MODIS Aerosol Retrieval Algorithm for Ocean

Over the ocean, the MODIS algorithm retrieves aerosol optical depth (OD) as well as information about the aerosol size distribution (the ratio between the two size modes and the mean particle size). Pre-assumptions on the general structure of the size distribution are required in the inversion of MODIS data. The volume-size distribution is described with two log-normal modes: a single mode to describe the accumulation mode particles (radius < 0.5 µm) and a single coarse mode to describe dust and/or salt particles (radius > 1.0 µm).

A. The Actual Measurement

Observed top of the atmosphere (TOA) radiances (L) at seven wavelengths (0.47, 0.55, 0.66, 0.87, 1.24, 1.64 and 2.13 µm) are measured by the MODIS instrument. From these, one calculates reflectances:

rlm = pL/cos(solar zenith angle)

where L = radiance, l = wavelength, m = measured.

B. The look-up table

A look-up table (LUT) is used in the aerosol retrieval. The LUT contains many sets of values, each set consisting of the pre-computed reflectances (expected r, or rexp) for each of the seven wavelengths listed above. The reflectances in the lookup table are computed using aerosol models that represent realistic possibilites for the aerosol properties of a vertical column of air, globally. The aerosol models are based on real world studies involving in-situ measurements and sun photometer data. Periodic revisions of the models are undertaken, and the data products distributed by NASA are revised when this occurs. Each model has somewhat different values for refractive indices at particular wavelengths, median particle radius, standard deviation of particle radius, and effective particle radius (see table below). Currently (2002), nine tropospheric aerosol models are used, including four "small" mode models (accumulation mode: dominated by chemical and combustion processes) and five "large" mode models (dominated by maritime particles (sea salt) and dust). Different lognormal models are used for each of the aerosol models. They are described by the geometric mean radius (rg), and the width of the distribution, s. The effective radius (reff) is the volume averaged radius of the distribution.

Pre-set Aerosol Properties for the Nine Models (as of July 2002)

Model type of aerosol
Refractive Index at l=0.47 to 0.87µm
Refractive Index at l=1.24µm
Refractive Index at l=1.64µm Refractive Index at l=2.13µm
median radius (rg)
standard deviation (s)
effective radius (reff)
1-small wet water soluble
1.45-0.0035i
1.45-0.0035i
 1.43-0.01i 1.40-0.005i
0.07
0.40
0.10
2-small wet water soluble
1.45-0.0035i
1.45-0.0035i
 1.45-0.0035i 1.40-0.005i
0.06
0.60
0.15
3-small water soluble w. humidity
1.40-0.0020i
1.39-0.005i
 1.39-0.005i 1.36-0.003i
0.08
0.60
0.20
4-small water soluble w. humidity
1.40-0.0020i
1.39-0.005i
 1.39-0.005i 1.36-0.003i
0.10
0.60
0.25
5-large sea salt, wet
1.45-0.0035i
1.45-0.0035i
 1.43-0.0035i 1.43-0.0035i
0.40
0.60
0.98
6-large sea salt, wet
1.45-0.0035i
1.45-0.0035i
 1.43-0.0035i 1.43-0.0035i
0.60
0.60
1.48
7-large sea salt, wet
1.45-0.0035i
1.45-0.0035i
 1.43-0.0035i 1.43-0.0035i
0.80
0.60
1.98
8-large dust-like
1.46-0.000i
1.46-0.001i
 1.46-0.000i 1.46-0.000i
0.60
0.60
1.48
9-large dust-like
1.46-0.000i
1.46-0.001i
 1.46-0.000i 1.46-0.000i
0.50
0.80
2.50
Each "entry" in the LUT is set of reflectances (rlexp for seven l) obtained by choosing and inserting into optical formulae (1) the properties that define the model (as shown above), and (2) different pre-set values for satellite observation angles and optical depth at 0.55 µm. The optical formulae used are the radiative transfer code developed by Ahmad and Fraser [1981]. Using this code, spectral reflectances are computed for each of the nine aerosol models for all possible combinations of preset values for the following four parameters: 1. Aerosol columnar depth (total aerosol loading) at 0.55 µm: 5 values (0.0, 0.2, 0.5, 1.0, 2.0)
Note: 0.0 corresponds to a pure molecular (Rayleigh) atmosphere, and 2.0 to a highly turbid atmosphere.
2. Solar zenith angles: 9 values 3. Satellite zenith angle: 16 values 4. Relative sun/satellite azimuth angles (16 values) The radiative transer code uses the aerosol properties associated with a given model, plus the combinations of values for the 4 parameters listed above (amounting to 2304 combinations for each optical depth at 0.55 µm), to compute hypothetical optical depths at the other five wavelengths (0.47, 0.66, 0.87, 1.24, 1.64 and 2.13 µm). Assumptions: For each model, the modeled satellite signal is assumed to be a combination of radiation from the atmosphere and reflection from the surface. The atmospheric calculation accounts for multiple scattering by molecules and the aerosol, as well as reflection of the atmosphere by the sea surface. The ocean surface calculation includes three contributions: the Fresnel ("sun glitter") reflection off the surface waves, reflection by whitecaps and foam and Lambertian reflectance coming from underwater scattering (sediments, chlorophyll, etc). The surface wind speed (for sunglitter and foam calculations) is assumed fixed at 6.0 m/s. Zero water leaving radiance is assumed at all wavelengths, except for at 0.55 µm, where a reflectance of 0.005 is used. Due to variable/unknown ocean color properties (from chlorophyll and sediments), the 0.47 µm channel is not used in the aerosol retrieval (i.e. only six channels are used). However, the look-up table still includes modeled reflectance values for the 0.47 band.

C. The aerosol retrieval

Over the ocean, the MODIS algorithm retrieves aerosol optical depth (OD) as well as information about the aerosol size distribution (the ratio between small and large particle size modes and the mean particle size). As we stated earlier, pre-assumptions about the general structure of the size distribution are required in the inversion of MODIS data. The algorithm assumes that the aerosol volume-size distribution can be described with two log-normal modes: a single mode to describe the accumulation mode particles (radius < 0.5 µm) and a single coarse mode to describe dust and/or salt particles (radius > 1.0 µm). To perform the aerosol retrieval, a combined spectral reflectance (rlc) must be computed assuming that the overall particle size distribution is a combination of one of the small size modes and one of the large size modes. The reflectance (r) from two log normal modes (one small, one large) can be approximated by the weighted average of the two modes ("s" for small, "l" for large, "c" for combined), calculated for the same optical thickness:

rlc = (h)rls +(1-h)rll

Where h equals the proportion of total reflectance due to the small size mode. The parameter h becomes important in the retrieval steps outlined below.

The steps in the algorithm go something like this:

1) Read the "look-up table" into computer memory.
As described above, the lookup table includes "modeled" satellite reflectance for a number of satellite/sun/surface angle combinations for each of the 4 "small" modes and 5 "large" modes. Remember that for each geometry + size mode combination the lookup table includes a set of expected reflectances (for each l) to go with each of the 5 hypothesized values of optical depth at 0.55 µm (OD = 0.0, 0.2, 0.5, 1.0 or 2.0).
2) Interpolate look-up table geometry to measured (satellite) geometry.
Read in the "observed" satellite/sun/surface geometry, and interpolate the "modeled" satellite reflectance in the LUT to the "observed" geometry. Now you are left with a smaller portion of the look-up table which includes one entry for each size mode, giving the predicted reflectance (rlexp) yielded by the model parameters, given the real world observation geometry for the satellite measurement.
3) Estimate AOD for all of the wavelengths based on 0.87 µm bandwidth.
For each geometry corrected "entry" take the satellite-observed reflectance at 0.87 and compare it to the set of r0.87exp values associated with the five OD0.55. Using interpolation, find the estimated actual OD0.87. Use this to derive estimated OD for the other wavelengths. These estimated ODs will yield (kind of going backwards) the rlexp for each wavelength.
4) For each combination of modes in the LUT (for example: "small mode #1 and large mode #5")
a. Pick a value of h b. Compute the expected combined (small + large) reflectance (rlc,sl) for that combination, for each wavelength (l) (remember that rl is now corrected for geometry):

rlc,sl = (h)rls + (1-h)rll

 
c. Compare with satellite measurement to calculate an error term (elsl) for each wavelength, using the following equation (where rlm = satellite measured radiance at l):

elsl = [rlm - rlc,sl]/[rlc,sl + 0.01]

and next, summing over all wavelengths (except 0.47 µm) to get the total relative error:

esl = square root[1/nS(elsl)2]

 
d. Try another value of h and see what the new error term is.
Selecting values of h: The computer program starts with "100% small mode" (h = 1) vs 0% small (h = 0) and computes the error esl for that choice. Then it picks a value (say 50% small h = 0.5 ), and computes the error again. Lets say, that h = 1 was better than h = 0 , so now the algorithm picks another value say, h = 0.75 , and so on.
e. Keep "tweaking" total OD0.87 and h, until you minimize the total relative error term above (summing for six wavelengths). Eventually, you have a best estimate of the size ratio, h, for that given combination of small and large mode.

5) Keep going, repeating the steps above until you have used all 20 combinations of small and large mode (now you see why we need computers!). The final answer ("best" solution) is the combination of (1) small+large modes, (2) OD0.87, and (3) h, that gives the minimum error.

6) Go back to the look-up table. Using the winning small and large modes, and the winning OD0.87, discover the optical depths at all other bandwidths (0.47, 0.55, 0.66, 1.24, 1.64 and 2.13 µm). Note: many of the publically distributed data products for MODIS report results for 0.55 µm, rather than for all of the wavelengths).

7) NOW GO TO THE NEXT PIXEL!!!!