EXTENSIONS TO A MODEL OF VARIABLE-WIDTH RIPARIAN BUFFERS, WITH
AN APPLICATION TO WOODY DEBRIS RECRUITMENT
John Van Sickle1
ABSTRACT: Effective management of riparian areas in watersheds requires that reach-scale
knowledge of riparian functioning be carefully 'scaled up' to provide models for entire stream
networks. Weller et al. (1998: Ecological Applications 8, 1156-1169) describe a useful
heuristic model for the network-scale average transmission of landscape runoff materials
through a variable-width riparian buffer. Their model demonstrates that a variable-width
buffer is likely to transmit more runoff materials, on average, from adjacent landscapes into a
stream than would a fixed-width buffer of the same mean width. By extending the Weller et
al. (1998) model, I show that analogous results are true for arbitrary distributions of buffer
widths and for other riparian functions such as woody debris input and stream shading.
I apply the extended model to woody debris recruitment from natural tree-fall in variable-
width riparian forest stands. The application suggests that the average number and volume of
tree boles falling into a stream network will be overestimated, if the estimate is based on the
average width of the network's riparian stands.
KEY TERMS: Riparian buffer, stream network, model, woody debris, sediment, nutrients,
material retention.
INTRODUCTION
Much of our quantitative understanding of the linkages between riparian areas and
streams is based on models developed from research on fairly short <500m), disconnected
stream reaches. To effectively manage riparian areas in whole watersheds, reach-scale
models must be 'scaled up' carefully so that assessments of riparian function and
'US Environmental Protection Agency, Western Ecology Division, National Health and
Environmental Effects Research Laboratory, Corvallis, OR 97333. Email: johnv@mail.cor.epa.gov
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structure, aggregated throughout the watershed, are reasonably accurate. Weller, Jordan,
and Correll (1998; hereafter WJC) have developed a valuable heuristic model for a
riparian buffer whose width varies along a stream network. Their key model prediction is
that such a buffer is likely to transmit more runoff materials (sediment and chemicals)
from adjacent landscapes into the network than would be predicted by applying a reach-
scale transmission model that ignores buffer width variation.
In this paper, I sketch the essential assumptions and components of the WJC
model. I then offer a more general formulation which allows the WJC model framework,
analytical methods, and key result to be extended to riparian functions such as large
woody debris delivery, stream shading, and bank stabilization (Gregory et al., 1991) - in
short, to any riparian function whose impact is strongly dependent on the width of the
buffer. Finally, the extended model is applied to the process of large woody debris
recruitment to a stream from a variable-width forested buffer.
MODEL ASSUMPTIONS AND RESULTS
The WJC Model
WJC model an idealized variable-width riparian buffer by using a grid adjacent to a
stream (Figure 1). The model buffer consists of adjacent columns of grid cells, with each
column extending unbroken from the stream edge out to a width of w cells (Figure 1).
Riparian columns of width w are assumed to be randomly and independently arrayed along a
stream network, so that the relative abundances of columns having various widths can be
specified by a single frequency distribution. WJC employ the Poisson distribution since it is
flexible, it appears reasonable for many real buffer width distributions, and allows it for a
positive frequency of zero-width gaps in the buffer. That is, they assume that the relative
frequency of buffer columns of width w is given by AVA/w!, for w = 0,1,2,..., where A is
the mean buffer width over the network.
Next, WJC give a simple conceptual model for the transmission of material, such as
sediment or nutrients, through a riparian buffer. Material from an upslope source area is
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assumed to enter a buffer column at its uphill edge and travel through the column directly
towards the stream (Figure 1). WJC assume that each unit of width in a riparian column will
Material Source Area
?
T
Riparian Buffer
ii 1 i i
:::\:>:\:j^^
W
Figure 1 - Model of a riparian buffer along a stream network, illustrating material
transmission from an upslope source area, through the buffer, and into the stream. Adapted
from WJC.
transmit a fixed fraction, t, of incoming material, where 0
-------
(1)
(See WJC for details of this derivation.). Equation 1 gives the mean network-scale material
retention, per column, for a variable-width buffer. It bears repeating that Equation 1 assumes
a spatially uniform input of runoff materials from all upslope source areas, and it also
assumes that all buffer columns have identical material transmission properties, regardless of
their longitudinal location in the network or proximity to other buffer elements (WJC).
1
— t§B:§
1Q ' 15
Buffer width, w
20
Figure 2. Material retention (l-tw) of a buffer, for low (f=0.1; solid line) and high (1=0.9;
dotted line) rates of transmissivity per unit buffer width w.
WJC then compare Rp with the mean retention RU that would be provided by a
uniform-width buffer which has the same width, A, as the average width of the variable-
width buffer. In this uniform-width buffer, 100% of the buffer columns have width w = A,
and the fractional retention of each column is (1-r). As a result, the mean retention over the
whole network for a constant -width buffer is simply RU - (1-?A). WJC then show that, for
Q
-------
In other words, a variable-width buffer will retain less material, on average over a
network, than will a fixed-width buffer having the same mean width. The difference
between Ry and RP can be substantial, for fairly narrow buffers having relatively high per-
unit retention (low per-unit transmissivity) (see Figure 4 of WJC). A practical implication
of this key result is that the network-scale material retention of a variable-width buffer may
be seriously overestimated, if it is derived by inserting the buffer's mean width into a model
of material retention which assumes a single value for buffer width (WJC).
Model Extensions
The WJC model can be extended in two ways. First of all, let/[w) be an arbitrary
function describing the relative frequency of buffer widths w = 0,1,2,... along the network.
The only restrictions are that/(w) > 0 for any w, and that I/(w) = 1. The Poisson distribution
assumed by WJC appears flexible, but other distribution functions might give more realistic
models for buffer width distribution in many cases.
Secondly, material transmission through a riparian buffer is only one of several
important processes that are strongly dependent on the width of the buffer. Define l(w) as a
generalized "Influence" function for a riparian buffer of width w. I(w) has the range (0,1)
and represents the influence of a buffer of width w, relative to the influence that would be
exerted by a buffer of infinite width. For the WJC model of material transmission, /(w) = tw,
and the corresponding influence functions for material retention are shown in Figure 2.
The effects of buffer width on other riparian processes, such as woody debris
recruitment, leaf litter input or stream shading, can similarly be represented by other
influence functions, sometimes called "riparian effectiveness" curves (FEMAT, 1993;
O'Laughlin and Belt, 1995). The shapes of these curves, and in particular the widths at which
effectiveness, or "influence" approaches 100% of its possible value, is a subject of some
importance for riparian management (Castelle et al, 1994).
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Mean Influences of Variable-Width and Uniform Width Buffers
We can now make the same set of assumptions as did WJC regarding the
independence and equal contribution of individual buffer columns throughout the network.
The mean influence of the buffer over the network is then given by
m«w[/(w)] = ]T/(w)/(w) (3)
m=0
In the language of probability theory, mean(I) is the expected value (theoretical mean) of
the function / of the random variable w, and_/(w) is a discrete probability distribution for w.
Equation 1 is thus a special case of Equation 3, for a specific influence function (material
retention), and a specific buffer width distribution (Poisson).
The key result of WJC (Inequality 2) can also be extended to other riparian processes
and buffer width distributions, by invoking the Jensen Inequality, a general property of
nonlinear transformations of random variables (Mood et al. 1974; Nedelman and Wellenius,
1986). This inequality states that, if 7(w) is any concave (downward-curving) function of a
random variable w, then
Mean[I(w)] < I(mean[w]) (4).
The material retention curve J(H>) = (\-tw) is a concave function (Figure 2), and Inequality 2
provides one example of the Jensen Inequality.
The Jensen Inequality implies that the mean, network-scale influence of a variable-
width riparian buffer will be less than the influence of a uniform-width buffer having the
same mean width. The inequality will be true to some extent for any riparian buffer function
having a downward-curving, nonlinear dependence on buffer width similar to that seen in
Figure 2. The magnitude of the difference between mean influences of variable-width and
uniform-width buffers will depend on the severity of nonlinearity in I(w), and also on the
prevalence of buffer widths that lie in the domain of the most nonlinear portions of the
riparian influence function.
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MODEL APPLICATION TO WOODY DEBRIS RECRUITMENT
Influence Functions
The extended model can be applied to the riparian buffer functions of stream shading
and large woody debris recruitment. For these two cases, a variable-width buffer containing
woody vegetation is assumed to be bordered by a land cover type, such as agricultural land or
recently-cleared forest, that is unable to provide canopy shading or large woody debris to the
stream. For riparian shading, solar radiation is the "material" that is progressively attenuated
as riparian canopy width increases, Steinblums et al. (1984), for example, assume an
exponential curve like those in Figure 2 for the influence of canopy width on stream shading.
Here, I focus on large woody debris delivery to the stream from sources of natural
tree fall such as windmrow, decomposition, and stream bank migration. Assume that a
riparian forest stand contains a single tree species having a constant density (trees/area) and
tree height h, for any stand width. Define IN(W) to be the number of tree boles delivered by a
stand of width w, as a fraction of the number delivered by a stand as wide or wider than the
stand height. Assuming that falling boles do not roll or slide towards the stream, the
maximum possible debris input is produced by stands of width w > h. Also assume a constant
probability of falling, per unit time, and assume that trees fall independently of one another
and are equally likely to fall in any direction, retaining their falling orientations when they
land.
With these assumptions, a tree bole located at perpendicular distance z from the
stream will, when it falls, have a probability of hitting the stream that is proportional to
cosl(z/h) (McDade, et al., 1990; Robison and Beschta, 1990; Van Sickle and Gregory,
1990). IN(W) is then the cumulative number of trees delivered from the uniform stand
between distances z = 0 and z = w, expressed as a proportion of the delivery from a stand of
width h:
(5).
-------
Equation 5 has been scaled so that IN = I for w = h, and IN(W) is illustrated in Figure 3 for
h = 50m.
01 .
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w
Id'
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8
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Q.
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.rr'
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v' .X ..rr-'"
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0 10 20 30 40 50
Buffer width
Figure 3. Influence functions for delivery of woody debris to a stream, from riparian
forest stands of height h = 50m and having widths between 5 and 50m. Curves are shown for
Number (N) and Volume (V) of boles, assuming trees are equally likely to fall in any
direction, and also for Number (n) of boles, assuming trees fall directly towards the stream.
On steeper slopes, trees may be much more likely to fall towards the stream (Van
Sickle and Gregory, 1990; Minor, 1997). Thus, as a counterpoint to the random-direction
case, I also explored the case in which all falling trees fall directly towards the stream, so that
any falling tree of height h at a distance z < A will hit the stream. In the directed-fall case, the
total number of boles delivered from a buffer stand of width w < h will be directly
proportional to the standing number in the buffer, relative to that in a buffer of width h, so
that /B(w) = w/h, for 0
-------
estimation is illustrated for a 50m-tall stand of Douglas-fir (Pseudotsuga menziesii (Mirb.))
bordering a 15m-wide stream, assuming equally-likely fall directions (Figure 3).
Mean Debris Recruitment into a Stream Network
The influence function In(w) for the number of tree boles delivered, assuming all
trees fall directly towards the stream, is linear in w (Figure 3). In this case, mean[In(w)]
= mean[wlh] = (l/K)mean[w]. That is, mean network recruitment can be calculated
directly from the mean stand width, regardless of the distribution of widths along the
network.
If trees are equally likely to fall in any direction, then the influence functions for
number and volume of delivered debris are both downward-curving (Figure 3), and the
Jensen Inequality will have some effect. I calculated the size of this effect for the random-
fall-direction influence functions of Figure 3.1 used Equation 3 for mean network debris
recruitment, with^w) assumed to be a Poisson distribution of stand widths, discretized in
5m width units. The difference in debris delivery rates between a uniform-width stand
and a variable-width stand of the same mean width were calculated for mean stand widths
ofX = 5, 10, 15,...50m (Figure 4).
£.0
Sffl
0) O
T3
go
!§j
,-v~ v.
.M'\
,N-" "V...
.
.-NT
10 20 30 40 50
Mean Buffer width
Figure 4. The difference I(mean[w\) - Mean[I(w)} in debris recruitment rates
between fixed-width and variable-width riparian buffer stands, as a function of mean
-------
buffer width X, for number (N) and volume (V) of tree boles delivered to stream. See
Figure 3 discussion for assumed stand characteristics.
For number of boles delivered, the overestimate is greatest for mean stand widths
close to h, because uniform-width stands having large mean widths can deliver tree boles
at near-maximum rates throughout the network (Figure 4). In contrast, variable-width
stands contain many buffer columns that are much narrower than h and hence deliver less
debris, so that the overall network average is reduced. For volume delivery, the difference
is greatest for small mean widths (Figure 4), because trees near the stream contribute a
much larger proportion of the total volume than of total number delivered (Figure 3), due
to bole taper and to the fact that volume delivery is greatest if the fallen bole can totally
span the channel. Thus, the loss of network-scale volume delivery from narrow, variable-
width buffers due to buffer gaps is not nearly compensated by other buffer columns that
are wider than average.
However, neither the mean number nor mean volume of tree boles are
overestimated by more than 0.1 of the maximum possible delivery rates, if the estimates
are derived directly from mean buffer width rather than from the full width distribution
(Figure 4; cf. Figure 4 of WJC). Thus, although the number and volume influence
functions appear distinctly curvilinear in Figure 3, this does not translate into a major
impact of the Jensen Inequality on mean network-scale debris recruitment.
The results in Figure 4 assume that riparian stands have a constant stand density
and probability of tree fall with respect to distance z from the channel. One could also
model stand density and tree fall probability as varying functions of z and incorporate
them as multiplicative factors in the integrand of Equation 5 (Van Sickle and Gregory,
1990), thus giving a much more realistic stand-width influence function for debris
delivery.
DISCUSSION
The central message of the extended WJC model is that reach-scale or site-scale
models of riparian effects on a stream must be carefully "scaled up" in order to give
10
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correct estimates for the riparian buffer effects on whole stream networks. In particular,
models that simplify a variable-width riparian buffer by treating it as one long strip of
uniform-width vegetation are likely to result in systematic overestimates of network-scale
buffer effects for processes such as material retention, debris recruitment, and stream
shading. Put another way, gaps and thin segments in buffers can have an overall
deleterious effect on network-scale riparian functioning that is out of proportion to their
relative occurrence in the buffer (WJC).
By employing only a probability distribution to describe buffer width variation,
the extended WJC model is neutral (Caswell, 1976) with respect to the absolute location
of buffer elements along the stream, as well as to their relative proximities. It thus
provides a null hypothesis against which riparian managers can test their ideas for
targeting riparian enhancement sites in order to yield maximum overall benefit to the
stream network. The extended WJC model proposes a spatially-simplistic watershed-
scale metric of riparian functioning — the mean buffer influence. In doing so, it
challenges riparian ecologists to demonstrate a clear need for, and to develop, more
realistic models of the riparian landscape.
REFERENCES
Castelle, A.J., A.W. Johnson and C. Conolly, 1994. Wetland and Stream Buffer Size
Requirements - A Review. Journal of Environmental Quality 23: 878-882.
Caswell, H,, 1976. Community Structure: A Neutral Model Approach. Ecological
Monographs 46: 327-354.
FEMAT, 1993. Aquatic Ecosystem Assessment: Riparian Ecosystem Components. In:
Forest Ecosystem Management: An Ecological, Economic and Social Assessment, p.V-
11
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25-V29. Report of the Forest Ecosystem Management Assessment Team, USDA Forest
Service et aL, Washington DC,
Gregory, S.V., F.J. Swanson, W.A. McKee and K.W. Cummins, 1991. An Ecosystem
Perspective of Riparian Zones. Bioscience 41: 540-551.
McDade, M.H., FJ. Swanson, W.A.McKee, J.F. Franklin, and J. Van Sickle, 1990.
Source Distances for Coarse Woody Debris Entering Small Streams in Western Oregon
and Washington. Canadian Journal of Forest Research 20: 326-330.
Minor, K.P. 1997. Estimating Large Woody debris Recruitment from Adjacent Riparian
Areas. M.S. Thesis, College of Forestry, Oregon State University, Corvallis, OR.
Mood, A.M., F.A. Graybill, and D.C. Does, 1974. Introduction to the Theory of Statistics
(3rd edition). McGraw-Hill, New York, 564 pp.
Nedelman, J. and T.Wallenius, 1986. Bernoulli Trials, Poisson Trials, Surprising
Variances, and Jensen's Inequality. The American Statistician 40:286-289.
O'Laughlin, J. and G.H. Belt, 1995. Functional Approaches to Riparian Buffer Strip
Design. Journal of Forestry 93(2): 29-32.
Robison, E.G. and R.L. Beschta, 1990. Identifying Trees in Riparian Areas That Can
Provide Coarse Woody Debris to Streams. Forest Science 36: 790-801.
Steinblums, I.J., H.A. Froelich and J.K. Lyons, 1984. Designing Stable Buffer Strips for
Stream Protection. Journal of Forestry 82: 49-52.
12
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Van Sickle, J, and S.V. Gregory, 1990. Modeling Inputs of Large Woody Debris to
Streams from Falling Trees. Canadian Journal of Forest Research 20: 1593-1601.
Weller, D.E., T.E. Jordan, and D.L. Correll, 1998. Heuristic Models for Material
Discharge from Landscapes with Riparian Buffers. Ecological Applications 8: 1156-
1169.
13
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TECHNICAL REPORT DATA
(Please read instructions on the reverse before completing)
1. REPORT NO.
EPA/600/A-00/022
2.
4. TITLE AND SUBTITLE Extensions to a model of variable-width riparian buffers,
with an application to woody debris recruitment.
7. AUTHORJS) John Van Sickle
9. PERFORMING ORGANIZATION NAME AND ADDRESS
US EPA NHEERL WED
200 SW 35th Street
Corvallis, OR i7333
12. SPONSORING AGENCY NAME AND ADDRESS
US EPA ENVIRONMENTAL RESEARCH LABORATORY
200 SW 35th Street
Corvallis, OR 97333
3. RECIPIENT'S ACCESSION NO.
5. REPORT DATE
6. PERFORMING ORGANIZATION
CODE
8. PERFORMING ORGANIZATION REPORT
NO.
10. PROGRAM ELEMENT NO.
1 1 . CONTRACT/GRANT NO.
1 3. TYPE OF REPORT AND PERIOD
COVERED
14. SPONSORING AGENCY CODE
EPA/600/02
15. SUPPLEMENTARY NOTES:
1 6. Abstract: Effective management of riparian areas in watersheds requires that reach-scale knowledge of riparian functioning be carefully
'scaled up' to provide models for entire stream networks. Weller et al (1 998: Ecological Applications 8, 1 1 56-1 1 69) describe a useful heuristic
model for the network-scale average transmission of landscape runoff materials through a variable-width riparian buffer. Their model
demonstrates that a variable-width buffer is likely to transmit more runoff materials through a variable-width riparian buffer. Their model
demonstrates that a variable-width buffer is likely to transmit more runoff materials, on average, from adjacent landscapes into a stream than
would a fixed-width buffer of the same mean width. By extending the Weller et al. (1998) model, show that analogous results are true for
arbitrary distributions of buffer widths and for other riparian functions such as woody debris input and stream shading. I apply the extended
model to woody debris recruitment from natural tree-fall in variable-width riparian forest stands. The application suggests that the average
number and volume of tree boles falling into a stream network will be overestimated, if the estimate is based on the average width of the
network's riparian stands.
17.
a. DESCRIPTORS
Riparian buffer, stream network, model,
woody debris, sediment, nutrients, material
retention
18. DISTRIBUTION STATEMENT
KEY WORDS AND DOCUMENT ANALYSIS
b. IDENTIFIERS/OPEN ENDED
TERMS
19. SECURITY CLASS (This Report)
20. SECURITY CLASS (This page)
c. COSATI Field/Group
21. NO. OF PAGES: 13
22. PRICE
EPA Form 2220-1 (Rev. 4-77} PREVIOUS EDITION IS OBSOLETE
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