Noise absorbance

Figure 42-2 Absorbance noise as a function of transmittance, for the exact solution (upper curve equation 42-32) and the approximate solution (lower curve equation 42-33). The noise-to-signal ratio, i.e., AE/Et was set to 0.01. (see Color Plate 3)...

We begin our investigation of the behavior of the absorbance noise by comparing it to the theoretical expectation from the low-noise condition according to equation 42-32 [3], This comparison is shown in Figures 44-11 a-1 and 44-1 la-2. These figures show what we might expect that as the S/N increases the computed value approaches the theoretical... 

Figure 44-lla-l Comparison of computed absorbance noise to the theoretical value (according to equation 44-32), as a function of S/N ratio, for constant transmittance (set to unity), (see Color Plate 12)... 

There is, however, something unexpected about Figure 44-1 la-1. That is the decrease in absorbance noise at the very lowest values of S/N, i.e., those lower than approximately Er = 1. This decrease is not a glitch or an artifact or a result of the random effects of divergence of the integral of the data such as we saw when performing a similar computation on the simulated transmission values. The effect is consistent and reproducible. In fact, it appears to be somewhat similar in character to the decrease in computed transmittance we observed at very low values of S/N for the low-noise case, e.g., that shown in Figure 43-6. 

Similarly, in Chapter 44, we have previously derived the absorbance noise and relative absorbance noise, and presented those as equations 44-24 and 44-77, respectively. 

Now that we have completed our expository interlude, we continue our derivation along the same lines we did previously. The next step, as it was for the constant-noise case, is to derive the absorbance noise for Poisson-distributed detector noise as we previously did for constant detector noise. As we did above in the derivation of transmittance noise, we start by repeating the definition and the previously derived expressions for absorbance [3],... 

Thus, in the constant-noise case the absorbance noise is again proportional to the N/S ratio, although this is clearer now than it was in the earlier chapter there, however, we were interested in making a different comparison. The comparison of interest here, of course, is the way the noise varies as T varies, which is immediately seen by comparing the expressions in the radicals in equations 47-94 - for Poisson noise and 47-96. 

Also, as equation 47-94 shows, the absorbance noise is again inversely proportional to the square root of the reference signal, as was the transmittance noise. And once again we remind our readers concerning the caveats under which equation 47-94 is valid. 

We present the variation of absorbance noise for the two cases (equations 47-94 - for Poisson noise and 47-96, corresponding to the Poisson noise and constant noise cases) in Figure 47-18. While both curves diverge to infinity as the transmittance —0 (and the absorbance - oo), the situation for constant detector noise clearly does so more rapidly, at all transmittance levels. 

Figure 47-18 Comparison between absorbance noise for the constant-detector noise case and the Poisson-distributed detector noise case. Note that we present the curves only down to 7 = 0.1, since they both asymptotically oo as r 0, as per equations 94 and 96.

We keep learning more about the history of noise calculations. It seems that the topic of the noise of a spectrum in the constant-detector-noise case was addressed more than 50 years ago [1], Not only that, but it was done while taking into account the noise of the reference readings. The calculation of the optimum absorbance value was performed using several different criteria for optimum . One of these criteria, which Cole called the Probable Error Method, gives the same results that we obtained for the optimum transmittance value of 32.99%T [2], Cole s approach, however, had several limitations. The main one, from our point of view, is the fact that he directed his equations to represent the absorbance noise as soon as possible in his derivation. Thus his derivation, as well as virtually all the ones since then, bypassed consideration of the behavior of noise of transmittance spectra. This, coupled with the fact that the only place we have found that presented an expression for transmittance noise had a typographical error as we reported in our previous column [3], means that as far as we know, the correct expression for the behavior of transmittance noise has still never been previously reported in the literature. On the other hand, we do have to draw back a bit and admit that the correct expression for the optimum transmittance has been reported. 

As usual, we will continue in the next chapter, where we will discuss the various aspects of absorbance noise that are of concern. 

Figure 51-28 Absorbance noise for Poisson-distributed data at low values of the reference signal.

Figure 51-29 Relative absorbance noise for Poisson-distributed data, determined by numerical computation using equation 51-77. Figure 51-29b is an ordinate expansion of Figure 51-29a. (see Color Plate 18)...

Let us now, as we normally do, continue to derive the expressions for absorbance noise again referring to our previous chapter [8], we can start with equation 52-29 ... 

Here again, in the low-noise case of scintillation noise, the absorbance noise is again independent of the reference signal level, and is now independent of the sample characteristics, as well, and depends only on the magnitude of the external noise source. 

In conformance with our regular pattern, we now derive the behavior of the relative absorbance noise for the low-noise case. Here we start with equation 52-100, the derivation of which is found in [9] ... 

Equation 52-149 presents a minor difficulty one that is easily resolved, however, so let us do so the difficulty actually arises in the step between equation 52-148 and 52-149, the taking of the square root of the variance to obtain the standard deviation conventionally we ordinarily take the positive square root. However, T takes values from zero to unity that is, it is always less than unity, the logarithm of a number less than unity is negative, hence under these circumstances the denominator of equation 52-149 would be negative, which would lead to a negative value of the standard deviation. But a standard deviation must always be positive clearly then, in this case we must use the negative square root of the variance to compute the standard deviation of the relative absorbance noise. 

In Figure 52-30 we plot the function -1 /ln(T) to complete this part of the analysis. We note that there is no minimum to the curve, and the noise from source continually improves as the transmittance decreases in this case the previous, conventional derivations agree with our results, although they do not indicate the V2 factor. Noting the transitions from equation 52-140 to 52-142 (and the corresponding portions of the derivation for absorbance noise and relative absorbance noise), we see that this factor arises from the equal noise contributions of the sample and reference channels therefore we conclude that in this case also, the missing factor is due to the neglect of the reference channel noise contribution. 

For a CCD detector the absorbance noise is independent of the spectral bandwidth, but it depends on the number of measurement pixels sam and reference pixels ref in such a way that sam should be as small as possible and rel should be larger than sam. The other component that influences the noise is the intensity I of the radiation source, in that the absorbance noise is inversely proportional to the square root of I [12]. As the intensity of the radiation source in CS AAS is in some cases up to two orders of magnitude higher than that of a typical LS for conventional AAS, an improvement in the SNR and limits of detection (LoD) by factors of 3-10 could be expected, unless other factors, such as flame noise, become dominant. The values given in Table 4.1 show that this expectation has in fact been realized for the majority of the elements. 

As a result of the almost perfect correlation of the spectral intensity values within the small range of observation, the minimum detectable absorbance signal is determined only by statistical variations of the intensity between the neighboring pixels (shot noise). This means that an increase in radiation intensity or of the measurement time by a factor of 4 will reduce the absorbance noise by a factor of 2 (square root of 4). 

Figure 4.15. Absorbance noise for different wavelength regions, using a measurement time of 5 s each (a) 1 mg l-1 Se at 196.026 nm (b) 0.01 mg l-1 Cd at 228.802 nm (from Heitmann et al. [19]).

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