Characterization of vehicle test courses by power spectra.

In the last several years, smaller robotic vehicles and all-terrain vehicles have introduced a need to re-evaluate some of the previous vehicle testing methodology that we use to test vehicles. The Mobility Systems Branch of the Geotechnical and Structures Laboratory has several vehicle ride courses used to evaluate the dynamic vibrational effect of cross-country of terrain on vehicles. This paper explains ways we characterize terrain for vehicle testing and generate performance analyses. Mathematical formulas have been derived for characterizing the terrain in terms of the dimension of its power spectral density curve. Also, the slope of the line relating the elevation profile detrending (1) length to some power of the detrended root mean squared (RMS) (2) value is used. The plots of some courses and their power spectral density curves are shown and then several different detrending values are considered in order to illustrate the application of the formulas. The paper also includes a short discussion of how the mathematical theory of wavelets is related to the averaging and detrending kernels.

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Purpose. The purpose of the research reported in this paper is threefold: (1) to assist the ERDC in quantifying micro-terrain for generating performance analysis with current test data sets, (2) to develop methodology to supply representative micro-terrain for vehicle dynamic analysis and simulators, and (3) to lay a mathematical foundation for further research into a better method of realistic automatic terrain generation for vehicle simulations.

Outline. In its introduction, this paper provides a short history of the work on cross-country terrain characterization for vehicle testing up to this time. In the introduction and appendix, there is a short discussion of the relation of various types of wave-lets to several types of detrending formulas that can be used to help smooth the terrain profiles. This smoothing process helps to better define the characterization of the terrain profile's power spectral representation. Wavelets are related to detrending in that they are averaging kernels that are summed over the profile data. In the appendix, several mathematical formulas (3) are derived for characterizing the terrain in terms of the dimension of its power spectral density and the slope of the line relating the detrending length to some power of the detrended RMS value. Then, in the main part of the paper, the validity of the current methodology of using the root-mean-squared (RMS) statistic associated with cross-country micro-terrain is re-evaluated in terms of the use of the frequency analysis of power spectrum transforms. In the main part of the paper we consider whether it is useful to use several different detrended lengths in the design of off-road vehicle test courses. The elevation profiles from three representative test courses were considered in order to characterize them using this mathematical methodology. We plot the original course and its power spectral density curves and then consider several different detrending values in order to apply the formulas. In the reference by Van Deusen (1996), the differences between natural and man-made terrain were evaluated from the viewpoint of measuring these terrain by surface profiling. In this report Van Deusen considered the short falls of using RMS alone for all man-made or modified terrain. We examine the cross-country and road surface profiles already measured and in use for testing at ERDC. After the profiles have been analyzed and studied, we will look at a new methodology and formulas derived in the appendix as a means of representing scaling effects in modeling and representing this terrain for vehicle testing.

Scope. As outlined in the Purpose section, this paper considers how how to quantify micro-terrain for use in performance analyses of test vehicles. It lays a mathematical foundation for further research into better methods of generating realistic automatic terrain for vehicle tests and simulations. The paper does not consider the relationship of this method of statistically determining the dimension of power spectral density curves to the theory of other methodologies related to statistically sampling fractal terrain (see Harte, 2001), nor does it consider how to relate the detrending length used to calculate the RMS to the expected normal modes of vibration in the test vehicles (see Harrell TR-01-16, 2001) as an introduction to this.

Background. As mentioned above, the problem of how to represent micro-terrain for vehicle studies was considered by Van Deusen (1966), in his report for the Chrysler Corporation on the design of the lunar vehicle for the National Aeronautics and Space Administration (NASA). He wrote programs using several different methods to detrend the radar elevation data from the Ranger program of lunar exploration. Some runway profile data and highway profile data were also considered. Van Deusen found when assuming a power spectral density curve proportional to negative two power, that the detrending parameter (4) needed could be related to the variance of the data for cross-country or natural terrain. In 1972, the Directorate of Research, Development and Engineering, AMC (5), authorized a vehicle mobility study by TACOM (6) and WES (7) for an assessment of mobility of wheeled vehicles already in the fleet and candidates for their replacement. One important focus of this study was to define the different vehicle mission levels and quantify the performance of military vehicles while operating at different mobility mission levels. These definitions and evaluations were developed using the Army Mobility Model (AMM, AMC-71) produced by WES. The approach was to evaluate the vehicle's speed as it traversed different terrain types, such as primary roads, secondary roads, trails, and cross-country, in a given mission and combine these predicted movement velocities over the various terrain in a quantitative manner. The resulting method was used to successfully rank the performance of different military vehicles while operating at different mission levels. In 1974, the US Army Training and Doctrine Command (TRADOC) initiated a study to determine future requirements for high-mobility vehicles in the Army inventory and determine and compare the effectiveness of high-mobility and standard-mobility (tactical) vehicles in combat support. The TRADOC mobility levels, which were generated from the 1972 study, were used to define the vehicle's mission. These mobility levels were defined as:

    Tactical high mobility: A level of mobility designating requirements
    for extensive cross-country maneuverability characteristics involved
    in operating in ground-gaining and fire-support environments.
    Tactical standard mobility: A secondary level of mobility
    designating the requirement for occasional cross-country movement.
    Tactical support mobility: A level of mobility designating the
    requirement for infrequent off-road operations. These operations
    occur over selected terrain with the preponderance of movement on
    primary and secondary roads.

The tactical mobility assessment required additional detail beyond the general description of terrain types (cross-country, primary and secondary roads) to include terrain classification and mission percent-ages over each terrain type. Standard terrain type definitions were used for primary and secondary terrain as well as for cross-country terrain.

    Primary Roads: Two or more lanes, all-weather, maintained, hard
    surface roads with good driving visibility used for heavy and
    high-density traffic. These roads have lanes with a minimum width of
    2.7 m (9 ft.) and the legal maximum GVW/gross combined weight for
    the country or state is assured for all bridges.
    Secondary Roads: Two lanes, all-weather, occasionally maintained,
    hard or loose surface (paved, crushed rock, gravel) roads intended
    for medium-weight, low-density traffic. These roads have lanes with
    a minimum width of 2.4 m (8 ft.) and no guarantee that the legal
    maximum GVW/gross combined weight for the country or state is
    assured for all bridges.
    Trails: One lane, dry weather, unimproved, seldom maintained, loose
    surface roads intended for low-density traffic. Trails generally
    have a minimum lane width of 2.4 m (8 ft.), no large obstacles
    (boulders, stumps, logs), and no bridging.
    Off-Road: Vehicle operations over virgin terrain that has had no
    previous traffic (cross-country) and over combat and pioneer trails.

METHODS AND MATERIALS

Transforms. Transforms are the result of transforming a function of one independent variable to that of another. A common example of a transform of f(t) is the cosine transform

f(t) = [[infinity].summation over (k=0)][A.sub.k] cos(2[pi][kt/T]),

or

f(t) = [[infinity].summation over (k=0)][A.sub.k]cos(2[pi][[omega].sub.k]t),

where [w.sub.k] = k/T is the frequency. The cosine transform of f(t) is literally the coefficients in the series [A.sub.k] = A([w.sub.k]) which are given by

[A.sub.k] = [2/T][T.[integral].0]f(t)cos(2[pi][[omega].sub.k]t)dt.

Thus the function in the time domain is transformed to a function in frequency domain. The cosine transform is an example of a more general class of linear transforms referred to as Fourier transforms. Note that the transform is a linear combination of the sequence of cosine functions, a property of the transform, and the constant [A.sub.k], a property of the data. The cosine functions are referred to as the basis functions or kernel of the transform. Other choices of basis functions give different transforms having different properties.

Consider a function as a list, or time record, of N+1 numbers that is a discrete sampling of a continuous function in time, f(t). The readings are equally-spaced over a total time of T = (N+1) [DELTA]t such that the position, k, in the list corresponds to a time equal to k[DELTA]t. In the discrete transform, the list of N+1 samples in time is transformed to a list of N+1 numbers, each corresponding to a discrete frequency. Each record can be computed from the summation

[f.sub.N] = [N.summation over (k=0)] [A.sub.k] cos(2[pi][[omega].sub.k]N[DELTA]t).

For the general discrete Fourier transforms the basis functions consist of both sines and cosines,

[f.sub.N] = [N.summation over (k=0)][[A.sub.k]cos(2[pi][[omega].sub.k]N[DELTA]t) + [B.sub.k]sin(2[pi][[omega].sub.k]N[DELTA]t)],

which is generally written in complex form as

[f.sub.N] = [k=N/2.summation over (k=-N/2][F.sub.k]exp(i[pi][[omega].sub.k]N[DELTA]t).

The algorithm for computing this transform efficiently is referred to as the Fast Fourier Transform.

Estimation of the Power Spectral Density. If the profile height, denoted by [Y.sub.i], is measured at equal increments [DELTA]x over a finite course length, the auto-correlation function can then be defined and computed by the formula:

[[phi].sub.[lambda]] = 1/n-[lambda][n-[lambda].summation over (i=1)][Y*.sub.i][Y.sub.i+[lambda]].

The corresponding definition and estimate of the power spectral density (PSD) is

P(n) = 2[DELTA]x([[phi].sub.0] + 2[m-1.summation over (i=1)][[phi].sub.i]cos(i[lambda][pi] / m) + [[phi].sub.m]cos(i[pi])).

Due to the inability to precisely resolve any frequency in a finite sample, it is necessary to smooth the spectral estimate from the above equation over neighboring frequencies.

[-.P(n)] = [[infinity].summation over (k=-[infinity])][w.sub.k]([lambda])P(n-k)

where [lambda] is the detrending length and w is a smoothing (detrending) window. The question then arises how does varying the detrending length affect the accuracy of the PSD estimation?

Wavelets. Wavelets are mathematical families of functional atoms or functional components. They can be generated from a single function [psi] by dilations and translations in the form: [[psi].sub.a,b] = |a|[.sup.1/2][psi](t-b)/a); where the parameters a and b are often restricted to a discrete set. This defines a discrete wavelet transform which provides one way to detrend elevation profiles. In this form they provide a way to decompose an arbitrary function similar to the way sines and cosines are used in the mathematical theory of Fourier analysis. Wavelet transforms are defined and used in wavelet methods to analyze signals in a way analogous to Fourier Transforms:

For a and b real numbers, a, b <> 0 the discrete wavelet transform is

(Tf)[.sub.m] = <[[psi].sub.m,n], f> = |[a.sub.0]|[.sup.-m/2] [integral]dt*[psi]([a.sub.0.sup.-m]t-n[b.sub.0])*f(t)

where a = [a.sub.0.sup.m] b = n[b.sub.0][a.sub.0.sup.m].

These methods provide an improved way to analyze and represent functions of which the spatial form has representations changing with time. The book by Krantz (1999) gives a good overview of some of the mathematical topics in the theory of real analysis and functional analysis, which played a part in their invention and development.

The uses of wavelets are varied. Fournier (1995) contains an introduction to their use in computer graphics and image compression, and Charles Chu (1997) explains the uses of them to improve the functional representation by interpolation of the spline (8) of the data for numerical partial differential equations. The papers by Newland and Butler (1999) use wavelets for the study of centrifuge experiments related to earthquake engineering. The reference Torrence and Compo (1998) provides data on using them to prove the statistical significance of El Nino related sea surface temperatures.

There are many different type of wavelets, corresponding to different uses: continuous wavelets (Meyer and Ryan, 1993), discrete wavelets (Chu, 1997), orthogonal wavelets, biorthogonal, and compact wavelets (Daubechies, 1992). The Daubechies scaling function, wavelet function, and filter coefficients are shown in the logistic test vehicle ride results report (Harrell, 2001). The definitions of the scaling function and wavelet function in this report are based on the mathematical algorithm used to compute the wavelet decomposition by Chu (1997). The wavelet filter coefficients are the multipliers used in the wavelet transform to extract the high and low frequency components of the signal. Chu (1997) has a good reference to help understand how wavelet function, scaling function, and filter coefficients work together to decompose, reconstruct, and approximate the signal. The particular wavelet that ERDC has been using to date to do its detrending of vehicle test course elevation profiles is: [w.sub.j] = [e.sup.-x*j/[lambda]]. The use of functions of this type to do detrending was first studied mathematically by Wigner (1932) and Ville (1948); Mallat (1998) explains the connection of this to wavelet theory. The exponential function decays fairly fast in terms of units of detrending length. The normal modes of vibrational response of vehicles to terrain can often be expressed in functions of this type (Harrell, 2002). The ERDC technical report by Harrell (2001) gives additional information about the relationship of wavelets to vehicle testing.

Some other possibilities for smoothing (detrending) windows are in Press et al. (1992):

[w.sub.j] = 1 - |(j - (1/2N))/(1/2N)| "Bartlett window"

[w.sub.j] = 1/(2(1 - cos(2[pi]j/N)) "Hann window"

[w.sub.j] = 1 - (j - (1/2N)/(1/2N))[.sup.2] "Welch window"

The amplitudes of the Fourier coefficients of the above spectral windows are shown in Table 1. Comparing the values gives some idea of how they average the data differently. But, according to Several Theorems of Wiener (Weiner, 1957) the extent of the different types of functions that these different detrending wavelets can represent is equivalent (9).

RESULTS

In 1971, the WES conducted a statistical analysis of terrain-vehicle-speed relative to dynamics of wheeled vehicles. One important conclusion to this effort was the acceptance of a single statistical measure of terrain micro-surface roughness developed by the Chrysler Corporation in 1966 for NASA. The NASA was interested in describing naturally occurring extraterrestrial terrain for the development of extraterrestrial land vehicles. The Chrysler Corporation concluded that micro elevations of a naturally-laying terrain could be described by the standard deviation of the detrended terrain elevations or the RMS, of the detrended terrain elevations. The term detrended terrain elevations refers to applying the various wavelet transforms to the terrain elevations. The formulas for these procedures will be explained in the following sections. Van Deusen's work in this area, using an exponential weighting factor, indicated that the RMS of the detrended course depended on the detrending parameter. This was a simplication of the earlier methodology which made it easier to construct a series of off-road test course using only a single parameter (the RMS). The terrain surface roughness (the RMS) was shown, in the 1971 WES study, to be a significant factor in vehicle speed when traversing natural cross-country terrain. The smoother the terrain surface, the faster the vehicle would travel and the rougher the terrain surface, the slower the vehicle would travel. This dynamic performance and terrain description technique was accepted by the TRADOC study to classify the different terrain types. Classes of surface roughness were assigned to the different terrain descriptions. Typical RMS values for off-road areas and roads are as follows:

    Primary Roads: Surface roughness values range from 0.1 inch RMS to
    0.3 inch RMS.
    Secondary Roads: Surface roughness values range from 0.1 inch RMS to
    0.6 inch RMS.
    Trails: Surface roughness values range from 0.1 inch RMS to 2.8 inch
    RMS.
    Off-Road: Surface roughness values ranges from 0.6 inch RMS to 4.5
    inch RMS.

In retrospect, a problem that arose with this decision was using an analytical method proven for natural terrain for man-made terrains like roadways. The Chrysler Corporation in its earlier reports had defined its conclusions based on natural terrain and proved that the RMS of a made-man terrain was not adequate to describe the frequency content of the changes in the micro-terrain elevations.

In the WES report by Murphy (1984) off-road elevation profile data from the ERDC ride courses at Letourneau, MS were studied. It was determined that a single RMS statistic with a detrending parameter of 10 feet could be used to adequately represent the observed power spectral density curves (of vehicle response to off-road terrain, but not roads). The implementation of how to design test courses based on this conclusion was followed by the study by Lessum (1971) in which he used Wiener-Bose theory to characterize the power spectrum in ride courses.

To summarize the previous history of research in this area:

* Military vehicle mission levels were defined to compare vehicle mission performance.

* Terrain descriptions were assigned to the different terrains encountered during military vehicle mission scenarios.

* Terrain micro-surface roughness was calculated on the different terrain types due to the vehicle-terrain/surface roughness performance relationships.

The TRADOC used the results of its study to assist in the creation of military vehicle operational requirements document. The requirement documents now state required vehicle mission levels, as defined in this report, associating percentages of terrain types for each mission level. Table 2 below summarizes the values that define this mobility classification system.

With the description of the different missions in measurable terms, the vehicle design and performance specifications developed by TACOM uses these same descriptions to determine how the vehicle would be physically tested. The decision to test new vehicles, using these terrain definitions, required the test community to monitor and maintain test courses to these classification levels. These criteria are also used to develop test courses for mission scenario testing and for durability testing.

In the last several years, smaller robotic vehicles and all-terrain vehicles have introduced a need to reevaluate some of the previous testing methodology. ERDC has expanded its research in this area to make use of some new techniques in mathematics which could possibly improve the characterization of vehicle response to micro-terrain. Some questions have arisen in the vehicle testing community about the adequacy of using a single RMS statistic to represent the terrain micro-geometry for vehicle testing. The general mathematical theory of wavelets, as they shed light on different detrending methods, can be considered as a more adequate solution to the problem than the more limited and specific Wiener-Bose theory. That is, the problem of how to improve the Fourier representation of time series is better understood in the more general mathematical theory of averaging kernels and wavelets. The formulas that arise from this approach are related mathematically to the previously used exponential detrending methodology. Several families of wavelets have already been used for characterizing the results of vehicle ride tests on the ERDC Logistics Test Vehicle (LTV). In the two ERDC reports listed in the references below, wavelet methodology was used to: (1) determine whether the power spectral densities resulting from vehicle drops of the ERDC LTV had higher normal modes than of the vehicle's frame vibration frequencies, (2) investigate the possibility of redesigning some of the Letourneau ride courses, and (3) test ways to represent the vehicle's response in terms of wavelet expansion coefficients resulting from the vehicle's response to the test courses.

In this paper we follow the overall approach in the reference by Van Deusen (1966). It assumes that the power spectral density of the elevation profile of the test course is of the form:

[P.sub.d] ([OMEGA]) = C* [[OMEGA].sup.-n],

where n = inverse power of the curve reprsenting the spectral density, which is constant for any given spectral estimate, and [OMEGA] = spatial frequency in cycles/foot. The formula to be derived for n = 2 is:

[[sigma].sup.2] = (C [[pi].sup.2] [lambda])/2

where [sigma] = RMS, [lambda] = detrending length, and C is a constant to be determined that characterizes the vehicle test course. It is due to Van Deusen (1966). This formula applies only in the case where the inverse power of the spectral density curve is 2. Not all the details of the mathematical derivation of this formula were given in Van Deusen's report. So, it is rederived in Appendix A along with a new corresponding formula for the case where the dimension of the inverse power of the spectral density curve is n = 4, i.e.,

[sigma] = C* (4[[pi].sup.4] [[lambda].sup.3])[.sup.1/2].

From the formula, derived from the assumption of an n = 2 power law, [[sigma].sup.2] = (C[[pi].sup.2][lambda])/2 and knowing the slope of the line from the above graph, we have that C = 2(.0059)/[[pi].sup.2] = 1.1955 X [10.sup.-3] (feet)[.sup.2] cycle/foot or 3.644 X [10.sup.-4] (meter)[.sup.2] cycle/meter.

The course is characterized by the PSD curve dimension of n = 2. From the formula [[sigma].sup.2] = (C[[pi].sup.2][lambda])/2 and knowing the slope of the line from the above graph, we have that C = 2(.092)/[[pi].sup.2] = 1.8643 X [10.sup.-2] (feet)[.sup.2] cycle/foot or 5.6824 X [10.sup.-3] (meter)[.sup.2] cycle/meter.

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From the formula [[sigma].sup.2] = (C[[pi].sup.2][lambda])/2 and knowing the slope of the line from the above graph, we have that C = 2(.63)/[[pi].sup.2] = 1.276 X [10.sup.-1] (feet)[.sup.2] cycle/foot or 3.8912 X [10.sup.-2] (meter)[.sup.2] cycle/meter. Looking at the formula we see that in this case the size of the square of the coefficient C that characterizes the test course is directly proportional to the surface roughness.

In the course of the analysis, plots were also made for each of the test courses of the linear RMS and the RMS to the 2/3 power versus detrending length. These were the other cases where, if the spectral density curve were of a certain inverse power, the analysis in the appendix indicated a possible formula relating these two variables. These results are shown in figures 10, 19, 20, 29. However, the goodness of linear fit of the relation of the detrending length to the RMS was not as good in these cases.

CONCLUSIONS

Considering the above graphs and the analysis that is given below in this paper, it seems reasonable to conclude that a dimension of n = 2 should be used for the inverse power of the spectral density curves of the test courses. There are two advantages of using this methodology over that of a simple RMS value. One is that the spectral power curve dimension along with the coefficient of the power spectral curve gives a more powerful means of differentiating test courses at a higher resolution. Another is that the parameters used do not depend on the detrending length used to smooth the elevation profile data. It seems reasonable to use these two parameters to characterize and differentiate the vehicle test courses that were considered: (1) the inverse power of the curve representing the course's spectral density, and (2) the coefficient C in the numerator of the curve representing the course's spectral density. Their computed values are given in Table 3 below.

According to the parameters and classification scheme on the National Automotive Test Center (NATC) website the upper (RMS UL) and lower limits (RMS LL) for the values of RMS which were computed for the MudLake Farming Road and the Forest Road place them in the category of a loose surface secondary roads. The values computed for the Letourneau Test Course place it in the cross-country category. Table 4 show these parameters, which were determined from performance specification information. The MudLake Farming Road has a 10 foot parameter detrended RMS of 0.3 and a 30 foot detrended RMS of 0.45 The Forest Road has a 10 foot detrended RMS of 1.21 and a 30 foot detrended RMS of 1.7. The Letourneau #1 course has 10 foot detrended RMS of 2.5 and a 30 foot detrended RMS of 4.3.

For comparison purposes, the above table by Van Deusen contains parameters measured from detrended data for aircraft runways.

DISCUSSION

Details of the Analysis of the Vehicle Test Course Profiles. In the following pages three different vehicle test courses are analyzed: (1) Mud Lake #1, a Delta, LA farm road, (2) Forest Road (right track), a road test section at the ERDC, Vicksburg, MS Geotechnical pavements test facility, and (3) Letourneau #1, at the Geotechnical Mobility Branch's specially constructed military vehicle ride course. Gravel was used to construct the Letourneau course in order to represent a wide spectrum of square wave frequencies. The difference of the frequencies occurring in this constructed course were replicated from those occurring in natural terrain. This can be seen by comparing the 1997 Letourneau course elevation profile readings and power spectrum plots (Figures 22, 23, 24) with those of the Mud Lake and Forest Road sites (Figures 1, 2, 10, 18). However, profile analysis of data measured in 1999 shows that some of these frequencies have disappeared from the Letourneau course. This is possibly due to vehicle trafficability passes, rain, and soil consolidation. In order to characterize the test courses for purposes of testing smaller vehicles, the elevation profile data were measured at 3-inch intervals. This compares to the previously used profile data station increments of 1 foot. Figures 4 through 10 show the Mud Lake elevation profile and power spectral density of the test courses both before and after they have been detrended (using three different detrending lengths). In Figures 1 and 11, the results of the detrending in terms of RMS and variance are plotted against the detrending length in order to test Van Deusen's theoretical formula (10).

Again, we assume that the power spectral density of the elevation profile of the test course is of the form:

[P.sub.d]([OMEGA]) = C*[[OMEGA].sup.-n],

where n = inverse power of the curve representing the spectral density, which is assumed to be constant for any given spectral estimate. [OMEGA] = spatial frequency in cycles/foot (11).

The results of the analysis in the rest of this section show, as Van Deusen theory claims, that for the Mud Lake road and Letourneau off-road course profiles a fairly exact linear relation exists between the detrending length squared (the variance) and the RMS. After the data have been plotted this way, the fitting constant C along with the dimension of the inverse power of the spectral density curve can be used to characterize the elevation profile for vehicle test purposes. The detrending data was plotted against a linear relation (n = 1), quadratic (n = 2), and cubic (n = 4) power of the detrending length. These three dimensions satisfy the conditions that allow the integral representing the exponential detrending wavelet (Wigner-Ville tranform) applied to the sine and cosine function to converge. Although it is not as clearcut as in the case of the other test courses, a look at the three figures (2, 20, 21) shows that the Forest Road data can also be best fitted linearly if the detrending length squared is plotted versus the square of the RMS. Thus, in all three cases, Mud Lake, Forest Road, and Letourneau, the detrended power spectrum is best characterized by a power spectrum of type represented functionally by a polynomial curve of the degree inverse power squared.

Mud Lake. After the profile has been detrended according to the formulas listed above, using a range of smoothing parameters, the results may be analyzed. The formula derived in the appendix relating the resulting RMS of the detrended profile to the smoothing (detrending) parameter can be used to test the assumption regarding the dimension of the power spectral density of the profile. The dimension computed using the formula can be used to characterize the test course. In the graph in Figure 1 a value of n = 2 is used as the inverse dimension of the power spectral density curve. The graph below shows a different value, n = 1, and clearly does not give as good a fit to the data as the value n = 2 in Figure 1.

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Forest Road. Turning now to a different test course, Figures 12 through 19 show the elevation profile and power spectral density of this course both before and after it has been detrended (using three detrending lengths). In Figures 2, 20, and 21, the results of the detrending in terms of RMS and variance are plotted against the detrending length in order to test Van Deusen's theoretically derived formula. Again, we assume that the power spectral density of the elevation profile of the test course is of the form:

[P.sub.d]([OMEGA]) = C*[[OMEGA].sup.-n],

where n = inverse power of the curve representing the spectral density, which is assumed to be constant for any given spectral estimate. [OMEGA] = spatial frequency in cycles/foot.

The detrending data was plotted against a linear relation (Figure 20, n = 1), and a fractional power (Figure 21, n = 0.66) of the detrending length. These three dimensions satisfy the conditions that allow the integral representing the exponential detrending wavelet (Wigner-Ville transform) applied to the sine and cosine function to converge. Although, it is not as clearcut as in the case of the other test courses, a look at the three figures (2, 20, 21) shows that the Forest Road data can also be best fitted linearly if the detrending length squared is plotted

versus the square of the RMS.

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By comparing the two graphs below, we can see the degree to which the power spectral density dimension n = 2 has been correctly fitted to the Forest Road data in Figure 2. Two different values n = 1 and n = 2/3 are tested in the figures below to compare how closely the data fits the line with the linear fit to the data in Figure 2.

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Letourneau. Figures 22 through 29 show the elevation profile and power spectral density of the test course both before and after it has been detrended (using three detrending lengths). In Figures 3 and 30 the results of the detrending in terms of RMS and variance are plotted against the detrending length in order to test Van Deusen's theoretical formula. Again, we assume that the power spectral density of the elevation profile of the test course is of the form:

[P.sub.d]([OMEGA]) = C*[[OMEGA].sup.-n],

[FIGURE 22 OMITTED]

where n = inverse power of the curve representing the spectral density, which is assumed to be constant for any given spectral estimate. [OMEGA] = spatial frequency in cycles/foot.

The results of the analysis show, as Van Deusen theory claims, that for the Letourneau off-road course profiles a fairly exact linear relation exists between the detrending length squared (the variance) and the rms. The detrending data is shown plotted against a quadratic (Figure 30, n = 2) power of the detrending length.

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APPENDIX A

Derivation of the formula relating the RMS to detrending length

(assuming the spectral density curve is an integral inverse power)

This appendix assumes that the power spectral density of the elevation profile of the test course is of the form:

(A1) [P.sub.d] ([OMEGA]) = C[[OMEGA].sup.-n],

where n = inverse power of the curve representing the spectral density, which is constant for any given spectral estimate. [OMEGA] = spatial frequency in cycles/foot (1). The formula to be derived for n = 2 is:

(A2) [sigma] = (C[[pi].sup.2][lambda]/2),

where [sigma] = RMS, [lambda] = detrending length, n = the inverse power of the curve representing the spectral density, and C is a constant to be determined that characterizes the vehicle test course (Van Deusen, 1966).

Weiss (1981) assumed the power spectral density to be of the form:

[P.sub.d] ([OMEGA]) = A[[sigma].sup.-2] + B[[sigma].sup.-3] + C[[sigma].sup.-4]

where A, B, C are positive or negative coefficients to be determined from [[sigma].sub.d], [[sigma].sub.c], [[sigma].sub.s] the detrended RMS of the information contained in the elevations, curvatures, and slopes respectively of the course elevation profiles.

Our approach is to first determine the best single inverse power fit, by plotting only the data profile elevations versus the detrending parameter. Then the spectral power curve coefficient is determined by calculating the slope of the linear fit of the logarithms of the RMS versus detrending length parameter.

Because of what we have learned about the fractal nature of terrain since Dr. Weiss wrote his paper in 1981, it makes sense to use the detrending parameter as a variable to calculate the spectral parameters instead of [[sigma].sub.d], [[sigma].sub.c], [[sigma].sub.s]. This is because the terrain is no longer considered to be approximated well at the microroughness level by the continuous polynomials (e.g., Pietgen and Saupe 1988; Mandelbroit, 1977; and Turcotte, 1992). The earlier approach calculated the coefficients of polynomials by interpolating values of slopes and curvatures measured at discrete intervals and inverting the determinant of a matrix. This does not work well for fractal terrain objects whose parameters are better estimated from looking at the effect of scaling or detrending transformations on parameters that characterize the data.

The detrending process that the exponential filter applies to the profile data can be represented by the discrete variable equation:

(A3) [-.F(x)] = [[infinity].summation over (n=0)](F(x + nu) + F(x - nu))[e.sup.-nu/[lambda]]))/(2[[summation].sub.n=0.sup.[infinity]][e.sup.-nu]/[lambda]).

In the limit, as n approaches infinity, this equation takes the continuous variable form:

(A4) [-.F(x)] = (1/(4[pi][lambda][OMEGA]))[[infinity].[integral].[u=0]](F(x + u) + F(x - u))[e.sup.-u/[lambda]]du.

Note that the integration in the transform is with respect to the variable u (2). We assume that the terrain profiles fall into the class of functions which are measurable (see Apostol [1974] for the mathematical definition of a measurable function) and for which this continuous variable transform exists. Detailed conditions under which the continuous variable form of this transform is equal to the discrete form were given by N. Wiener (1957). By Theorem 28, page 160, in this reference, the two are equal for a class of measureable functions if, and only if, the power spectrum is continuous. The assumption that the power spectrum of the terrain can be given in the form of an integral inverse power satisfies this condition of Wiener except at the point zero. We must then make the further assumption for the purposes of this analysis that there is an upper limit and finite cutoff of the frequencies considered which affect the vehicle's response to the terrain. The frequencies of the vehicle frame axis normal modes of vibration, generated by terrain test courses, and measured over the years at ERDC for test vehicles have been less than a 100 cycles per second (3). Therefore, this assumption will not cause a problem for our application area. An example of an application area in which this assumption would not be justified would be in the use of a similar methodology to make measurements to characterize atmospheric conditions from satellite or ground-based radar reflections.

Following Van Deusen (1966), in order to understand this transform, we calculate its effect on functions of the form:

(A5) F(x) = sin(2[pi][OMEGA]x).

To calculate the result we need to use the indefinite integral formula from calculus:

(A6) [integral]([e.sup.ax] sin(bx + c)dx = ([e.sup.ax]/([a.sup.2] + [b.sup.2]))(a sin(bx + c) - b cos(bx + c))

where a, b, c are arbitrary constants.

We now can proceed with the integration if we let a = -1/2[pi][OMEGA][lambda], b = 1, and c = 2[pi][OMEGA]x in the above formula:

(A7) [integral](sin(2[pi][OMEGA]x + u)*[e.sup.-1/2[pi][OMEGA][lambda]u]du = ([e.sup.-1/2[pi][OMEGA]u]/((1/2[pi][OMEGA][lambda])[.sup.2] + 1)((-1/2[pi][OMEGA][lambda])sin(2[pi][OMEGA]x + u) - cos(2[pi][OMEGA]x + u)) and

(A8) [integral](sin(2[pi][OMEGA]x - u)*[e.sup.-1/2[pi][OMEGA][lambda]u]du = ([e.sup.-1/2[pi][OMEGA]u]/((1/2[pi][OMEGA][lambda])[.sup.2] + 1)((-1/2[pi][OMEGA][lambda])sin(2[pi][OMEGA]x - u) + cos(2[pi][OMEGA]x - u)).

To complete the full calculation indicated in formula (B4), formulas (B7) and (B8) need to be added together.

To simplify this part of the calculation, we use the elementary trigonometric identities:

sin(2[pi][OMEGA]x + u) + sin(2[pi][OMEGA]x - u) = 2 sin(2[pi][OMEGA]x)cos(u), and

cos(2[pi][OMEGA]x + u) + cos(2[pi][OMEGA]x - u) = 2 sin(2[pi][OMEGA]x)*sin(u).

We obtain:

[-.F(x)] = 1/(4[pi][OMEGA][lambda])([e.sup.-1/2[pi][OMEGA]u]/((1/2[pi][OMEGA][lambda])[.sup.2] + 1)((-1/2[pi][OMEGA][lambda])*2(sin(2[pi][OMEGA]x)cos(u) + sin(2[pi][OMEGA]x)sin(u))|[.sub.0.sup.n[pi]]

where the indefinite integral is to be evaluated at the midpoint and end of the averaging interval: 0 and n[pi]. Doing this, we finish the evaluation of the definite integral:

(A9) [-.F(x)] = 1/(4[pi][OMEGA][lambda])(1/((1/2[pi][OMEGA][lambda])[.sup.2] + 1)((-1/2[pi][OMEGA][lambda])*2(sin(2[pi][OMEGA]x) - [e.sup.-n[OMEGA]/2](sin(2[pi][OMEGA]x))

And, as n (which measures the size of the averaging interval in increments of 2[pi]) approaches infinity this goes to:

(A10) [-.F(x)] = -1/(2[OMEGA][pi][lambda])[.sup.2] (1/((1/2[pi][OMEGA][lambda])[.sup.2] + 1)(sin(2[pi][OMEGA]x))).

Now the amplification factor of the detrending transform on the function (A5) F(x) = sin(2[pi][OMEGA]x) can be determined as:

(A11) F(x) - [-.F(x)] = A sin(2[pi][OMEGA]x)

where A = the amplitude ratio.

Where, we have derived that:

(A12) A = (1 - (1/2[pi][OMEGA][lambda])[.sup.2](1/(1/2[pi][OMEGA][lambda])[.sup.2] + 1) = 1 - (1/2[pi][OMEGA][lambda])[.sup.2]/(1/2[pi][OMEGA][lambda])[.sup.2] + 1) = 1/(1/2[pi][OMEGA][lambda])[.sup.2] + 1).

Now, the following theorem is needed to go further in the derivation:

Theorem (4). Assume that the variance, [[sigma].sup.2] of the detrended increment (5)

[I.sub.H,[DELTA]] = ([-.F(x)] - [-.F(x - [DELTA]))] = [[sigma].sup.2]|[DELTA]|[.sup.2H]

of a time series (6) is proportional to the increment [DELTA] (7). Let [g.sub.[DELTA]](t) = [delta](t) - [delta](t - [DELTA]), then the power spectrum of the detrended increment is a stationary stochastic process of the form (8):

(A13) [P.sub.I.sub.H,[DELTA]]([OMEGA]) = [[sigma].sup.2.sub.H]*[[OMEGA].sup.-(2H+1)]|[^.g.sub.[DELTA]]([OMEGA])|[.sup.2].

This theorem says that although the original elevation profile may not be statistically stationary (9), the detrended increment of it is. Proof: By applying the assumptions of the Theorem twice, the covariance of [I.sub.H,[DELTA]] is equal to:

(A14) E{[I.sub.H,[DELTA]](t)*[I.sub.H,[DELTA]](t - [tau])} = ([[sigma].sup.2]/2)(|[tau] - [DELTA]|[.sup.2H] + |[tau] + [DELTA]|[.sup.2H] - 2|[tau]|[.sup.2H]).

If f([tau]) = |[tau]|[.sup.2H], then, [^.f]([omega]) = -[[lambda].sub.H]*|[omega]|[.sup.-(2H+1)] (where [[lambda].sub.H] > 0 is some constant) by the standard formulas from Fourier transform theory. So, if [g.sub.1]([tau]) = |[tau] - [DELTA]|[.sup.2H] and [g.sub.2]([tau]) = |[tau] + [DELTA]|[.sup.2H], then [^.g.sub.1]([omega]) = - ([e.sup.-i[DELTA][omega]])[[lambda].sub.H]|[omega]|[.sup.-(2H+1)] and [^.g.sub.2]([omega]) = -([e.sup.i[DELTA][omega]])[[lambda].sub.H]|[omega]|[.sup.-(2H+1)] by the standard formulas for the Fourier transform of the translation of a function. By noticing that [e.sup.i[DELTA][omega]] + [e.sup.-i[DELTA][omega]] - 2 = 2(1 - cos([DELTA][omega])) = 2 [sin.sup.2] ([DELTA][omega]/2), this gives the formula for the power spectrum

(A15) [P.sub.I.sub.H,[DELTA]]([omega]) = 2[[sigma].sup.2.sub.H][[lambda].sub.H][[omega].sup.-(2H+1)][sin.sup.2]([DELTA][omega]/2)

which proves the Theorem for the substitution [[sigma].sub.H.sup.2] = [[sigma].sup.2][[lambda].sub.H]/2.

Now in order to prove the formula (A2), assuming the PSD (power spectral density) is of the form (A1) we use the fact that by the above Theorem, [sigma] the variance or RMS squared, is equal to the PSD integrated over all frequencies; which by formulas (A1), (A11), (A12), and (A15) equals

(A16) [[sigma].sup.2] = [[infinity].[integral] [0]]C*[[OMEGA].sup.-n]d[OMEGA]/((1/2[pi][OMEGA][lambda])[.sup.2] + 1)[.sup.2].

If we let v = 2[pi][OMEGA][lambda] this can be written as

[[sigma].sup.2] = C(2[pi][lambda])[.sup.n-1][[infinity].[integral] [0]][v.sup.-n]dv/((1/v)[.sup.2] + 1)[.sup.2] = C(2[pi][lambda])[.sup.n-1][[infinity].[integral] [0]][v.sup.-n+2]dv/([v.sup.2] + 1)[.sup.2]

for n = 2, 3, 4, 5, respectively, this indefinite integral is equal to C(2[pi][lambda])[.sup.n-1] times a factor which is:

-1/2(v/([v.sup.2] + 1)) + 1/2(arctan(v)), n = 2;

-1/2(v/([v.sup.2] + 1)), n = 3;

1/2(v/([v.sup.2] + 1)) + 1/2(arctan(v)), n = 4;

1/2(1/([v.sup.2] + 1) + 1/2(ln([v.sup.2] + 1)), n = 5.

These formulas can be verified taking derivatives and using the fact that (arctan(v))' = 1/(1 + [v.sup.2]). Using the fact that arctan(0) = 0, arctan(([infinity]) = [pi]/2, ln(1) = 0, ln([infinity]) = [infinity] these indefinite integrals can be evaluated as equal to: [pi]/4 for n = 2, 0 for n = 3, [pi]/4 for n = 4, divergent for n = 5. We have:

(A17) [[sigma].sup.2] = (C[[pi].sup.2][lambda])/2 for n = 2, and [[sigma].sup.2] = C(4[[pi].sup.4][[lambda].sup.3]) for n = 4.

This result indicates that the RMS should be linearly proportional to the detrending length (10) for a PSD of curve of inverse power -2 and also for one of inverse power -4.

Endnotes for Appendix

(1) We shall use feet instead of meters for units in this paper because the WES/ERDC elevation profiles are measured in English units.

(2) This differs by a constant factor from the transform used by Van Deusen. The calculation to follow will show this constant factor is needed to make the amplification factor be equal to that stated in his report.

(3) See Harrell's 2001 ISTVS paper for an example and further discussion of how test vehicle normal modes of vibration are calculated.

(4) Mallat (1998) page 213.

(5) Here E means mathematical expectation of the function inside of the parenthesis. It is equal to the integral of the function divided by the measure of the region it is integrated over.

(6) H is a parameter called the Hurst exponent which relates to the case of fractional Brownian motion. See the references by Mandelbroit (1977), Mallat (1998), Harte (2001) for more information on its properties and definition.

(7) This is more general than assuming it to be independent of [DELTA] and what is called a stationary stochastic process. This includes the case of what is sometimes called Fractional Brownian Motion stochastic processes. Mallat (1998) page 212.

(8) For a given function g, [^.g] denotes its Fourier transform.

(9) Having statistical properties varying with the sampling interval.

(10) Assuming, of course, that the wavelets used in the averaging or detrending functions are exponentials such as the ones we are using.

Table 1. Coefficients of Spectral Windows. Values measured as the
smoothing operator moves aways from the center point--computed in units
of the detrending length.

Coefficient             Hann      Bartlett   Welch   Exponential
                        Values    Values     Values  Values

[A.sub.0]               1         1          1        1
[A.sub.1]=[A.sub.-1]    0.97553   0.9        0.99     0.36788
[A.sub.2]=[A.sub.-2]    0.90451   0.8        0.96     0.13534
[A.sub.3]=[A.sub.-3]    0.77389   0.7        0.91     0.04978
[A.sub.4]=[A.sub.-4]    0.65451   0.6        0.84     0.01832
[A.sub.5]=[A.sub.-5]    0.50000   0.5        0.75     0.00673
[A.sub.6]=[A.sub.-6]    0.34549   0.4        0.64     0.00247
[A.sub.7]=[A.sub.-7]    0.20611   0.3        0.51     0.00091
[A.sub.8]=[A.sub.-8]    0.09549   0.2        0.36     0.00033
[A.sub.9]=[A.sub.-9]    0.024471  0.1        0.19     0.00010
All others             <0.02      0         <0.15    <0.0001

Table 2. Definition of Mobility Levels.

                          Operation Mix
                        Percentage of Mission

Mission           Primar  Secondar  Trail  Off-
                    y        y        s    Road
                  Roads   Roads

          Central Europe Scenario Areas
Tactical-High      10       30       10     50
Tactical-          20       50       15     15
Standard
Tactical-Support   30       55       10      5

          Mid-East Scenario Areas
Tactical-High       5       20       25     50
Tactical-          15       35       35     15

Table 3. Characterization of Two Road Sections and One Vehicle Test
Course by the Spectral Characteristics of Their Elevation Profiles.

                       Spectral Density Coefficient   Spectral Curve
Course Name            (feet)[.sup.2] cycle/foot      Inverse Dimension


Mud Lake Farming Road    1.2 X [10.sup.-3]                 2
Forest Road              1.9 X [10.sup.-2]                 2
Letourneau #1            1.3 X [10.sup.-1]                 2

Table 4. Estimates of Parameters Defining Military Roads, and Trails.

Primary Roads         RMS  RMS  Spectral Density    Spectral Curve
                      LL    UL  Coefficient         Inverse Dimension

high quality          0.1  0.1  1.4 X [10.sup.-7]      2.5
secondary pavement    0.2  0.2  1.9 X [10.sup.-7]      2.5
rough pavement        0.3  0.3  8.0 X [10.sup.-7]      2.5
Secondary Roads
loose surface         0.3  0.6  3.0 X [10.sup.-5]      2
loose surface w/pot-  0.3  0.6  4.0 X [10.sup.-6]      2.4
holes
Belgian block         0.3  0.6  5.0 X [10.sup.-4]      1.4
Trails
one lane, unimproved  0.6  2.8  5.0 X [10.sup.-4]      1.9

Table 5. Representative Values Previously Used to Characterize Terrain
and Roads.

Course Name           Spectral Density Coefficient   Spectral Curve
                      (feet)[.sup.2] cycle/foot      Inverse Dimension


Aircraft Runway 12       2.5 X [10.sup.-6]           2
Smooth Highway           1.2 X [10.sup.-6]           2.1
Aircraft Runway 3        2.2 X [10.sup.-7]           2
Highway with gravel      1.1 X [10.sup.-5]           2.1
Lunar profile            1.2 X [10.sup.-3]           2
Aberdeen, MD vehicle     1.6 X [10.sup.-3]           2
test course

ACKNOWLEDGMENTS

The author acknowledges Mr. Dennis Moore and Mr. Wendell Gray of the Mobility Systems Branch, GSL, ERDC, who provided the test course elevation profile data files. Randolph Jones of the Mobility Systems Branch generously shared reports and information and wrote a short summary about past work in this area. This written material was incorporated in the outline section of the paper's introduction and parts of the methods and materials section. Dr. William Willoughby of the Mobility Branch GSL, Dr. John Peters of GSL, and Dr. Joseph Kolibal of the Mathematics Department of the University of Southern Mississippi read an earlier version of the manuscript and provided many useful questions and comments that improved its exposition.

Endnotes for Paper

(1) Detrending is a method of filtering the terrain elevation profiles by taking a weighted average of the elevation profile value at which the trend is being computed. There are several possible mathematical approaches and formulas which can be used. Details are given later in the method and materials section of the paper.

(2) RMS is an acronym used in the characterization of the surface roughness of terrain, meaning root mean squared. It is determined by first detrending the surface elevation measurements taken at one foot intervals in a terrain profile and then computing the ordinary square root of the variances of the measurements from the detrended values of the terrain profile data set. Let the detrended data be denoted F(x) and its mean [-.F(x)] then

RMS = ([summation over (i)](F([x.sub.i]) - [-.F([x.sub.i])])[.sup.2]/n)[.sup.1/2]

(3) Some originally given in the Van Deusen reference without proofs.

(4) The value used in the weighted average of the coefficient in the exponent used to do the profile detrending.

(5) US Army Materiel Command.

(6) US Army Tank Automotive Command.

(7) US Army Waterways Experiment Station.

(8) A spline of interpolated data assumes the data comes from a continuous function. The spline of the interpolated data is then a polynomial approximation to the assumed function whose coefficients are determined to agree with the value of the function and a certain number of its derivatives at the points which are specified.

(9) This statement can be made for precise using functional analysis and the theory of Hilbert spaces along with their different infinite dimensional basis functions and the degrees of differentiability of the functions that are represented.

(10) See Appendix A for a mathematical derivation of this formula.

(11) Van Deusen uses meters, but because the WES/ERDC elevation profiles are measured in English units, feet are used here.

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Andrew W. Harrell

USA Engineer Research and Development Center (ERDC), Vicksburg, MS 39180