**Is Technology Ahead of Science?**

**Joseph K. Berry
Berry & Associates // Spatial Information Systems, Inc. (BASIS), Fort Collins,
Colorado**

**Abstract**

**Agriculture is inherently a spatial endeavor. The newly
emerging technologies of Geographic Information Systems (GIS), Global Positioning System
(GPS) and Intelligent Devices and Implements (IDI) are enabling farmers to more
effectively organize, update, query and analyze mapped data about their fields. The
melding of these technologies marks a turning point in the collection and analysis of
field data— from a "whole-field" to a "site-specific" perspective
of the inherent differences throughout a field. In many ways, the new technological
environment is as different as it is similar to traditional mapping and data analysis. For
agriculture research it appears to propagate as many questions as answers. Is the
"scientific method" relevant in the data-rich age of knowledge engineering? Is
the "random thing" pertinent in deriving mapped data? Are geographic
distributions a natural extension of numerical distributions? Can spatial dependencies be
modeled? How can "on-farm studies" augment agriculture research? This paper
explores the conceptual differences between spatial and non-spatial data, its analysis and
the opportunities and challenges it poses.**

**Introduction**

**Site-specific management, commonly referred to as precision
farming, is about doing the right thing, in the right way, at the right place and time .
It involves assessing and reacting to field variability and tailoring management
actions, such as fertilization levels, seeding rates and variety selection, to match
changing field conditions. It assumes that managing field variability leads to both cost
savings and production increases. Site-specific management isn’t just a bunch of
pretty maps, but a set of new procedures that link mapped variables to appropriate
management actions. This conceptual linkage between crop productivity and field conditions
requires the technical integration of several elements.**

**Site-specific management consists of four basic elements:
global positioning system (GPS), data collection devices, geographic information systems
(GIS) and intelligent implements. Modern GPS receivers are able to establish positions
within a field to about a meter. When attached to a harvester and connected to a data
collection device, such as a yield/moisture meter, these data can be "stamped"
with geographic coordinates. A GIS is used to map the yield data so a farmer can see the
variations in productivity throughout a field. **

**The GIS also can be used to extend map visualization of yield
to "map-ematical" analysis of the relationships among yield variability and
field conditions. Once established these relationships can be used to derive a
"prescription" map of management actions required for each location in a field.
The final element, intelligent implements, reads the prescription map as a tractor moves
through a field and varies the application rate of field inputs in accordance with the
precise instructions for each location. The combining of GPS, GIS and IDI (intelligent
devices and implements) provides a foothold for both the understanding and the management
of field variability.**

**To date, most analysis of yield maps have been visual interpretations. By
viewing a map, all sorts of potential relationships between yield variability and field
conditions spring to mind based on a farmer’s indigenous knowledge. Data
visualization can be extended through GIS analysis linking digital maps of yield to field
conditions. This "map-ematical" processing involves the same three levels
required by visual analysis: cognitive, analysis and synthesis. At the cognitive level
(termed desktop mapping) computer maps of variables, such as crop yield and soil
nutrients, are generated. These graphical descriptions form the foundation of
site-specific management. The analysis level uses the GIS’s analytical toolbox to
discover relationships among the mapped variables. This step is analogous to a
farmer’s visceral visions of relationships, but uses the computer to establish
mathematical and statistical connections. The synthesis level of processing uses spatial
modeling to translate the newly discovered relationships into management actions
(prescriptions). The result is a prescription map linking management actions, such as
variable rate control of inputs, to the unique pattern of field conditions. **

**Differences Between
Spatial and Non-Spatial Data**

**Data is fundamental to site-specific management. All of these
data are geographical in their occurrence (collected in real-world space), but the nature
of the data how it is processed determines its spatial character. Some data, such as the
number of times an aspiring assistant professor is cited in the literature, is inherently
non-spatial. However, if you simultaneously recorded the location in the library of the
publications containing the citations, the data would take on a geographic component. The
utilization of the spatial component during processing determines the extent of the
spatial nature of the data.**

**The coupling of descriptive information (what, how much, etc.)
with location information (where) identifies geographical data. Descriptive information,
when expressed numerically, forms four basic data types: nominal (values are merely
exclusive), ordinal (values imply a hierarchical ordering, such as small, bigger,
biggest), interval (values are ordered within a scale containing a constant interval, such
as 42, 50 and 54 degrees centigrade), and ratio (values are ordered, contain a constant
interval and have an absolute reference, such as zero degrees Kelvin). **

**Traditionally, location information has been depicted
graphically as map features comprised of point, lines and polygons. Within a computer,
these features are defined by a sequence of numeric strings identifying geographic
coordinates (X,Y,Z) and a descriptive value, such as an ID number linking the feature to
tables of descriptive information. This additional data type, termed choropleth,
characterizes discrete objects, such as property lines, roads and pipelines. **

**However, some things do not exhibit sharp boundaries in
geographic space. This data type, termed isopleth, forms a continuous distribution in
geographic space. Gradients result from the nature of things, such as air temperature,
chemical concentrations and terrain elevation, or from how we characterize them, such as
soil nutrient distribution from point samples. **

**A final data type, termed binary, identifies things that exist
only in two states, such as land/water, present/absent and good/bad. Sharp borders are
formed, but they often reflect interpretation and conditions more than the inherent nature
of the data (e.g., "suitable areas" and a reservoir’s
"shoreline").**

**Discrete objects can be directly digitized into a computer from
existing maps, air photos, GPS records, or other sources and their descriptive information
can be expressed in any of the four basic data types. Spatially defined binary data most
often result from reclassification of exiting spatial data and is usually expressed as
nominal data. Spatial gradients are derived from discrete measurements through
aggregating, smoothing and interpolation of interval or ratio data types. Map contouring,
reverses the effect by dividing continuous data into a set of discrete polygons, or
response zones, implying sharp boundaries. **

**Surface Modeling**

**By its very nature, site-specific management primarily involves
continuous spatial variables. While much of the infrastructure is discrete, such as roads,
fences, and ditches, the focus of a farm database is on expansive fields with production
factors that vary with geographic space. As a result, surface modeling plays a dominant
role in site-specific management.**

**Surface Modeling Using
Continuously Logged Data**

**Map surfaces, also termed spatial gradients, can be derived by
summarizing continuously logged data and assigning the summary value to regularly spaced
grid spaces. For example, yield mapping systems summarize point measurements (i.e., yield
monitor readings) falling within each grid cell of an imaginary grid laid over a field.
The average, standard deviation and other statistics are used to characterize the yield
within the "spatially aggregated unit." There are two main advantages to this
approach: 1) it provides consistent geographic referencing among various mapped data
layers, and 2) it smoothes out measurement "noise" while reporting statistics on
localized yield variation. **

**However, there are some technical issues that need to be
considered. First, the positioning of the samples is assumed to be exact. In the case of
yield monitors, there are several sources of error, such as mass flow time lags, that
contribute to imprecise positioning. Also, the gridding resolution (cell size) can greatly
affect summary calculations. If the grid pattern is too large, some of the information in
the data will be lost ("averaged-over"). If it is too small, undo credence might
be attached to differences arising simply from measurement and positioning errors. **

**Other technical issues involve the configuration of the summary
window and the summary procedure employed. A single cell window design uses only those
measurements that actually fall within a cell for the summary calculations. An inline
design uses the direction of travel to "filter" surrounding data, thereby
helping to smooth out measurement errors resulting from the "coughs and spits"
of uneven grain flows through a combine. The technique is analogous to "moving
averages" used in analyzing times series data, such as commodity prices or the stock
market indices. Both the single and inline techniques have been used for
"on-the-fly" data compression—simply keep the summary statistics and
discard the pile of raw measurements (not recommended). The nearest-neighbors technique
involves post-processing of the raw data. It moves a window around the field sequentially
centered on each grid cell (spatial summary unit). At each stop, it calculates a summary
of the points falling within the window and assigns that value to that grid cell. In
addition to window design, the summary procedure can vary— such as simple or weighted
averaging. For example, a distance-weighted average is influenced more by nearby
measurements than those that are farther away.**

**Surface Modeling Using Point
Sampled Data**

**Whereas continuously logged data is analogous to a
"census" of spatial units, point sampling derives a statistical estimate for
each spatial unit based on a set of dispersed measurements. Sampling design is critical to
interpolation success and involves four distinct considerations: 1) stratification, 2)
sample size (intensity), 3) sampling grid, and 4) sampling pattern. It is important to
note that traditional non-spatial sampling considerations (a representative number of
random samples) are inappropriate for surface modeling as they focus on assessing the
typical response (average) in numeric space. Surface modeling, on the other hand, seeks to
map the geographic distribution (variance) and requires a sampling design that proportions
samples as a function of spatial extent. **

**Two broad approaches are used in deriving geographic
distributions of spatial variables: map generalization and spatial interpolation. Map
generalization fits a functional form to an entire set of sample points. The simplest is a
flat X,Y plane (geographic axes) that has half the points above it and half below it as
viewed along the Z-axis (measurement axis). The Z value corresponds to the arithmetic
"average" in non-spatial statistics that is assumed to have a uniform
distribution in geographic space (flat plane). **

**If the plane is allowed to tilt while minimizing its deviations
to the data points (best fit of a 1 ^{st} degree polynomial in three-space), it
will identify the spatial trend in the data. This procedure is analogous to fitting a
"regression line" in non-spatial statistics. Relaxing the assumption of a plane
(linear relationship) involves fitting N^{th} degree polynomials as curved
surfaces which is similar to non-linear regression and other prediction equation fitting
techniques used in traditional data analysis. **

**Instead of fitting a functional form to an entire data set,
spatial interpolation fits a localized function within a "roving window" moved
throughout a spatial extent. Thiessen polygons are formed by assigning the value of the
nearest sample point to each spatial unit resulting in a map of the perpendicular
bisectors between neighboring sample points. Nearest-neighbors technique simply averages
the set of sample points within a specified radius of each spatial unit. If done
repeatedly, the technique results in consistent smoothing of the geographic distribution
of the data and ultimately approaches the flat "average" plane.
Inverse-distance-weighted procedures use the distance from the spatial unit to each sample
within the summary window to weight-average with closer samples having more influence. **

**Kriging techniques determine window configuration and weighting
factors as a function of the spatial autocorrelation in the sample set. Although a
detailed discussion of spatial autocorrelation is beyond the scope of this paper, it
should be noted that it relates the similarity among sample points (i.e., the inverse of
the variance) to the distance between samples. In essence, it expresses to what degree
"nearby things are more related than distant things" (Tobler’s first law of
geography), the assumption behind all spatial interpolation techniques. Whereas the Geary
and Moran indices report an overall measure of spatial autocorrelation, a variogram plot
captures its functional form over a range of inter-sample distances.**

**Surface Analysis**

**There are numerous procedures for analyzing relationships within (univariate)
and among (multivariate) map surfaces. Surface analysis techniques provide insight into
subtle patterns contained in the surfaces, such as areas of significantly different high
and low yields from typical levels in a field. In addition, other techniques can summarize
the degree of coincidence among surfaces and/or test hypotheses, such as which wheat
variety performs best within areas of high nematode concentrations. Surface analysis also
can be used to derive spatially responsive prediction equations, such as a regression
equation relating yield (independent variable) to soil nutrient surfaces (independent
variables). **

**Underlying all of these techniques is the realization that a map surface is a
set of spatially organized numbers first, colorful image (traditional map) later. The
important outcome is a "map-ematical" treatment of the numbers that respects the
spatial autocorrelation and spatial dependencies captured in the surfaces. **

**Many of the traditional mathematical and statistical procedures in non-spatial
data analysis translate to surface analysis. The accompanying tables identify several of
the procedures for univariate (within a single surface, Table 1) and multivariate (among
two or more surfaces, Table 2) mapped data.**

**Spatial Analysis**

**Spatially organizing data provides an additional set of
analytical operations beyond the extensions of traditional data analysis techniques.
Geographically dependent procedures, such as optimal path delineation, inter-visibility
among map features, binary masking, and effective proximity are examples of an entirely
new set of map-ematical operations. Although a detailed discussion of spatial analysis is
beyond the scope of this paper, it should be noted that these procedures arise from the
spatial character of mapped data and do not have direct lineage to non-spatial mathematics
and statistics.**

**For example, the distance between two points is easily measured
with a ruler. In mathematics, the manual procedure is replicated by evaluating
Pythagora’s theorem (c ^{2}= a^{2}+b^{2}) using the X,Y
coordinates to define the sides of a right triangle, then solving for the hypotenuse. Both
approaches, however, assume the "shortest, straight line distance between two
points"— a constrained definition of distance that is rarely the actual path of
movement connecting things in the real world. **

**The traditional concept of distance is expanded to one of
proximity by relaxing the assumption that connections are always "between two
points," but can be among sets of points (e.g., current location to everywhere within
a spatial extent). Further expansion of the concept relaxes the assumption of
"straight line" connectivity by introducing absolute and relative barriers that
must be respected in determining the ‘shortest" movement path connecting
locations. **

**In site-specific management, effective proximity can be used to
identify locations that are "uphill," "up-wind," or "up along
surface or ground water flows." Intervening terrain can be considered in determining
sediment loading potential, resulting in "effective environmental buffers"
allowing farming activity close to streams when conditions warrant. Current laws
delineating a "fixed" buffer, such as 100 feet around class 2 streams, might be
easy to draft, but serves neither fish nor farmer as real world conditions affecting
sediment loading potential vary along a stream.**

**Most resource and environmental processes do not adhere to
simple geographic concepts, such as Euclidian distance. The GIS modeling
"toolbox" contains a myriad of new spatial statistics and spatial analysis
procedures that promise to revolutionize agricultural research, as much as it impacts farm
management and operations. As technology relaxes simplifying assumptions, understanding of
complex spatial relationships increases. The technology is in place, but the science
supporting its application is not. The transition from non-spatial science to
spatially-driven science is the missing link needed for successful site-specific
management.**

**Opportunities and
Challenges**

**The new spatial technologies mark a turning point in the
collection and analysis of field data— from a "whole-field" to a
"site-specific" perspective of the inherent differences throughout a field. In
many ways, this new technology is as different as it is similar to traditional mapping and
data analysis. For agriculture research it appears to propagate as many questions as it
answers.**

*Is the "scientific method" relevant in the data-rich
age of knowledge engineering?*

**The first step in the scientific method is the statement of a
hypothesis. It reflects a "possible" relationship or new understanding of a
phenomena. Once a hypothesis is established, a methodology for testing it is developed.
The data needed for evaluation is collected and analyzed and, as a result, the hypothesis
is accepted or rejected. Each completion of the process contributes to the body of
science, stimulates new hypotheses, and furthers knowledge.**

**The scientific method has served science well. Above all else,
it is efficient in a data constrained environment. However, technology has changed the
nature of that environment. Yield monitors, for example, can "census" an entire
field easier and at less cost than manually sampling a few plots. Surface modeling can map
the spatial autocorrelation within a set of soil samples. Effective proximity to
influential features can be derived. Terrain conditions, such as slope and aspect, can be
mapped. Localized variation in a variable, such as moisture content, can be spatially
characterized. Remotely sensed data encapsulates the conditions and characteristics of a
scene as spectral response values. **

**The result is a robust database composed of thousands of
spatially-registered locations (spatial units) relating a diverse set of variables. In
this data-rich environment, the focus of the scientific method shifts from efficiency in
data collection and analysis to the derivation of alternative of hypotheses. Hypothesis
building results from "mining" the data under various spatial and thematic
partitions. The radical change is that the data collection and initial analysis steps
proceed the hypothesis statement— in effect, turning the traditional scientific
method on it’s head.**

**Is the "random
thing" pertinent in deriving mapped data? **

**A cornerstone of traditional data analysis is randomness. In
data collection it seeks to minimize the effects of autocorrelation and dependence among
variables. Historically, "census" of a variable was prohibitive and randomness
provided an unbiased sample set for estimating the typical state of a variable (i.e.,
average). Calculation of the variation within the sample set (i.e., variance) establishes
how typical the typical is. In multivariate analysis, the mean vector and covariance
matrix are used to assess the dependency among variables (i.e., correlation).**

**For questions of central tendency and non-spatial dependency in
data, randomness is essential, as it supports the basic assumptions about analyzing data
in numeric space (devoid of spatial interactions). However, in geographic space,
randomness rarely exists. The ambient temperatures for neighboring locations are not
random. Nor is crop yield. The spatial relationships among mapped variables, such as crop
yield and soil nutrient levels, rarely display random patterns of coincidence. **

**Spatial interactions are fundamental to site-specific
management and research. Adherence to the "random thing" runs counter to
continuous spatial expression of variables. This is particularly true in sampling design.
While efficiently establishing the central tendency, random sampling fails to consistently
exam the spatial pattern of variations in a variable. An underlying systematic sampling
design, such as systematic unaligned, is needed to insure a consistent distribution of
samples over the spatial extent.**

**Are geographic distributions
a natural extension of numerical distributions? **

**To characterize a variable in numeric space, density functions,
such as the standard normal curve, are used. They translate the pattern of discrete
measurements along a "number line" into a continuous numeric distribution.
Statistics describing the functional form of the distribution determine the central
tendency of the variable and ultimately its probability of occurrence. If two variables
are considered simultaneously, a three-dimensional probability surface is derived.
Consideration of additional variables results in an N-dimensional numerical distribution.**

**The geographic distribution of a variable can be derived from
discrete sample points positioned in geographic space. The map generalization and spatial
interpolation techniques described earlier can be used to form a continuous distribution,
in a manner similar to deriving a numeric distribution. In effect, the Gaussian, Poisson
and binomial density functions used in non-spatial statistics are analogous to the
polynomial, inverse-distance-squared and krigging density functions used in spatial
statistics. **

**Although the mechanical expressions are similar, the
information contained in numeric and geographic distributions is different. Whereas
numeric distributions provide insight into the central tendency of a variable, geographic
distributions provide information about the pattern of variations. Generally speaking,
non-spatial characterization supports "whole-field" management, while spatial
characterization supports "site-specific" management. It can be argued that
research using non-spatial techniques provides minimal guidance for site-specific
management— in fact it might even be dysfunctional. **

**Can spatial dependencies be
modeled? **

**Non-spatial modeling, such as linear regressions derived from
point sampled data, assume spatially independent data and seeks to implement the
"best overall" action everywhere. Site-specific management assumes spatially
dependent data and seeks to evaluate "IF <spatial condition> THEN <spatial
action>" rules for the specific conditions at each location throughout a field.
The underlying philosophies the two approaches are at odds. However, the
"mechanics" of their expression spring from the same roots.**

**Within a traditional mathematical context, each map represents
a "variable," each cell or polygon represents a "case" and the value
at that location represents a "measurement." In a sense, each cell or polygon
can be conceptualized as a sample plot— it is just that sample plots are everywhere.
A yield monitor (and remotely sensed data for that matter) provides a direct measurement
for each spatial unit (average of several yield readings or integration of electromagnetic
energy). Point sampling, such as for soil nutrients and elevation, uses surface modeling
techniques to statistically estimate a response for each spatial unit. **

**The result is a data structure that tracks spatial
autocorrelation and spatial dependency. The structure can be conceptualized as a stack of
maps in which a vertical pin spears a sequence of values defining each variable for that
location— sort of a data shishkebab. Regression, or similar techniques, can be
applied to the data vectors uncovering a spatially-dependent model of the relationships. **

**Admittedly, imprecise, inaccurate or poorly modeled surfaces,
may incorrectly track the spatial relationships. But, given good data, the map-ematical
approach has the capability to model the spatial character inherent in the data. What is
needed is a concerted effort by the scientific community to identify guidelines for
spatial modeling and develop techniques for assessing the accuracy of mapped data and
results of its analysis.**

**How can "on-farm
studies" augment agriculture research?**

**Agriculture research has historically focused on intensive
investigations implemented at experimental fields. These studies are well-designed and
methodically executed by researchers who are close to the data. As a result, the science
performed is both rigorous and professional. However, it is extremely limited in both time
and space. The findings might accurately reflect relationships for the experimental field
during the study period, but offer minimal information for a farmer 70 miles away under
different biological agent, soil and climatic conditions. **

**Farmers, on the other hand, manage large tracks of land for
long periods of time, but are generally unaccustomed to administering scientific projects.
As a result, farm operations and on-farm studies are often incompatible. On-farm variety
trials, to a limited degree, bridge this gap. The growing popularity of site-specific
management has the potential to fill the gap. Overhead, on-board and proximal sensors are
posed to collect detailed farm-wide data that a couple of years ago would have required an
army of graduate students. **

**It is recognized that sophisticated instrumentation and the
databases they generate are required to implement site-specific management. But, often
overlooked is the reality that these data form the scientific fodder needed to build the
spatial relationships demanded by the process. Site-specific management has changed
farming operations, now it must change farm research. A close alliance between researchers
and farmers is fundamental to this change. Without it, constrained research (viz.
esoteric) mismatches the needs of evolving farm technologies, and heuristic (viz.
unscientific) rules-of-thumb are substituted. The farmer has the data and the researcher
has the methodology— both are key to successfully implementing site-specific
management.**

**Summary**

**Agriculture is inherently a spatial endeavor. Emerging
technologies enable farmers to effectively organize, update, query and analyze mapped data
about their fields. The melding of these technologies marks a turning point in the
collection and analysis of field data— from a "whole-field" to a
"site-specific" perspective of the inherent differences throughout a field. In
many ways, the new technological environment is as different as it is similar to
traditional mapping and data analysis. It provides new capabilities for characterizing
spatial relationships, such as spatial statistics, surface modeling and spatial analysis.
These techniques allow better understanding of the spatial dependencies within and among
mapped data. In addition to new opportunities, the techniques pose new challenges for
farmers and researchers alike. In a sense, technology is ahead of science— sort of
the cart before the horse. Site-specific management can map spatial patterns and reactions
to a meter (technological cart), but our historical science base has been calibrated for
the entire field (scientific horse). **

**_________________________________**

**General References**

**Berry, J.K., 1993. Beyond Mapping: Concepts, Algorithms and Issues in GIS,
published by GIS World Books. A compilation of columns on map analysis
considerations published in GIS World from 1989 to 1993. Companion software tMAP contains
"hands-on" exercises in map analysis capabilities that are cross-referenced to Beyond
Mapping and Spatial Reasoning books.**

**Berry, J.K., 1995. Spatial Reasoning for Effective GIS, published by GIS
World Books. A compilation of columns on map analysis considerations published in GIS
World from 1993 to 1995. Companion software gCON contains digital slide shows on GIS
concepts that are cross-referenced to Beyond Mapping and Spatial Reasoning
books.**

**Berry J.K., in press. Precision Farming Primer. A compilation of columns
on site-specific crop management considerations published in Successful Farming’s
ag/INNOVATOR newsletter from 1994 to 1998. Companion software pfMAP contains
"hands-on" exercises and digital slide shows in precision farming data analysis
that are cross-referenced to the Precision Farming Primer book.**

**Cressie, N.A., 1993. Statistics for Spatial Data, published by John
Wiley and Sons. Aimed at scientists and engineers, the book uses spatial data to
illustrate spatial theory and methods. Includes highly detailed treatments of such
integral areas as geostatistical data, models of spatial lattice data, asymptotics and
spatial point patterns.**

**Fotheringham, S. and P. Rogerson, 1994. Spatial Analysis and GIS,
published by Talyor and Francis. The book focuses on the relative lack of research into
spatial analysis and GIS integration and its potential benefits. It examines GIS and
spatial analysis integration issues, research emphasizing methods of spatial analysis, and
mathematical modeling and GIS issues.**

**Shaw, G. and D. Wheeler, 1994. Statistical Techniques in Geographical
Analysis, published by Halsted Press. Covers a range of techniques, from simple
descriptive to parametric and nonparametric methods, in bivariate and multivariate
settings. It sequentially introduces topics and reinforces them with appropriate
application examples.**

**Note ^{1}**

**Note ^{2}**

**__________________________________________________________________**

**Table 1. Annotated Listing of Example
Univariate Surface Analysis Techniques**

**Slope ~ rate of change of each surface element (1**^{st}derivative)

**Aspect ~ orientation of each surface element (direction)**

**The slope and aspect of an elevation surface (altitude derived from a surveyed points or rectified aerial photos) identifies terrain steepness and orientation; example uses include road-building and water runoff modeling****The slope and aspect of a barometric surface (air pressure gradient derived from a set weather station data) estimates wind speed and direction****The slope and aspect of a thermal gradient in a lake (generated from remote sensing data of surface temperature) identifies rate and direction of cooling from a thermal input (nuclear power station ponds)****The slope and aspect of a total revenue surface (generated by summing the cash flow stream for each surface element) identifies a marginal revenue surface which shows the spatial distribution of relative cash flow****The slope and aspect of a proximity surface determines the speed and direction of the optimal movement in traversing each surface element****What would the slope of a slope map show? Or, the aspect of a slope map?**

**Aggregation ~ sum of the values for all or a portion of the surface elements (integral); example uses include cut/fill calculations in road building and total yield estimates in site-specific management**

**Coefficient of Variation ~ localized variation surrounding each surface element (roughness)**

**Mathematical Translations ~ scalar arithmetic, logarithmic, trigonometric, and logical operations; example use of taking the cosine of the zenith angle formed between the sun’s position and each elevation surface element to calculate insolation (sun energy at each location)**

**Statistical Operations ~ describe and characterize surface**

**Descriptive statistics (min, max, range, median, mode, mean, skewness, etc.)****Similarity assessment (spatial autocorrelation)****Predictive statistics (map generalization and interpolation)****Accuracy assessment (residual analysis of how well surface fits a data set)**

**Other "Stuff" (Standard Normal Variable surface; pattern recognition filters)**

**__________________________________________________________________**

**Table 2. Annotated Listing of Example
Multivariate Surface Analysis Techniques**

**Mathematical Translations ~ arithmetic, logarithmic, trigonometic, and logical operations among two or more maps; for example,**

**NewSurface - OldSurface = DifferenceSurface (***arithmetic*)**( ( NewSurface - OldSurface ) / OldSurface ) * 100 = %ChangeSurface(***equation*)**b**_{0}+ b_{1}* X_{1}Surface + b_{2}* X_{2}Surface = PredictedSurface (*prediction*)**If (SomeSurface) > 200 Then NewSurface = 1 Else NewSurface= 0(***logic tests*)

**Statistical Operations ~ describe and characterize coincidence among surfaces**

**Descriptive statistics (cross-tabular tables; coincidence statistics)****Similarity assessment (comparison tests; clustering)****Predictive statistics (regression; induction; genetic modeling)****Accuracy assessment (residual analysis of how well multivariate modeled surface fits reality; error propagation modeling)**