Theory 



Multiexponential relaxation and local diffusion cells distribution 





In bulk water the relaxation of the ^{
1}H nuclei magnetization is well described
by a single exponential function. For water in high S/V systems the
relaxation is enhanced, and shorter relaxation times
are observed. The semilogarithmic plots of the decay of the nuclear magnetization in two watersaturated model systems, each one characterized by a narrow distribution of pore sizes, are straight lines, with faster decay for the sample with the smaller pores. Both systems exhibit faster decay than for bulk water. 





Relaxation
represent an average over regions that can be
much larger than an “individual pore”. If diffusion is fast enough to
maintain magnetization nearly constant within distances of diffusion in a
local relaxation time, that is, within the local diffusion cell,
the local relaxation rate is proportional to an
S/V averaged over this local volume. It is often useful to regard the
local V/S as an operational definition of
“pore size”. 

If the
porespace is heterogeneous in ρ and/or S/V only
on a scale shorter than local diffusion lengths, which implies
uniform local diffusion lenghts and local relaxation times,
the relaxation of the sample will be singleexponential. Otherwise, the
distribution of local S/V will be reflected by a multiexponential
relaxation [J. Appl. Physics 1996, 36563664]. 







Inversion from signal time to relaxation times


The problem is then to invert the
multiexponential function in order to obtain the distribution of
relaxation times.
The function to minimize is the squared error of fit plus a penalty term. Usually some smoothing is implemented in the distributions by means of a penalty function in order to avoid excess variation. We have proposed an algorithm (UniformPENalty, UPEN)) to obtain distributions of relaxation times, tested on rocks and other porous materials, as well as artificial data, where we could see distributions more than three decades wide. UPEN uses a smoothing coefficient varying with the relaxation time and determined by iterative feedback in such a way that the smoothing penalty, rather then the coefficient, is roughly uniform. UPEN: Uniform PENalty inversion of multiexponential decay data


The first
line is weighted (error of fit)^{2}; the second is the
penalty. 



ß_{p} is the slope (pendenza) compliance parameter. ß_{c} is the curvature compliance parameter. ß_{a} is the amplitude compliance (for outside data range). ß_{0} is the compliance floor (not critical, but not zero). ß_{00} (normally = 1) multiplies the other ß 's. D_{Q} is data point time spacing in Nepers. 

UPEN tends to
avoid more maxima and minima in the distribution than demanded by the
data, minimizing the appearance of separate populations in the computed
distributions to the extent permitted by the data. This is an example of a carbonate rock saturated by water. UPEN gives a sharp line that is not broadened more than is consistent with the noise and in the same distribution shows a tail decades long without breaking it up into several peaks not required by the data to be separate. On the contrary, other solutions obtained by fixed smoothing coefficient or by singularvalue decomposition (with substantially the same rms error of the fit) in any case tend to broaden the major peak and break the tail into more peaks, which might be interpreted as physically meaningful resolved compartments.






Errors of Fit, Noise,
Quality of Fit


The E_{i}
's are the errors of fit. For T_{2} data, signal amplitude
s_{i} may represent a window average of B_{i}_{
}echoes. R_{r} (R for noise: Rumore, Rauschen)
is an estimate of individualecho error of fit, using a compromise
weighting to give equal weights at long and short times, rather than
Σ(B_{i}E_{i})^{2}/ΣB_{i}_{
}to give the best statistical estimate of the ensemble average
echo noise R. This compromise weighting is used for all the R's. If there is a slowlyvarying error of fit in addition to random noise, one may get a better estimate of the noise from R_{v} (v for variation), computed from differences of the E 's for secondnearest neighbors, since a nonrandom slowlyvarying error of fit would not greatly affect these differences. 

If there are random errors only, R_{r} should be somewhat less than R_{v} , because the fitting minimizes the errors, rather than differences of errors. Having separate estimates of the random noise and the overall error of fit can warn us if the data are not, or cannot be, adequately fit. Thus, the qualityoffit parameter 

R_{rv }= ln ( R_{r} / R_{v} ) 

s 







E_{2} is the least maximum absolute error of fit to a rectangular distribution of relaxation times by 2 sharp lines, a factor of Y apart. The data time spacing in Nepers is Dq, which for T_{2} data without discarded points is T_{EE}/T, where T is relaxation time of the minimum between the peaks, and T_{EE} is the spacing of the echoes employed. If the peaks are of comparable size and are the only significant features on the distribution, SNR is the signaltonoise ratio. If the peaks are disparate, only the signal in the smaller peak is counted. Additional major features make resolution more difficult. The above SNR value is for marginal resolution; several times higher SNR is required for firm resolution. If the two tentative peaks do not have zero inherent line width, Y must be reduced by about the mean halfwidth (on a logarithmic time scale). 







where w is the halfwidth of the line and SNR counts only the signal in the line. Nearby features widen the line.  






Under some conditions, to suggest a significant change in the quality of a fit, the relative change in error of fit should be at least  


with the same adaptation to T_{2} data as mentioned above. N is the number of (possibly windowed) data points used in the computation. 
