1156
Ecological Applications, 8(4), 1998, pp. 1156?1169
q 1998 by the Ecological Society of America
HEURISTIC MODELS FOR MATERIAL DISCHARGE FROM LANDSCAPES
WITH RIPARIAN BUFFERS
DONALD E. WELLER, THOMAS E. JORDAN, AND DAVID L. CORRELL
Smithsonian Environmental Research Center, 647 Contees Wharf Road, P.O. Box 28,
Edgewater, Maryland 21037-0028 USA
Abstract. For landscapes with riparian buffers, we develop and analyze models pre-
dicting landscape discharge based on material release by an uphill source area, the spatial
distribution of riparian buffer along a stream, and retention within the buffer. We model
the buffer as a grid of cells, and each cell transmits a fixed fraction of the materials it
receives. We consider the effects of variation in buffer width and buffer continuity, quantify
the relative contributions of source elimination and buffer retention to total discharge
reduction, and develop statistical relationships to simplify and generalize the models. Width
variability reduces total buffer retention, increases the width needed to meet a management
goal, and changes the importance of buffer retention relative to source elimination. Variable-
width buffers are less efficient than uniform-width buffers because transport through areas
of below-average buffer width (particularly gaps) dominates landscape discharge, especially
for narrow buffers of highly retentive cells. Uniform-width models overestimate retention,
so width variability should be considered when testing for buffer effects or designing buffers
for water quality management. Adding riparian buffer to a landscape can decrease material
discharge by increasing buffer retention and by eliminating source areas. Source elimination
is more important in unretentive or wide buffers, while buffer retention dominates in narrow,
retentive buffers. We summarize model results with simpler statistical relationships. For
unretentive buffers, average width is the best predictor of landscape discharge, while the
frequency of gaps was best for narrow, retentive buffers. Together, both predictors explain
.90% of the variance in average landscape transmission for any value of buffer reten-
tiveness. We relate our results to ecological theory, landscape-scale buffer effects, buffer
management, and water quality models. We recommend more empirical studies of buffer
width variability and its effects on material discharge. Landscape models should represent
width variability and the nonlinear interactions between buffers and source areas.
Key words: grid cell; landscape ecology; landscape index; model; nonpoint source pollution;
nutrient discharge; raster; riparian buffer; riparian management; scaling; sediment discharge; water
quality.
INTRODUCTION
Landscape heterogeneity is one focus of the emerg-
ing discipline of landscape ecology (Forman 1982, Ris-
ser et al. 1984, Turner 1989), which seeks to understand
spatial patterns, their causes, and their ecological ef-
fects (Forman and Godron 1987, Forman 1995). A body
of theory for some aspects of landscape ecology has
been derived from first principles (Forman 1995) or
other theoretical constructs (Gardner et al. 1987, Milne
1988, Gardner and O?Neill 1990, Milne 1992, O?Neill
et al. 1992). The effects of heterogeneity on material
transport are also a major interest for landscape ecology
(Risser et al. 1984, Jordan et al. 1986, Correll et al.
1992), but less progress has been made in developing
an empirical and theoretical foundation in this area.
This may restrict the application of landscape ecology
principles to some practical issues, such as managing
nonpoint-source pollution (Loehr 1974, Duda 1982).
Landscapes with riparian buffers are one case where
Manuscript received 2 June 1997; revised 15 December
1997; accepted 22 December 1997.
strong effects of landscape heterogeneity on material
fluxes have been demonstrated. A riparian buffer is a
strip of relatively undisturbed vegetation positioned
along a stream and downhill from a source of material
release. Riparian forests can take up large amounts of
water, sediment, and nutrients from surface and
groundwater draining uphill agricultural areas (Lowr-
ance et al. 1984, Peterjohn and Correll 1984, Jacobs
and Gilliam 1985, Jordan et al. 1993). Buffers can also
retain materials from other sources (Chescheir et al.
1991, Schellinger and Clausen 1992, Hanson et al.
1994, Bren 1995). With effective retention, small areas
of riparian buffer can greatly reduce land discharges
of nutrients to aquatic systems (Jordan et al. 1986), so
there has been strong interest in managing riparian sys-
tems to reduce nonpoint-source water pollution (Lowr-
ance et al. 1985, Correll et al. 1994, Correll 1997,
Haycock et al. 1997, Lowrance et al. 1997).
Riparian buffers have been studied by using transects
to observe material uptake from water traversing the
buffer (Lowrance et al. 1984, Peterjohn and Correll
1984, Jacobs and Gilliam 1985, Jordan et al. 1993),
November 1998 1157RIPARIAN BUFFER LANDSCAPE MODELS
FIG. 1. Conceptual model of a landscape with a riparian
buffer. The landscape is divided into a grid, and cells along
the stream are occupied by the buffer ecosystem. Water and
materials flow downhill from the source ecosystem, through
the buffer, and to the stream. Models exploring the relative
roles of buffer retention and source ecosystem replacement
to total nutrient reduction also include a maximum landscape
width, wmax.
and buffer models have also focused on the transect
scale (Tollner et al. 1982, Williams and Nicks 1988,
Flanagan et al. 1989, Phillips 1989, Nieswand et al.
1990, Altier et al. 1994). Success in scaling transect
results to landscape-level analyses and models has been
mixed. Some statistical comparisons among watershed
landscapes have found little effect of streamside veg-
etation on water quality (Omernik et al. 1981, Hunsaker
and Levine 1995, Johnson et al. 1997). Other statistical
analyses (Osborne and Wiley 1988) and spatial models
(Levine and Jones 1990, Hunsaker and Levine 1995,
Soranno et al. 1996) have concluded that near-stream
areas do have a disproportionate effect on water qual-
ity. The importance of the riparian zone within the
whole catchment remains poorly understood (Johnson
et al. 1997).
In this paper, we develop and analyze a suite of math-
ematical models for material discharge from landscapes
with riparian buffers. The models predict landscape
discharge based on material release from a source eco-
system, the spatial distribution of riparian buffer, and
material retention within the buffer. We analyze the
models to explore how variability in buffer width in-
teracts with buffer retentiveness to yield overall land-
scape discharges. We consider the effects of buffer
width and continuity, quantify the contributions of
source elimination and buffer retention to total dis-
charge reduction, and suggest some statistical simpli-
fications for scaling results to larger landscapes. We
relate the results to the theory of landscape ecology,
to the design of riparian buffers for reducing nonpoint-
source pollution, and to nonpoint-source models for
predicting total discharge from real landscapes.
MODELS AND RESULTS
Conceptual model
Our model considers a hypothetical landscape con-
taining two ecosystems: an uphill source ecosystem
that releases waterborne materials and a downhill ri-
parian buffer that can take up those materials before
they reach a stream. We implement the spatial model
as a grid of buffer cells grouped along a stream, with
water transporting materials downhill through the buff-
er and into the stream (Fig. 1). We model the retention
of materials within the buffer by letting each cell dis-
charge a fixed fraction, t, of the materials it receives.
The discharge dw from a column of buffer cells of width
w would then be the fraction tw of the input, i, to that
column:
dw 5 itw 5 iewlnt. (1)
Thus, material uptake is a simple first-order process,
and material flux decreases exponentially with the
width of buffer traversed. Eq. 1 treats a buffer cell as
a ??black box?? because we want to focus on the higher
level interaction of ecosystem retention with landscape
structure, not retention mechanisms. The linear reten-
tion function (Eq. 1) contributes to a mathematically
tractable landscape model, and first-order response
functions have been observed in buffer systems (Ches-
cheir et al. 1991, Haycock and Pinay 1993) and used
in spatial models of material flux (Hunsaker and Levine
1995, Soranno et al. 1996).
Assumptions and analyses
We analyze a suite of related models that all share
some assumptions. We assume that material release
from a buffer receiving no source inputs is effectively
zero, so we model only the fate of materials from the
source system. We represent the landscape as a grid of
cells. We assume that water enters a cell along its uphill
edge and leaves along its downhill edge. We model
total retention from both surface water and ground-
water with a single, first-order function. We assume
that all buffer cells have identical retention capabilities.
We supplement this basic framework to address spe-
cific questions. We explore the consequences of vari-
ability of buffer width by comparing uniform-width
buffers to buffers that have a Poisson width distribu-
tion. In choosing the Poisson distribution, we assume
that the riparian buffer is a minor component of the
landscape and buffer cells are arrayed randomly and
independently along the stream channel (Pielou 1977,
Sokal and Rohlf 1981). We also assume that the output,
i, from the source ecosystem is independent of buffer
width. Then, we drop this last assumption and let
source ecosystem area and material output decrease
with increasing buffer width. This complicates the
model, but allows us to evaluate the relative importance
of source elimination and buffer retention in reducing
total landscape discharge. Finally, we examine buffers
that follow any width distribution to test the generality
of conclusions about width variability from analysis of
the Poisson distribution and to suggest some useful
simplifications of the mathematical model.
1158 DONALD E. WELLER ET AL. Ecological Applications
Vol. 8, No. 4
FIG. 2. Landscape buffer transmission, T, vs. average
buffer width for three values of cell transmission, t. The solid
curves are for uniform-width buffers (Eq. 3), while the dashed
curves are for buffers with Poisson width distributions (Eq.
10). T is a fraction of the total material export from the source
ecosystem to the buffer ecosystem.
FIG. 3. The Poisson distribution. Frequency distributions of buffer width (Eq. 6) are shown for three values of average
width, l.
Uniform-width buffer
For a riparian buffer of uniform width w, the average
per column discharge from the entire landscape, D, is
given by Eq. 1:
D 5 dw 5 itw. (2)
We can divide D by the input from the source ecosys-
tem to derive a relationship for the fraction of input
discharged by the buffer. The average buffer transmis-
sion, T, for the landscape is then
T 5 D/i 5 tw 5 ewlnt. (3)
The quantity T is a landscape index that summarizes
the combined effects of buffer retentiveness and buffer
width. This index is independent of material input from
the source ecosystem and describes the fraction of any
material load from the source ecosystem that would
reach the stream. The fraction of materials transmitted
to the stream decreases exponentially with buffer
width, and the rate of decrease with increasing width
is higher when individual buffer cells are more reten-
tive (Fig. 2).
The width, w9, required to keep landscape discharge
below some target threshold D9 is given by D 5 itw9
# D9, which yields
ln D9/i ln T9
w9 $ 5 (4)
ln t ln t
where T9 is the target threshold expressed as average
buffer transmission. Note that if the chosen threshold
is zero, then cell transmission, t, must also be zero
(L?Hopital?s rule for indeterminate forms shows that
Eq. 4 approaches 0 as t ? 0). Values of t between 0
and 1 allow landscape transmission to approach zero
asymptotically with increasing width (Eq. 3). The
width needed to remove half the materials entering the
buffer (half-distance) is given by Eq. 4 with T9 5 0.5:
w1/2 5 ln 0.5/ln t 5 20.693/ln t. (5)
Buffer with Poisson width distribution
Width distribution.?Natural riparian buffers seldom
occur at uniform widths, and the widths could follow
different statistical distributions. We explored the con-
sequences of one particular distribution, the Poisson,
which gives variable buffer width along the stream
channel, including some zero-width gaps (Fig. 3). The
distribution may be appropriate for many riparian buf-
fers and provides a mathematically tractable way to
explore the effects of variability in buffer width. The
probability of any integer buffer width w $ 0 is
w 2l
l e
p 5 (6)w
w!
where l $ 0 is the average width of the buffer. Initially,
we let the output, i, from the source ecosystem be in-
dependent of buffer width. This is reasonable if buffer
width is small relative to the width of the source eco-
November 1998 1159RIPARIAN BUFFER LANDSCAPE MODELS
FIG. 4. The difference between Poisson landscape trans-
mission, TP, and uniform-width landscape transmission, Tu,
for three values of cell transmission, t (see Eq. 11).
system (wmax k l), as already assumed in choosing the
Poisson distribution.
Landscape discharge.?The fraction of average ma-
terial discharge, D, contributed by all columns of width
w is the frequency of width w (Eq. 6) times the dis-
charge width w:
w 2l
l e
wp d 5 it . (7)w w
w!
Summing over all possible buffer widths gives average
per column discharge:
` w(lt)
2lD 5 ie . (8)
O
w!w50
The summation is a power series in lt and converges
to elt (Beyer 1976), so Eq. 8 simplifies to
D 5 ie2l(12t). (9)
and the average transmission (T 5 D/i) for the entire
buffer is
T 5 e2l(12t). (10)
For the linear Poisson model, the landscape index T
(Eq. 10) summarizes the combined effects of buffer
retentiveness and the buffer width distribution. The in-
dex describes the fraction of any source ecosystem dis-
charge that would reach the stream. This fraction de-
creases exponentially with average buffer width, and
the rate of decrease with width is higher when buffer
cells are more retentive (lower t, Fig. 2). Landscape
transmission (Eq. 10) is equally sensitive to changes
in average width and cell transmission. To illustrate
this more clearly, define cell retention, r 5 1 2 t, so
that T 5 e2lr. An equal change in l or r would alter T
by the same amount, but l has a wider range (l $ 0)
than t or r, which must be between 0 and 1.
Comparison to uniform-width buffer.?For a given
average buffer width, the Poisson distribution will give
some stream frontage with less than average buffer
width (more material discharge), but also some front-
age with greater than average buffer width (less dis-
charge). How will these two effects balance? Eqs. 3
and 10 can be used to calculate the difference between
landscape transmission for Poisson buffers (TP) and the
uniform-width buffers (Tu) of the same average width
w?:
TP 2 Tu 5 e2w?(12t) 2 tw? 5 e2w?(12t) 2 ew?lnt. (11)
The two exponential terms will be equal (TP 5 Tu)
whenever 2w?(1 2 t) 5 w? ln t. This occurs for three
parameter values: w? 5 0 (no buffer, TP 5 Tu 5 1), w?
? ` (infinitely wide buffer, TP and Tu both ? 0), and
t 5 1 (completely unretentive buffer, TP 5 Tu 5 1). For
any other parameter values, the Poisson buffer trans-
mits more materials than the uniform-width buffer (Fig.
2), because 1 2 t is greater than ln t (see Eq. 11) for
all 0 # t , 1. The difference between the uniform-
width and Poisson buffers is greatest for narrow (low
w?) but highly retentive (t near 0) buffers (Fig. 4). As
average width increases but cell transmission is fixed,
the difference between Poisson and uniform-width buf-
fers first increases, then peaks and declines. The av-
erage width at which the difference (TP 2 Tu) is greatest
decreases as cell transmission declines (Fig. 4).
We can also compare uniform- and variable-width
buffers by asking what minimum buffer width (either
l9 or w9) is needed to keep landscape transmission T
below some threshold T9. For a Poisson buffer, the
average width, l9, such that T 5 e2l9(12t) # T9 is
2ln T9
l9 $ . (12)(1 2 t)
Dividing Eq. 12 by Eq. 4 gives the ratio of the minimum
average width for a Poisson buffer divided by the min-
imum width for a uniform-width buffer:
minimum Poisson average width 2ln T9/ln(1 2 t)
5
minimum uniform width ln T9/ln t
ln t
5 . (13)
t 2 1
The ratio is greater than 1 for all values of 0 # t , 1,
so a given material reduction would require a wider
average width for a Poisson buffer than for a uniform-
width buffer. The ratio of needed widths is greater for
more retentive buffers (lower t). The Poisson buffer
must average about 1.5 times wider at t 5 0.4, about
two times wider at t 5 0.2, and about four times wider
at t 5 0.02 (Fig. 5). L?Hopital?s rule shows that the
ratio ? 1 as t ? 0.
Variability in buffer width has important effects on
landscape discharge. When buffer width varies within
the landscape, the buffer retains less material than a
uniform-width buffer of equivalent average width. In-
ferring landscape discharge from average width only
would overestimate material retention, and the error
1160 DONALD E. WELLER ET AL. Ecological Applications
Vol. 8, No. 4
FIG. 5. Comparison of the average Poisson width, l9, and
the width, w9, of a uniform-width buffer needed to achieve
any target landscape transmission, T9. The ratio of Poisson
width to uniform width (Eq. 12) is plotted against cell trans-
mission, t.
FIG. 7. Fraction of material discharge through gaps in
buffers with Poisson width distributions. The solid lines are
the fraction of discharge through gaps (Eq. 16) for different
values of cell transmission, t. The dashed line is the frequency
of gaps (Eq. 15).
FIG. 6. Width distribution of material transmission for Poisson buffers of average width l 5 4 at three values of cell
transmission, t (Eq. 14).
involved would be greatest for narrow but highly re-
tentive buffers.
Width distribution of material loss.?The extra ma-
terial discharge through areas of below-average buffer
width outweighs the extra material retention where the
Poisson buffer is wider than average. We can examine
this more completely by comparing the frequency dis-
tribution of buffer widths (Eq. 6) to the width distri-
bution of material transmission. Material transmission
through all columns of stream frontage of width w (Eq.
7) divided by input, i, is
transmission through all columns of width w
w 2l(lt) e
w
5 p t 5 (14)w
w!
(note that pwtw 5 T). For unretentive buffers (t`Sw50
near 1), the width distribution of material transmission
(Fig. 6) is quite similar to the distribution of column
width (Fig. 3, center), but becomes increasingly skewed
as retentiveness increases (Fig. 6). When buffer cells
are highly retentive (t ? 0), almost all landscape dis-
charge comes for areas with little or no buffer (Fig. 6).
The frequency of gaps (zero buffer width) is given by
Eq. 6 with w 5 0, such that
p0 5 e2l (15)
while Eq. 14 with w 5 0 divided by Eq. 10 gives
proportion of landscape discharge through gaps
5 e2lt. (16)
For unretentive buffers, the proportion of discharge
through gaps is the same as the proportion of gaps (Fig.
7). In contrast, almost all the discharge from retentive
buffers comes through gaps (e2lt ? 1), even when gaps
are a very small proportion of stream frontage (Figs.
3 and 6). As buffer retentiveness increases, gaps in the
November 1998 1161RIPARIAN BUFFER LANDSCAPE MODELS
FIG. 8. Buffer retention for uniform-width buffers as a
function of buffer fraction. The vertical axis shows fractions
relative to maximum discharge from a landscape completely
covered by source ecosystem (no buffer). The horizontal axis
is the fraction of the landscape occupied by streamside buffer
(f). The dashed line shows how export from the source eco-
system (1 2 f) declines with decreasing source ecosystem
area. This line defines an upper limit for buffer retention.
Retention capacity (1 2 tf; upper solid line) increases with
increasing buffer area (see Eq. 19). Realized buffer retention
(the product of source export and retention capacity; lower
solid line) is highest for intermediate buffer fractions.
buffer are increasingly the sites of material delivery to
the stream (Fig. 7).
Source elimination vs. buffer retention
Adding a riparian buffer to a landscape can reduce
material discharge in two ways. The buffer retains ma-
terials, and some of the source ecosystem is replaced
with buffer ecosystem that does not release materials
(Staver et al. 1988). The previous models did not in-
clude the trade-off between source area and buffer
width, so they could not represent the effect of source
elimination. Now, we enhance the model so that the
buffer width in a column of stream frontage reduces
the width and material release of the uphill source area.
Uniform-width buffer.?For unbuffered stream front-
age, the width of source area is wmax, and its material
release is swmax, where s is material release per source
ecosystem cell. When a buffer of width w is added, the
width of source ecosystem drops to (wmax 2 w) and its
material release drops by sw to s(wmax 2 w). Buffer
retention further reduces this flux by tw to give the final
landscape discharge:
D 5 s(wmax 2 w)tw. (17)
Average buffer transmission, T, for the landscape is
average discharge (Eq. 17) divided by average input,
I 5 s(wmax 2 w), from the source ecosystem. This gives
the same expression for T derived earlier (Eq. 3). The
expression for minimum buffer width to achieve a de-
sired landscape discharge is Eq. 4. These correspon-
dences between the enhanced and simpler models sup-
port the utility of the simpler representation.
The discharge reduction from source ecosystem re-
placement is the reduction in source output (sw), di-
vided by swmax to express the reduction as a fraction
of the maximum possible discharge when w 5 0:
sw w
source elimination 5 5 5 f (18)
sw wmax max
where f 5 w/wmax is the fraction of the landscape oc-
cupied by the buffer. This reduction occurs whether or
not there is material retention in the buffer.
Discharge reduction by buffer retention is the input
from the source ecosystem s(wmax 2 w) minus final
landscape discharge s(wmax 2 w)tw, again divided by
swmax to obtain a fraction of maximum possible dis-
charge:
buffer retention
ws(w 2 w) 2 s(w 2 w)tmax max
5
swmax
w f fmax
5 (1 2 f )(1 2 {t } ) 5 (1 2 f )(1 2 t ) (19)
where w is replaced with fwmax (Eq. 18) and t 5 .wmaxt
Eq. 19 has a clear interpretation. The term (1 2 f) is
the fraction of land occupied by the source and also
the fraction of maximum material export that is actually
released by the source. This fraction falls linearly from
1 at f 5 0 to 0 at f 5 1 (Fig. 8). The second term, 1
2 t
f
, is the material retention capacity for a buffer of
actual width w 5 fwmax. The quantity t 5 is thewmaxt
fraction of material transmitted through a buffer with
cell transmission t and width wmax. Thus, t is the min-
imum possible buffer transmission for the landscape,
and 1 2 t is the maximum possible retention. Raising
t to the fth power gives the fraction of material trans-
mitted when the buffer occupies only part of the land-
scape (w , wmax). Note that tf . t unless t 5 0, t 5
1, or f 5 1. The difference (1 2 tf) is the realized
retention capacity for a buffer of width w 5 fwmax. This
difference rises with increasing buffer width, starting
at 0 when f 5 0 (no buffer present) and curving upward
to its maximum value of (1 2 t) when f 5 1 (Fig. 8).
The initial rise is steeper and the curvature is greater
for lower values of t (potentially more retentive land-
scapes). Multiplying the source export fraction (1 2 f)
by buffer retention capacity (1 2 tf) yields the actual
reduction in discharge from retention (Eq. 19). As f
increases from 0 to 1, buffer retention (Eq. 19) is zero
at f 5 0 (no buffer), rises to a maximum at intermediate
f, then drops back to zero at f 5 1 (no source to release
materials) (Fig. 8). The shape of this curve depends on
t 5 . The curve rises more steeply with increasingwmaxt
f and reaches a higher peak at lower f when t is low
(high retention potential) (Fig. 9).
Total material reduction is the sum of source elim-
ination (Eq. 18) and buffer retention (Eq. 19):
1162 DONALD E. WELLER ET AL. Ecological Applications
Vol. 8, No. 4
FIG. 9. (Left) Discharge reduction from buffer retention in a uniform-width buffer for three values of minimum possible
buffer transmission, t. The vertical axis gives fractions relative to the maximum possible material discharge from a landscape
completely occupied by source ecosystem. The horizontal axis is the fraction of the landscape occupied by streamside buffer
(f). Buffer retention cannot exceed source ecosystem export (1 2 f; dashed line). (Right) Isopleths for the same three values
of minimum possible buffer transmission as a function of cell transmission, t, and maximum landscape width, wmax.
FIG. 10. Fractions of material reduction from source elimination and buffer retention in a uniform-width buffer. The
vertical axis gives fractions of the maximum possible material discharge from a landscape completely occupied by the source
ecosystem. The horizontal axis is the fraction of the landscape occupied by streamside buffer (f). The solid curves show the
fraction of material reduction from source elimination (Eq. 18), the fraction from buffer retention (Eq. 19), and their sum
(Eq. 20). (Left) Landscape with relatively high retention potential (t 5 0.0001). (Right) Landscape with low retention potential
(t 5 0.5).
total reduction 5 f 1 (1 2 f)(1 2 tf)
5 1 2 (1 2 f)tf. (20)
The total discharge reduction is always 0 at f 5 0 (no
buffer) and 1 at f 5 1 (no source). As the fraction of
buffer, f, increases from 0 to 1, the importance of buffer
retention to total reduction depends on the retention
potential of the landscape. In landscapes with high re-
tention potential (Fig. 10, left), buffer retention rises
steeply as f first increases above 0 so that buffer re-
tention is a large fraction of total reduction. However,
as f increases further, buffer retention peaks and ap-
proaches a declining curve of slope 21. In this region,
total material reduction changes little and further in-
creases in buffer width merely raise source elimination
while reducing buffer retention. For unretentive buffers
(Fig. 10, right), buffer retention (Eq. 19) may be less
than source elimination (Eq. 18) for all values of f. For
such ineffective buffers, buffer retention is always a
minor component of total reduction, and more of the
benefit of adding the buffer comes from eliminating
the source ecosystem.
Buffer with Poisson width distribution.?We also ex-
amined the relative importance of source ecosystem
elimination and buffer retention in landscapes with buf-
fers following a Poisson width distribution. We re-
placed the fixed source input term, i, in Eq. 7 with the
expression, s(wmax 2 w), for variable source input to
obtain
w 2l
l e
wp d 5 s(w 2 w)t . (21)w w max
w!
As before (Eq. 8), we sum over all possible widths:
` ` `w w(lt) w(lt)
2lD 5 p d 5 se w 2 .
O O Ow w max
1 2w! w!w50 w50 w50
(22)
The two summations are power series in lt converging
to elt and ltelt, respectively (Beyer 1976). Eq. 22 sim-
plifies to
November 1998 1163RIPARIAN BUFFER LANDSCAPE MODELS
FIG. 11. Graphical comparison of buffer retention in buffers with Poisson width distributions and uniform-width buffers.
(Left) Buffer retention for the four combinations of cell transmission t and maximum width wmax indicated by the pairs of
numbers on the figure. The vertical axis shows fractions relative to maximum discharge from a landscape completely covered
by source ecosystem (no buffer). The horizontal axis is the fraction of the landscape occupied by streamside buffer (f). The
upper curve in each pair is for a uniform-width buffer, while the lower is for a Poisson buffer. (Right) The difference between
uniform-width buffering and Poisson-width buffering for each pair of curves in the left panel.
D 5 s(wmax 2 lt)e2l(12t). (23)
The average input to the buffer is
` ` ` wwl
2lI 5 s(w 2 w)p 5 s w p 2 e .
O O Omax w max w
1 2w!w50 w50 w50
(24)
The first summation (the Poisson distribution function)
converges to 1, and the second is a power series con-
verging to lel. Average input then is
I 5 s(wmax 2 l) (25)
and average landscape transmission is
2l(12t)(w 2 lt)emaxT 5 D/I 5 . (26)(w 2 l)max
Results for the Poisson model including source elim-
ination do not differ greatly from the earlier Poisson
model. For example, average buffer transmission T in
the enhanced model is the expression from the simpler
model (Eq. 10) times (wmax 2 lt)/(wmax 2 l). The pro-
portion of total discharge through gaps is the expres-
sion from the simpler model (Eq. 16) multiplied by
wmax/(wmax 2 lt). Both of these multiplying ratios are
very close to 1 as long as l K wmax, a constraint we
have already accepted in choosing the Poisson distri-
bution. The close correspondence between the simple
and enhanced models again supports the utility of the
simpler representation.
Discharge reductions from source elimination and
buffer retention can be derived as before. The reduction
from source elimination is the same as for a uniform-
width buffer (Eq. 18) with w 5 l. The reduction from
buffer retention is average input from the source eco-
system (Eq. 25) minus final landscape discharge (Eq.
23), divided by maximum possible discharge:
buffer retention
2l(12t)s(w 2 l) 2 s(w 2 lt)emax max
5
swmax
2 f w (12t)max
5 (1 2 f ) 2 (1 2 ft)e . (27)
Total reduction is the sum of Eqs. 18 and 27:
total reduction
2l(12t) 2 f w (12t)max
5 1 2 (1 2 ft)e 5 1 2 (1 2 ft)e . (28)
Eq. 27 for Poisson buffer retention cannot be fac-
tored into source export and buffer retention capacity
terms, and the effects of cell transmission and maxi-
mum width cannot be combined into a single parameter.
These complexities preclude the simple interpretations
achieved for Eq. 19, but graphical analysis demon-
strates the similarities to and differences from uniform-
width systems. Eq. 27 gives curves of buffer retention
vs. buffer fraction like the curves for uniform-width
buffers (Eq. 19, Fig. 8). Buffer retention is always less
for a Poisson buffer than for a uniform-width buffer of
equivalent average width (Fig. 11, left), except for spe-
cial parameter values (t 5 1 or f 5 0) where buffer
retention is zero. The decrease in buffer performance
due to width variability is consistent with earlier ob-
servations (Figs. 2 and 4). The difference between Pois-
son and uniform-width systems depends on the buffer
fraction, f, and the landscape parameters cell trans-
mission, t, and maximum landscape width, wmax. As the
fraction of buffer increases from 0, the curves for Pois-
son buffer retention rise less steeply and reach a lower
peak retention at higher f than do curves for uniform-
width buffers with identical landscape parameters (Fig.
11, left). Higher cell retention (lower t) and narrower
landscapes (lower wmax) increase the shortfall in Pois-
son buffer retention relative to uniform-width buffers
(Fig. 11, right). The difference is greatest for values
1164 DONALD E. WELLER ET AL. Ecological Applications
Vol. 8, No. 4
TABLE 1. Correlations of landscape transmission, T, with
statistics of the buffer width distribution for three values
of cell transmission t. Statistics are ranked in order of de-
creasing zrz for t 5 0.5. Correlations with zrz . 0.7 (roughly,
r2 . 0.5) are in italics, and the highest zrz for each value
of t is shown in boldface type. Thirteen variables (variance,
standard deviation, kurtosis, and the frequencies of widths
1 through 10) are omitted because they correlated weakly
(r2 , 0.25) with landscape transmission.
Statistic
Cell transmission t
0.1 0.5 0.9
Evenness
CV
Frequency of gaps
(width 0)
Mean
Median
Mode
Skewness
20.909
0.794
0.995
20.547
20.434
20.410
0.198
20.967
0.942
0.871
20.775
20.664
20.569
0.419
20.763
0.835
0.579
20.989
20.893
20.701
0.669
of f where buffer retention is most important (Fig. 11,
right). For low values of wmax, Eq. 27 may give negative
buffer retention at higher f. This unreasonable result
occurs when terms for w . wmax of the Poisson infinite
series represent a significant fraction of stream front-
age. Numerical analysis shows that more than 99.9%
of the stream frontage will have buffer widths less than
wmax as long as average width, l, is less than about half
of wmax. In choosing the Poisson, we have already ac-
cepted l K wmax, so avoiding l . 0.5 wmax does not
further limit the model.
Other width distributions
Landscape models with buffers following uniform or
Poisson width distributions were mathematically trac-
table, but many buffers may have less convenient dis-
tributions. We statistically analyzed models for other
distributions to check the generality of our findings.
We considered all the ways to distribute buffer up to
10 cells wide along a stream frontage 10 cells long.
There are 1110 possible distributions, but some are re-
dundant because the order of widths along the channel
is unimportant. We wrote a computer program to iden-
tify the unique frequency distributions for stream
lengths up to l units long and buffers up to wmax cells
wide. The program uses a set of wmax nested loops. The
outermost loop steps through all possible values for the
frequency of width wmax. This frequency can take values
from 0 through l, where l is the length of stream front-
age. The next loop steps through all possible values (0
through l) for the frequency of width wmax 2 1. The
next loop does the frequency of width wmax 2 2, and
so on until the innermost loop handles the possible
frequencies of width 0. Each loop is exited when the
sum of frequencies exceeds the available stream front-
age, so that invalid distributions are discarded. The
innermost loop saves only those frequency distribu-
tions where the sum of frequencies equals available
stream frontage l. The number of possible distributions
increases geometrically with both length of stream
frontage and maximum buffer width, and there are
184 756 unique frequency distributions (ignoring order
along stream) for the 10 by 10 landscape. For each
distribution, we calculated ??true?? landscape transmis-
sion, T, from the full width distribution and discharge
model by averaging estimates from d/i 5 tw for all
widths in the distribution.
We explored statistical models for predicting the
??true?? landscape transmission from simpler statistics
of the width distribution. We considered the mean, me-
dian, mode, variance, standard deviation, coefficient of
variation (CV), skewness, and kurtosis of the width dis-
tribution and the frequencies of all width classes. We
also considered the evenness (Pielou 1977) of buffer
width, E, calculated as
l1
E 5 p ln p (29)
O c cl c51
where l is the number of columns of stream frontage,
c is an index for those columns, and pc is the fraction
of all buffer cells that are in column c.
The strength of correlation of landscape transmission
T with these predictors varied with cell transmission t
(Table 1), when the correlations were calculated sep-
arately for different values of t. Six of the predictors
(mean, median, mode, evenness, CV, and the frequency
of zero width) were highly correlated (r . 0.7) with
landscape transmission T for some values of t (Table
1). The best predictor depended on the value of t. For
retentive buffers (t 5 0.1 in Table 1), landscape trans-
mission T was most strongly correlated with the fre-
quency of gaps. For unretentive buffers (t 5 0.9 in
Table 1), mean buffer width was the best predictor of
T, while evenness was best at predicting T for mod-
erately retentive buffers (t 5 0.5 in Table 1). Because
each predictor works best for a unique range of t values,
combining predictors in multiple regression models
gave extremely good estimates of landscape transmis-
sion for any value of t. Multiple regression models
predicting T from mean buffer width and the frequency
of gaps always gave r2 . 0.91 for any value of t. Adding
evenness further raised the minimum r2 for any t to
0.97 (Fig. 12). The results of the statistical analysis
suggest that the mathematical model might be replaced
by simpler statistical relationships for some applica-
tions. This should be useful in testing the model or in
modeling larger landscapes.
DISCUSSION
Effects of variability in buffer width
Variability in buffer width across a landscape has
important effects on landscape discharge. A variable-
width buffer retains less material than a uniform-width
buffer of equivalent average width (Figs. 2 and 4).
Conversely, to achieve a target efficiency, a variable-
width buffer must have a greater average width than a
uniform-width buffer (Fig. 5). The difference in effi-
November 1998 1165RIPARIAN BUFFER LANDSCAPE MODELS
FIG. 12. Coefficients of determination for predicting land-
scape transmission, T, from statistics of the buffer width dis-
tribution. Models were fit separately for different values of
cell transmission, t. Light solid lines are for the indicated
univariate relationships. The heavy solid line is for models
using two independent variables, mean buffer width and gap
frequency. The dashed line is for a three-variable model that
adds evenness.
ciency between uniform- and variable-width buffers
increases with buffer cell transmission and decreases
with buffer width (Figs. 2 and 4), so the effect of vari-
ability is most important for efficient buffers that are
narrow but highly retentive. Variable-width buffers are
less efficient because the width distribution includes
some below-average widths, and material transmission
through those smaller widths dominates overall land-
scape discharge (Figs. 6 and 7). The dominance of
smaller widths increases as buffer cell retention in-
creases (Fig. 6). Gaps in riparian buffers are important
sites of material delivery, particularly in narrower or
more retentive buffers (Figs. 7 and 12, Table 1). Elim-
inating gaps should be a high priority for buffer man-
agement and will yield greater benefit than widening
the buffer elsewhere.
Variability in buffer widths should be considered
when evaluating or designing buffers for water quality
management. Recommended widths have been devel-
oped from empirical transect studies or from models
designed to represent transects (Phillips 1989, Nies-
wand et al. 1990, Barling and Moore 1994, Castelle et
al. 1994). Real landscapes feature winding streams,
variable topography, and complex boundaries, so it will
generally not be possible to maintain buffers of truly
uniform width. Inferring landscape discharge from the
average buffer width only (ignoring width variability
and gaps) overestimates material retention, and the er-
ror is greatest for narrow, retentive buffers (Figs. 4 and
11). Buffers designed without considering width vari-
ability will probably not meet management goals.
More empirical data are needed to quantify buffer
width variability and its effects on material discharge.
Many studies have considered how width affects ma-
terial concentration in water traversing a single transect
through a buffer (see review in Lowrance et al. 1997).
However, we are aware of no published studies that
have quantified the distribution in buffer widths across
a landscape or determined the effects on watershed
discharge of among-watershed variations in the buffer
distribution. This lack of information limits efforts to
test model predictions, such as the effects of mean
buffer width and gaps on landscape discharge (Fig. 12,
Table 1), or to verify the choice of distribution function
(e.g., Eq. 6) for modeling width variation.
Some management models have added a complexity
in modeling buffer width not considered in our model:
buffer retention is assumed to vary with slope and soil
properties. These models have been used to recommend
the width needed at each streamside position to achieve
a uniform retention at all positions (Phillips 1989, Xi-
ang 1996). Even with such enhancements, it is still
important to acknowledge nonuniformity in achieved
buffer retention and to integrate that variability across
the landscape.
Source reduction and buffer retention
We separated the reduction in material discharges
from establishing a buffer into two components: elim-
ination of source areas and buffer retention (Staver et
al. 1988). Buffer retention is most important relative
to source elimination when narrow bands of highly
retentive buffer are present (Figs. 9?11), and variability
in buffer width reduced the relative importance of buff-
er retention in total discharge reduction (Fig. 11). For
unretentive buffers, much of the benefit of having a
buffer comes from removing some of the source area
that supplies materials rather than from retention within
the buffer. Source elimination works to reduce land-
scape discharge even where buffer retention might fail,
as when flow bypasses the riparian zone (Denver 1991,
Jordan et al. 1993, Altman and Parizek 1995). Where
material reductions are due to source elimination rather
than buffer retention, adding other land uses that nei-
ther release nor retain materials would have the same
effect on material discharge as adding buffer ecosys-
tem. Understanding the importance of source elimi-
nation relative to buffer retention is necessary for cor-
rectly interpreting evidence of buffer retention at the
landscape scale (Omernik et al. 1981, Osborne and Wi-
ley 1988, Hunsaker and Levine 1995) and for weighing
the costs and benefits of different buffer designs.
Simplification and scaling
Our results suggest some simplifications that may be
useful for predicting the effects of riparian buffers. The
need to generalize more detailed data or models to
larger systems (the ??scaling?? problem) has been em-
phasized in recent articles on the theory and practice
of ecology (Meentemeyer and Box 1987, Turner et al.
1989, King et al. 1991, Rastetter et al. 1992). The ef-
fects of buffer distribution were predictable from sim-
ple statistical models, which explained much of the
variance in landscape transmission without using the
mathematical model or the complete width distribution
1166 DONALD E. WELLER ET AL. Ecological Applications
Vol. 8, No. 4
(Table 1, Fig. 12). The fraction of gaps was the best
predictor of overall discharge for retentive buffers,
while mean buffer width was best for unretentive buf-
fers (Table 1, Fig. 12). Mean width and the fraction of
gaps can be efficiently estimated by sampling (Sokal
and Rohlf 1981), so it may be possible to predict buffer
effects without completely mapping the buffer distri-
bution. Tests with landscape and water quality data are
needed to determine whether results for different pre-
dictors (Table 1, Fig. 12) are robust in real landscapes
where some model assumptions may not apply. If so,
our results may help efforts to use statistical models
to test for buffer effects at the landscape scale (Omernik
et al. 1981, Osborne and Wiley 1988, Hunsaker et al.
1992, Johnson et al. 1997) as well as practical efforts
to predict nonpoint-source pollution. The existence of
simple relationships describing variation across many
buffer distributions also supports the idea that simpler
models apply at the landscape scale (O?Neill et al.
1986, Meentemeyer and Box 1987, Turner 1989).
Landscape indices
The development of indices for describing land-
scapes has been an active focus of landscape ecology.
Most of this work has sought to describe the spatial
patterning of ecosystems or habitats with measures like
dominance, diversity, contagion, or fractal dimension
(Milne 1988, O?Neill et al. 1988, Gardner and O?Neill
1990, Gustafson and Parker 1992). Some of these in-
dices have been compared to water-quality data (Hun-
saker et al. 1992), and there is a need for new methods
to characterize landscape attributes that influence water
quality (Hunsaker and Levine 1995, O?Neill et al.
1997). Our results suggest some indices, such as mean
buffer width and the frequency of buffer gaps, that may
be useful for predicting the function of landscapes with
riparian buffers. We have also defined a new landscape
index, average buffer transmission T, that differs from
other indices in combining information on landscape
pattern and ecosystem function in a single index. It
may be possible to develop more such hybrid indices
for describing other landscape functions.
Water-quality modeling
Our analysis has implications for effectively includ-
ing riparian buffers in discharge and water-quality
models. A riparian buffer receives a flux of materials
from the source ecosystem and modifies that flux before
it reaches a stream. Models that add the separate out-
puts from landscape elements cannot capture such im-
portant nonlinear interactions between ecosystems.
Simple loading models (Correll and Dixon 1980, Beau-
lac and Reckhow 1982, Frink 1991) may yield incorrect
predictions or inferences in landscapes with riparian
buffers (Jordan et al. 1986). More complex, spatially
lumped simulations of nonpoint-source pollution (e.g.,
Knisel 1980, Haith and Shoemaker 1987, Bicknell et
al. 1993) also share this limitation. In theory, spatially
distributed models (see review in Tim and Jolly 1994)
could represent the source?buffer transfer, but have not
been applied to the issues considered in this paper
(Merot and Durand 1997). Moreover, it is often difficult
to parameterize, interpret, and verify complex distrib-
uted models (Beven and Binley 1992, Bloschl and Si-
vapalan 1995). Some simulations have modeled spatial
interactions along lines of flow (Levine and Jones 1990,
Hunsaker and Levine 1995, Soranno et al. 1996). En-
hancement and further integration of such models with
landscape maps and water quality data could incor-
porate or test many of the results and hypotheses gen-
erated in this paper.
Several studies have included riparian zones in mul-
tivariate statistical analyses relating watershed char-
acteristics to water quality in the associated streams.
Some have reported that land use in riparian zones had
no unusual effect on watershed discharges (Omernik
et al. 1981, Hunsaker and Levine 1995, Johnson et al.
1997), while others concluded that streamside area did
have a disproportionate influence on water quality (Os-
borne and Wiley 1988, Levine and Jones 1990, Hun-
saker and Levine 1995). None reported the overriding
effects of riparian buffers seen in transect studies
(Lowrance et al. 1984, Peterjohn and Correll 1984,
Jacobs and Gilliam 1985, Jordan et al. 1993). Our mod-
els suggest some reasons for these ambiguous results
and some possible model improvements. The multi-
variate statistical models used in the landscape analyses
may not effectively represent the nonlinear interactions
between buffers and their source areas. Second, the
landscape analyses incorporate riparian buffers by es-
timating land use proportions within a fixed distance
from the stream. This is a poor measure of riparian
geography (Johnson et al. 1997) and gives little infor-
mation on the distribution of buffer widths. Measures
like mean buffer width and the frequency of gaps (Table
1, Fig. 12) might be more useful predictors and can be
efficiently measured by sampling (Sokal and Rohlf
1981) rather than complete mapping. Finally, data on
riparian geography must have adequate spatial reso-
lution (Hunsaker and Levine 1995). Stream networks
from 1:24 000 scale maps miss smaller streams and are
not adequate to quantify the presence and extent of
riparian areas (Barling and Moore 1994, Bren 1995).
Landscape theory
The present analysis follows others that have used
simple mathematical models to develop the body of
theory and general principles underlying landscape
ecology (Gardner et al. 1987, Milne 1988, 1992, Gard-
ner and O?Neill 1990, O?Neill et al. 1992). Many im-
portant complexities of riparian buffers were not con-
sidered in our models, including: differences between
surface and subsurface retention (Peterjohn and Correll
1984); effects of cover type (Correll 1997); retention
mechanisms (Altier et al. 1994, Weller et al. 1994, Gold
and Kellogg 1997); saturation of retention over time
November 1998 1167RIPARIAN BUFFER LANDSCAPE MODELS
(Weller et al. 1994); variations with slope, soil type,
and surface characteristics (Phillips 1989, Xiang 1996);
and channelized or deep subsurface flow (Staver et al.
1988, Dillaha et al. 1989, Denver 1991, Jordan et al.
1993, Altman and Parizek 1995, Bohlke and Denver
1995, Lowrance et al. 1997). Instead, we used a simple,
first-order model of retention and focused on the effects
of landscape structure and the interactions among land-
scape elements. The approach of analyzing the behav-
ior and practical implications of simple models has
been instrumental in developing ecological theory in
other areas, such as population and community ecology
May 1976). We used our model to explore one of the
basic themes of landscape ecology, the effects of pat-
tern on process (Forman and Godron 1987, Turner
1989, Forman 1995). In particular, we considered how
the structure of the landscape (the distribution of source
and buffer ecosystems) interacts with ecosystem func-
tion (the release or retention of materials) to give over-
all landscape function (net release of materials from
the landscape). Our model was simple enough to focus
on this interaction and yield some basic understanding
and testable hypotheses. Our analysis adds simple mod-
els for material transport to the body of general theory
describing the ecological effects of spatial patterning.
ACKNOWLEDGMENTS
We thank Zhi-Jun Liu, Michelle Coffee, two anonymous
reviewers, and Ecological Applications editor Monica Turner
for helpful comments on the manuscript. This research was
supported by the Smithsonian Institution, the Smithsonian
Environmental Sciences Program, National Science Foun-
dation grants BSR-9085219 and DEB-9317968, and the Na-
tional Oceanic and Atmospheric Administration?s Coastal
Oceans Program Office.
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