be helpful to improve the quality of the results, especially in the visible wave length range.

5 [1] [2] [3] [4] [5] [6]

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