Recent advances in statistical methods for the estimation of sediment ...

REVIEWS OF GEOPHYSICS, SUPPLEMENT, PAGES 1117-1123, JULY 1995 U.S. NATIONAL REPORT TO INTERNATIONAL UNION OF GEODESY AND GEOPHYSICS 1991-1994

Recent advances in statistical methods for the estimation of sediment and nutrient transport in rivers T. A. Colin Branch of Systems Analysis, U.S. Geological Survey, Reston, Virginia

Introduction This paper reviews advances in methods for estimat­ ing fluvial transport of suspended sediment and nutri­ ents. Research from the past four years, mostly dealing with estimating monthly and annual loads, is empha­ sized. However, because this topic has not appeared in previous IUGG reports, some research prior to 1990 is included. The motivation for studying sediment transport has shifted during the past few decades. In addition to its role in filling reservoirs and channels, sediment is in­ creasingly recognized as an important part of fluvial ecosystems and estuarine wetlands. Many groups want information about sediment transport [Bollman, 1992]: Scientists trying to understand benthic biology and catchment hydrology; citizens and policy-makers con­ cerned about environmental impacts (e.g. impacts of logging [Beschta, 1978] or snow-fences [Sturges, 1992]); government regulators considering the effectiveness of programs to protect in-stream habitat and downstream waterbodies; and resource managers seeking to restore wetlands. Developing better methods for estimating sediment transport is important for a second reason: The meth­ ods can also be used to estimate nutrient transport. Nutrients play a critical role in determining the ecol­ ogy of rivers, lakes and estuaries. Scientists, regulators, and resource managers increasingly want to understand nutrient fluxes. Tools for estimating sediment and nutrient transport are changing. Recent research has been directed at de­ veloping methods for reducing the number of samples required to obtain acceptable load estimates. In par­ ticular, rating curve estimators and statistically-based sampling [Thomas, 1985; Thomas, 1991; Thomas and Lewis, 1993; Thomas and Lewis, 1995], requiring fewer samples at lower cost, have attracted the most atten­ tion. A motivation for developing these methods is eco­ nomics. Traditional methods for estimating sediment loads [Porterfield, 1972] call for near-daily sampling

over many years. Such sampling programs appear to be increasingly vulnerable to government cost-cutting. For example, most long-term daily sediment stations in Canada were recently discontinued in the belief that data needs could be met with lower cost alternative sampling methods [Spiizer, 1991; Smith, 1991; Ongley, 1992; Yuzyk et al, 1992; Day, 1988; Day, 1992]. At the same time, users of data are demanding higher quality information. Results need to be objective and reproducible. The precision of results must be reported. While traditional "worked records" based on daily data may produce good estimates of sediment and nutrient transport, the quality of these estimates is difficult to assess [Gilroy, 1991]. In contrast, the precision of "rat­ ing curve" [Gilroy et al, 1990] and statistically-based sampling estimators [Thomas, 1985], are readily com­ puted.

The Problem The underlying problem in load estimation involves evaluating an integral. The amount (or load) of a con­ stituent (e.g. sediment or nutrients) transported past a river cross-section during a time interval {t , t&} is given by: a

L

= =

L(t)dt

(1)

/ " K • Q(t) • C{t)dt

(2)

where L is the integrated load during

{t ,t }] a

b

L(t) is the instantaneous load at time t; K is a units conversion factor; Q(t) is streamflow at time t, (usually relatively accu­ rately known); C(t) is the average concentration of the constituent in the cross-section at time t.

Copyright 1995 by the American Geophysical Union. Paper number 95RG00292. 8755-1209/95/95RG-00292$15.00

Direct application of equation (2) is usually impossi­ ble because C(t), the continuous time trace of concen­ tration, is known only at those times when concentra­ tions were measured. 1117

COHN: FLUVIAL TRANSPORT ESTIMATION

1118

M e t h o d 1: E s t i m a t e a C o n t i n u o u s C o n c e n t r a ­ tion Trace

mated from:

C c(0=exp[^o + A-lnQ(t)] (5) The load integral can be evaluated if one develops an estimator, C(t) (throughout this paper, tildes are where used to identify estimators), for the concentration at Po,Pi are ordinary least squares regression coefficients. arbitrary time t. There are several ways to do this. Because this model provides a convenient statistical T h e W o r k e d R e c o r d Traditionally, a graph of framework, much recent work has been devoted to cor­ sediment concentration versus time, C (t), was drawn recting its deficiencies and expanding its applicability. by hand [Porterfield, 1972]. After sample values were plotted on the graph, the line was drawn on the basis M o d i f i c a t i o n 1: R e t r a n s f o r m a t i o n a n d B i a s of hydrologic judgment, sediment transport curves, and C o r r e c t i o n F a c t o r s The rating curve estimator of experience. The load estimator, L , was then evalu­ equation (5) is not statistically consistent [Lane, 1975; ated numerically [Press et al, 1986, Chapter 4]: DeLong, 1982; Thomas, 1985; Jansson, 1985; Ferguson, 1986; Koch and Smillie, 1986; Cohn et al, 1989]. Its results are biased, in general systematically underesti­ L= j K • Q(t) • C (t)dt (3) mating loads. In studies with field data [Walling et al, 1981; Fenn et al, 1985], this bias sometimes ex­ The quality of L as an estimator depends on the hy- ceeded 50%. Thomas [1985], Ferguson [1986], and Koch drologist's skill and training. and Smillie [1986] note that the bias arises when model results computed using the logarithm of C are " ^trans­ R a t i n g C u r v e s Campbell and Bander [1940] ob­ formed" into real units. Three methods for compensat­ served that the relation between the logarithm of sedi­ ing for this bias are now commonly employed. Thomas ment concentration and the logarithm of discharge was [1985], Ferguson [1986] and Koch and Smillie [1986] de­ approximately linear. They suggested that this relation scribe an approximately-unbiased estimator: could be used as a "rating curve." For periods when no sediment data had been collected, sediment con­ s centrations could be estimated from water discharge. Cqmu (i) = exp(p + Pi . In Q(i) + —) (6) Although the rating curve remains an empirical re­ where sult without physical justification, it has come into widespread use. It appears to be adequate for many s is the estimated standard deviation of the residu­ purposes [Miller, 1951; Colby, 1955]. The rating curve als. is simple, and, by including additional regressor vari­ ables (for example, see Cohn et al [1992a]), can be eas­ Thomas [1985], and Koch and Smillie [1986] also recom­ ily modified to account for variability associated with mend Duan's [1983] "smearing estimate," a nonpara­ non-linear flow dependence and time trends. One can metric retransformation function appropriate for nonstratify data (e.g. by season, discharge or other vari­ normal error distributions: able) or use multiple rating curves to describe more complicated concentration/discharge relations [Colby, 1 1955]. The rating-curve's shortcomings, some of which are discussed below, are also well-documented [ Walling C (t) = exp0 + fa • lnQ( which is the minimum variance unbiased estimator if the assumed Po,Pi are model coefficients; linear model is valid. e is residual error; the regression residuals are com­ The three bias-correction methods yield nearly iden­ monly assumed independent and identically dis­ tical results under the following conditions: tributed (iid) normal random variables, with a 1. the assumed linear model is approximately cor­ mean of zero and variance denoted by

>

l s

t

t l i e

average of the sampled loads;

SLQ

n

e

a v e r a

e

t

u

e

discharges

= T^T £ £ i ( Q < "