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# Signal-to-noise ratio in diffusion-ordered spectroscopy: how good is good enough?

### Jamie Guest

### Peter Kiraly

### Mathias Nilsson

### Gareth A. Morris

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.

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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 *D*. This is obtained by fitting experimental measurements of
signal attenuation as a function of pulsed field gradient amplitude to a
theoretical model, usually some variation on the Stejskal–Tanner equation (Stejskal and Tanner, 1965; Sinnaeve, 2012). The value added by the DOSY approach over
simple PFGSE measurements is that since all signals from spins within a
given species should show the same diffusion, in favourable cases
cross sections through the DOSY spectrum at different *D* values give separate spectra – which can be interpreted just as normal 1D spectra – for each of the components of a mixture. This paper examines the impact of one crucial determinant of the success or failure of a DOSY experiment, the
signal-to-noise ratio (SNR) of the experimental data.

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 *t*_{1}. The frequency
*F*_{1} at which a given signal appears is determined directly by the
frequency of evolution in *t*_{1}: while the phase or amplitude of a signal
may behave unexpectedly, the frequency is determined directly by the quantum
mechanics, so signals should always appear at the “correct” frequency. In
pseudo-2D methods such as DOSY (and relaxation-based analogues, often
referred to as relaxation-ordered spectroscopy, ROSY; Lupulescu et al., 2003;
Gilard et al., 2008; Nishiyama et al., 2010; Dal Poggetto et al., 2017), this is not the case: the diffusion dimension is a statistical construct, and the positions of
signals in the diffusion dimension are scattered about the true *D* values.
When a DOSY spectrum is constructed, peaks in the diffusion domain are
conventionally given Gaussian shapes with widths that reflect the
uncertainty in *D* estimated from the fitting statistics. Thus, in COSY spectra, peaks with the same chemical shift are exactly aligned; in DOSY spectra,
peaks with the same diffusion coefficient have Gaussian shapes that should
overlap but are not coincident. This is just one reason why the
interpretation of DOSY spectra demands more of the spectroscopist's skill
and judgment than most other types of NMR spectrum; others include the
effects of signal overlap and of systematic errors introduced by imperfect
experiments.

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 *D*, are to be resolved, however, high-quality experimental data are essential. One of the key determinants of the utility of a DOSY spectrum is
its diffusion resolution, the minimum difference in *D* that can safely be
distinguished. In an ideal experiment, this is determined by the
signal-to-noise ratio of the experimental data. Here we use theory, empiricism and simulated and experimental data to answer some key questions. How good do our experimental data need to be to resolve a given difference in *D*? How is the uncertainty in *D* related to the signal-to-noise ratio (SNR) of raw experimental data, and can this relationship be expressed in a simple form? At what point do improvements in SNR stop translating into improved resolution in the diffusion domain?

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 *D*. A series of *N* otherwise identical
experiments is carried out in which the amplitudes *G* of diffusion-encoding field gradient pulses are varied to map out the decay of signal amplitude as a function of *G*. In the great majority of experiments, a simple fit to a single
exponential is used; multiexponential fitting is possible but is extremely demanding of SNR (Nilsson et al., 2006) and is not considered here. The diffusional
attenuation ${S}_{i}/{S}_{\mathrm{0}}$ in successive measurements takes the form

where the form of *b*_{i} is determined by the pulse sequence used (Sinnaeve,
2012). In the simple case of a pulsed field gradient spin or stimulated echo
in which spatial encoding and decoding are performed by two gradient pulses
of duration *δ* a time Δ−*δ* apart,

if the gradient pulses are rectangular in shape, or

if half-sine-shaped gradient pulses are used. In the former case the effective gradient *G*_{i} is equal to the peak gradient applied in a given
pulse; in the latter *G*_{i} is equal to the peak gradient multiplied by
$\mathrm{2}/\mathit{\pi}$. These expressions assume that the field gradient is constant
across the sample, which is not always a good approximation; the effects of
field gradient non-uniformity can be taken into account by replacing the
term *G*^{2} by an appropriate power series in *G*^{2} (Connell et al., 2009).

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 *S* of a signal, whether in
terms of peak height or of signal integral, is well described by the
addition of a Gaussian distribution of standard deviation *σ*_{S}.
In the case of peak height, the SNR is by convention defined as $S/\left(\mathrm{2}{\mathit{\sigma}}_{S}\right)$ in NMR spectroscopy. In a DOSY dataset using
*N* different gradient strengths *G*_{i}, each of the *N* measurements *S*_{i} of
the amplitude of a given peak will have the same standard deviation
*σ*_{S}. The effect of this uncertainty on the value of *D* determined
by nonlinear least squares fitting can easily be found by brute force Monte
Carlo simulation or directly from the Cramér–Rao lower bound (CRLB). The latter has been extensively used in NMR, notably for selecting “optimum” sampling patterns *G*_{i} for the simultaneous determination of
the diffusion coefficients of species of different *D* or for the estimation of diffusion distributions *S*(*D*) (see e.g. Brihuega-Moreno et al., 2003; Franconi et al., 2018; Reci et al., 2019; note that the derivations given in the first two references
contain some minor typographical errors). The question of optimum sampling
is considerably complicated by the presence of multiple sources of
systematic error in diffusion NMR experiments and the need to allow for the
likelihood of signal overlap, and is largely avoided here; rather, we use the CRLB for the much more pedestrian purpose of quantifying limiting diffusion resolution in DOSY.

A convenient measure of resolution *R*_{D} in the diffusion dimension of the
DOSY spectrum is the inverse of the coefficient of variation of *D*, that is
the ratio of the estimated *D* to its estimated standard deviation *σ*_{D}. Using the conventional definition of SNR given above, expression (10) of Franconi et al. (2018) becomes

where

and

For a given diffusion coefficient *D* and choice of *N* gradient values *G*_{i},
therefore, the dependence of the resolution *R*_{D} on the signal-to-noise
ratio of a given signal can be calculated. Here *R*_{D} was evaluated as a
function of the number *N* of gradient values sampled, the maximum exponent
*ϵ*_{max}, and the form of the sampling scheme.

Expressions (4) and (5) allow direct calculation of *R*_{D}. Equivalent
results can be obtained easily by Monte Carlo methods, constructing an
attenuation table ${e}^{-{\mathit{\u03f5}}_{i}}$ and then repeatedly adding Gaussian
noise *n* of standard deviation ${\mathit{\sigma}}_{S}=\mathrm{1}/\left(\mathrm{2}\mathrm{SNR}\right)$ to each point of the
table and fitting it to a function of the form $\mathit{\alpha}\phantom{\rule{0.125em}{0ex}}{e}^{-\mathit{\beta}{\mathit{\u03f5}}_{i}}$. The standard deviation *σ*_{β} of the
parameter *β* is then the inverse of *R*_{D}. Again, *R*_{D} was evaluated as a function of the number *N* of gradient values sampled, the maximum
exponent *ϵ*_{max}, and the form of the sampling scheme.

Experimental ^{1}H DOSY data were acquired for a 100 mM solution of
quinine in DMSO-d_{6}, with 50 mM sodium trimethylsilylpropionate (TSP) as a reference, using the Oneshot pulse sequence (Pelta et al., 2002) on a 500 MHz Varian VNMRS spectrometer equipped with a 5 mm triaxial gradient probe at 25 ^{∘}C nominal temperature. Twelve quadratically spaced (equally spaced in gradient squared) nominal gradient amplitudes from 12.5 to 52.8 G cm^{−1} were used, with a net gradient-encoding rectangular pulse width of 1 ms and a diffusion delay Δ of 0.16 s. Eight transients of 16 384 complex points were acquired for each gradient value in a total experiment time of 5 min. The data were subjected to standard DOSY processing in VnmrJ, consisting of zero-filling, reference deconvolution (Morris et al., 1997) with a target Lorentzian linewidth of 1.3 Hz, baseline correction, peak picking, fitting to a Stejskal–Tanner equation modified to compensate for the measured gradient non-uniformity of the probe used (Damberg et al., 2001; Connell et al.,
2009), and construction of the DOSY spectrum using the fitted signal
amplitude, diffusion coefficient *D*, and standard error *σ*_{D}. The
signal decay for the quinine methoxy peak at 3.9 ppm, which had a SNR of
14 400 : 1 at the lowest gradient used, was extracted, and the Stejskal–Tanner fit repeated with different additions of synthetic Gaussian noise to
investigate the influence of SNR on *R*_{D}.

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 *R*_{D} will be, but just how good depends on what parts of the attenuation
curve those points sample. If only the early part of the curve is sampled
(*ϵ*_{max} < 1, where *ϵ*_{max} is the maximum
value of *ϵ*), then the effect of diffusion on the measured points will be small, or if too wide a range of gradients *G* is sampled (${\mathit{\u03f5}}_{max}\gg \mathrm{1}$), then many of the measured points will contain very little signal, and in both cases *R*_{D} will suffer. In a
typical high-resolution DOSY experiment, the sample will contain species of different sizes with a range of diffusion coefficients *D*. Where the range is
not too wide, it is common practice to use a simple sampling scheme in which the field gradient pulse amplitude is incremented either linearly, from some
minimum value *G*_{min} to a maximum *G*_{max} in equal steps of *G*:

or, quadratically, from *G*_{min} to *G*_{max} in equal steps of *G*^{2}:

Because the diffusion-encoding gradient pulses *G* also play a part in
determining coherence transfer pathways in many NMR methods for measuring
diffusion and complementing and reinforcing the effects of phase cycling, it is important in practice that small values of *G*_{min} be avoided. This is
particularly important if experiments such as Oneshot (Pelta et al., 2002) that employ unbalanced bipolar gradient pulse pairs are used with low numbers of transients (and hence incomplete phase cycling). Common practice is
therefore to use a constant ratio ${G}_{min}/{G}_{max}=\mathit{\kappa}$, where
*κ*=0.05–0.25, so that *G* varies from *κ**G*_{max} to
*G*_{max}. Linear and quadratic sampling give similar diffusion resolution,
as is shown below. Quadratic sampling can make it easier to detect
systematic deviations from exponential decay as a function of gradient
squared, and hence to identify peaks in which the signals of species of different *D* overlap.

For a given set of experimental delays and pulse durations, linear and
quadratic spacing in *G* will give different sets of Stejskal–Tanner exponents *ϵ*_{i}. Different diffusion coefficients *D* will give different maxima *ϵ*_{max}, and because the attenuation caused by the
minimum gradient *G*_{min} depends on *D*, the minimum Stejskal–Tanner exponent *ϵ*_{min} will vary slightly with *ϵ*_{max}. Thus for linear sampling the Stejskal–Tanner exponents are

and for quadratic sampling

Figure 1 compares the results of Monte Carlo simulations (small filled
circles) of exponential fits for the two sampling schemes, with SNR = 100 and
*κ*=0.05 in both cases, as a function of *N* and *ϵ*_{max}
with the Cramér–Rao upper bounds (open circles) for *R*_{D}. As expected, there is excellent agreement between the Monte Carlo and analytical results.
The lines for linear regression confirm that there is a direct
proportionality with $\sqrt{(N-\mathrm{1})}$ for low
*ϵ*_{max} but that for higher *ϵ*_{max}, where the signal is strongly attenuated for greater *ϵ*_{i} values, the line
of best fit is displaced. The slope of the line of best fit for *R*_{D} as a
function of $\sqrt{(N-\mathrm{1})}$ rises as *ϵ*_{max} increases until it reaches a maximum at around ${\mathit{\u03f5}}_{max}=\mathrm{2.1}$, after which it decreases again. This is again as expected: for low
*ϵ*_{max} the data are dominated by points that have high
precision but low attenuation, while for high *ϵ*_{max} the
converse is true.

The predicted diffusion resolution *R*_{D} is a function of the sampling
scheme, SNR, maximum Stejskal–Tanner exponent *ϵ*_{max}, and number of gradient values used *N*. Given the nature of Eqs. (4) and (5), it is clear that no simple analytical form exists for ${R}_{D}(\mathrm{SNR},{\mathit{\u03f5}}_{max},N)$. Equally, it is known that *R*_{D} is directly proportional to SNR, and it is reasonable to expect *R*_{D} to be proportional to the square root of *N*−1, since (a) increasing *N* will decrease the effects of random errors in proportion to the square root of the effective number of independent measures of *D* and (b) that number will be dependent on *N*−1, since it is the *change* in signal amplitude that provides information on *D*, reducing
the number of degrees of freedom by 1. In general the effective number will be less than *N*−1 for all but low values of *ϵ*_{max},
because signal attenuation will reduce the information content for higher
values *ϵ*_{i}. It is thus reasonable to seek an approximate
analytical representation of the form

Figure 2 shows the variation of *f* as a function of *ϵ*_{max},
calculated numerically using Eqs. (4), (5), (8), and (9) for values of *N* between 10 and 200 for linear and quadratic sampling, together with fits to
a three-parameter function of the form

The quality of fit is more than adequate for practical use, establishing a simple relationship between diffusion resolution, signal-to-noise ratio, and experimental parameters; fit parameters are given in Table 1.

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 *t*_{1} noise in multidimensional spectra (Mehlkopf et al., 1984; Morris, 1992). In general, the accuracy and
reproducibility of NMR data tend to deteriorate as the number of pulses used
in a sequence increases (because of pulse phase and amplitude jitter caused
by limited radio-frequency spectral purity), as the durations of the delays used increase (because of the cumulative effect of field-frequency
fluctuations), and as the overall duration of an experiment increases (because of slow changes in environmental factors such as room temperature and air pressure). Most such perturbations are at least semi-systematic in nature, but many (particularly pulse-phase instability) have effects that
can appear random and can therefore decrease, at least to some extent, with time averaging. Other sources of distortion in the measured signal decay are
both systematic and reproducible and therefore do not decrease with time
averaging. These include changes in signal attenuation caused by convection
(never wholly absent in practical NMR experiments on liquids; Swan et al., 2015; Barbosa et al., 2016) and by the presence of signals from unwanted coherence transfer pathways. Distortions caused by spatial non-uniformity of the field gradient can be corrected for if appropriate calibration is performed (Damberg et al., 2001; Connell et al., 2009).

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
*R*_{D}. If the contributions from sources other than noise to the errors in
the experimental peak height as a function of gradient strength are normally
distributed and have a root mean square deviation which is a fraction $\mathrm{1}/\left(\mathrm{2}{\mathrm{SNR}}_{\mathrm{lim}}\right)$ of the initial peak amplitude, then the effect on fitting, and
hence on diffusion resolution, of adding uncorrelated noise is to degrade
the effective signal-to-noise ratio SNR in Eq. (10) to

This gives a final predicted diffusion resolution for given experimental conditions of

where *f*(*ϵ*_{max}) can be approximated by Eq. (11). Thus, if the noise contribution to the experimental uncertainty is dominant, the
effective signal-to-noise ratio is the actual SNR, but at high SNR the
effective signal-to-noise ratio for the purposes of Stejskal–Tanner fitting is the limit SNR_{lim} imposed by other error sources.

To investigate the effect of signal-to-noise ratio on diffusion resolution,
synthetic noise was added to the experimental data used to construct the
^{1}H DOSY spectrum of quinine shown in Fig. 3. Figure 4 shows the effect
of SNR on the measured *R*_{D} for experimental data for the quinine methoxy
peak, found by titrating in extra noise as described above. The experimental
signal-to-noise ratio of the first gradient increment was 14 400 : 1, but the diffusion resolution *R*_{D} found when the raw experimental data were fitted was only 420, a small fraction of the predicted value of almost 15 000. As Fig. 4 shows, at low SNR the observed diffusion resolution follows the line expected for the unmodified Cramér–Rao limit of Eq. (11), but as SNR increases the improvement in *R*_{D} levels off, slowly approaching the
limit seen for the data with no noise added. Fitting of Eq. (13) to the
noise-supplemented experimental data gave a value of just over 300 for
SNR_{lim}. To put this value in context, it corresponds to a respectably
small root mean square uncertainty in the signal amplitudes measured of
$\mathrm{1}/\mathrm{600}\sim \mathrm{0.17}$ % of the original peak intensity, typical of
good-quality results obtained with multiple pulse sequences on a modern spectrometer. With extended time averaging and appropriate precautions and
instrumental interventions, it is possible to obtain data with significantly smaller uncertainties than this (see e.g. Power et al., 2016), but the cost in time
and effort can be considerable.

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 https://doi.org/10.17632/d7bdxz9hsk.1 (Morris, 2021).

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 *Magnetic Resonance*.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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.

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