I think a summary is that:
Background levels and errors thereof can be estimated very precisely as a
percentage of the level, but as an absolute number, it nevertheless swamps out
the signal contained in a Bragg peak. For example, if background = 10^6 photons
+/- 1000 (very good estimate as a percentage--0.1%), that still adds an
absolute number of +/- 1000 photons to a Bragg peak which might represent 1000
photons or so, so this noise is a big problem. One would, however, gain a lot
by having many pixels in each spot and profile-fitting them, as you mentioned,
as this would improve sampling and reduce error.
What about this comparison, though: either measure a photon count of 10^6 once
on a background of 10^7 or measure the same reflection 1000 times independently
at 10^3 photons on a background of 10^4. Assuming no readout noise or other
noise source, wouldn't the latter obviously be better? In a sense, each pixel
in the Bragg peak in the first case would really be 1000 pixels in the second,
and the noise from the background could be cancelled much more effectively....?
Admittedly this is an orthogonal issue to the background subtraction one, since
obviously as you demonstrated the background makes I/sig worse, but this does
make background subtraction markedly better, maybe even enough to warrant
always erring on the side of too much non-crystal stuff?
And definitely always arguing to measure data at low intensities multiple times
rather than once at high intensity, for a given total dose!
Jacob
-----Original Message-----
From: James Holton [mailto:jmhol...@lbl.gov]
Sent: Thursday, January 15, 2015 12:00 PM
To: Keller, Jacob;CCP4BB@JISCMAIL.AC.UK
Subject: Re: [ccp4bb] X-ray Source Differences (WAS: RE: [ccp4bb] How far does
rad dam travel?)
Jacob,
Background subtraction is unfortunately not as forgiving as you think it is.
You can subtract the background, but you can't subtract the noise.
This is because noise (by definition) is never the same twice. Yes, the "average" or
"true" background under a spot may be flat, but any given observation of it will be
noisy, and there is no way to separate the noise that came from the Bragg-scattered photons from
the background-scattered photons that hit the same pixel. Each photon is an independent event,
after all.
Consider an example: if you see 4 photons in an isolated spot after 1 second and there is zero
background then sigma(I) = sqrt(I) = sqrt(4) = 2, and your I/sigma is most likely 2. I write
"most likely" because the "true" photon arrival rate (the thing that is
proportional to F^2) doesn't have to be 4 photons/s just because you counted four in one
observation. The long-term average could easily be something else, such as 3 photons/s, 5
photons/s or even 3.2 photons/s (on average).
Observing 4 photons is not unlikely in all these scenarios. However, if you consider all possible
"true" rates, simulate millions of trials and isolate all the instances where you counted
4 photons you will find that the "true" rate of 4.0 photons/s turns up more often than
any other, so that's your best guess.
Nevertheless, if your "true" rate really is 4.0 photons/s, then the probability of actually seeing
4 photons is only ~20%! The other 80% of the time you will see something else. 20% of the time you will
count 3, there is a 15% chance you will only see 2, and a ~2% chance you will see zero. There is also a 2%
chance of seeing 11 photons in 1 second when the "true" rate is really 4. You just don't know. Of
course, with more observations you can narrow it down. If you do 100 trials and see an average of 4.0, then
you are obviously a lot more confident in the "true" rate of 4 photons/s. But this isn't observing
4 photons, this is observing 400 photons and dividing that number by 100. Because the error in counting 400
photons is sqrt(400)=20, your signal-to-noise is
400/sqrt(400) = 20! This is why multiplicity is a good thing. However, if all
you have is one sample of 4 photons your best guess is I = 4 and
sigma(I) = 2.
Now consider the case where there is background. Say the "true rate"
for the background is 10 photons/pixel/s, and for simplicity lets say your 4 photon/s spot lies entirely within one pixel. What
is your signal-to-noise now? Well, if you take 100 pixels in the neighborhood around your spot you will count ~1000 photons,
giving you an excellent estimate of the "true" background rate: 10 photons/pixel/s with a sigma of sqrt(1000)/100 =
0.3, or a "signal-to-noise" of 31.6 for the estimate of the background level. Looking at the one pixel containing your
spot, let's say you saw 14 photons in it, that means you have a "sigma" of 2 from the 4 "spot photons" and a
"sigma" of 0.3 from the background photons for a total sigma of sqrt(2^2+0.3^2) = 2.02 and I/sigma = 1.97, right?
Wrong.
The reality of the situation is the 14 photons that landed in your spot pixel were
independent events from all the photons that landed in the nearby background pixels.
Yes, you know that there "should" be 10, but the probability of actually
getting 10 exactly is only 12.4%. You are just as likely to see 11 or 9, and there is a
5% chance of 14 background photons hitting the spot area. In that case the Bragg
intensity could easily be zero, and the 14 photons you saw were just a random fluctuation
from the average of 10. You just don't know! And your error bars should reflect that.
The correct value for sigma(I) turns out to be the square root of the TOTAL number of
photons that hit the spot area: sqrt(14) = 3.7 and the I/sigma of your 4-photon spot is
now 1.07. With 100 background photons/pixel, your I/sigma = 0.4. This is how background
degrades your resolution limit.
The bright side of it is that the degradation of I/sigma rises only with the
square root of the background level, not the background level itself. As a
general rule: it takes 3x the Bragg photons to cut I/sigma in half, 15x the
Bragg photons to cut it to 1/4, etc. So, a factor of 2 in background is at
worst a 40% hit in I/sigma. Not bad, but not good either.
Now, since the noise from background is proportional to the square root of the
background level, the background level is proportional to the illuminated area,
and the illuminated area, in turn, is the square of the beam size (for a square
beam), the extra noise from a beam-crystal size miss-match is equal to the
excess linear dimension of the beam.
That is, if you have a situation where you would get 16 Bragg photons and 48
background photons into a given spot from a 10 micron crystal with a 10 micron
beam, that will give you I/sigma = 2. But if you use a
20x20 micron beam you have quadrupled the illuminated area. Assuming your 10
micron crystal is embedded in a block of vitrified water, for the same dose to
the crystal (photons/area) you will still get 16 Bragg photons but 192
background photons in the spot area pixels, reducing your I/sigma to 1.1. Most
people would rather not do that.
So, yes, smaller beams are better if your crystal is actually small, and it is surrounded by
"stuff" of similar thickness, density and elemental composition. You also need to bear
in mind the "stuff" that is in the beam path before and after your crystal because this
is part of the illuminated volume too. Ideally, you want your crystal sticking out into the air,
then any beam size is more or less equivalent because
air/N2 scatters 1000x less than the solid stuff in the loop. The only problem with that
is a lot of crystals hate surface tension. This is why I recommend having surrounding
"stuff", but no too much. A factor of 2 in volume is a good compromise.
Yes, there is such a thing as big crystals with a "sweet spot" that can only be
accessed with a small beam and there is definitely a lot of excitement about that. What
I keep wondering is: what went wrong in the rest of that crystal?
Brighter beams are better for getting your experiment over more quickly,
but if you're attenuating then those extra photons are going to waste. Faster
detectors can help with this, but if they are too fast they will start picking
up high-frequency noise in the beam. This is a problem for anomalous, but not
for resolution. 5% error when you are only counting 4 photons is a minor
consideration. Background is the primary enemy of resolution. After disorder,
of course!
As for where to put resources, I try not to think about which aspect of structural
biology should be cut in favor of some other part because I think structural
biology as a whole is important! Especially when you consider how much money is
wasted on <insert random political statement
here>.
Detectors can always be better, but at the moment low-end noise levels and
speed are not limiting factors. The challenges are either 1) detecting weak
spots (aka resolution) or 2) accurately measuring small differences between
strong spots (aka anomalous differences). 1) is limited by pixel count and 2)
by calibration. I say pixel count because larger active areas are always
better for background reduction (inverse square law), but only if your spots
take up more than a few pixels. If your spots are all smaller than a pixel
then your pixels are too big.
Colin Nave (JSR, 2014) has calculated that the ideal MX detector would have
about 1e9 pixels in it. Only problem with that is the going rate for a pixel
these days is ~$0.25 each. For anomalous, the biggest problem with detectors
is calibration, which is a lot harder to deal with than you might think. The
best evidence of this fact is that if you simulate data with every kind of
noise you can think of you still get low-resolution R-meas values of ~0.5%
(Holton et al, FEBS 2014, Diederichs, 2009). I have never seen a real dataset
like that.
Nevertheless, if you count 1,000,000 photons, the sigma of that count is 1000,
or 0.1% error. Something else is getting in the way. Unfortunate really,
because if we could routinely get R-meas = 0.1% we would never need to make
metal derivatives again.
-James Holton
MAD Scientist
On 1/8/2015 9:47 AM, Keller, Jacob wrote:
Yes, this is great info and thoughts. What I still do not understand, however,
is why the noise from air/loop scattering is so bad--why not make sure only the
top of the Gaussian is engulfing the crystal, and the tails can hit air or
loop? Isn't the air scattering noise easily subtractable, being essentially
flat over time, whereas uneven illumination of the crystal is highly difficult
to correct?
Also, in light of these considerations, it would seem to me a much better use
of resources not to make brighter and smaller beams but instead concentrate on
making better low-intensity big beam profiles (top-hats?) and lower-noise,
faster detectors (like Pilatus and the new ADSC).
Jacob
-----Original Message-----
From: James Holton [mailto:jmhol...@lbl.gov]
Sent: Tuesday, December 30, 2014 3:57 PM
To: Keller,Jacob;CCP4BB@JISCMAIL.AC.UK
Subject: Re: [ccp4bb] How far does rad dam travel?
Yes, bigger is okay, and perhaps a little better if you consider the effects of
beam/crystal vibration and two sharp-edged boundaries dancing over each other. But
bigger is better only to a point. That point is when the illuminated area of
non-good-stuff is about equal to the area of the good stuff. This is because the total
background noise is equal to the square root of the number of photons and equal volumes
of any given "stuff" (good or non-good) yield about the same number of
background-scattered photons. So, since you're taking the square root, completely
eliminating the non-good-stuff only buys you a gain of 40% in total noise. Given that
other sources of noise come into play when the beam and crystal are exactly matched
(flicker), 40% is a reasonable compromise. This is why I recommend loop sizes that are
about 40% bigger than the crystal itself. Much less risk of surface-tension injury, and
the air around the loop scatters 1000x less than the non-crystal stuff in the
loop: effectively defining the "beam size".
As for what beam profiles look like at different beamlines, there are some
sobering mug-shots in this paper:
http://dx.doi.org/10.1107/S0909049511008235
Some interesting quirks in a few of them, but in general optimally focused beams are Gaussian. Almost by
definition! (central limit theorem and all that). It is when you "de-focus" that things get really
embarrassing. X-ray mirrors all have a "fingerprint" in the de-focused region that leads to
striations and other distortions. The technology is improving, but good solutions for "de
focusing" are still not widely available. Perhaps because they are hard to fund.
Genuine top-hat beams are rare, but there are a few of them. Petra-III is particularly proud of
theirs. But top-hats are usually defined by collimation of a Gaussian and the more x-rays you have
hitting the back of the aperture the more difficult it is to control the background generated by
the collimator. If you can see the shadow of your pin on the detector, then you know there is a
significant amount of "background" that is coming from upstream of your crystal! My
solution is to collimate at roughly the FWHM. This chops off the tails and gives you a tolerably
"flat" beam in the middle.
How much more intense is the peak than the tails? Well, at the FWHM,
the intensity is, well, half of that at the center. At twice that
distance from the center, you are down to 6.2%. The equation is
exp(-log(16)*(x/hwhm)**2) where "hwhm" is 1/2 of the FHWM.
HTH!
-James Holton
MAD Scientist
On 12/30/2014 12:10 PM, Keller, Jacob wrote:
Yes, it gets complicated, doesn't it? This is why I generally
recommend
trying to use a beam that matches your crystal size.
...or is bigger, right? Diffuse scattering, yes, but more even illumination
might be worth it?
Generally, James, I have a question: what is the nature of the intensity
cross-sections at most beamlines--are they usually Gaussian, or are some
flatter? Or I guess, if Gaussian, how much more intense is the peak than the
tails?
JPK
