A Spatially Referenced Regression Model (SPARROW) for Suspended Sediment
in Streams of the Conterminous U.S.

Gregory E. Schwarz, Richard A. Smith, Richard B. Alexander, and John R.
Gray, U.S. Geological Survey, Reston, Virginia

Mailing address: U.S. Geological Survey, 413 National Center, 12201
Sunrise Valley Dr., Reston, VA 20192, Phone (703) 648-5718, Fax (703)
648-5295, Email gschwarz@usgs.gov.

INTRODUCTION

Suspended sediment has long been recognized as an important contaminant
affecting water resources. Besides its direct role in determining water
clarity, bridge scour and reservoir storage, sediment serves as a
vehicle for the transport of many binding contaminants, including
nutrients, trace metals, semi-volatile organic compounds, and numerous
pesticides (U.S. Environmental Protection Agency, 2000a). Recent efforts
to address water-quality concerns through the Total Maximum Daily Load
(TMDL) process have identified sediment as the single most prevalent
cause of impairment in the Nation's streams and rivers (U.S.
Environmental Protection Agency, 2000b). Moreover, sediment has been
identified as a medium for the transport and sequestration of organic
carbon, playing a potentially important role in understanding sources
and sinks in the global carbon budget (Stallard, 1998).

A comprehensive understanding of sediment fate and transport is
considered essential to the design and implementation of effective plans
for sediment management (Osterkamp and others, 1998, U.S. General
Accounting Office, 1990). An extensive literature addressing the problem
of quantifying sediment transport has produced a number of methods for
estimating its flux (see Cohn, 1995, and Robertson and Roerish, 1999,
for useful surveys). The accuracy of these methods is compromised by
uncertainty in the concentration measurements and by the highly episodic
nature of sediment movement, particularly when the methods are applied
to smaller basins. However, for annual or decadal flux estimates, the
methods are generally reliable if calibrated with extended periods of
data (Robertson and Roerish, 1999). A substantial literature also
supports the Universal Soil Loss Equation (USLE) (Soil Conservation
Service, 1983), an engineering method for estimating sheet and rill
erosion, although the empirical credentials of the USLE have recently
been questioned (Trimble and Crosson, 2000). Conversely, relatively
little direct evidence is available concerning the fate of sediment. The
common practice of quantifying sediment fate with a sediment delivery
ratio, estimated from a simple empirical relation with upstream basin
area, does not articulate the relative importance of individual storage
sites within a basin (Wolman, 1977). Rates of sediment deposition in
reservoirs and flood plains can be determined from empirical
measurements, but only a limited number of sites have been monitored,
and net rates of deposition or loss from other potential sinks and
sources is largely unknown (Stallard, 1998). In particular, little is
known about how much sediment loss from fields ultimately makes its way
to stream channels, and how much sediment is subsequently stored in or
lost from the streambed (Meade and Parker, 1985, Trimble and Crosson,
2000).

This paper reports on recent progress made to address empirically the
question of sediment fate and transport on a national scale. The model
presented here is based on the SPAtially Referenced Regression On
Watershed attributes (SPARROW) methodology, first used to estimate the
distribution of nutrients in streams and rivers of the United States,
and subsequently shown to describe land and stream processes affecting
the delivery of nutrients (Smith and others, 1997, Alexander and others,
2000, Preston and Brakebill, 1999). The model makes use of numerous
spatial datasets, available at the national level, to explain long-term
sediment water-quality conditions in major streams and rivers throughout
the United States. Sediment sources are identified using sediment
erosion rates from the National Resources Inventory (NRI) (Natural
Resources Conservation Service, 2000) and apportioned over the landscape
according to 30-meter resolution land-use information from the National
Land Cover Data set (NLCD) (U.S. Geological Survey, 2000a). More than
76,000 reservoirs from the National Inventory of Dams (NID) (U.S. Army
Corps of Engineers, 1996) are identified as potential sediment sinks.
Other, non-anthropogenic sources and sinks are identified using soil
information from the State Soil Survey Geographic (STATSGO) data base
(Schwarz and Alexander, 1995) and spatial coverages representing
surficial rock type and vegetative cover. The SPARROW model empirically
relates these diverse spatial datasets to estimates of long-term, mean
annual sediment flux computed from concentration and flow measurements
collected over the period 1985-95 from more than 400 monitoring stations
maintained by the National Stream Quality Accounting Network (Alexander
and others, 1998), the National Water Quality Assessment Program, and
U.S. Geological Survey District offices (Turcios and Gray, in press).
The calibrated model is used to estimate sediment flux for over 60,000
stream segments included in the River Reach File 1 (RF1) stream network
(Alexander and others, 1999).

SPARROW uses statistical methods to calibrate a simple, structural model
of riverine water quality, one that imposes mass balance in accounting
for changes in contaminant flux. As applied here, the mass-balance
approach facilitates the interpretation of model results in terms of
physical processes affecting sediment transport, and makes possible the
estimation of various rates of sediment generation and loss associated
with stream channels and features of the landscape. The statistical
approach provides a basis for assessing the error of these inferred
rates and of the error in extrapolated estimates of sediment flux made
for streams in the RF1 network.

An important implication of the holistic modeling approach adopted in
this analysis is that estimates of sediment production and loss are
based on, and therefore consistent with, measurements of in-stream flux.
Other ancillary information, such as direct measurements of long-term
sediment storage and release from reservoirs (Steffen, 1996), is
incorporated into the analysis by specifying additional equations
explaining these ancillary variables. The imposition of cross-equation
constraints affords this information a statistically consistent weight
in explaining in-stream sediment flux. Thus, the methodology described
here represents a general framework for synthesizing a wide spectrum of
available information relevant to the understanding of sediment fate and
transport.

METHODOLOGY

The SPARROW methodology (Smith and others, 1997) has been modified to
incorporate greater spatial resolution. The primary spatial reference
frame for the model continues to be the RF1 reach network: all point
sources and landscape features are referenced to a particular RF1 reach.
However, considerable internal structure has been added to each reach.
Reach watersheds are delineated using the 1-kilometer HYDRO 1K digital
elevation model (DEM) (U.S. Geological Survey, 2000a), and explicit
pathways are defined between landscape features and their adjacent RF1
streams. The delineation method uses a "burn-in" process whereby the RF1
reach is first digitized in the 1-kilometer grid and then the elevations
of RF1 grid cells are artificially lowered to insure that simulated flow
from surrounding cells moves into them. Flow directions based on the
steepest descent determine the extent of the reach watershed and the
undefined tributary flow paths leading from the landscape to the RF1
channel cells. To insure the accurate determination of in-stream travel
time, RF1 stream pathways continue to be defined by the line work of RF1
channels rather than by the grid-cell representation.

A schematic of a typical reach watershed, illustrating its spatial
structure and associated features, is given in figure 1. Flow
directions, represented by the arrows crossing each adjacent grid cell,
define the movement of water in undefined tributaries leading to the RF1
stream. The "burn-in" method insures that all flow paths intersect a
reach cell at some point within the watershed, although inconsistencies
between the RF1 reach and the DEM-defined stream channel may
artificially lengthen "off-RF1" flow paths and shorten "on-RF1" paths
(see figure 1 for an example). The length of the flow path provides a
rough estimate of the distance sediment must travel in smaller
tributaries before reaching the larger streams included in the RF1
network. Travel time in small streams versus large rivers has been shown
to be an important factor affecting the in-stream delivery of nutrients
(Alexander and others, 2000) and could be of similar importance for
sediment.

The enhanced spatial structure afforded by the DEM facilitates the
incorporation of spatially integrating features into the model.
"Off-channel" reservoirs, located on the grid net according to their
geographic coordinates provided by the NID, act as potential sinks for
sediment emanating from cells with flow paths that intersect the
reservoir grid cell. Similarly, "off-RF1" monitoring stations can be
located on the grid and given a basin representation. Although these
stations are not useful for calibrating the delivery process within RF1
channels, they offer a high-resolution view of other processes affecting
the movement of sediment across the landscape.

Other important spatial features identified in the model include point
sources, located relative to RF1 streams based on geographic coordinates
(S. Rubin, Environmental Protection Agency, written commun., 1999), and
land associated with uses that serve as likely sources or sinks for
sediment. Point-source loadings of total suspended solids are determined
by methods developed by the National Oceanic and Atmospheric
Administration for the National Coastal Pollutant Discharge Inventory
(National Oceanic and Atmospheric Administration, 1993). Land use is
taken from the 21-class, 30-meter resolution NLCD, and summarized
according to the number of 30-meter cells of a given land-use class that
are mapped to a corresponding 1-kilometer cell. NLCD land use is used to
refine the areal extent of the various sediment erosion rates associated
with different land covers identified in the NRI.

![](media/image1.png){width="5.372222222222222in"
height="3.2736111111111112in"}

Figure 1. Schematic of a typical reach watershed illustrating the grid
cell structure and identified attributes.

The mean annual suspended-sediment flux generated within and leaving
reach watershed *j*, referred to as the incremental reach flux *F~j~*,
can be expressed as

\(1\)
$F_{j} = \sum_{c = 1}^{N_{j}}{e^{- \delta^{'}d_{c,j} + \alpha^{'}Z_{c,j}}\beta^{'}}S_{\text{c,j}}$,

where *N~j~* is the number of 1-kilometer grid cells, indexed by *c*, in
reach watershed *j*, **d***~c,j~* is a vector of factors describing the
pathway from cell *c* to the outlet of reach *j*, **δ** is a vector of
coefficients associated with the pathway variables, **Z***~c,j~* is a
vector of landscape and climatic characteristics affecting the delivery
of sediment within cell *c*, **α** is a vector of coefficients
associated with the **Z** variables, **S***~c,j~* is a vector of
sediment sources, and **β** is a vector of associated source
coefficients.

The vector **d** consists of (1) variables representing the landscape
flow-path distance traversed to reach the RF1 stream, (2) the mean slope
of the "off-RF1" flow path, the time of travel incurred along the RF1
stream, (3) variables affecting the retention of sediment in any
reservoir located along the landscape or RF1 flow path, such as
streamflow, reservoir age, and NID estimates of surface area or storage
volume, and (5) other variables identifying possible sinks along the
flow path, such as forested land or land classified by STATSGO as
wetlands or alluvium. Variables included in the **Z** vector include
runoff, overland flow, slope, and indicators of soils or other factors
affecting the movement of sediment off the field to channels. The source
vector, **S**, includes sediment erosion from the NRI and point-source
loadings.

The 1-kilometer spatial detail used to determine *F~j~*, corresponding
to nearly 8 million grid cells for the more than 60,000 reaches in the
conterminous U.S., places a heavy computational burden on the iterative,
non-linear least-squares, calibration method. To reduce the number of
computations, the reach model is simplified by assuming the **Z**
variables take a single mean value ${\overline{Z}}_{j}$ for all cells in
the reach and, for the **d** variables, by substituting a second-order
Taylor approximation about the reach-level mean ${\overline{d}}_{j}$.
The imposition of a common ${\overline{Z}}_{j}$value for all cells in a
reach is not restrictive given the spatial coarseness of existing
information. The resulting approximation is

\(2\)
$F_{j} \approx e^{- \delta^{'}{\overline{d}}_{j} + \alpha^{'}{\overline{Z}}_{j}}\sum_{c = 1}^{N_{j}}\left\{ \left( 1 - \delta^{'}\left( d_{c,j} - {\overline{d}}_{j} \right) \right)S_{c,j}^{'}\beta + (\beta \otimes \delta)^{\prime}\left( S_{c,j} \otimes \left( d_{c,j} - {\overline{d}}_{j} \right)\left( d_{c,j} - {\overline{d}}_{j} \right)^{\prime} \right)\delta \right\}$.

This approximation effectively converts the unit of observation in
equation (1) from a 1-kilometer grid cell to a reach segment, replacing
the non-linear terms dependent on individual cell values with non-linear
and linear terms dependent on reach-level means, variances and
covariances of the **d** and **S** variables.

To complete the model structure, individual reaches are combined to form
a nested basin. Each nested basin *i* consists of the set *J*(*i*) of
reaches upstream from monitoring station *i* and below any monitoring
station located further upstream (if such stations exist) (see figure
2). The sediment load for nested basin *i*, denoted *L~i~*, is equal to
the sum of the incremental fluxes from the nested reach segments *j* ∈
*J*(*i*), plus the monitored sediment discharged from the set *U*(*i*)
of nested basins bounding the upper drainage of nested basin *i* (there
may be more than one) and delivered to monitoring station *i*. The
sediment load *L~i~* is related to the upstream incremental fluxes,
*F~j~*, and monitored loads, *L~u~*, according to a log-linear relation

\(3\)
$\text{ln}\left( L_{i} \right) = \text{ln}\left( \sum_{j \ni J(i)}^{}e^{- \delta^{'}d_{j,i}}F_{j} + \sum_{u \ni U(i)}^{}{e^{- \delta^{'}d_{u,i}}L_{u}} \right) + \varepsilon_{i}$,

where **d***~j,i~* represents a vector consisting of the same variables
in **d***~c,j~*, but corresponding to the RF1-reach path extending from
the downstream-end of reach *j* to the *i*^th^ monitoring station
(accordingly, **d***~j,i~* has values of 0 for all variables pertaining
to "off-RF1" flow paths). In equation (3), an independent error term
*ε~i~* has been added to represent the combined effect of measurement
and model error introduced at nested basin *i*.

Data on reservoir storage can be incorporated directly into the model by
introducing an additional storage equation. Let **d**\* and **δ**\*
pertain to the subset of path variables and associated coefficients
determining the rate sediment is stored in reservoirs, and define *R~k~*
as the annual amount of stored sediment measured at a reservoir on reach
*k* (a similar analysis can be done for "off-RF1" reservoirs). The
reservoir storage equation takes the form

\(4\)
$\text{ln}\left( R_{k} \right) = \text{ln}\left( \sum_{j \ni J(k)}^{}\left( e^{\delta^{{}^{\prime}}d_{j,k}^{}} - 1 \right)F_{j} + \sum_{u \ni U(k)}^{}\left( e^{\delta^{{}^{\prime}}d_{u,k}^{}} - 1 \right)L_{u} \right) + w_{k}$,

where *w~k~* is a random error.

Joint estimation of equations (3) and (4), with the *F~j~* and
corresponding **α**, **β**, and **δ** parameters defined by equation
(2), is by non-linear three-stage least squares. To insure robust
estimates and to facilitate the estimation of prediction error, the
calibration of the model is repeated 200 times employing a bootstrap
estimation algorithm (see Smith and others, 1997).

![](media/image2.png){width="3.4854166666666666in"
height="2.283333333333333in"}

Figure 2. Schematic of a nested basin defined by upstream and downstream
monitoring stations.

The flexible mathematical structure used in equations (1) - (3) is
capable of accommodating a number of hypotheses concerning sediment fate
and transport. Sites of sediment storage, identified in the model as a
subset of the **d** variables, can act as sediment sources or sinks,
depending on the sign of corresponding **δ** coefficients. A random
coefficient form of the model allows storage sites to serve as sources
in some regions and sinks in others. Such behavior can be inferred
statistically by relating the prevalence of storage sites in nested
basins to the magnitude of the squared residual *ε* in these basins
(Godfrey, 1988). Non-point sources of sediment, such as soil erosion
included under **S**, are distinguished from sediment losses from
storage (e.g., an alluvial plain) identified with **d**, on the
assumption that the former is a primary process due to weathering
whereas the latter is a consequence of the accumulation of previously
weathered material which is later released to streams under changing
hydraulic conditions. Accordingly, the potential for storage loss in the
model depends on the extent of accumulated upstream soil erosion due to
weathering. The empirical validity of the USLE estimate of soil erosion
can be evaluated through statistical hypothesis tests conducted on the
relevant **β** coefficients. Alternative measures of soil erosion can
also be empirically evaluated in the model by substituting variables
serving as determinants of the USLE for the USLE erosion estimate.

The estimation of long-term suspended-sediment load at a monitoring
station is based on the regression of the natural logarithm of
instantaneous suspended-sediment concentration on current and lagged
values of the natural logarithm of daily flow and other variables
representing seasonal and trend effects. If the station has
concentration data collected more frequently than a weekly basis, the
regression model is modified to account for serial correlation. To be
included in the analysis, a station must have at least 3 years of data
between 1985 and 1995. Only data within the period 1985-95 are included
in the regression.

Mean-annual suspended-sediment load is estimated by first simulating
load for each day over the 1985-95 period and then averaging daily
values on an annual basis. Simulated loads are obtained by taking the
exponential of the sum of the predicted daily load given by the
calibrated regression model with the time trend variable set to a base
year of 1992 and a randomly selected residual from the regression model.
For days having actual monitoring data, the daily load is computed by
multiplying the measured instantaneous concentration by the daily flow.
If a station has a data record with sufficient frequency to estimate a
serial correlation parameter, the simulated daily load is based on the
conditional prediction associated with past and future observed loads,
plus a normally distributed random error having a correlation structure
consistent with the conditional prediction and with the variance
estimated by the regression model. The Monte Carlo process used to
estimate simulated daily loads for the 1985-95 period is repeated 200
times, providing 200 values for estimating the mean and standard
deviation of the average annual sediment load for a site.

SUMMARY

The model described here is intended to empirically evaluate
regional-scale processes affecting the long-term (i.e., decadal)
transport of sediment in rivers. Additionally, the model will provide
estimates of sediment mean annual flux for every reach included in the
RF1 network. Error estimates for these process evaluations and stream
predictions are determined using robust bootstrap methods. Future work
will address the dynamic behavior of sediment flux associated with
non-steady state streamflow conditions.

REFERENCES CITED

Alexander, R.B., Slack, J.R., Ludtke, A.S., Fitzgerald, K.K., and
Schertz, T.L., 1998, Data from selected U.S. Geological Survey national
stream water-quality monitoring networks, *Water Resources Research*,
34(9), pp. 2401-2405.

Alexander, R.B., Brakebill, J.W., Brew, R.E., and Richard A. Smith,
1999, *ERF1---Enhanced River Reach File 1.2* (U.S. Geological Survey
Open-File Report 99-457, Reston, Virginia).

Alexander, R.B., Smith, R.A., and Schwarz, G.E., 2000, Effect of channel
size on the delivery of nitrogen to the Gulf of Mexico, *Nature*, 403,
pp. 758-761.

Cohn, T. A., 1995, Recent advances in statistical methods for the
estimation of sediment and nutrient transport in rivers, U.S. National
Report to International Union of Geodesy and Geophysics, 1991-1994,
*Review of Geophysics*, 33 Supplemental, available on the World Wide Web
at URL <http://webserver.cr.usgs.gov/sediment/> accessed on 11/13/00.

Godfrey, L.G., 1988, *Misspecification Tests in Econometrics*,
Cambridge, 252 p.

Meade, R. H., and Parker, R. S., 1985, Sediment in rivers of the United
States, in *National Water Summary 1984*, U.S. Geological Survey
Water-Supply Paper 2275, U.S. Government Printing Office, pp. 49-60.

National Oceanic and Atmospheric Administration, 1993, *The National
Coastal Pollutant Discharge Inventory, Point Source Methods Document*,
Pollution Sources Characterization Branch, Strategic Environmental
Assessments Division, Office of Ocean Resources Conservation and
Assessment, National Ocean Service, 247 p.

Natural Resources Conservation Service, 2000, *National Resources
Inventory, 1992*, U.S. Department of Agriculture, data available on the
World Wide Web at URL <http://www.nhq.nrcs.usda.gov/NRI/>, accessed on
11/15/00.

Osterkamp, W. R., Heilman, P., and Lane, L. J., 1998, Economic
considerations of a continental sediment monitoring program,
*International Journal of Sediment Research*, 13(4), pp. 12-24.

Preston, S.D. and Brakebill, J.W., 1999, Application of spatially
referenced regression modeling for the evaluation of total nitrogen
loading in the Chesapeake Bay watershed, *U.S. Geological Survey Water
Resources Investigations Report 99-4054*, 12 p.

Robertson, D. M. and Roerish, E. D., 1999, Influence of various water
quality sampling strategies on load estimates for small streams, *Water
Resources Research*, 35(12),

> pp. 3747-3759.

Schwarz, G.E. and Alexander, R.B., 1995, *State Soil Geographic
(STATSGO) Data Base for the Conterminous United States*, U.S. Geological
Survey Open-File Report 95-449, Reston, Virginia.

Smith, R.A., Schwarz, G.E., and Alexander, R.B., 1997, Regional
interpretation of water-quality monitoring data, *Water Resources
Research*, 33(12), pp. 2781-2798.

Soil Conservation Service, 1983, *National Engineering Handbook*,
Section 3 Sedimentation, U.S. Department of Agriculture.

Stallard, R. F., 1998, Terrestrial sedimentation and the carbon cycle,
*Global Biogeochemical Cycles*, 12(2), pp. 231-257.

Steffen, L. J., 1996, A reservoir sedimentation survey information
system -- RESIS, in *Proceedings of the Sixth Federal Interagency
Sedimentation Conference*, March 10-14, 1996, Las Vegas, NV: Sponsored
by the Subcommittee on Sedimentation, Interagency Advisory Committee on
Water Data, p. 29-37.

Trimble, S. W. and Crosson, P., 2000, U.S. Soil Erosion Rates -- Myth
and Reality, *Science*, 289, pp. 248-250.

Turcios, L.M., and Gray, J.R., in press, U.S. Geological Survey Sediment
and Ancillary Data on the World Wide Web: Proceedings of the 7th Federal
Interagency Sedimentation Conference, March 25-28, 2001, Reno, Nevada, 6
p.

U.S. Army Corps of Engineers, 1996. National Inventory of Dams, data
available on the World Wide Web at URL <http://crunch.tec.army.mil/>
accessed 11/13/00.

U.S. Environmental Protection Agency, 2000a, National water quality
inventory, 1998 report to Congress, available on the World Wide Web at
URL <http://www.epa.gov/305b/98report/>, accessed 9/15/00.

U.S. Environmental Protection Agency, 2000b, The quality of our nation's
water, 1998: EPA841-S-00-001, available on the World Wide Web at URL
<http://www.epa.gov/305b/98report/98brochure.pdf>, accessed 9/27/00.

U.S. General Accounting Office, 1990, Water pollution -- Greater EPA
leadership needed to reduce non-point source pollution, *U.S. General
Accounting Office Report* GAO/RCED-91-10, 56 pp.

U.S. Geological Survey, 2000a, Hydro 1k elevation derivative database,
data available on the World Wide Web at URL
<http://edcdaac.usgs.gov/gtopo30/hydro/namerica.html> accessed 11/13/00.

U.S. Geological Survey, 2000b, National Land Cover Data, data available
on the World Wide Web at URL
<http://edcwww.cr.usgs.gov/programs/lccp/natllandcover.html>, accessed
11/13/00.

Wolman, M. G., 1977, Changing Needs and Opportunities in the Sediment
Field, *Water Resources Research*, 13(1), pp. 50-54.