Submitted to Computers in GeoSciences in September, 1996. Accepted in October, 1996. Will be published in a special issue of Computers in GeoSciences "about June" (1997). The special issue is to include a CD Rom containing all the animations and graphics. Be the first in your research lab, company, or university to get one!
Key words: animation, uncertainty, spatial data, digital elevation model, optimal route, random fields
Stochastic approaches to modeling spatial uncertainty result in the creation of many potential realizations for the spatial dataset of interest (for examples, see Openshaw, 1979; Goodchild et al., 1992; Fisher, 1993a; Ehlschlaeger & Goodchild, 1994b; Ehlschlaeger & Shortridge, 1996). Examining these realizations (which may number in the hundreds) and exploring the similarities and differences between them can be a major challenge. The authors' efforts to accomplish this prompted an exploration of non-traditional forms of cartographic representation, including animation. A particular problem in the generation of smooth animations from a series of "stills" is the creation of intermediate images to blend from one still to the next. These intermediate images are critical for a smooth blending from one realization to the next, but if they lack the statistical characteristics of the actual realizations then the animation will misrepresent the data and the form is inappropriate for analysis or communication.
The following section discusses the goals and methodology involved in the creation of the simulated surface realizations. The third section covers the application of surface realizations to a least-cost path algorithm, which provides a measure for the expected distribution of path costs. The fourth section describes the role that animation can play in visualizing uncertainty: the user may develop a better understanding for the impact of generalized spatial data on the outcome of the least-cost path algorithm. Then, the fifth section examines conceptual and practical issues for the interpolation between realizations necessary for generating an animated sequence. Finally, the paper addresses the contribution animations may make in the analysis and communication of uncertainty in spatial data. Animation, if correctly produced, may offer an alternative to the usual examination of tens or hundreds of static maps during the exploratory phase of data analysis, or to the usual slides and transparencies developed for a presentation. It is worth investigating whether animation could provide any additional benefits for uncertainty analysis beyond simply keeping one's audience awake during a presentation.
For the purposes of this paper, we assume that the corridor location problem to be solved requires a DEM of 30m sampling interval. However, such data is not always available, and thus we examine the effects of replacing it with coarser 3 arc second data, using an error model calibrated in regions for which both are available. We assume that the differences between the two resolutions are consistent between both areas: the area where the problem must be solved, and the area where both data sets are available. The relationship between 30m and the coarser 3 arc second data may be characterized by examining the distribution of differences between the two data sets at a large number, J, of randomly selected spatially uncorrelated locations on the surface. The difference is modeled using the mean and standard deviation of 96 sets of randomly drawn, spatially independent points scattered across the surface. Unconditional stochastic simulation can then be employed to define a probability density function (p.d.f). The p.d.f. generates random surfaces with a gaussian distribution matching the mean and standard deviation observed in the difference maps and is represented by:
where E(u) is a realization of the higher quality
30m elevation data using the more generalized 3 arc second data m(u),
T is a group of sets of spatially uncorrelated sample points, is
a random variable with mean 0.0 and variance 1.0 perturbing the realizations
standard deviation, and Z(u)is a random field perturbing
all points u within the realization. The following expressions
define the remaining terms:
and
where m(m(T)) is the average mean for all sets;
m(s2(T)) is the average
variance for all sets; s2(s2(T))
is the variance of the variances for all sets; and Z(u)
specifies the random field with spatial dependence parameters where u
is a point on the random field with a theoretical mean of 0.0 and theoretical
variance of 1.0, v is the set of points affecting u,
wu,v is the spatial autocorrelative
effect between points u and v, is
a random variable with a mean of 0.0 and variance of 1.0, du,v
is the distance between u and v, D
is the minimum distance of spatial independence, E is the distance
decay exponent, and F is a parameter that adds flexibility to the
probability distribution function model fitting process. Matching the spatial
autocorrelation using F is important in order to capture the terrain
texture observed in the higher resolution dataset (Goodchild, 1986; Theobald,
1989), and was implemented as the GRASS command r.random.surface
(Ehlschlaeger & Goodchild, 1994a). The
result of performing this analysis is a set of parameters defining the
p.d.f. for modeling the uncertainty of the 3 arc second elevation surface:
(6)
where D = 4600, E = .07, and F = 200 for the parameters of Z(u). Each surface created from these parameters is a potential realization of what the actual difference surface might be. By adding each difference field to the 3 arc second surface, a large collection of alternative, equally probable models of the elevation surface are created.
This study's digital elevation models cover a large study area around Santa Barbara, California, as depicted in Figure 1. The 3 arc second data set, USGS quad los_angeles-w, extends from 119 to 120 degrees west of Greenwich, and from 34 to 35 degrees north. The terrain is characterized by mountainous topography extending inland (brown and black, to the north and east) from the Pacific Ocean, punctuated by river valleys and narrow coastal plains (green). A test area (in black and white) several kilometers on each side was defined within this DEM. It extends from the coastal plain at the town of Goleta in the south across the Santa Ynez mountains to the Santa Ynez river valley in the north. For the purposes of this paper, we assumed that no 30m data existed for this test area, and that we would have to determine the location of a path within this area using 30m quality data.
Figure 1 - Study Region: Los_Angeles-w DEM (larger image link: 29k).
Six 30m data sets outside of the test area were compared to collocated data from the Los_Angeles-w DEM to develop parameters for the p.d.f. The p.d.f. was used to generate 250 realizations simulating 30 meter quality elevation data for the test area. These elevation surfaces were processed to create 250 corridor realizations. The next section describes how the corridor realizations may be used to demonstrate uncertainty in the route location application caused by the coarseness of the 3 arc second elevation data.
(7)
Second, two locations in the test area were chosen to be the endpoints for a hypothetical path. For each of the 250 realizations created in the first step, an accumulated cost surface was generated. The value in each cell of the accumulated cost surface represents the accumulated cost to travel to this cell from both of the endpoints across the cost surface produced in step one (Church et al., 1992). The resulting cost pattern in any single realization gives a visual indication of the degree to which the character of the terrain restricts the corridor of the least-cost path.
The authors have explored two methods for displaying realization results. The first method, used in an earlier study, condenses optimal routes from all realizations into a single, static map. The second method presents route cost information in animations of cost surface realizations.
Figure 2 - Optimal Paths (larger image link: 102k).
Using method one, as visualized in Figure 2, routes were generated connecting the two white spheres. This figure portrays the terrain of the study area (at a vertical exaggeration of 1.5) draped with a representation of the 250 optimal paths. Because the density of paths across certain portions of the test area was so great, displaying each path individually on the same image was impossible. Instead, we have colored the raster based on the number of optimal routes passing through each cell. Out of the 250 realizations, forty or more optimal paths traveled through blue cells, five or more paths through green cells, one path through yellow cells, and no optimal paths through gray cells. The white line on the surface outlines the optimal route traced on the original 3 arc second data set, which had a cost of 56,561 units. Using the actual 30 meter data for the test area, the optimal path cost 61,368 and followed the red line in Figure 2. The distribution, as seen in the histogram in Figure 3, was unimodal; the mean cost of the 250 optimal paths was 64,034 units with a standard deviation of 2,991 units.
Figure 3 - Histogram of Potential Optimal Paths
While this information provided a good measure of expected costs, it left several questions unanswered. Since a major goal of spatial data uncertainty research is to understand impacts of generalized map data on applications, these questions include: Under what conditions does a realized optimal path deviate from the optimal path on the 3 arc second data set? While it is obvious that the quality of DEM necessary depends on the choice of start and stop locations, does the large spatial variability of optimal paths indicate that 3 arc second elevation data are inadequate for determining the optimal path between these two locations? Method two attempts to answer these questions through the generation and dynamic visualization of accumulated cost surface realizations, rather than individual optimal paths.
We refer to each of these realizations as accumulated cost surfaces, because they portray the aggregate cost to travel to any particular cell on the surface. These surfaces may be preferred to a simple calculation of the optimal route for a given elevation realization. In addition to showing a single specific optimal path, accumulated cost surfaces provide a better idea of the general optimal path corridor. By comparing the different patterns from all of these realizations, the researcher can gain insight into how uncertainty in the spatial data affects the result of the path algorithm. While 250 realizations were used to calculate the statistics of DEM uncertainty for the optimal route problem, a maximum of 35 were used in these animations (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, and 35; links 18-25k).
Figure 4 - Cost surface for realization 20
The colors in both static and dynamic depictions of these accumulated cost surfaces (see the animations in "Optimal Route Movie", as well as Figure 4) attempt to maximize the contrast between the least expensive optimal route and the most expensive optimal route of the accumulated cost surfaces. The color scheme employed is different from the one used for the path image. The least expensive route (or routes if multiple paths have the same cost) within a realization are represented by white cells. Black cells indicate the cells that fall within one percent of the least expensive optimal route cost in the 250 realizations. The colors then ramp from black through blue and green to yellow (yellow cells have accumulated costs 10% greater than the least expensive optimal path in the 250 realizations). Gray cells have costs greater than 10% of the least expensive optimal path. The red line encompasses the set of cells with costs below that of the most expensive optimal path, 83,809 units, of any realization.
We were interested in determining the effectiveness of non-traditional methods of visualization for illustrating the impact of uncertainty on the application. The following section discusses the methods we employed to generate a smooth animation to accomplish this.
Spatial autocorrelative characteristics can play a significant role in understanding the impact of uncertainty during the research process. This information can be especially critical in a spatial application for which the relative locations of objects are important (e.g., optimal path routes and viewshed analysis). The video "Random Fields and their use in Representing Spatial Autocorrelation" (Ehlschlaeger, 1994) communicates the importance of spatial autocorrelation in representing spatial uncertainty. The video includes two animations, both of which show the impact of potential sea level rise on the shoreline of Boston Harbor. The first animation, "Ignoring Autocorrelation Movie", assumes the uncertainty term has a constant value for each realization. While the magnitude of error is represented correctly at every cell, the shoreline shapes do not represent potential results. The second animation, "Spatial Autocorrelation Included Movie", incorporates the spatial autocorrelative characteristics of uncertainty. In both animations, the amount of time a section of land remains underwater represents the probability of submergence given a 1.9 meter rise in sea level. However, the second animation details the effect of ocean level rising on contiguous regions (e.g., "What is the probability that this road will not be covered with water?").
The spatial autocorrelative characteristics described in the second section are used in the realization for the animation developed in this study. Each realization possesses different landscape elevation values. As a result of the changes to each elevation surface, the accumulated cost surface is also unique for each realization. In an attempt to portray both simultaneously, the main approach adopted in this paper employed a 2.5-dimensional perspective view of the test area with the accumulated cost surface draped over the elevation surface. While visually appealing, the use of the perspective view is not immune from criticism because of the problems of displaying three-dimensional data on a flat, two-dimensional screen. People may have trouble perceiving perspective models (due to distortion and obscured sections) and gathering useful information from them (Dorling & Openshaw, 1992). Therefore, for comparative purposes, a 2-D animation using elevation contours to portray the terrain was produced; it is viewable at the main animation page. In either instance, however, the goal of the visualization is not to provide precise rendering of the detail of a single scene, but to promote understanding of the magnitude of change between images.
The spatial pattern of any single realization is not of particular interest, for no one realization is more likely than any other to approximate the actual elevation surface. Instead, more interesting information may be gleaned from the details of the relationships between the realizations. The changes in the width and route of the corridor illustrate the impact of uncertainty. If there is little change from realization to realization, one can be fairly certain that the least-cost path lies along a rather well defined corridor. On the other hand, if change is dramatic, and a variety of differing corridors are suggested, then error in the spatial data has translated into a great deal of uncertainty in the application. Much can be gleaned from comparing static images of multiple realizations, but developing an implicit understanding of the changes between each image can be quite time consuming. Researchers may benefit from animation's ability to dynamically depict the range of uncertainty inherent in their data; they may spot relationships in the data, identify errors in their assumptions, or consider new directions for research (Fisher, 1993b; van der Wel et al., 1994). The particular animation method employed may affect how the viewer perceives the data and the relationships between variables. While this work does not address the impact of various animation technique on viewer perception, it does present a method of interpolation which avoids smoothing the intermediate images.
Another role for animation during the exploratory phase of research is the generation of additional information about the uncertainty of data and how data uncertainty affects the application. The process of interpolation employed here creates a very large number of statistically valid dependent realizations. Animating these images provides a natural way to view this massive influx of data in a time-effective manner (Dorling, 1992). Viewers also see the simultaneous movement of elevation and accumulated cost surface, providing greater understanding for that relationship. As mentioned in the third section, there were two questions we wished to answer with this animation: Under what conditions does a realized optimal path deviate from the 3 arc second optimal path? And, does the large spatial variability of optimal paths indicate that 3 arc second data sets are inadequate for determining the optimal path between these two locations? This study's animation, "Optimal Route Movie", helps answer these questions.
Observing the animation, the viewer is bound to notice several factors. The most obvious fact is that optimal paths often change location and cost for reasons not easily perceived simply by viewing the elevation surface. While the overall shape of the 3 arc second data set doesn't change, many smaller ridges and valleys appear and disappear within the realizations. By comparing the images of the 3 arc second data set and the 30m data set side-by-side, one notices that the 3 arc second data set is missing many ridges and valleys apparent in the 30m data. This missing terrain texture is apparently why optimal routes on the realizations of the fine resolution 30m DEM are approximately 12% more expensive than the optimal route on the coarse resolution 3 arc second DEM. The viewer will also notice realizations for which the optimal path shifts to a dramatically new location. The animation makes it clear that there is no simple relationship between the optimal routes on different realizations, or between the optimal routes computed at different spatial resolutions. Some fine resolution optimal paths are similar to the coarse resolution optimum, but some are very different. Clearly, the uncertainty introduced by resorting to coarse resolution data is propagated and amplified in the optimal routes. To answer the two questions posed earlier, the animations demonstrate that there are no simple relationships between optimal paths at the two resolutions; and coarse resolution data is indeed inadequate for the original purpose of finding an optimal path between the two endpoints originally chosen.
Figure 5- A similar application for which the coarse 3 arc second DEM is adequate
While this study's animations show a particular case for which uncertainty in coarse data renders that data inadequate, conditions in other cases may be such that uncertainty does not invalidate the same coarse dataset. For example, Figure 5 illustrates optimal paths between two different endpoints using the same cost equation and the same 250 surface realizations (graphic representation is the same as for Figure 2). The spatial distribution of these optimal paths demonstrate that there is little difference between corridor solutions for the second set of endpoints. In this case, 3 arc second data is adequate.
A central issue for the production of a smooth animation is the generation of intermediate images to ensure that the transition of images is gentle and cohesive (MacEachren & DiBiase, 1991). Allowing for a transition permits the viewer to see the magnitude and pattern of the differences in elevation and cost surface between the realizations. These intermediate images are interpolations between the original realizations; we generated eight interpolated images between each of the 35 realizations to develop a smooth transition. Nearly 90 percent of the frames in the animation, then, are not original realizations; they are interpolated images. In Figure 6, the upper timeline represents the animation sequence for "Optimal Route Movie". The horizontal lines represent independent realizations. The dark line represents the flow of "Optimal Route Movie". While there are only 35 independent realizations, the 61 second animation portrays 306 different realizations of the optimal path.
Figure 6 - Timelines of Animations: Upper from Optimal Route Movie, lower from Spatial Autocorrelation Included
For this study of uncertainty, the statistical and spatial characteristics of the interpolated surfaces must match the error model or the resulting animation will become a misleading visualization tool, or at least it will be much more difficult to achieve desired results. For example, "Spatial Autocorrelation Included Movie" used a linear interpolation for the transitions. In order for the viewer to best view actual realizations, the authors developed the lower timeline in Figure 6. The dark line represents the flow of the "SAI Movie", with the horizontal sections representing pauses in the animation flow for two seconds at each independent realization. Between each independent realization, the diagonal lines represent intermediate frames morphing from one realization to the next. Because these intermediate frames were not independent realizations, the goal of the "SAI Movie" was to allow half the time to be spent showing actual realizations while the other half created a transition between realizations. Therefore, the animation was only able to show 29 realizations in 113 seconds. In addition to only half of the time being spent viewing realizations, terrain motion was not providing accurate visual clues of uncertainty in terrain estimation.
The interpolation method is clearly quite important for animations of stochastic surfaces. Linear interpolation results in intermediate images with different characteristics than the independent realizations they tie together. Mean uncertainty is modeled correctly, but variance of uncertainty is lower for the linearly interpolated images. And, the spatial autocorrelative characteristics of uncertainty are also not representative. A non-linear interpolation presented here solves this problem. The following equations are used to interpolate between independent realizations in Optimal Route Movie:
(8)
where ix,y
is an "interpolated" random value between random values
x
and
y,
with a mean of 0.0 and standard deviation of 1.0; and:
(9)
where Zix,y(u)
is an "interpolated" surface between surfaces Zx(u)
and Zy(u), with every point
having a mean of 0.0 and a standard deviation of 1.0. We draw attention
to the word "interpolated" because we are trying to create values of Zix,y(u)
and ix,ythat
are appropriate for our p.d.f. and have values similar to nearby values
of i. In place of a formal proof, imagine the two endpoints x
and y as orthogonal unit vectors. Since each vector in Zix,y(u)and
ix,y
has a mean of 0.0 and standard deviation of 1.0, we can retain their statistical
properties by locating intermediate realizations along a circle between
them centered at the origin (which explains the sin() and cos() functions).
By using the functions for Zix,y(u)
and
ix,y
, interpolations of independent realizations are also (dependent) realizations
in their own right. Since these interpolated images are valid representations
of the surface, the employment of the nonlinear function allows one to
generate visually accurate animations.
Viewing animations of spatial data uncertainty as they affect applications also provides a good mechanism for allowing a greater visualization of the probabilistic nature of uncertainty (Beard et al., 1991). In a deterministic world, we expect answers to questions such as: "How long will the optimal route be?" or "Where are the possible locations of optimal routes?". Probabilistically, we may never know what the actual answers will be with very generalized data sets, but we may learn what factors will determine the actual answers and how the actual answers relate to the information we don't have. And, on occasion, we will find that very generalized data may be useful for meeting the needs of specific applications requesting precise data.
Several directions present themselves for further exploration. This example is simple, as complex interactions go. The researcher is confronted with only one independent variable, that being elevation, and one dependent variable, that being the cost surface. Many spatial applications involve considerably more inputs than this. Extending the methodology presented here may produce a method for visualizing the role uncertainty plays for each input layer individually upon the outcome of the analysis. This research focused primarily upon the interpolation method, but frame rate and duration are other key technical factors which affect perception of the animation. These were examined in earlier work on the Boston Harbor data, but relationships between all three factors merit continued exploration. Additional research is also warranted to assess the relative ease (or difficulty) people have in making sense of animations of abstract concepts like uncertainty (see Evans, 1996 for current research into viewers' perception of simultaneous displays of spatial data and its associated reliability). This direction is especially important for understanding the role for animations of uncertainty to communicate of results to the community. Understanding how people perceive spatial animation representations should provide a stronger basis for developing more effective animations.
The implementation of the non-linear interpolation method allows for the portrayal of uncertainty as the image shifts smoothly through a series of realizations. The resulting animation is a complex visualization tool for perception of a complex spatial phenomenon. Dynamic visualization may prove valuable for developing understanding and appreciation for the role data uncertainty plays in spatial analysis.
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1. Randomly determine 250 samples of independent random points from the 3 arc second data set using makeInd.csh
2. Develop statistics from these points by comparing the difference between the 3 arc second data set and available 30m data.
3. Find the parameters that best fit the random surface model by fitting the difference of 3 arc second data to available 30m data. So far, every time the random surface model parameters D, E, and F were used to describe the p.d.f., the 3D solution space was valley shaped. The following shell scripts test various combinations of model parameters, moving closer to the optimal combination: mC1.csh, mC2.csh, mC3.csh, and mC4.csh. The goal of each shell script was to test 27 locations surrounding the best solution identified by that stage in the analysis. If a better solution was found, the next shell script would choose locations around the new optimal parameters. Otherwise, the next shell script would test 27 locations closer to the previous optimal solution.
4. Employ the spatial statistical parameters from step 3 to generate 250 realizations using makeVis.csh.
5. Build non-linear interpolations between realizations. Building an interpolation between realizations takes less time than building the random surface. makeInterp.csh
6. (Optional) Check to see that interpolations of random surfaces have similar statistical and spatial statistical characteristics as the original realizations. checkInterp.csh, InterpSS.txt
7. Generate optimal path for 3 arc second data and render it. Generate optimal path for 30m data in the study area and render it. In an actual application, we wouldn't have this data. But, it is important to check and determine whether results meet expectations. visDted.csh, visDEM.csh
8. Determine range of optimum path values. Load data into spreadsheet for histogram. Develop color scheme to demonstrate this range, and give the color scheme to all the realizations. describeLCP.csh, makeColor.csh
9. Generate the script that runs the animation creation software. Finally, run the animation creation software script. makeMovie.csh, mAllMovie.csh.