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\noindent{\bf Harold Edwin HURST}\\
b. 1 January 1880 - d. 7 December 1978
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\noindent{\bf Summary.} Harold E. Hurst, English hydrologist 
and ``Father 
of the Nile," is principally known for a remarkable power-law 
relation that was later interpreted as symptom of 
``long-run", or ``global" statistical dependence in the Earth sciences. 
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Hurst, hailed as perhaps
the foremost Nilologist of all time and spoken
of as ``Abu Nil," the Father of the Nile, spent
the bulk of his career in Cairo as a civil
servant of the British Crown, then of
Egypt. ({\it Who's Who}, 1979, p. 1257,
and {\it Who's Who of British
Scientists 1969/70}, pp. 417-418.)


His early training is worth recounting, mostly
from his own words and those of his wife Mrs. Marguerite
Brunel Hurst. The son of a village builder of
limited means, whose family had lived near
Leicester for almost three centuries, he left
school at age 15. He had been trained mostly in
chemistry, and also in carpentry by his father.
He then started as a pupil teacher at a
school in Leicester, attending evening classes to
continue his own education.


At age 20, he won a scholarship that enabled
him to go to Oxford as a noncollegiate student.
After a year, he became an undergraduate at the
recently reestablished Hertford College, and
soon switched to a major in physics and worked
at Clarendon Laboratory.


His lack of preparation in mathematics was a
handicap, but thanks to the interest
that Professor Glazebrook took in an unusual
candidate who was very strong in practical work,
he won a first-class
honors degree, to everyone's surprise, and
was asked to stay for three years as a
lecturer and demonstrator.


In 1906, Hurst went to Egypt for a short
stay that was to last 62 years, of which
the most fruitful were after he turned 65.
His first duties included transmitting standard
time from the Observatory to the Citadel of
Cairo, where a gun was to be fired at
midday.  However, he became increasingly
fascinated with the Nile, and his study
and exploration of the Nile basin made him
well known internationally.  He traveled
extensively by river and on land, first
on foot with porters, then using a bicycle,
later by car, and later still by plane.  The
low Aswan Dam had been built in 1903,
but he realized how important
it was to Egypt that provision should
be made not only for the dry
years but for a series of dry years.
Irrigation storage schemes
should be adequate for every situation,
very much, as in the Old
Testament, Joseph stored grain for the
lean years.  He was one
of the first to realize the need for
the ``Sudd el Aali,"
that is, the High Dam and Reservoir at Aswan.


Hurst's name is linked to a statistical
method he initiated, without at all realizing
the scope of what he was doing.  He
used this technique to discover a major
empirical law concerning
the form statistical dependence takes in
geophysics. At first, it seems surprising that
anything of the kind could come from an author
so poorly prepared in mathematics and working so
far from any major center of learning, but
at second thought these circumstances
may have been vital to both the
birth of his idea and its survival. He
investigated the Nile using a
peculiar method of analysis that follows very
literally a recipe found in ancient
books on optimum dam design and consists
in a ratio $R(d)/S(d)$.  The numerator $R$
was defined, in current terminology
as the range over a
time span $d$ relative to the cumulative
sum of river discharges.  The
denominator $S$ was defined as a standard
deviation of yearly river
discharges.  Critics termed $R/S$ narrow,
undocumented, and ad hoc,
but in fact this ratio - quite unexpectedly - turned
out to be eminently
intrinsic to the problem at hand and to have
unparalleled qualities.
With the computers of today, the calculation
of $R/S$ is easy, but in
the mid-1960s,
it was viewed
as very time-consuming. Therefore,
one can imagine the amount of hard work
implied in such research before
the advent of computers.  But of course
the Nile is sufficiently
important to Egypt to justify comparatively
large
expenditures (and to preclude
forcing Hurst to retire).
Not being pressed by time and
having exceptionally abundant
data at his disposal, Hurst was
in a position to evaluate their effect
upon the design of the then-future
High Dam.  He showed that
the dams the data demanded must be
far larger than suggested by the
standard model of stochastic variability
of river discharges.
Indeed, all earlier writings on this topic
led to the expectation that in
the long run $R(d)$ should
increase like the square root of $d$,
whereas Hurst's extraordinarily
well-documented empirical law shows
an increase like the power $H$ of
$d$, where $H$ is about 0.7.  Hurst
adamantly maintained  that this finding was
significant, despite the fact that no test
existed by which its statistical
significance could be assessed
objectively. Finally, at the
ages of 71 and 75, he read two long papers
that attracted attention. He was awarded the
Telford Gold Medal by the Institution of
Civil Engineers in 1957.

In the 1950s and 1960s the statistical profession
was not at all prepared to tackle
``power-law" relations like
Hurst's. Scaling was an esoteric notion. Hydrologists were
advised to manage with
Markov or ARMA processes.
Unfortunately, the memory
those processes required increased
with the length of the sample.

The significance of Hurst's work was
revealed through contributions of B.B.
Mandelbrot starting in 1965. He first
interpreted the power-law as
a symptom of underlying
scaling, more precisely, of self-affinity
with an anomalous exponent $H$.  The standard
Brownian case $H=0.5$ corresponds to
independent increments, hence no statistical
dependence whatsoever.  But when $H$
is not 0.5, the dependence is very strong, positive 
or negative.  It can be called, ``long-range," or 
perhaps better, ``global."  Furthermore,
to show that $H$ differing from 0.5
is not inconceivable,  Mandelbrot proposed
that, instead of the Wiener Brownian
motion, the cumulative river discharge might
be represented by the fractional
Brownian motion model, which he devised
for this purpose.

The division by $S$ began as a traditional
normalization, not deserving a careful
look.  But as Mandelbrot showed, it turned out,
serendipitously, to be extraordinarily
important.  The undivided $R$ is not
robust at all, being confined
to measure global dependence
when the margins are Gaussian
or follow other very
short-tailed random variables.
After division by $S$, to the contrary,
the ratio $R/S$ is extraordinarily robust.
It gave rise to a new technique,
``$R/S$ analysis," which can reveal and
estimate global dependence independently
of the length of the marginal
distribution's tails and remain useful
even when the data has an infinite variance.

This is an instance of when
a result is truly unexpected,
is hard to comprehend, even by
those best disposed to listen.
New tools are not accepted without struggle.
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\begin{thebibliography}{3}
\bibitem{1}  Hurst, H.E. 1951.  Long-term storage capacity of
reservoirs.  {\it Transactions of
the American Society of Civil Engineers}, {\bf 116}, 770-808.

\bibitem{2}  Hurst, H.E. 1955.  Methods of using long-term 
storage in reservoirs.
{\it Proceedings of the Institution of Civil 
Engineers}, part 1, 519-577.

\bibitem{3}  Hurst, H.E. Black, R.P. AND Simiaka, Y.M. 1965.  
{\it Long-term Storage,
an Experimental Study.} Constable, London.

\bibitem{4}  Mandelbrot, B.B. 2000.  {\it Gausssian 
Self-Affinity and Fractals: Globality, the Earth, $1/f$,
and $R/S$}. (Selected Papers Vol. H). Springer, New York.  
Reproduces and
makes easily available the author's and his associates'
papers from the 1960s and 1970s.
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\hfill{Benoit B. Mandelbrot}

\end{thebibliography}

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