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<br />,
<br />,
<br />
<br />modeling paradigm, which turned its back on the Hortonian
<br />spirit and his specific warning that "[while] the best possible
<br />use should be made of all the available data . .. [this] rule
<br />must often be tempered according to the necessities of econ-
<br />omy of labor and with due regard to the avoidance of metic-
<br />ulous accuracy" (Horton 1931). Hydrological modelers often
<br />did the very opposite: the "necessities of economy of labor"
<br />being drastically reduced by the computer, they not only
<br />stopped paying "due regard to the avoidance of meticulous
<br />accuracy," but also made the cultivation of it their main pre-
<br />occupation. And, as if to flaunt their disdain for the Hortonian
<br />spirit, hydrologic modelers have pursued this "meticulous ac-
<br />curacy" most vigorously in the fitting of exactly those "sta-
<br />tistical and empirical formulas ... which if extended would
<br />lead to absurdities."
<br />1 never cease to be amazed by how otherwise reasonable
<br />people fall into the trap of believing that, by embellishing
<br />purely empirical models with refined mathematical formal-
<br />isms, they somehow can breeze a Uhydrological soul" into
<br />them and transform them into "theoretical hydrological mod-
<br />els" that make sense even when extended far beyond the range
<br />of observation. In the past, I even used to test empirically the
<br />validity of such beliefs when oppoJ;lunities offered themselves,
<br />which only reinforced my conviction that such beliefs are mis-
<br />guided and futile. Two examples will illustrate my experience.
<br />Almost 20 years ago, a noted stochastic-hydrologist friend
<br />asked me for a set of flood peak discharge data to test his new
<br />regional flood-frequency model; he explicitly requested that 1
<br />give him no other information except the numbers. For a hy-
<br />drologic modeler to tell me he wants no hydrological infor-
<br />mation on the hydrologic prototype he is modeling has always
<br />been one of the best ways to stir my blood-and so I sent to
<br />my friend a set of fabricated numbers. As expected, his model
<br />duly produced from them a "theoretical(!) regional(!!)
<br />f100d(!!!) distribution." Since I still have the computer
<br />printout with the results in my files, I can reveal in confidence
<br />that, for example, the 10,OOO-year flood for the region is
<br />2.5181. While the units are not known (as requested, I sup-
<br />plied numbers only), the magnitude of the flood must be me-
<br />ticulously accurate because the parameters of the theoretical
<br />distribution model are given to eight decimal places. To the
<br />credit of the investigator, I must say that he detected some
<br />peculiarities in my data, namely that some of the computed
<br />model parameters pointed to a "small homogenous region,"
<br />while others to a "large heterogenous region." I assured him
<br />he was right on both counts: the small region was my desk,
<br />the large region was my imagination.
<br />About IS years ago, another hydrologist friend tried to con-
<br />vert me to his belief (which he called basic postulate) that if
<br />a model provides a really accurate approximation then its
<br />structure and parameters must correspond to the physical prop-
<br />erties of the prototype. "Hence," he wrote, "the structure and
<br />parameters of well fitted equations can clarify the hidden phys-
<br />ical structure of the natural process. . . . [to achieve this, it is]
<br />only necessary to have an exact fit [based on] long data series
<br />to meet strict statistical requirements. . . ," To prove his point,
<br />he offered to infer the "hidden physical structure" of some;
<br />unknown to him, basin just by filling his model to a reasonably
<br />long record of monthly precipitation and runoff. I supplied him
<br />with such data from a small research basin in the Canadian
<br />shield, with the total area of 0.35 km', very little soil, and with
<br />its steep granite slopes circling a good-size lake whose outflow
<br />represented the total basin runoff. I made this choice in order
<br />to have a basin whose physical structure was clear-in this
<br />case, and for the requested monthly data, it was an almost
<br />perfect example of a single, slightly nonlinear, reservoir.
<br />Based on a very good fit of his model, the hydrologistiden-
<br />tified the following three physical components in the runoff:
<br />
<br />(I) surface flow with lag time less than one month; (2) shallow
<br />subsurface flow with lag time about two months; and (3) deep
<br />ground-water flow with lag time of about seven months. Of
<br />course, only the first component was physically realistic. The
<br />second had nothing to do with an shallow subsurface flow
<br />(which was negligible and all safely in the lake within a few
<br />days at most), but was in fact a part of the recession limb of
<br />the surface outflow hydrograph, the part of the precipitation
<br />delayed by the lake routing effect. However, by far the most
<br />interesting was the third component, which beautifully falsi-
<br />fied the hydrologist's basic postulate and compromised the al-
<br />leged physical meaning of his model. There was absolutely no
<br />"seven-month delayed outflow," whether surface or ground
<br />water, all the carryover being completed within one or two
<br />months, depending on whether precipitation occurred at the
<br />beginning or end of a calendar month. However, there were
<br />mild autumn rains, which usually occurred about seven months
<br />after the main runoff event, the spring snowmelt. These rains
<br />caused a small bump on the outflow hydrograph, which the
<br />model wrongly identified as runoff delayed by deep ground-
<br />water seepage from the spring. I then proposed to my friend
<br />an exact test based on synthetic data simulated by an exact
<br />model of a hypothetical system of the prescribed form but my
<br />offer was declined. '
<br />Both of the foregoing models, by the way, are good empir-
<br />ical models and can be useful in practical applications where
<br />nothing more than a concise representation of known facts is
<br />required. However, it is the modeler's responsibility to know
<br />that facts and not nonsense are being represented. I perceive
<br />avoidance or repudiation of this responsibility as misguided,
<br />whether it is in theoretical or empirical modeling.
<br />Such models are meaningless if we want to gain hydrolog-
<br />ical insight into what is not yet known-if we are interested,
<br />say, in the shape of the upper tail of the flood distribution of
<br />Horton's Rock Creek per se, in a way similar to an astron-
<br />omer's interest in the distribution of frequencies in the spec-
<br />trum of the Aldebaran. What is there to be learned about this
<br />tail if, based on some log-Pearson III best fit to a few observed
<br />floods barely representing the body of the distribution, we pre-
<br />sume the answer to be known? What are the hydrological rea-
<br />sons for the belief that this model will faithfully describe the
<br />shape of the tail all the way into the nebulous heights of
<br />10,OOO-year and million-year floods-as one distinguished
<br />American hydrology professor once insisted to me, dismissing
<br />my scepticism as a "complete disregard of the very founda-
<br />tions of the theory of mathematical statistics?"
<br />What an irony! When Karl Pearson developed his system
<br />of frequency distributions in response to the appeal of his bi-
<br />ologist colleague (Weldon) for mathematical tools to help him
<br />analyze his morphological measurements of shrimps and shore
<br />crabs, Pearson always consulted Weldon about the probable
<br />biological limits to which his curves could be reasonable ex-
<br />trapolated. Now, more than a century later, hydrologists are
<br />consulting the extrapolated Pearson's curves when they seek
<br />answers regarding the probable hydrological limits of floods!
<br />If the deterministic modelers should be tempted to applaud
<br />me for rightly debunking their stochastic counterparts, they
<br />should pause. One does not have to look any further than, for
<br />example, the unit hydrograph, to see that they have little rea-
<br />son for complacency. Let's just recall how this useful empir-
<br />ical concept has been redefined as "unit response of a linear
<br />system," which was then almost beaten to death by all kinds
<br />of rigorous theory (of linear systems, of course, not of hydro-
<br />logical systems), including Fourier and Laplace transforms,
<br />Laguerre analysis, time-series analysis, matrix methods. mo-
<br />ment matching, cumulants, etc. All this prestidigitation has
<br />contributed next to nothing to the understanding of hydrolog-
<br />ical systems, which, as some rudimentary hydrological inves-
<br />
<br />JOURNAL OF HYDROLOGIC ENGINEERING / APRIL 1997/47
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