Forecasting off-road trafficability

Applied Geography (1983), 3, 239-253

Forecasting off-road trafficability Malcolm G. Anderson Geography Department, Bristol University, Bristol BS8 IS& England

University Road,

Abstract Forecasting off-road trafficability represents an interesting science field, since variables ranging from solar insolation

problem

in the earth

to soil shear strength are involved. However, existing trafficability models do not include precipitation-soil moisture deterministic submodels. Instead, they frequently commence with soil moisture and invoke empirical relationships to predict trafficability. This investigation couples a deterministic soil moisture submodel to a principal empirical soil moisture-trafficability model. Sensitivity analysis shows the impact cloud cover, as well as other variables, can have on the trafficability types, and the complexity of the post-storm trafficability response.

of three soil

Introduction There exist demands from a wide variety of agencies to be able to predict the offroad performance of specified types of vehicles. Such demands emanate from agricultural, civil and military requirements. A review of soil trafficability prediction was undertaken by the US Army Corps of Engineers (1967). In this review soil moistureesoil strength relationships received considerable attention (e.g. Collins 1967a; Molthan 1967), and it is clear that the estimation of these two variables, and their conjoint relationship, is a central element in establishing successful trafficability models. Collins (1971) was able to establish relationships of soil strength with other soil properties. Specifically, Collins successfully specified relationships of the form : In RCI= 4.605 + where

2.123 +0~008 (C)-O.693 0~149+0~002

In M

(1)

(C)

M = moisture

content (percentage dry weight) C = percentage of clay in the soil under study RCI = rating cone index

which was shown to hold for USDA soil classes ranging from clay to sand. The RCI is defined as the product of the cone index (cone penetrometer test) and the remoulding index (cone penetrometer test undertaken on remoulded samples). Full test procedures are outlined in Collins (1971: Appendix B). Table 1 illustrates the minimum RCI (Vehicle cone index, VCI) found to exist, below which specified vehicles could not complete 4&50 passes. Empirically derived soil moisture prediction equations taken from 23 sites, together with empirical relationships of the form of Equation (l), have been combined to provide a model capable of predicting soil moisture and soil strength (Smith and Meyer 1973), and thus the trafficability (Table 1). In this model the soil moisture prediction is undertaken by establishing 0143~~6228,‘83/030239-15$03.COi

1983 Butterworth

& Co (Pubhshers)

Ltd

240

Forecasting

ojfroad

trajficahility

Table 1. Vehicle cone index (VCI): ranges for different vehicle types VCI”

Vehicle type

20-29 3049

Canadian snowmobile High speed tractors with wide tracks

50~59 60-69 7om79 go-99 > 100

Tractors with average contact pressure All-wheel drive trucks and tractors with comparatively low contact pressure All-wheel drive trucks and dump trucks Rear-wheel drive trucks (e.g. half-ton pick-up VCI = 88) Rear-wheel drive vehicles generally not expected to operate off roads

a VCI is that minimum RCI below which the vehicle could not complete 40 50 passes Source: Collins (1971)

the relationship between precipitation and the soil moisture depletion and accumulation in different soil layers. This suite of curves is empirical, being represented by up to sixth-degree polynomials. To improve the applicability of forecasting trafficability to as wide a range of environments as possible, it is desirable that the empirical elements in such models are minimized. Not only would this reduce initial model calibration, but it also would achieve the equally important goal of rendering such models operational either in areas where initial data were sparse, or in areas in which, for a number of different reasons, the operator may be denied access after having had the opportunity to take only some initial data relevant to the model. There are three principal areas to the problem of developing a more deterministically based trafficability model that can be identified. The first such area of trafficability forecasting that may be modelled deterministically is that of the soil moisture response to precipitation. Considerable empirical evidence is now available to show the changing soil moisture conditions under a variety of controls. Weyman (1973) showed the complex reversal of directions of soil-water flux in a slope section that typically takes place during a storm, whilst Biswas rt (11.(1966) have shown the typical redistribution of moisture in a vertical soil column following the cessation of infiltration. Changes of soil moisture in plan are no less dramatic and somewhat more complex. Anderson and Kneale (1980, 1982) have shown that, on shallow, undulating topography, the soil-water potential surface rarely mirrors the topographic surface, and the soil-water flow paths vary significantly both spatially and temporally during storm events. In a more comprehensive simulation study Anderson (1982a) showed that only in the case of relatively steep slopes (>25 degrees) and relatively permeable soils (hydraulic conductivity > 1 x 10 4 cm s ’ ) do soil-water flow paths always converge into topographic hollows. This evidence regarding soil-water changes suggests the strong desirability of incorporating a deterministic soil-water model into the trafficability forecasting system in order to free the model design from empirical precipitationsoil moisture relationships, which cannot be reasonably expected to accommodate the known, often rapid and varied, soil moisture responses. The extreme sensitivity of the RCI, and hence trafficability (Table l), to soil moisture is illustrated by the utilization of Equation (1). For a soil with 49 per cent clay and moisture content of 45 per cent the RCl is 61. whilst reducing the moisture content to 30 per cent is sufficient to increase the RCI to 189. Whilst the initial simulation investigation undertaken in this study is one-

Malcolm G. Anderson

241

dimensional as regards soil-water movement, there is evidence therefore to commend its application to incorporate three-dimensional soil-water movement in the context of trafficability forecasting. The second area is that of establishing a deterministic model which relates the tractive forces between the terrain and the vehicle track. Baladi and Rohani (1979) outline such a model which utilizes engineering properties of soil and commences with derivations from the shear-stress/shear-deformation characteristics of soil. This deterministic mathematical formulation for the prediction of steering performance of tracked vehicles thus represents a significant departure from mobility cone index as a measure of soil strength. Whilst Baladi and Rohani are utilizing cohesive strength and angle of internal friction of the material, they have demonstrated that these parameters can be successfully related to the mobility cone index (Rohani and Baladi 1981) so the large pre-existing cone index database can be utilized. The third area for deterministic modelling is the link between soil moisture and the fundamental engineering properties required as inputs to the model of Baladi and Rohani outlined here. Wroth and Wood (1978) and Schofield and Wroth (1968) provide experimental evidence of the correlation of index properties, proposed by Atterberg, with the undrained shear strength of a soil. Of potentially greatest use to the analysis considered above, however, are the derived relationships which Wroth and Wood (1978) provide for water content (w). w + (A/Gs) In cu = constant where

Gs=specific gravity of the soil particles cu = cohesive strength (undrained) A= C’c/ln 10 C’c = compression index of the remoulded

(2)

sample

Whilst this type of relationship is not pursued further here, it is important to appreciate that predictions of soil moisture can enter into equations of the form of (1) or (2) to make predictions of trafficability. The ultimate goal is to enter those predictions into analytical relationships of the form in Equation (2) which can then provide estimates of the engineering soil properties which are themselves the entry point to deterministic models of the form described by Baladi and Rohani (1979). It is not the role of this investigation to attempt a complete deterministic linkage from precipitation to trafficability. Rather, the more limited objective of this paper is to outline the deterministic soil moisture model which may be used in this context and to couple this to existing and proven empirical rating cone indices. This will facilitate a detailed examination of the sensitivity of trafficability forecasts to input environmental factors, as well as estimating actual cone index changes for specific soils over storms. Thus, although full development of the complete deterministic approach requires further work, it is argued that in the interim substantial insights can be gained by the results from a coupled soil-water deterministic model and empirical moistureetrafficability model. It is, however, considered important to place this work in the likely future development of research in this field. Whilst the ultimate model criterion for operational trafficability is undoubtedly that of parsimony, it is necessary initially to evaluate somewhat fuller analytical models to ensure the appropriate design of such forecasting procedures. This paper therefore seeks to explore the more analytical aspects in the hope that the resulting findings will provide a firmer basis for parsimonious model design.

242

Forecasting

Description

ojJroad

trajficuhilit!,

of the simulation

model

The geometric structure of the soil profile used was of finite depth (2.5 m) and divided into 10 compartments of different thickness (top 3 cells=0.2 m thick, cells 447, 0.25 m thick, and cells 8-20, 0.3 m thick). The role of water movement between compartments obeys Darcy’s Law in finite difference form, and consists of solving the vertical flow equation: (3) where

t= z= U= Y= k=

time depth volumetric moisture content matric suction hydraulic conductivity

The soil surface incorporated a detention capability, the maximum for which must be specified. When no precipitation occurs, evaporation takes place according to an energy balance method which is described in more detail below. Runoff takes place

0

0.1

0.2

0.3 (1, MOISTURE

Figure 1. Soil moisture-soil study.

suction

0.4 CONTENT,

0.5

0.6

CWCM’

relationships

used in the simulation

Malcolm G. Anderson

243

when the surface detention capacity is filled and rainfall intensity exceeds the infiltration rate. The average hydraulic conductivity for flow through the boundary between adjoining cells I and J is weighted according to this thickness K,, = ((KPT,) + (K,.T,))/(T,

(4)

+ T*)

where T = cell thickness The flux through the bottom boundary of the 10 cells was taken to be equal to the hydraulic conductivity of the bottom cell. The flux between each cell then followed darcy’s Law in discrete form: 41= (A - WW(TJ

where 4 is the total potential To make defined :

this

(5)

+ q)

basic

for the respective

framework

operational

cell three

elements

have

to be further

Soil retention functions The soil moisture-hydraulic conductivity and soil moisture-matric suction relationships have to be defined to allow modelling of the water flux under both saturated and unsaturated conditions. Either empirical curves can be used for this purpose, or a suction-moisture curve can be assumed and the hydraulic conductivity-soil moisture relationship derived. It was considered sensible to avoid any experimental errors in this area, and thus the K-0 curve was derived from the following relationship [Campbell (1974), Millington and Quirk (1959) and Jackson (1972) provide details of the theory, whilst Hillel (1977) exemplifies its use in a simulation study directed to ecological problems] : f Ki = K,(O@,)

((2j+l-2i)Y’J’)

j= im

(6)

jzi ((Y-

l)y;‘)

where K, corresponds to Bi; 8, and K, refers to the respective parameter values at saturation; and the summations are made over the j0 increments for which the calculation is made. Use of this procedure, Equation (6) thus allows physically consistent K-B curves to be obtained from given 0-V curves

Evaporation

estimating

procedures

Evaporation can either be assumed to be a sine function operational during daylight hours (e.g. Hillel 1977) or, if sufficient information is available, the energy balance method can be used to estimate evaporation. This latter procedure is much the more satisfactory for the purpose here, since the impact of the various input environmental factors on trafficability can be assessed individually. The basic energy balance used here is thus: R = H + LE + G (neglecting where

storage

terms)

R = the net radiation received by the surface H = the sensible heat transfer from air to surface

(7)

244

Forecusting

tr&uhilit_v

off+&

0

0.1

0.2 MOISTURE

0.3 CONTENT,

0.4

0.5

0.6

W/W

Figure 2. Hydraulic conductivity-soil moisture relationships used in the simulation study. These curves are physjcally consistent with the curves shown in Fig. 1 (Equation 6). LE= the heat used in converting liquid to vapour evaporation) G= the heat flux into (or out of) the ground

(L is the latent

heat, E

This equation can thus be solved to estimate E. A number of solutions are available which have been tested against measured data, for example Van Keulen (1975) and Ballick et aE. (1981). It is beyond the scope of this paper to detail the full mathematical basis of the Ballick et al. model which is utilized here. The input requirements are given in the section below, and the reader is referred to the referenced report for the mathematical derivations of the extensions to Equation (7) which are used. Soil strength-moisture

reluti~nship,~

Collins (1971) provides one of the most detailed empirical analyses of soil strength relationships to soil properties. Data collected from 95 test sites across the United

Malcolm G. Anderson

245

States formed the basis of the study. This study derived relationships for the cone index and RCI, but it is the latter that is of greatest interest in terms of the VCI (Table 1) and trafficability. Equation (1) obtained in the study reported by Collins, is the relationship used in this investigation. It has to be stated that, despite the widespread sampling frame used in the study, at RCI values in excess of 300 the standard deviation of the estimate increases dramatically from a value in the approximate range 20-25 to a standard deviation in excess of 150. This is probably explained by the observation, already noted above, that the RCI is very sensitive to the soil moisture content. The principal justification for use of equation (1) in this study [as well as in the SMSP model of Smith and Meyer (1973) discussed above], apart from its comprehensive empirical base, is the fact that it remains acceptably accurate over the range of RCI values of principal interest, that is values < 150 (see Collins 1971: plate 30). In any case, different gearing of the RCI-soil moisture relationship will not change the relative effects on RCI of altering the input evaporation and soil parameters. However, it is considered here that the use of equation (1) currently provides the most satisfactory way of estimating the absolute effect that such changes may have on the RCI. It must be emphasized that the soil-water forecasting model coupled to the empirical RCI index (Equation 1) is an attempt to improve elements of the rainfall-soil moisture-strength-trafficability predictive capability. The current scheme must eventually be integrated with parallel investigations on shear strength-trafficability modelling, to which reference has already been made. Sensitivity

analysis of soil-water model components

Given the input data in Table 2 the appropriate simulation time increment was evaluated that was both conservative of computer time, but which also gave acceptably accurate results when compared to a simulation increment time of one second. The increment selected as most suitable was that of 10 seconds, effecting a net reduction of computer costs of a factor of 20, but retaining accuracy such that the soil-water estimates and the total column water balance had discrepancies less than 0.0001 per cent compared to a 1 s increment (a full report of this aspect is contained in Anderson 1982b). Figure 3 illustrates the standard data (Table 2) run over 48 hours, with the evaporation variables repeated for the second day. Also shown in this figure is the effect of an increased initial moisture content (0.50) on the RCI. Whilst the minimum RCI is unaffected (since the surface in both cases is saturated), the subsequent values portray a significant departure. Taking an RCI of 120 (see Table l), then with the standard moisture start value the minimum RCI is achieved at 1500 hours. With 8=0.50 the RCI value of 120 is achieved seven hours later at 2200. Altering the cloud cover to 0 throughout (clear sky) and also to 1 (complete cloud cover), and retaining all other data as Table 2, provides an assessment of this variable (Fig. 4). The essential fact of importance here is that the cloud cover changes have effect only after 30 hours (i.e. commencement of daylight on the day following the precipitation). This emphasizes the fact that the increase in RCI from the minimum value has two elements: an initial relatively shallow rise in RCI dominated by the soil hydrologic characteristics, and a following steeper rise not now dominated by soil dranage, but by evaporation controls. This phenomenon was apparent for all trials undertaken, with the two elements becoming more distinct as the initial water content was reduced, as Fig. 3 illustrates. The particularly important role of evaporation can be illustrated in the case where

246

Forecusting o@road tr@icabiiitJ Table 2. ‘Standard’

input data for the simulation

(a) Site and storm puru~eters Precipitation: 80 mm

of RCI

Duration 090&1200 hours Latitude 49” Slope Orientation 0” (South =O”)

Julian Calendar day: 265 Slope angle: 12 degrees Met. instrument height: 175 cm Atmospheric pressure: 9726 mb Cloud type: 3 (altocumulusf” (b) Meteorological variables

Time (h)

Air temperature (“C) Relative humidity (%) Cloud cover (Cl) Ground temperature Wind speed (m s-l)

(“C)

0000

0300

0600

0900

1200

1500

1800

2100

2400

7.0 76.4

6.9 76.4

7.1 81.8

9.2 81.8

IO.1 71.7

9.8 76.6

8.8 81.8

8.3 81.8

7.1 76.4

0.8 7-3 0.6

0.8 6.2 0.5

0.9 6.6 2.1

1.0 15.6 3.1

1.0 20.3 2%

1.0 17.9 2.9

1.0 10.6 2.9

0.9 9.3 2.1

0.8 7-3 0.6

(c) Soil properties

Initial water content (cm3 cmm3) Surface detention capacity (cm) Surface hydraulic conductivity (cm s- ‘) Soil-water retention curves (see Figs 1 and 2) Clay percentage (for equation 1) “Range l-8; l=cirrus;

Clay

Loam

Sand

0.38 1.0 2,E-6

O-18 1.0 2-E-5

0.16 1.0 l,E-4

49

17.66

3.5

8=Cog

the precipitation event is delayed until 1800 hours (Fig. 5). In the case of an 80 mm storm (cloud cover=O), the minimum RCI remains the same (see Fig. 4), but the time taken to achieve an RCI of 200 is some 19 hours after precipitation commences at 1800 hours (Fig. S), whilst with precipitation at 0900 it takes 25 hours before a similar RCI value is reached. Thus, for the clay soil analysed here, an early morning storm event is shown to be rather more restrictive in terms of trafficability. For totally overcast skies (cloud cover = l), however, the respective times to achieve RCI = 200 become close: 25 hours for precipitation starting at 1800 and 28 hours for a storm starting at 0900 (see Figs 4 and 5). Analysis of the loam soil (Figs 1 and 2 and Table 2) under similar precipitation, reveals a more rapidly changing trafficability capability as would be expected from soils of greater permeability (Fig. 6). The response of the loam is characterized by a more rapid initial rise in RCI (better drainage than the clay) and a much steeper rise in RCI occasioned by evaporation. This latter effect is due to the flux balance between evaporation and drainage in the surface simulation cell, as controlled by the

Figure 3. Simulations of RCI for clay soil with: (a) input data as shown in Table 2; (b) initial soil moisture content at 050.

CLAY

L!

5

10

15

20

25

nm+

Figure 4. Simulations

30

35

40

45

50

HOURS

of RCI for clay soil. All input data as Table 2 except cloud cover change to clear sky (0) and total cloud cover (1).

Forecasting ofiroad trajficahiliq

Figure 5. Simulation of RCI for clay soil. Input data changed with respect to Table 2: (a) time of precipitation (1800-2100 hours); (b) cloud cover-both 0 and 1 conditions simulated.

Figure 6. Simulations

of RCI for loam soil, with cloud cover conditions

0 and 1 simulated.

Malcolm

G. Anderson

249

appropriate entry points on the soil retention curves. In all trials of the loam soil these higher rates of change of RCI, with respect to the clay, were typical. From Table 2 it can be seen that there are too many variables to be presented here for which both the individual sensitivity and combined sensitivity (with other variables) can be analysed. Accordingly, Table 3 summarizes the sensitivity of the model to changes in the variables listed, taken singly. One point worthy of emphasis here is that initial moisture content can be interpreted as a sensitive parameter initially, but after precipitation its relative sensitivity diminishes in all cases, and its subsequent effects can be less pronounced than cloud cover, for example. With this exception (together with that of evaporation rendering the time of the storm occurrence an important parameter as discussed) the other evaporation elements have a relatively minor impact on the model sensitivity. By contrast, the model is of course very sensitive to soil type, as defined by soil retention curves. The sensitivity plots for the sand are not presented here since, in all cases, the rise to RCI > 200 was only three hours after the cessation of precipitation, and it is only values of RCI somewhat less than this value that pose a traf~cability restriction (Table 1). The prediction of the minimum

RCI

The sensitivity analysis emphasized the differences in RCI recovery rates following the cessation of precipitation. However, of specific interest also is the type of vehicle that can operate successfully (as defined by the RCI in Table 1) throughout the entire storm event. There is thus a requirement to predict the minimum RCI for given conditions. In this context, initial soil conditions and total storm precipitation were considered the most useful parameters to change in explorations of the respective changes in minimum RCI values (see Table 3). From a practical forecasting standpoint, it is particularly appropriate that the most sensitive elements of the model are those which are among the most readily definable (we have already observed that the model has the capacity of collapsing the evaporation subroutine to a simple sine function for which only the daytime maximum rate need be specified). The data given in Table 2 were run with changes made to the storm precipitation and initial soil-water conditions. In all, 28 simulations were undertaken for each of the three soils, and the minimum resulting RCI values are shown contoured in Figs 7-9. Interpolation of the simulation results in these figures were undertaken Table 3. Sensitivity of variables in trafficability prediction model Very sensitive

Moderate sensitivity

Insensitive

Start MC” Precipitation Soil type Time of precipitationb

Cloud cover Latitude Air temperature Ground temperature

Wind speed Relative humidity Atmospheric pressure Soil detention capacity Surface slope Surface ~rmeability

’ Moderate sensitivity following initial drainage. b Evaporation can be very significant when storm occurs several daylight hours after simulation start time

250

Forecasting

qflhad

trafficahilit~~

Figure 7. Minimum RCI values differing associated with Drecioitation and initial soil&ate; conditions for clay soil (other input data as Table 2). The minimum RCI simulated was 96.9. 0, simulations undertaken; --loOP, contours of RCI.

3 HOUR

PRECIPITATION,

MM

0

20

40

60

o-

r

I

I

60 I

-100

.

.

.

-900

Figure 8. Minimum RCI values simulated for loam soil (see Table 2 for input data). 0, undertaken ; simulations -50 -, contours of RCI.

3 HOUR 0 0

-100

20

PRECIPITATION, 40

MM 60

60

Malcolm 3 HOUR 0

20

PRECIPITATION. 40

MM Ml

80

G. Anderson

251

Figure 9. Minimum RCI values simulated for sand soil (see Table 2 for input data). 0,

simulations undertaken ; contours of RCI.

-1CKh,

-700

-800

using the method of splines incorporating a 24th-order polynomial to smooth the estimates. Initial soil-water potential is used to specify the start soil-water conditions. This procedure was adopted only because it was felt that, from a practical standpoint, this variable is much more easily determined in the field (with Soil Test ‘quick draw’ tensiometers for example) than is volumetric moisture content. Figures 7-9 illustrate a commonality of precipitation-initial soil-water status controls on minimum RCI. For initial soil-water potentials less than approximately -300 cm, the storm precipitation and initial soil-water status are both seen to affect the RCI values. In excess of approximately - 300 cm, by contrast, the storm precipitation is of greater significance and provides the principal discrimination. This general pattern is shown to apply to all three soils. The mechanism for this shift of minimum RCI controls as shown in Figs 7-9 is that, for low initial suctions, increased precipitation above the minimum required for saturation cannot further affect the RCI. For correspondingly high initial suctions, surface saturation is not achieved with the three-hour duration storms simulated here; hence for these cases the RCI is predominantly a function of precipitation throughout the full range of precipitation used in these simulations. It is further to be noted that all three figures exhibit a marked degree of similitude. This is related to the soil retention curves (Figs 1 and 2) and to the extent to which they are considered, or can be shown to be, affine. This aspect is explored much more fully in a general context by Su and Brooks (1976) and Anderson (1982b), but it is important to note the similitude of response in the three soil types, since this intrinsic mathematical relationship, if further developed, may be able more simply to relate soil type to trafficability performance. It is noteworthy that the 100 RCI contours (good vehicle capability-Table 1) for both the clay and the loam are associated with almost identical precipitation-initial

252

Forecasting

@-road

rrqfficuhilit~~

soil-water conditions. Either side of this curve, the rate of decrease (increasing precipitation) and increase (decreasing precipitation) in RCI with rainfall is greater in the case of the loam than the clay soil (Figs 7 and 8). The same precipitation volumes were also run with durations of 12 and 24 hours. An exactly similar pattern of response to that for three-hour duration storms was noted. These lower intensity storms yielded generally higher RCI values (clay, loam and sand minima were 167. 142 and 136, respectively, to be compared to Figs 7-9). Since the intensity of 80 mm in three hours was sufficient to induce saturation in all soifs (with low initial soil-water potential) then it is these worst traf~cability conditions that are of special interest. Figures 7--9 were selected for presentation for this reason. Discussion The prediction of trafficability in soils touches on a broad spectrum of environmental variables, from cloud cover and solar insolation to soil hydrologic properties and soil shear strength reIationships. This paper has sought to affirm the need to articulate this prediction process by the use of deterministic models. Evidence has been provided in the sensitivity analysis undertaken to show the sometimes marked and complex interaction that exists between evaporation and soil moisture, and the RCI. The complex temporal dependencies of RCI upon such factors indicate the restricted capability of empirical approaches. Current deterministic modelling efforts in the areas linking moisture to engineering soil properties and trafficability (reviewed above) should be able to be combined with the soil-moisture model outlined here. In the meantime, the use of the empirically derived and statistically significant equation (1) provides perhaps the soundest method of estimating RCI. When combined with the soil-moisture model outlined, this provides the means of predicting RCI for a wide range of environmental conditions affecting off-road trafficability. In general terms, a potentially important similitude of response has been illustrated (Figs 7-9) and this deserves further attention. In specific terms the importance of cloud cover, and the temporally varying importance of initial moisture conditions have been demonstrated. Acknowledgements The work Experiment

reported Station,

in this paper was undertaken Vicksburg, Mississippi, USA.

at the US Army

Waterways

References Anderson, M. G. (1982a) Modelling hillslope soil water status during drainage. Trunsacrions Institute

of British

Geographers

Anderson, M. G. (1982b) Modelling

7, 337-353. hillslope soil water movement.

Final report on US Army

Corps of Engineers Contract DAE~O-78-G-1~4. Anderson, M. G. and Kneale, P. E. (1980) Topography and hillslope water relationships in a catchment of low relief. Journal ofHydrology 47, 115-128. Anderson, M. G. and Kneale, P. E. (1982) The influence of low angled topography on hilislope soil water convergence and stream discharge. Journal of Hydrobgy 57, 65-80.

Baladi, G. Y. and Rohani, B. (1979) A terrain-uehic~e

~nferactio~ model ,for analysis qf steering

Malcolm G. Anderson

253

performunce uf~r~c~-luying vehicles. US Army Waterways Ex~riment Station, Technical Report GL-79-6. Ballick, L. K., Link, L. E. and Scoggins, R. K. (1981) Thermal modeling of terrain surface elements. Vicksburg, MS: US Army Waterways Experiment Station, Technical Report EL-8 1-2. Biswas, T. D., Nielsen, D. R. and Bigger, J. W. (1966) Redistribution of soil water after infiltration. Water Resources Research 2, 5 133524. Campbell, G. S. (1974) A simple method for determining unsaturated conductivity from moisture retention data. Soil Science If 7, 31 l-314. Collins, J. G. (1967a) A tentative soil strength prediction system. In Report on Conference on Soil Trafficuhility Prediction, Appendix C. Vicksburg, MS: US Army Waterways Experiment Station. Collins, J. G. (19678) Influence of water tables on soil moisture and soil strength. In Report on Conference on Soil Trufhcahility Prediction, Appendix D. Vicksburg, MS: US Army Waterways Experiment Station. Collins, J. G. (1971) Forecasting traf~cability of soils. In Technical Memorffndum 3-331, Report 10. Vicksburg, MS: US Army Waterways Experiment Station. International Hillel, D. (1977) Computer simulation soil-water dynamics. Ottawa: Development Research Center, Canada. Jackson, R. A. (1972) On the calculation of hydraulic conductivity. Soil Science Society of America Proceedings 36, 380-383. Millington, R. J. and Quirk, J. P. (1959) Permeability of porous media. Nature 183, 387-388. Molthan, H. D. (1967) Influence of soil variability on soil moisture and soil strength predictions. In Report on Con&rence on Soil Tru~~cubil~ty Prediction, Appendix E. Vicksburg, MS: US Army Waterways Experiment Station. Rohani, B. and Baladi, G. Y. (1981) Correhztion of mobility cone index with fundamental engineering properties ofsoil. Vicksburg, MS: US Army Waterways Experiment Station, Miscellaneous Paper SL-81-4. Schofield, A. N. and Wroth, C. P. (1968) Critical state soil rne~hun~~s.New York: McGrawHill. Smith, M. H. and Meyer, M. P. (1973) Automation of a model for predicting soil moisture and soil strength (SMSP model). Vicksburg, MS: US Army Waterways Experiment Station, Miscellaneous Paper M-73-l. Su, C. and Brooks, R. H. (1976) Hydraulic functions of soils from physical experiments and their applications. Corvallis: Oregon State University. Report WRRI-41. US Army (1967) Report on Conference on Soil Tra~~cab~lity Prediction. Vicksburg, MS: Waterways Experiment Station. Van Keulen, H. (1975) Simulation of water use and herhage growth in acid regions. Wageningen: Elsevier. Weyman, D. R. (1973) Measurements of the downslope flow of water in a soil. ~ournuZ of Hydrology 20, 267-288. Wroth, C. P. and Wood, D. M. (1978) The correlation of index properties with some basic engineering properties of soils. Canadian Geotechnical Journal 15, 137-145. (Revised manuscript received 15 November 19B.2)