Fundamentals of GIS ------Using GIS-- Lecture 10: Part 1: GPS Part 2: Interpolation and geostatistics By Austin Troy 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Part 1:
Global Positioning System Many materials for this part of the lecture adapted from Trimble Navigation Ltds GPS Web tutorial at http://trimble.com/gps/index.html Lecture Materials by Austin Troy, Weiqi Zhou and Jarlath ONeil Dunne 2008 ------Using GIS-- Fundamentals of GIS GPS Stands for Global Positioning System GPS is used to get an exact location on the surface of
the earth, in three dimensions. GPS is a very important data input source, used for surveying, military operations, engineering, vehicle tracking, flight navigation, car navigation, ship navigation, unmanned vehicle guidance, agriculture, and of course, mapping For mapping, a GPS tells us where and allows us to input what 2008. Austin Troy. All rights reserved Fundamentals of GIS GPS GPS is a worldwide radio-navigation system formed from 24 satellites and
their ground stations. Uses satellites in space as reference points for locations here on earth Ground stations help satellites determine their exact location in space. There are five monitor stations: Hawaii, Ascension Island, Diego Garcia, Kwajalein, and Colorado Springs. 2008. Austin Troy. All rights reserved Source: Wikipedia Fundamentals of GIS
How does GPS work? GPS derives position relative to satellite reference points, using triangulation The GPS unit on the ground figures its out distance to each of several satellites using the time it takes for a radio signal to travel to the satellite To do this, the exact position of the satellites at a given time, must be known; otherwise they cant serve as reference points 2008. Austin Troy. All rights reserved Fundamentals of GIS How does GPS work?
y km z km x km 2008. Austin Troy. All rights reserved Fundamentals of GIS How does GPS work? We need at least 3 satellites as reference points to triangulate our position. Based on the principle that where we know our exact distance from a satellite in space, we know we are somewhere on the surface of an imaginary sphere with
radius equal to the distance to the satellite. With two satellites we know we are in the plane where the two intersect. With three or more, we can get two possible points, and one of those is usually impossible from a practical standpoint and can be discarded 2008. Austin Troy. All rights reserved Fundamentals of GIS How does GPS work? Heres how the sphere concept works A fourth satellite narrows it from 2 possible points to 1 point Source: Trimble Navigation Ltd. 2008. Austin Troy. All rights reserved
Fundamentals of GIS How does GPS work? This method assumes we can find exact distance from our GPS receiver to a satellite. How does that work? Simple answer: see how long it takes for a radio signal to get from the satellite to the receiver. Since we know speed of light, we can answer this This gets complicated when you think about the need to perfectly synchronize satellite and receiver. A tiny error in synchronization can result in hundreds of meters of positional error 2008. Austin Troy. All rights reserved
Fundamentals of GIS How does GPS work? The difficult part is measuring travel time, because the amount of time elapsed is tiny (about .06 seconds for an overhead satellite), and we require a way to know precisely WHEN the signal left the satellite To do this requires comparing lag in exactly similar patterns, one from satellite and one from receiver. Analogy, going to a stadium, sitting 1000 feet from the speaker and pressing play on your handheld tape player containing REO Speedwagon at exactly the same time as the guy in the sound booth presses play for that same song. 2008. Austin Troy. All rights reserved
Fundamentals of GIS How does GPS work? delayed:I cant fight this feeling any more, Local: I cant fight this feeling any more, Only, instead of using cheesy eighties rock power ballads, GPS uses something called pseudo-random code. This code has to be extremely complex (hence almost random), so that patterns are not linked up at the wrong place on the codethat would generate the wrong time delay and hence the wrong distance Source: Trimble Navigation Ltd.
2008. Austin Troy. All rights reserved Fundamentals of GIS How does GPS work? So how do we know that the two Speedwagon fans are pressing play at exactly the same time? Do Speedwagon fans all think alike? Hardly. We must assume that satellite and receiver generate signal at exactly the same time; if theyre off by 1/1000th of a second, that means 200 m of error The satellites have expensive atomic clocks that keep perfect timethat takes care of their end. But what about the ground receiver? 2008. Austin Troy. All rights reserved
Fundamentals of GIS How does GPS work? Here is where the fourth satellite signal comes in. While 3 perfect satellite signals can give a perfect location, 3 imperfect signals cant, but 4 can Imagine time to receiver as distance, with each distance from each satellite defining a circle around each satellite of that radius If receiver clock is correct, 4 circles should meet at one point. If they dont meet, the computer knows there is an error in the clock: They dont add up 2008. Austin Troy. All rights reserved
Fundamentals of GIS How does GPS work? Dotted lines represent real distance, and solid lines represent erroneous distance, based on clock error they dont meet. Notice here we used three circles, because were looking in 2D, but in reality (3D) this represents four satellites, or four circles Assuming the clock error affects all measurements equally, the computer can then simply apply a correction factor that makes circles meet in one place
2008. Austin Troy. All rights reserved Source: Trimble Navigation Ltd. Fundamentals of GIS How does GPS work? The receiver then knows the difference between its clocks time and universal time and can apply that to future measurements. Of course, the receiver clock will have to be resynchronized often , because it will lose or gain time This is one reason why a GPS receiver needs at least four channels to get four signals
2008. Austin Troy. All rights reserved Fundamentals of GIS How does GPS work? So now we know how far we are from the satellites, but how do we know where the satellites are?? We cant use them as a reference otherwise. Because the satellites are ~ 20,200 km up they operate according to the well understood laws of physics, and are subject to few random, unknown forces. This allows us to know where a satellite should be at any given moment.
2008. Austin Troy. All rights reserved Fundamentals of GIS How does GPS work? There is a digital almanac on each GPS receiver that tells it where a given satellite is supposed to be at any given moment. While the positions can be predicted very accurately based on simple mathematics, the DOD does monitor them using precise radar, just to make sure. These errors are called ephemeris and are caused by gravitational pull of other celestial bodies That info is relayed to the satellite, which transmits
the info when it sends its pseudo random code. 2008. Austin Troy. All rights reserved Fundamentals of GIS GPS sources of error Even after all this, there are still many factors that can generate errors and reduce positional accuracy One of the biggest error sources is the fact that the radio signal does not travel at the exact speed of light in different parts if the atmosphere as it does in the vacuum of space. Source: Trimble Navigation Ltd.
This can be partly dealt with using predictive models of known atmospheric behavior 2008. Austin Troy. All rights reserved Fundamentals of GIS GPS sources of error Signals also can bounce off features, like tall buildings, cliffs and mountains, resulting in multipath error, where a direct signal hits, followed by a bunch of bounced signals which can confuse the receiver. Good receivers have algorithms that can deal with this by determining what counts as a multi-path
signal and choosing the first one as the signal to use There are other errors as well, resulting from things like ionospheric distortions and satellite inaccuracies 2008. Austin Troy. All rights reserved Fundamentals of GIS GPS: selective availability Until May of 2000, the DoD intentionally introduced a small amount of error into the signal for all civilian users, calling it selective availability, so non- US military users would not have the same positional accuracy as the US military. SA resulted in about 100 m error most of the time Turning off SA reduced error to about 30 m radius
Here is Clintons letter: http://www.ngs.noaa.gov/FGCS/info/sans_SA/docs/statement.html 2008. Austin Troy. All rights reserved Fundamentals of GIS Differential GPS This is a way to dramatically increase the accuracy of GPS positioning to a matter of a few meters, using basic concepts of geometry This was used in the past to overcome SA, but with that gone, is now used for reducing the 30m error DGPS uses one stationary and one moving receiver to help overcome the various errors in the signal
By using two receivers that are nearby each other they are getting essentially the same signals; since position of one is known, clock error can be calculated 2008. Austin Troy. All rights reserved Fundamentals of GIS How does DGPS work? The stationary receiver must be located on a control point whose position has been accurately surveyed: eg. USGS benchmarks The stationary unit works backwardsinstead of using timing to calculate position, it uses its position to calculate timing
It determines what the GPS signal travel time should be and compares it with what it actually is Can do this because, precise location of stationary receiver is known, and hence, so is location of satellite 2008. Austin Troy. All rights reserved Fundamentals of GIS How does DGPS work? Can do this because, precise location of stationary receiver is known, and hence, so is location of satellite Once it knows error, it determines a correction factor and sends it to the other receiver.
2008. Austin Troy. All rights reserved Fundamentals of GIS How does DGPS work? Since the reference receiver does not know which satellites the mobile receiver is using, it sends a message to it telling the correction factor for all It used to be that only big companies and governments could use DGPS because they had to set up their own reference receiver station Now there are many public agencies that maintain them, especially the Coast guard; these stations broadcast on a radio frequency, which GPS receivers
with a radio receiver can pick up 2008. Austin Troy. All rights reserved Fundamentalsto of GIS Introduction GIS Differential GPS DGPS improves accuracy much more than disabling of SA does This table shows typical errorthese may vary Source: http://www.furuno.com/news/saoff.html
2008. Austin Troy. All rights reserved Fundamentals of GIS Surveyor DGPS There are even more accurate types of DGPS that surveyors use These are accurate to a matter of millimeters This uses a very involved method that wont be discussed here One of the techniques they use though, carrier-phase GPS is beginning to make its way into consumer GPS Use carrier-phase signal, which is much smaller cycle widths than the standard code phase signal
2008. Austin Troy. All rights reserved Fundamentals of GIS Aviation DGPS FAA is implementing DGPS for the continent, so all planes can get extremely accurate GPS navigation, called Wide Area Augmentation System (WAAS) They have installed 25 ground reference stations as well as a master ground station that almost instantaneously processes and sends out satellite errors Improves error to 7 m and, when finished, will allow GPS to be used as primary navigational tool for Category I landings, where there is some visibility.
Soon, it will allow zero-visibility landing navigation 2008. Austin Troy. All rights reserved Fundamentals of GIS GPS Uses Trimble Navigation Ltd., breaks GPS uses into five categories: Location positioning things in space Navigation getting from point a to point b Tracking - monitoring movements Mapping creating maps based on those positions Timing precision global timing You can learn about all these applications at these web links, but we mainly care about mapping
2008. Austin Troy. All rights reserved Fundamentals of GIS GPS Uses The uses for GPS mapping are enormous. Here are just a few examples: Centerlines of roads Hydrologic features (over time) Bird nest/colony locations (over time) Fire perimeters Trail maps Geologic/mining maps Vegetation and habitat
2008. Austin Troy. All rights reserved Fundamentals of GIS ------Using GIS-- Part 2: Introduction to interpolation, geostatistics and spatial sampling Lecture Materials by Austin Troy, Weiqi Zhou and Jarlath ONeil Dunne 2008 Fundamentals of GIS What is interpolation?
Three types: 1. Resampling of raster cell size 2. Transforming a continuous surface from one data model to another (e.g. TIN to raster or raster to vector). 3. Creating a surface based on a sample of values within the domain. 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS How does it Look Let say we have our ground water pollution samples
This gives us 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS How does it work This can be displayed as a 3D trend surface in 3D analyst 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS
Requirements of interpolation Interpolation only works where values are spatially dependentthat values for nearby points tend to be more similar Where values across a landscape are geographically independent, interpolation does not work 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Interpolation examples Elevation: Elevation values tend to be highly spatially autocorrelated because elevation at location (x,y) is
generally a function of the surrounding locations Except is areas where terrain is very abrupt and precipitous, such as Patagonia, or Yosemite In this case, elevation would not be autocorrelated at local (large) scale, but still may be autocorrelated at regional (small scale) 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Interpolation examples Elevation: Source: LUBOS MITAS AND
HELENA MITASOVA, University of Illinois 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS How does interpolation work In ArcGIS, to interpolate: Create or add a point shapefile with some attribute that will be used as a Z value Click Spatial Analyst>>Interpolate to Raster and then choose the method
2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Three methods in Arc GIS IDW SPLINE Kriging 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Inverse Distance Weighting
IDW weights the value of each point by its distance to the cell being analyzed and averages the values. IDW assumes that unknown value is influenced more by nearby than far away points, but we can control how rapid that decay is. Influence diminishes with distance. IDW has no method of testing for the quality of predictions, so validity testing requires taking additional observations. IDW is sensitive to sampling, with circular patterns often around solitary data points 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Inverse Distance Weighting
IDW: assumes value of an attribute z at any unsampled point is a distance-weighted average of sampled points lying within a defined neighborhood around that unsampled point. Essentially it is a weighted moving avg n ^ z ( x0 ) i z ( xi ) i 1 Where i are given by some weighting fn and n
i 1 i 1 Common form of weighting function is d-p yielding: n ^ z ( x0 )
z ( x i ) d ij p i 1 n d ij p i 1
2010All lecture materials by Austin Troy except where noted Fundamentals of GIS IDW-How it works Z value at location ij is f of Z value at known point xy times the inverse distance raised to a power P. Z value field: numeric attribute to be interpolated Power: determines relationship of weighting and distance; where p= 0, no decrease in influence with distance; as p increases distant
points becoming less influential in interpolating Z value at a given pixel 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS IDW-How it works There are two IDW method options Variable and fixed radius: 1. Variable (or nearest neighbor): User defines how many neighbor points are going to be used to define value for each cell 2. Fixed Radius: User defines a radius within which every point will be used to define the value for each cell
2010All lecture materials by Austin Troy except where noted Fundamentals of GIS IDW-How it works Can also define Barriers: User chooses whether to limit certain points from being used in the calculation of a new value for a cell, even if the point is near. E.g. wouldn't use an elevation point on one side of a ridge to create an elevation value on the other side of the ridge. User chooses a line theme to represent the barrier 2010All lecture materials by Austin Troy except where noted
Fundamentals of GIS IDW-How it works What is the best P to use? It is the P where the Root Mean Squared Prediction Error (RMSPE) is lowest, as in the graph on right To determine this, we would need a test, or validation data set, showing Z values in x,y locations that are not included in prediction data and then look for discrepancies between actual and predicted values. We keep changing the P value until we get the minimum level of error. Without this,
we just guess. 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS IDW-How it works This can be done in ArcGIS using the Geostatistical Wizard You can look for an optimal P by testing your sample point data against a validation data set This validation set can be another point layer or a raster layer Example: we have elevation data points and we generate a DTM. We then validate our newly created DTM against an existing DTM, or against another existing elevation points data set. The computer determine what the optimum P is to minimize our error
2010All lecture materials by Austin Troy except where noted Fundamentals of GIS IDW-How it works 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Optimizing P value 2010All lecture materials by Austin Troy except where noted
Fundamentals of GIS Plot of model fits The blue line indicates degree of spatial autocorrelation (required for interpolation). The closer to the dashed (1:1) line, the more perfectly autocorrelated. Mean pred. Error near zero means unbiased
2010All lecture materials by Austin Troy except where noted Where horizontal, indicates data independence Fundamentals of GIS Plot of model errors 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Spline Method
Another option for interpolation method This fits a curve through the sample data assign values to other locations based on their location on the curve Thin plate splines create a surface that passes through sample points with the least possible change in slope at all points, that is with a minimum curvature surface. Uses piece-wise functions fitted to a small number of data points, but joins are continuous, hence can modify one part of curve without having to recompute whole Overall function is continuous with continuous first and second derivatives. 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS
Spline Method SPLINE has two types: regularized and tension Tension results in a rougher surface that more closely adheres to abrupt changes in sample points Regularized results in a smoother surface that smoothes out abruptly changing values somewhat 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Spline Method Weight: this controls the tautness of the curves. High weight value with the Regularized Type, will result in an increasingly smooth output surface.
Under the Tension Type, increases in the Weight will cause the surface to become stiffer, eventually conforming closely to the input points. Number of points around a cell that will be used to fit a polynomial function to a curve 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Pros and Cons of Spline Method Splines retain smaller features, in contrast to IDW Produce clear overview of data Continuous, so easy to calculate
derivates for topology Results are sensitive to locations of break points No estimate of errors, like with IDW Can often result in over-smooth surfaces 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Kriging Method Like IDW interpolation, Kriging forms weights from surrounding measured values to predict values at unmeasured locations. As with IDW interpolation, the closest measured values usually have the most influence. However, the kriging weights for the
surrounding measured points are more sophisticated than those of IDW. IDW uses a simple algorithm based on distance, but kriging weights come from a semivariogram that was developed by looking at the spatial structure of the data. To create a continuous surface or map of the phenomenon, predictions are made for locations in the study area based on the semivariogram and the spatial arrangement of measured values that are nearby. --from ESRI Help 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Kriging Method In other words, kriging substitutes the arbitrarily chosen p from IDW with a probabilistically-based weighting
function that models the spatial dependence of the data. The structure of the spatial dependence is quantified in the semi-variogram Semivariograms measure the strength of statistical correlation as a function of distance; they quantify spatial autocorrelation Kriging associates some probability with each prediction, hence it provides not just a surface, but some measure of the accuracy of that surface Kriging equations are estimated through least squares 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Kriging Method
Kriging has both a deterministic, stochastic and random error component Z(s) = (s) + (s)+ (s), where (s) = deterministic component (s)= stochastic but spatially dependent component (s)= spatially independent residual error Assumes spatial variation in variable is too irregular to be modeled by simple smooth function, better with stochastic surface Interpolation parameters (e.g. weights) are chosen to optimize fn 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS
Kriging Method Hence, foundation of Kriging is notion of spatial autocorrelation, or tendency of values of entities closer in space to be related. This is a violation of classical statistical models, which assumes that observations are independent. Autocorrelation can be assessed using a semivariogram, which plots the difference in pair values (variance) against their distances. 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Semivariance
Semivariance(distance h) = 0.5 * average [ (value at location i value at location j)2] OR n 2 { z ( x ) z (
x h )} i i (h) i 1 2n Based on the scatter of points, the computer (Geostatistical analyst) fits a curve through those points The inverse is the covariance matrix which correlation over space
2010All lecture materials by Austin Troy except where noted shows Fundamentals of GIS Variogram Plots semi-variance against distance between points Is binned to simplify Can be binned based on just distance (top) or distance and direction (bottom) Where autocorrelation
exists, the semivariance should have slope Look at variogram to find where slope levels Binning based on distance only Binning based on distance and direction 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Variogram SV value where it flattens
out is called a sill. The distance range for which there is a slope is called the neighborhood; this is where there is positive spatial structure The intercept is called the nugget and represents nugget random noise that is range spatially independent 2010All lecture materials by Austin Troy except where noted sill
Fundamentals of GIS Steps Variogram cloud; can use bins to make cloud plot of all points or box plot of points Empirical variogram: choose bins and lags Model variogram: fit function through empirical variogram Functional forms? Fundamentals of GIS Functional Forms
From Fortin and Dale Spatial Analysis:A Guide for Ecologists Fundamentals of GIS Kriging Method We can then use a scatter plot of predicted versus actual values to see the extent to which our model actually predicts the values If the blue line and the points lie along the 1:1 line this indicates that the kriging model predicts the data well 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Kriging Method
The fitted variogram results in a series of matrices and vectors that are used in weighting and locally solving the kriging equation. Basically, at this point, it is similar to other interpolation methods in that we are taking a weighting moving average, but the weights () are based on statistically derived measures. z ( xautocorrelation ) 0 s are chosen so that the estimate is unbiased and the estimated variance is less than for any other possible linear combo
of the variables. 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Kriging Method Produces four types of prediction maps: Prediction Map: Predicted values Probability Map: Probability that value over x Prediction Standard Error Map: fit of model Quantile maps: Probability that value over certain quantile 2010All lecture materials by Austin Troy except where noted
Fundamentals of GIS Kriging output: prediction 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Kriging: Ordinary vs. Universal Known as Kriging in the presence of universal trends. Universal kriging is used where there is an underlying trend beyond the simple spatial autocorrelation Generally this trend occurs at a different scale Trend may be fn of some geographic feature that occurs on one part of the map
2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Example Here are some sample elevation points from which surfaces were derived using the three methods 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Example: IDW Done with P =2. Notice how it is not as smooth as Spline. This is
because of the weighting function introduced through P 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Example: Spline Note how smooth the curves of the terrain are; this is because Spline is fitting a simply polynomial equation through the points 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Example: Kriging
This one is kind of in betweenbecause it fits an equation through point, but weights it based on probabilities 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Other methods of interpolation Thiessen polygons This method builds polygons, rather than a raster surface, from control points grows polygons around sample points
that are supposed to represent areas of homogeneity Source: Jens-Ulrich Nomme http://www.tu-harburg.de/sb3/pssd/GISMethods/thiessen.html 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Sampling Approaches An intermediate approach is the stratified random sample Create geographic or non-geographic subpopulations, from each of which random sample is taken
Proportional or equal probability SRS: enforce a certain sampling rate, hj= nh/Nh for each stratum h and obs j. Simple SRS: enforce a certain sample size nh Disproportionate SRS: where hj varies such that certain strata are oversampled and certain undersampled. 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Sampling Approaches DSRS is advantageous when subpopulation variances are unequal, which is frequently the case when stratum sizes are considerably different. In DSRS we sample those strata with higher variance at higher rate. We may also
use this when we have an underrepresented subpopulation that will have too few observations to model if sampled with SSRS. Proportional samples are self-weighting because the rates are the same for each stratum The other two have unequal sampling probabilities (unless a simple SRS has equal Nh) and may require 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Sampling Approaches When the stratifying unit is geographical (e.g. county, soil polygon, forest stand), this is called a cluster sample. In a one stage cluster sample (OSCS) a series of
geographic units are sampled and all observations within are sampled: obviously this does not work for interpolation More relevant is a two stage cluster sample (TSCS) in which we take a sample of cluster units and then a subsample of the population of each cluster unit. In this type of sample, variance has two components, that between clusters and that between observations 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Sampling The number of samples we want within each zone depends on the statistical certainty with which we
want to generate our surface Do we want to be 95% certain that a given pixel is classified right, or 90% or 80%? Our desired confidence level will determine the number of samples we need per strata This is a tradeoff between cost and statistical certainty Think of other examples where you could stratify. 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Sampling A common problem with sampling points for interpolation is what is not being sampled? Very frequently people leave out sample points that are
hard to get to or hard to collect data at This creates sampling biases and regions whose interpolated values are essentially meaningless 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Sampling example Say we were looking at an inland area, far from any ocean, and we decided that precipitation varied with elevation. How would we set up our sampling design? In this case, flat areas would need fewer sample points, while areas of rough topography would need more In our sampling design we would set up zones, or strata,
corresponding to different elevation zones and we would make sure that we get a certain minimum number of samples within each of those zones This ensures we get a representative sample across, in this case, elevation; 2010All lecture materials by Austin Troy except where noted Fundamentals of GIS Sampling The number of zones we use will determine how representative our sample is; if zones are big and broad, we do not ensure that all elevation ranges are represented
2010All lecture materials by Austin Troy except where noted
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