Statistics describing ground surface slope have geomorphic, hydrologic, engineering, and military applications, because slope steepness influences rates of runoff, soil creep, and soil flowage and the ease of cross-country movement of men and vehicles. Slope maps can be drawn to show the areal distribution of (1) degree of slope and (2) magnitude of the downslope component of gravitational acceleration. The first uses lines of equal slope tangent (isotangents) and is therefore a form of first-derivative map. Isotangents are drawn from a large number of slope readings determined from a large-scale contour map or by field measurement with surveying instruments. Planimeter measurement of areas between successive isotangents yields a percentage-frequency distribution from which slope-tangent mean, variance, and standard deviation are computed as population parameters.

The second type of map uses lines of equal sine of slope (isosines) and shows the distribution of the downslope component of gravitational acceleration, which determines the intensity of shear stresses acting upon soil, surface water, and any surface objects. Measurement of areas between successive isosines yields a frequency distribution from which sine mean, variance, and standard deviation are computed as population parameters.

Isotangent and isosine maps were drawn and associated frequency distributions computed for drainage basins in three areas unlike in size and stage of erosional development: (1) a small, youthful basin in weak clays of badlands at Perth Amboy, New Jersey; (2) two mature basins strongly controlled by massive sandstones of the Catskill plateau; (3) two basins with smooth slopes in the vicinity of Emporium, Pennsylvania, where Carboniferous strata are maturely dissected. Frequency distribution of sine of slope is strongly skewed and has a high mean in the Perth Amboy locality because of immaturity of stage of basin development. It is bimodal in the Catskill locality, where structural benching is pronounced, but is smoothly symmetrical with almost normal form in the Emporium locality, which may be representative of typical mature erosional topography developed on a homogeneous rock mass.

Because of time and labor needed to prepare slope maps and measure frequency distributions by planimeter, two methods of rapid point sampling of slope were tested for reliability: (1) location of sampling points by drawing random pairs of rectangular coordinates; (2) use of sampling points uniformly spaced on a rectangular grid. A test area of about 1 square mile was selected near the center of the Emporium, Pennsylvania, topographic quadrangle. Population parameters of tangent of slope were computed by planimeter measurements of a detailed isotangent map of the test square. Samples of 100 points each were then taken directly from the topographic map of the test square by both random co-ordinate and grid methods and compared with the slope population. In tests using the statistic χ2/d.f. and the statistic z the hypotheses that random co-ordinate sample and population variances are equal and that sample and population means are equal were accepted at the 5 per cent level. Confidence limits based on population variance were determined for distribution of means for samples of various sizes. A chi-square test of goodness of fit required acceptance of the hypothesis that random-coordinate sample distribution is similar to the population distribution.

Operator variance was tested by means of a t-test of paired differences in tangent values read by two operators at the same points. A low mean of differences, not significantly different from zero, established that variations introduced by operator judgment are small compared with natural slope differences. It is concluded that the random-coordinate method of point sampling can be applied to determine slope mean, variance, standard deviation, and general form of slope distribution within predictable small limits.

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