Error budgetPosted by Peter Thejll Aug 11, 2013 11:13AM
We have some observationally based estimates of B-V for earthshine. We are working on various models of the same that would allow us to check our assumptions and set error limits.
A somewhat sophisticated model of the color (B-V) of earthshine has been made. It consists of a Monte Carlo simulation where scattering events are decided by random number generators and probabilities bazsed on albedos.
I draw photons from the wavelength distribution given by the Wehrli 1985 Solar spectrum. For each photon we allow a Rayleigh scattering event to occur with probabilities depending on the wavelength. If the photon is not Rayleigh scattered its further fate is given by factors describing the fraction of the surface covered by land (0.3), and the albedos of clouds, land (either Soil or Vegetation), and ocean. Some photons are absorbed while others are scattered into space. The scattered photons
are collected and the emergent spectrum built up. If repeated for a large number of Monte Carlo trials a smooth spectrum results. The B-V colour of this is then calculated, using Bessel transmission curves for the Johnson B and V filters. The trials are done assuming that all land is covered by either vegetation or is bare soil. Cloud albedos
are wavelength-independent, while ocean albedos are zero above 750 nm. Tables of albedos for land types and ocean is given by "Spectral signatures of soil, vegetation and water, and spectral bands of LANDSAT 7.Source: Siegmund, Menz 2005 with modifications." found at this link:http://bit.ly/14jD3qm.
The B-V of the incident solar spectrum is calculated as a check. The expected solar value of 0.623 (+/- 0.001) is recovered, where the uncertainty is the S.D. of the mean over the trials. The B-V values of the emergent spectra are subject to stochastic noise, so a number (20, each with 2 million photons) of trials is performed and the mean of derived photometric values, with standard deviation of the mean, calculated.
The results for earthlight are:
B-V of earthlight, assuming all land is vegetation-covered:
0.5692 +/- 0.00057
where the uncertainty is the S.D. of the mean.
B-V of earthlight, assuming all land is bare soil:
0.5759 +/- 0.00068
where the uncertainty is the S.D. of the mean.
The difference is 0.0067 +/- 0.00088 with earthshine due to the soil-only model being redder than the vegetation-only model. The difference in B-V is small, but significant.
A validating test is to look at how much flux is taken out at 550 nm - we know that extinction at that wavelength is near k_V = 0.1 so about 10% of the flux is removed. Since Rayleigh scattering is symmetric in the forward and backward direction (for single scattering events) we estimate that half of the light removed from the beam is scattered into space. Thus we expect the flux of Rayleigh scattered light to be about 5% of the incoming flux. We measure the Rayleigh/Incoming flux ratio to be 0.0521 +/- 0.0003. That is, as expected, about 5% of the sunlight is scattered by Rayleigh scattering into space, a similar amount being scattered downwards and giving us the blue sky, and totalling an extinction at 550 nm of near k=0.1. This assumes all absorption is scattering due to Rayleigh. At 550 nm that is not a bad approximation.
For the B band at 445 nm we find that on average 2.3 times more flux is taken out by Rayleigh, corresponding to an extinction coefficient 2.3 times larger than in V. That is a bit steep, surely? Chris has measured k_B=0.15 from extinction data.
So, the model suggests that when you switch all land areas from vegetation only to soil only the B-V will redden by less than 0.01. The color of both modes is near B-V=0.57, with realistic land-ocean, and cloud fractions, as well as albedos. The model indicates a colour very close to the cruder model from the paper (B-V=0.56 at k=0.1). We use the exercise here to estimate that errors due to omission of land- and ocean- albedo is of the order 0.01 mags, and that B-V of earthlight is near 0.56.
Now, what did we find observationally?
We found DS B-V is 0.83. We also know that the sunlight has B-V=0.64 but appears as Moonlight at B-V=0.92. so that we can infer that one reflection on the Moon reddens light by 0.28 in B-V. The earthlight is also reddened by striking the Moon and we can infer that our observation of the DS implies earthlight that is 0.28 bluer, or B-V=0.83-0.28=0.55. That is very close to what our model suggests!
Note that all uncertainties here are governed by photon statistics -
not underlying physics or observational skills. Note also that we use
the phrase 'earthlight' when we mean the light scattered by Earth and
observed in space. By 'earthshine' we shall mean the color of the DS of
the Moon, which is illuminated by the earthlight. We have not even tried
to model the effects of hemispheric geometry, or any effects of
reflectance-behaviour on angles.
--------- added later -------------
This was done under an assumption that corresponds to 'airmass is 1
everywhere' - this is wrong, of course, since airmass for those surface elements near the edge of the terrestrial disc is larger than
1. It turns out that if you assume that airmass goes as sec(z) then the
area-weighted effective airmass for a hemisphere viewed from infinity is
2. We repeat the above trials with twice as much Rayleigh scattering.
The colours we get now are:
Assuming all land is plants : B-V = 0.5063 +/- 0.0009
Assuming all land is bare soil: B-V = 0.5152 +/- 0.00075
The difference is: 0.0089 +/- 0.0012
The Earthlight is now bluer than under the airmass=1 assumption (good), and the difference in turning land pixels from bare soil to vegetation is still small.
If we assume complete ignorance of the soil/vegetaion mix we get:
B-V earthlight, for airmass=1 : 0.57 - 0.58
B-V earthlight, for airmass=2 : 0.51 - 0.52.
The difference between airmass=1 (not realistic) and airmass=2 (realistic) is 0.06 mags.
I conclude that the model indicates a B-V of earthlight, at the most realistic assumption of the effective or average airmass involved in Rayleigh scattering, near 0.515 with a small dependence on wheather the land is vegetation or bare soil on the order of 0.01 at most.
I should repeat the above with VE1 and VE2 filters to see what change in NDVI vegetation index we might expect.
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