FractalsA Bridge to the Future for
Soil scientist Yakov Pachepsky (left) and hydrologist Walter Rawls examine soil
samples from fields where yields and soil pore geometry are different. The
ability of soils to retain and transport water is closely related to fractal
parameters of pore space.
Soil scientists have at least one thing in common with stockbrokers: Both
deal with subjects so complex they often seem unmanageable. It should be no
surprise that both are turning to a mathematics of chaosfor solutions.
The mathematician Benoit Mandelbrot is largely responsible for the current
interest in fractal geometry, a math that shows the irregular shapes of nature.
He first attempted to use it to master the commodities market in the 1950s. In
the early 1960s, he went on to work for IBM, where he developed the computer
power that fuels today's fractal frenzy.
Fractal geometry is particularly suited to advanced computer graphics
packages. It has gained widespread attention of soil scientists and agronomists
over the past decade. For fractal geometry may open a gate in a wall many of
them run into as they gather data on several different scales.
It used to be that soil scientists interested in soil hydrology gathered
data only on a small scaleeither 2- by 2-inch soil samples for lab
studies or 6- by 6-foot plots in the field, says Yakov Pachepsky. He is an
Agricultural Research Service
cooperating soil scientist from Duke University in Durham, North Carolina. [For
more things that fractal mathematics can be used for, see "A Geometric
Language for the Universe."
"But approach that was 20 years ago," says Pachepsky, who is
currently located at the ARS Remote Sensing and Modeling Laboratory in
"Now, with precision agriculture, we are asked to deal with
combine-mounted yield monitors that churn out data on about 20-square-yard
grids as fast as the farmer's combine crosses the field," he says.
"And satellites send images of Earth on grids of a square mile or more.
Thin slices of soil reveal a realm of jagged pore boundaries. This unevenness
is similar at different magnifications, indicating the fractal nature of soil
pore space. Computer software helps to both visualize and quantify differences
among pore spaces in soils. Surprisingly, pore space geometry appears to be an
indicator of soil productivity.
"We'll be out of business if we can't relate data between all these
scales. Fractal geometry offers the potential of bridging them."
Fractal geometry gets its name from the irregular fragments it deals with.
It is scale-independent, which means that a basic shape stays the
samealong with any objective measurements of itno matter how much
you enlarge or reduce the size of its image.
Traditional measurements are made in dimensions of line, area, volume, or
mass. Fractal dimensions range from 1 for a straight line to almost 2 for a
chaotic, unpredictable squigglelike a day of extreme highs and lows on
the stock market.
Fractal Dimensions Can Indicate Plant Stress
In an ARS-funded study, soil scientist Bahman Eghball at the University of
Nebraska at Lincoln used fractal geometry to identify corn roots stressed by
lack of nitrogen fertilizer. He dyed and photographed the roots. Then he
projected the photographic slides on three different-sized grids.
"If one or more roots fell into a grid, we counted the grid as one
intersected by roots," Eghball says. By plotting the logarithm of the
number of grids the roots intersected against the logarithm of grid sizes,
Eghball obtained a line. The slope of that line is the fractal dimension.
Eghball found this dimension could be used to spot plant stress. When corn
plants were grown without nitrogen fertilizer, roots stopped branching out,
intersecting fewer grids. The result was a lower fractal dimension. That
dimension, once fed into a computer graphics program with a fractal package,
could be used to generate a computer image of the roots.
Eghball has used the perspective of fractal geometry to see some surprising
things in the world of agriculture. For example, by analyzing 60 years of USDA
crop yield statistics1930 to 1990he found that oats and soybeans
were the riskiest crops in terms of having the most year-to-year yield
fluctuations in response to weather. He also found that the Green Revolution of
the 1950s and 1960s not only raised yields, but also raised risks of
year-to-year yield variation.
Eghball explains that the comparison of 10 different crops with wide
variations in yields was possible only with a tool like fractal geometry.
Scientists can project a photograph of a plant's root system onto grids of
different sizes and count the number of root intersects within each grid. By
plotting the logarithm of the number of intersected grids against the logarithm
of grid sizes, a line is obtained, the negative slope of which is the fractal
dimension. Scientists can use this dimension to spot plant stress.
"The fractal dimension is unaffected by the fact that oat yields are
much lower than corn or soybean yields," he says. "For that matter,
it can compare 'apples and oranges'or a fiber crop like cotton with a
grain crop like corn. This is possible because we are comparing patterns of
behavior of objects, not the objects themselves."
Eghball says another strength of fractal geometry is its ability to
determine if a phenomenon is predictable. Generally, a fractal dimension close
to 1 means it is more predictable. So rice yield (1.2) is more predictable than
oat (1.47) or soybean (1.45) yields.
Fractals and the Chaos of Spatial Variability
One of the chaotic messes in the agronomy world is what scientists call
spatial variability. That is, whatever you talk about in a farm fieldcrop
yield, erosion, soil moisture, soil temperature, drainage, waterflow, chemical
movement, soil fertilityall can vary widely, even in a space as small as
10 feet. So how can we extrapolate to thousands of acres of land without costly
measurements every few feet or so? Laj Ahuja wants to see if fractal analysis
can help. He uses it to quantify spatial variability so it can be plugged into
computer models at his Fort Collins, Colorado, research unit. The models do not
currently account for this variability.
"Precision farming has an interest in quantifying and mapping this
spatial variability on landscapes so farmers can manage different parts
differently," says Ahuja, who heads ARS' Great Plains System Research Unit
in Fort Collins.
With the help of ARS colleagues and scientists at Colorado State University,
Ahuja is trying to bridge plot, field, and farm scales so he can predict these
processes across entire watersheds encompassing many farms and ranches.
Pachepsky and microbiologist Lawrence J. Sikora at Beltsville, working with
soil scientist Martin Rabenhorst at the nearby University of Maryland at
College Park, want to use fractal dimensions of physical aspects of
soilsuch as the volume and inner surface roughness of soil poresto
characterize soil quality relative to its ability to grow plants.
"These are easy-to-measure characteristics," Pachepsky says. The
scientists make their measurements from thinly sliced sheets of soil samples,
the structure of which is preserved by the addition of resin.
Walter J. Rawls, who heads Beltsville's ARS Hydrology Laboratory, is working
with soil scientist Raymond R. Allmaras at St. Paul, Minnesota, and others to
predict soil water movement based on these pore measurements, along with
measurements of the soil's density and ability to hold water. They use fractal
geometry to relate difficult-to-measure soil hydraulic properties to other soil
variables readily available from soil surveys.
This corn plant was grown in a wooden box containing metal pegs so its
root-branching pattern could be studied. Soil has been removed to expose the
Soil scientist Jerry C. Ritchie, who is also with the Hydrology Laboratory,
uses fractal geometry to analyze heights of range plants traced by airborne
lasers. The heights reveal vegetation typegrass, shrub, or transition
zone from grass to shrub. Vegetation type is important to predicting soil
moisture, because it affects roughness of land surface. The roughness
influences windspeed which in turn greatly affects evaporation of water from
soil and water uptake by plants.
G. LeRoy Hahn, former head of the Biological Engineering Research Unit at
the U.S. Meat Animal Research Center at Clay Center, Nebraska, and his
colleagues Roger Eigenberg and John A. Nienaber used fractal geometry with
cattle body temperatures to determine that steers begin to suffer heat stress
when the air temperature reaches 77oF. Hahn recently retired but
continues his work as a collaborator.
Yud-Ren Chen, who was in charge of the unit at Clay Center before Hahn, also
contributed to the animal stress study. Chen found mathematical equations to
compute the fractal dimensions of animal temperatures. He now heads the ARS
Instrumentation and Sensing Laboratory in Beltsville.
One study was done with six young steers kept in indoor stalls. Sensors in
their ear canals automatically recorded their temperatures every 30 seconds for
at least 2 weeks, while their stalls ranged from a cool 39oF to
63oF to a hot 80oF to 105oF.
By comparing body temperature fluctuations in the hot and cool chambers, the
scientists calculated the fractal dimension. It stayed level at 1.7 until the
air temperature reached 77oF. After that, the dimension dropped
precipitously to 1.2, indicating that the steers were so heat-stressed they
could no longer control their temperatures.
Hahn says that fractal analysis provided a way to assign a number to the
degree of fluctuation in body temperature. As the chamber became too warm, the
steers became stressed and their temperatures fluctuated less, making the
fractal dimension lower. "They were losing their ability to regulate their
temperatures. Normally, cows, like people, have marked, random fluctuations in
body temperature," Hahn says.
Hahn and Nienaber have since done similar experiments with sheep and pigs.
"Using the fractal dimension of 1.7 as a stress threshold, we can now
tell feedlot managers that it's best to turn on their sprinklers when the air
temperature approaches 80oF," Hahn says. "We've found this
stress threshold correlates well with temperatures at which steers begin to
lose interest in feeding."
Hahn also says that observations of individual steers with higher
temperature thresholds raise the possibility of using the stress threshold to
help breed more heat-tolerant cattle.
Whether data collection is done with livestock or with landscapes, up close
or far away, the agricultural research world is finding that fractal geometry
may be a way to find order in chaos.By
Comis,Agricultural Research Service Information Staff, 6303 Ivy Lane,
Greenbelt, Maryland 20770, phone (301) 344-2748.
Yakov Pachepsky is at the
USDA-ARS Remote Sensing and
Modeling Laboratory, Bldg. 007, 10300 Baltimore Ave., Beltsville, MD
20705-2350; phone (301) 504-7468, fax (301) 504-5823.
Bahman Eghball is in the
USDA-ARS Soil and Water
Conservation Research Unit, University of Nebraska, 119 Keim Hall, Lincoln,
NE 68583-0934; phone (402) 472-0741, fax (402) 472-0516.
Lajpat R. Ahuja is in the
USDA-ARS Great Plains Systems Research
Unit, P.O. Box E, Fort Collins, CO 80522-0470; phone (970) 490-8315, fax
(970) 490-8310, e-mail
Lawrence J. Sikora is at the
USDA-ARS Soil Microbial
Systems Laboratory, Bldg. 318, 10300 Baltimore Ave., Beltsville, MD
20705-2350; phone (301) 504-9384, fax (301) 504-8370.
Raymond R. Allmaras is in the
and Water Management Research Unit, University of Minnesota, 439 Borlaug
Hall, St. Paul, MN 55108; phone (612) 625-1742, fax (612) 649-5175.
Walter J. Rawls and
Jerry C. Ritchie are at the
USDA-ARS Hydrology Laboratory, Bldg.
007, 10300 Baltimore Ave., Beltsville, MD 20705-2350; phone (301) 504-7490, fax
John A. Nienaber, G. LeRoy
Hahn, and Roger A. Eigenberg are at the USDA-ARS
U.S. Meat Animal Research Center, P.O.
Box 166, Clay Center, NE 68933-0166; phone (402) 762-4270, fax (402) 762-4273.
Yud-Ren Chen is at the USDA-ARS
Sensing Laboratory, Bldg. 303, 10300 Baltimore Ave., Beltsville, MD
20705-2350; phone (301) 504-8450, fax (301) 504-9466.
"FractalsA Bridge to the Future for Soil Science" was
published in the April 1998 issue of Agricultural Research magazine.
Click here to
see this issue's table of contents.