## Signal-to-Noise (SNR) and Uncertainty Estimates

As simple as this may sound, error estimate is quite complicated. Most significant errors might be from
systematic errors unrelated to noise. If the systematic error (such as recycle delay is too short
for spin to fully recover between scans) is very small, the following may be used to estimate noise
contribution to the peak height ratio.

SNR is measured as the ratio of the peak height over the

root-mean-square deviation (RMSD or standard deviation) value of the noise floor.
Good SNR is typically > 10, especially for publication. A SNR of 3 (3σ) is minimally acceptable with a probability,
also referred to as the confidence level, of the
peak not being random noise at ~ 99.7%. SNR and the noise level also put a limit on the uncertainties of a peak
integral, height or ratios between different peaks. However, SNR doesn't account for any systematic noise and artifacts.

Due to uneven intensities of the peaks, SNR is measured peak by peak against a common region of the noise floor.

Note that
SNR depends on the processing parameters as well as the instrument. More line-broadening with a window function
increases SNR and peak sharpening decreases SNR. More (but not too much) line-broadening can be used to suppress noise as
long as this does not introduce signal overlapping or broad peak base. Too much broadening, however, pushes the
intensity into the base which makes the base intensity difficult to integrate, leading to errors. Different amount of line broadening
with a window function is commonly used for different purposes with the same data sets.

**In VNMRJ:** First, process data with line-broadening (check parameter

**lb**), phasing, and basic baseline
correction (

**dc**). To estimate the SNR of a peak,
first zoom in to one peak and leave sufficient noise floor on either side or both sides of the peak, then enter

**dsn**
in the command line. If the region
contains more than one peak,
the SNR calculated is for the tallest peak.

**In Mnova:** Mnova has an SNR routine in a script but it is awkward to use. To estimate SNR:

- Process the spectrum to its final form.
- Go to
**Tools → Load Scripts → NMR Tools → SNR Calculation ...** and follow the instruction. You
will be asked to zoom into a noise region, then a signal region and pick peaks. SNRs for all picked peaks will be displayed.

### Estimate Uncertainty Originating from Random Noise in Ratio of Two Peak Heights

Given two peak heights A and B, and RMSD noise level Δp, the uncertainty in measured peak height ratio R = A/B is:

**ΔR = (ΔA*B - A*ΔB)/B**^{2}
= ΔA/B - A*ΔB/B^{2}

where ΔA and ΔB are the uncertainties in the peak heights.

We may subsitute both ΔA and ΔB with Δp, and use the measured SNR values for the two peaks,
SNR

_{A} = A/Δp and SNR

_{B} = B/Δp. But ΔA and ΔB can be either positive or negative,
requiring the two terms to be added as absolute values (positive). We take the absolute value and
get the uncertainty:

**
|ΔR| = 1/SNR**_{B} + R/SNR_{B}

The relative uncertainty over the ratio R is:

**|ΔR|/R = [1/SNR**_{B} + R/SNR_{B}]/R

For example, if the ratio of measured peak heights between A and B is 1.5, SNR of B is 10.0, the absolute
uncertainty in the ratio is: ΔR = 1/10.0 + 1.5/10.0 = ±0.25. The relative uncertainty is: ΔR/R = 0.25/1.5 = ±16.7%.
If SNR of B peak is 30.0, the uncertainty drops to ±0.083 with the relative uncertainty at ±5.5%.

### Another Approach to Error Estimate

The approach above gives the "maximum" uncertainty from noise contribution by defining the maximum range of the error spread.
Another approach,

as outlined here,
is to estimate the error involving the two variables (peak heights) according to the Pythagorean theorem, which leads
to the following uncertainty range for the peak height ratio:

**
(ΔR/R)**^{2} = (ΔA/A)^{2} + (ΔB/B)^{2} = (1/SNR_{A})^{2} + (1/SNR_{B})^{2}

Using this formula, the relative uncertainty of the example above, with SNR

_{A} = 15, SNR

_{B} = 10, is:
ΔR/R = ±0.12 or ±12%. The absolute uncertainty is: ΔR = 0.12*1.5 = ±0.18.

This error estimate is smaller than the approach above.

**For integral ratios**, it's trickier. It depends also on the width of each integral. If both peaks are integrated
with roughly the same width, the uncertainty in two integral ratio roughly matches that for the peak height ratio estimate above.

*Updated, 2019, H. Zhou*

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