ELSEVIER
l
Short Communication
Magnetic Resonance Imaging, Vol. 14, Nos. 718, pp. 895897, 1996 Copyright 6 1996 Elsevier Science Inc. Printed in the USA. All rights reserved 0730725X/96 15.00 + .OO
PI1 SO730725X( 96) 00175S
A ROBUST METHOD FOR CALCULATING GEOMETRIC MEAN TIMES FROM MULTIEXPONENTIAL RELAXATION DATA, USING ONLY A FEW DATA POINTS AND ONLY A FEW ELEMENTARY OPERATIONS G.C.
BORGIA,* V.
BORTOLOTTI,*
R.J.S.
BROWN,? AND
P.
FANTAZZINI~ *University of Bologna, Dept. of
ICMA, Bologna, Italy, tClaremont, CA 917113721, USA, and University of Bologna, Dept. of Physics, Bologna, Italy
A method is presented to compute values of geometric mean time Tg that uses only a few data points equispaced in the logarithm of time (or equispaced in time and weighted by l/t) and a few elementary operations for the computation. The method has been tested on a large number of synthetic relaxation data and on actual NMR relaxation measurements in porous samples, using as few as four points (including the two points needed to normalize the relaxation for decay from 1.0 to 0) on each relaxation curve. This computation of the geometricmean rate very adequately matches the synthetic data and the results of multiexponential inversion of many (or only a few) data points from NMR measurements. When many computations are needed in short times, as for voxelbyvoxel computations in magnetic resonance imaging (MRI) or for depthbydepth computation in nuclear magnetism logging (NML) of oilwells, a very quickly computed estimate of Tg should be useful.
Copyright 0 1996 Elsevier Science Inc.
Keywords: Multiexponential NMR relaxation; Geometric mean time; Fast acquisition and computation.
Watersaturated porous media can show distributions of NMR relaxation times covering ranges of a thousand or more. Various single parameters or kinds of averages have been computed from such distributions. Geometric mean time
Tg
is defined by means of amplitudes ci and times
Ti
of a multiexponential analysis, ln
Tg = I: ( ci
In r )/C ci . The time
Tg
has been demonstrated in sand stones to be a good parameter for correlating relaxation behavior with properties of the pore space important for fluid flow, such as permeability and irreducible water saturation. Furthermore,
Tg
is a parameter that is robust with respect to both measurement and computation. ln some cases the only use made of multiexponential analy sis is to compute
Tg
values. When many computations are needed in short times, as for voxelbyvoxel computa tions in magnetic resonance imaging (MRl) or for depth bydepth computation in nuclear magnetism logging (NML) of oilwells, a very quickly computed estimate of
Tg
should be useful. The algorithm described in this paper for computing
Tg
uses only a few elementary operations for the com putation. The method has been tested both on actual NMR relaxation data from water saturated sandstones and on a large number of synthetic relaxation data. Let us consider the integral
G=
S(f) = C cieef’q, C
Ci
= 1;
G = c ci s”” er f TminfTi
where
Tmin G Ti < T,,.
We need
dr m m s s

= Zco(a> + Z,(b), a
b
Address correspondence to G.C. Borgia,
Dept. of
Bologna, Italy.
ICMA, University of Bologna, Viale Risorgimento 2,
895
896 Magnetic Resonance Imaging
l
Volume 14, Numbers 718, 1996
where
a
=
Tti,/Ti
and
b
=
Tm,,ITi
. By means of integration by parts,
s m
Z,(a) = [e’
In 71: + e+ In 7d7
0
a  s 6’ In rd7 = &(a) + Z, + Z,(a),
0
where
Z2(a) = P
In
a, Zx = y = 0.5772157 s * .
(EulerMascheroni constant), and I( a, b) = Zi (b) + Z2
a)
+ I3 + Z4 a). Once again, by integration by parts,
Z,(b)  Ep 1 ; +; ; + . * * ( 1 *
For
b
% 1, and taking half the last term kept from a slowly converging alternating series, this expression can be approximated as
Z,(b) ( 1
k . For a < 1
e
z,(a) = 
s (I
e’ In
TdT = 
0
s
In 7 i F d7. 0 .
12(a) + &(a> 2
=ha+a:+
Finally,
Z(a, b) M *na+ays  . . . + . . . . (2)
By substituting eq. (2) in eq. ( 1):
G M InT,i~+(lnT)+T,i, ’ y 0 T
+z+. u 1)
By the definition of geometric mean time,
Tg = exp(ln T),
Ttin exp(G +
Y)
. The amount ( 1
/T)
is the unknown average rate. In any case, ( 1 /T) 1 1 /
Tg
so a partial correction is done by substituting ( 1 /T) for 1 /
Tg
By considering that
Tg
%
T,,,
one gets exp(
T&T,)
= 1 +
T&T,.
Thus, the correction is simply the time
T,,
. The last correction is very small when
T,,,=
>
5Ti.
In this case
Tg
= Tmin
xp(G + y) 
Tti,. (3)
Let us consider now a set of experimental magneti zations Sj, tj, with j = 1, 2, 3 . * * n, and
tj
=
tj1
ezq. The amount analogous to G is:
g= 2q Zyi: (Sj  Sn) s,s, . (4)
If ri is very short compared with the shortest relaxation time,
t,
is very long compared with the longest relaxation time, and
q 4
1, then g = G. When these conditions are only partially satisfied, some corrections can be made. First of all, the equivalent interval of integration is from
tl eeq to tneq,
because S’ corresponds to an interval of ln~fromlntjqtolntj+q: Tti,
E t,e9, T,, = t,,eq. (5)
In the case of G, we used normalized values of S( t) ; the normalization for g is given by the denominator in Eq. (4). But S1 is not at t = 0. For single exponential relaxation, the extrapolation of Si to
t
= 0 can be done by using the factor exp( ti
/Tgo)
, where
Tgo = t,
exp( G + y ) is the first estimate of
Tg.
For a distribution of times, this factor will give an undercorrection due to the short components. Now we have a reduced g,
Calculating geometric mean imes 0 G.C.
BORGIA, ETAL.
897
l
fig Not Extrapolated) 0 “, Extrapolated)
0 100 200 300 400 500
T* (127p& sum) (ms)
Fig. 1. Scatterplot of the ratios between Tg values, computed
by 5components its to relaxation curves, and the time T, computed by eq. (7) using all the 127 points as a function of this 7’, value. The data refer to 77 sandstone amples saturated y brine. This reduces Tgo by a factor exp(  gtt/T,,,) ,
Tg = Tgo a.
(6)
By using Eq. (6) and Eq. (5) in Eq. (3), we have
Tg
= T,. =
tJexp(g + y 
q)  g  emq]. (7)
The approximation Eq. (7) has been tested on
T1 relaxation data from 77 sandstone samples covering many decades of permeabilities and a large interval of
porosities, with
T,,
values ranging from 10 to 500 ms. Relaxation curves were acquired at 127 tpoints logarithmically equispaced in the range 0.4 ms to 6 s. Figure 1 shows the ratios between
Tg
computed by 5 component fits to relaxation curves and
T,
computed by Eq. (7) using all the 127 points. The values of
Tg
are computed both for amplitudes extrapolated to t = 0 (open circles) and for amplitudes at t = 0.4 ms (not
extrapolated). The values of Tg and T, match very well
for notextrapolated amplitudes, while the extrapolated data overemphasize the contribution of short compo nents. So the algorithm works well, using only elemen tary operations, but another interesting result is that
much fewer than 127 points are sufficient for a very
good estimation of
Tg.
Figure 2 shows the ratios between T, values computed using only very few data points and those computed from all the 127 data points. Using only 7 points, all but three samples, with short
T,,
give values that match the 127 point values within 2%. Going down to Cpoint computa tion 90% of the samples have an error within 5%.
Artificial data simulated the relaxation due to rectan
gular distributions of relaxation times giving 50
Tg
values from 1 ms to 1 s. Five components were used to represent width ratios of the distribution ranging from a factor of 1 to 100. Values of
T,
were also computed using from 127 down to as few as 4 points, assuming a weighting factor for the correction tl(g + e“) in Eq. (7), in order to compensate for ( 1 IT)
>
1
Tg Also
for these artificial data the accord between
Tg
and
T, is very good.
The error is negligible for
T,.
7points when
Tg
s larger than a few ms, and once again
T,
computed from 4 points approximates
Tg
very well, giving an error of the order of a few percent at lower
Tg
values, and becoming negli
gible at slightly higher Tg
values. We have noted that
T,
127~points and
T,
7points differ very little not only from
T8
5components, but also from
Tg
twocomponents. The geometric mean itself is very stable for at least two computed components. Various single parameters or kinds of averages have been computed from distributions or sums of exponen
tial terms.
Average rates, average times, and mean square times are more vulnerable to both measurement errors and computational variability than the ‘ ‘compro mise” between mean rate and mean time, namely
Tg
=
T,,
which appears to be robust, both with respect to measurement and computation.
AcknowledgmentsInvestigation supported by University of Bolo gna (funds for selected research topics) and by MURST Grants.
1.15 . . . . , . . . . , . . . . , . . . . , . . . . 0.95 ...BQ.n..I....t....'..*. 0 100 nx 3al 400 500
T, (127~point um) (ms) Fig. 2. Scatterplot of the ratios between T, values, computed
by using only few data points, and
T,
computed by using all the 127 points as a function of this
T,
value. The experimental
data are the same as in Fig.
1.