Diffusion-ordered NMR spectroscopy (DOSY) constructs multidimensional spectra displaying signal strength as a function of Larmor frequency and of diffusion coefficient from experimental measurements using pulsed field gradient spin or stimulated echoes. Peak positions in the diffusion domain are determined by diffusion coefficients estimated by fitting experimental data to some variant of the Stejskal–Tanner equation, with the peak widths determined by the standard error estimated in the fitting process. The accuracy and reliability of the diffusion domain in DOSY spectra are therefore determined by the uncertainties in the experimental data and thus in part by the signal-to-noise ratio of the experimental spectra measured. Here the Cramér–Rao lower bound, Monte Carlo methods, and experimental data are used to investigate the relationship between signal-to-noise ratio, experimental parameters, and diffusion domain accuracy in 2D DOSY experiments. Experimental results confirm that sources of error other than noise put an upper limit on the improvement in diffusion domain accuracy obtainable by time averaging.

The utility of pulsed field gradient spin or stimulated echo (PFGSE)
experiments for distinguishing between the NMR signals of different species
was first pointed out by Stilbs (Stilbs, 1981), but practical applications
of this principle only became common with the introduction of
diffusion-ordered spectroscopy (DOSY) by Morris and Johnson (1992).
In DOSY (Johnson, 1999; Morris, 2007), a pseudo-2D (or higher-dimensional) spectrum is synthesized in which the signals of an NMR spectrum are
dispersed into an extra dimension according to the estimated diffusion
coefficient

One common analogy is that DOSY is akin to performing chromatography within
an NMR tube, separating spectra rather than physically separating analytes.
The name DOSY is, however, misleading in some respects. In conventional 2D
NMR methods such as COSY, NOESY, and TOCSY, the 2D spectrum can be obtained by direct Fourier transformation of signals that are phase or amplitude
modulated as a function of an evolution period

In simple mixtures in which the NMR signals are well resolved and the
individual species have very different diffusion coefficients, even a crude
DOSY experiment will work well. Where species of similar size, and hence
similar

While it is to be hoped that a clearer understanding of the role that signal-to-noise ratio plays in limiting the quality of DOSY spectra will prove useful, it should be stressed that SNR is just one of many factors involved. In particular, the analysis presented here takes no account of the effects of the systematic and reproducible experimental imperfections that all DOSY experiments are affected by. These include for example the spatial non-uniformity of pulsed magnetic field gradients (Damberg et al., 2001; Connell et al., 2009) and the effects of peak overlap (Botana et al., 2011). Questions such as choosing the optimum balance between time averaging and the number of different field gradient values to be used require many different factors to be taken into account, of which SNR is just one.

In its commonest (“high-resolution”) form, DOSY uses least squares fitting of the amplitudes of peaks in pulsed field gradient echo spectra to
determine diffusion coefficients

Experimental data are imperfect, most notably because of the presence of a
background of random electronic noise. In a well-conducted experiment the
effect of this on the measurement of the amplitude

A convenient measure of resolution

Expressions (4) and (5) allow direct calculation of

Experimental

Equation (4) shows that, as is intuitively reasonable, the diffusion
resolution is directly proportional to SNR (provided that systematic sources of
error are negligible). The proportionality constant is, however, a
complicated function of the choice of sampling function and its relation to
the diffusion coefficient: the more data points are measured, the better

For a given set of experimental delays and pulse durations, linear and
quadratic spacing in

Diffusion resolution

The predicted diffusion resolution

Relative diffusion resolution

Fitted parameters for Eq. (11) obtained from the data of Fig. 2. No error estimates are given as the data fitted are not normally distributed.

In principle, diffusion accuracy should increase indefinitely as the
signal-to-noise ratio of the experimental data increases. (“Accuracy” is
used here in the sense of the reliance that can be placed on the positions
of peaks in the diffusion dimension of a DOSY spectrum, i.e. the
“trueness” of the diffusion dimension.) In practice diffusion accuracy does not increase indefinitely, because spectral noise is far from the only
source of uncertainty in the signal attenuations measured in DOSY
experiments. Radio-frequency pulse irreproducibility, field-frequency ratio instability, gradient noise, temperature variation, and a range of other
sources all limit the reliability of signal intensity measurements in NMR,
limiting resolution in DOSY and causing

There is thus a practical limit to the benefits to be gained by increasing
SNR, whether by time averaging, increasing the signal strength (e.g. by
increasing sample concentration), or reducing the noise (e.g. by using a
cold probe and preamp). This is illustrated here with experimental data
obtained as described earlier for the methoxy signal from a sample of
quinine. The starting SNR of the quinine methoxy peak in the lowest gradient
spectrum was 14 400 : 1; successively greater amounts of synthetic Gaussian
noise were added and fitting repeated, averaging the results of 100
additions, to show the influence of SNR on the diffusion resolution

To investigate the effect of signal-to-noise ratio on diffusion resolution,
synthetic noise was added to the experimental data used to construct the

500 MHz Oneshot

Diffusion resolution

It is well known that the signal-to-noise ratio of diffusion-weighted experimental NMR data plays a critical role in determining the diffusion resolution of a DOSY spectrum constructed from it. There is thus a temptation to conduct very long experiments with extensive time averaging in order to obtain the best possible results. Conversely, in dilute systems the temptation is to conduct equally long experiments in the hope of obtaining results with sufficient diffusion resolution to shed light on speciation, etc. In both cases it is possible, and indeed common, to waste a great deal of instrument time for no good result, either because sources of error other than noise dominate the fitting statistics, or because the final signal-to-noise ratio is insufficient. Here it is shown that a trivial calculation with Eq. (11) will show both whether or not such experiments may be worth attempting in the first place, and what limiting diffusion resolution is achievable.

Raw experimental data for Fig. 3 and the Mathematica code used to generate Figs. 1, 2, and 4 can be downloaded from DOI

GAM and MN designed the experiments and simulations. JG and PK carried out the experimental work. JG and GAM performed the simulations and analysis. GAM prepared the manuscript with contributions from all the co-authors.

Mathias Nilsson is an editor of

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This article is part of the special issue “Geoffrey Bodenhausen Festschrift”. It is not associated with a conference.

This work was supported by the Engineering and Physical Sciences Research Council (grant numbers EP/N033949/1 and EP/R018790/1).

This research has been supported by the Engineering and Physical Sciences Research Council (grant nos. EP/R018790/1 and EP/N033949/1).

This paper was edited by Fabien Ferrage and reviewed by two anonymous referees.