The National Academies Press

Currently Skimming:

Pages 168-279

The Chapter Skim interface presents what we've algorithmically identified as the most significant single chunk of text within every page in the chapter.
Select key terms on the right to highlight them within pages of the chapter.

From page 168... ... 168 The main goal of this research was to develop equations correlating soil erosion parameters defined in previous chapters (i.e., EC, Ev, Et, vc, and tc) with the common soil engineering properties. Read the entire page →
From page 169... ... Development of Correlation Equations 169 the basis of the USCS classification for fine-grained soils, and it is reasonable to think that the USCS categories therefore have good potential for distinguishing between erosion categories for fine-grained soils as well. As discussed in the previous chapters, nearly 330 EFA test results compiled in the NCHRPErosion spreadsheet were divided into different USCS classification groups. Read the entire page →
From page 170... ... 170 Relationship Between Erodibility and Properties of Soils Figure 131. Velocity–erosion rate and shear stress– erosion rate plots for fat clay (CH) Read the entire page →
From page 171... ... Development of Correlation Equations 171 Figure 133. Velocity–erosion rate and shear stress– erosion rate plots for poorly graded gravel (GP) Read the entire page →
From page 172... ... 172 Relationship Between Erodibility and Properties of Soils Figure 135. Velocity–erosion rate and shear stress– erosion rate plots for high-plasticity silt (MH) Read the entire page →
From page 173... ... Development of Correlation Equations 173 Figure 137. Velocity–erosion rate and shear stress– erosion rate plots for low-plasticity silty clay (ML-CL) Read the entire page →
From page 174... ... 174 Relationship Between Erodibility and Properties of Soils Figure 139. Velocity–erosion rate and shear stress– erosion rate plots for clayey silty sand (SC-SM) Read the entire page →
From page 175... ... Development of Correlation Equations 175 Figure 141. Velocity–erosion rate and shear stress– erosion rate plots for poorly graded sand (SP) Read the entire page →
From page 176... ... Figure 143. Velocity–erosion rate and shear stress– erosion rate plots for poorly graded sand with silt (SP-SM) Read the entire page →
From page 177... ... Development of Correlation Equations 177 Figure 145. Erosion category charts with USCS symbols. Read the entire page →
From page 178... ... 178 Relationship Between Erodibility and Properties of Soils 7.2 Plots of Critical Velocity and Shear Stress Versus Mean Particle Size Briaud et al. Read the entire page →
From page 179... ... Development of Correlation Equations 179 U.S. Army Corps of Engineers Figure 147. Read the entire page →
From page 180... ... 180 Relationship Between Erodibility and Properties of Soils This section is dedicated to the step-by-step process of frequentist regression. This approach was implemented in three major steps: 1. Read the entire page →
From page 181... ... Development of Correlation Equations 181 3. TAMU/Coarse data set, 4. Read the entire page →
From page 182... ... EC Ev (mm-s/ m-h) Read the entire page →
From page 183... ... EC Ev (mm-s/ m-h) Read the entire page →
From page 184... ... EC Ev (mm-s/ m-h) Read the entire page →
From page 185... ... EC Ev (mm-s/ m-h) Read the entire page →
From page 186... ... EC Ev (mm-s/ m-h) Read the entire page →
From page 187... ... EC Ev (mm-s/ m-h) Read the entire page →
From page 188... ... 188 Relationship Between Erodibility and Properties of Soils Figure 149. PDF, ECDF, and histogram plots for critical velocity in TAMU/Global data set. Read the entire page →
From page 189... ... Development of Correlation Equations 189 Figure 150. PDF, ECDF, and histogram plots for critical velocity in the TAMU/Coarse data set. Read the entire page →
From page 190... ... 190 Relationship Between Erodibility and Properties of Soils Figure 151. PDF, ECDF, and histogram plots for critical velocity in the TAMU/Fine data set. Read the entire page →
From page 191... ... Development of Correlation Equations 191 Similar figures are included in Appendix 3 for all five erodibility parameters as well as the 12 major geotechnical properties for each subgroup (see Figure 148) Read the entire page →
From page 192... ... 192 Relationship Between Erodibility and Properties of Soils 7.3.2 Second Order Statistical Analysis The second order statistical analysis dealt with constructing correlation matrices for the different parameters defined in each group and discussed in the previous step. Before the regression relationships could be generated, it was necessary to learn about any potential interparameter relationships between the model variables (geotechnical parameters) Read the entire page →
From page 193... ... Figure 152. Correlation matrix for EFA/Fine data. Read the entire page →
From page 194... ... Figure 152. (Continued) Read the entire page →
From page 195... ... Development of Correlation Equations 195 7.3.3.2 Measures of Statistical Significance The best models were selected after passing through a four-filter process: Filter 1. R2, Filter 2. Read the entire page →
From page 196... ... 196 Relationship Between Erodibility and Properties of Soils where SSE(reg) = sum of square errors for the regression values, SSE(res) Read the entire page →
From page 197... ... Development of Correlation Equations 197 These equations were obtained through a deterministic statistical analysis, meaning that they reflect the predicted value as a single number, with no quantification of the possible error associated with predicting the future event. One of the major goals of this study was to provide the engineer with a reliability-based approach to also assess whether the predicted values are conservative enough for the purpose of engineering design. Read the entire page →
From page 198... ... 198 Relationship Between Erodibility and Properties of Soils associated with q = 1.0 are shown as black open circles, and the data points associated with q = 0.6 are shown as red plus signs. The POU is then calculated by counting the number of data points above the black solid line (the data points in which Zm > Zp) Read the entire page →
From page 199... ... Development of Correlation Equations 199 7.3.4.1 Critical Shear Stress The first filter was R2. The study began with the first subgroup, TAMU/Global (see Figure 148) Read the entire page →
From page 200... ... 200 Relationship Between Erodibility and Properties of Soils 135 combination groups in the EFA/Fine data set. Figure 158 and Figure 159 show the results for R2 for each combination group for the linear and power models, respectively. Read the entire page →
From page 201... ... Development of Correlation Equations 201 Figure 158. R2 results for the linear models in EFA/Fine data set: critical shear stress. Read the entire page →
From page 202... ... 202 Relationship Between Erodibility and Properties of Soils earlier, cross-validation is a technique used to assess the estimator performance by folding the data set into k random folds. The model is trained by using the k-1 fold and then validated by using the remaining folds. Read the entire page →
From page 203... ... Development of Correlation Equations 203 if the engineer desires a higher confidence level, say, near 90%, then he or she needs to multiply the predicted value by 0.6. Figure 163, however, shows that by using the Group 124 correlation equation (see Table 48) Read the entire page →
From page 204... ... 204 Relationship Between Erodibility and Properties of Soils Figure 164. Number of data points in each of 105 combination groups for the EFA/Coarse data set: critical shear stress. Read the entire page →
From page 205... ... Development of Correlation Equations 205 were very low as compared with the EFA/Fine data set. That is, many regression groups might meet the requirements of the first two filters, R2 and MSE, but only marginally pass through the F-test. Read the entire page →
From page 206... ... 206 Relationship Between Erodibility and Properties of Soils (with 0.3 Pa offset) Read the entire page →
From page 207... ... Development of Correlation Equations 207 Group No. Independent Variables Model Expression R2 MSE F-value/ F-stat CrossValidation Score 34 PI, VST, PF, 0.99 0.191 140 –15.44 44 PI, , VST, PF, 0.99 0.198 88 –1.4 46 PI, WC, VST, PF, 0.99 0.197 70.5 0.08 47 PI, , WC, VST, PF, 0.99 0.169 56.1 0.09 51 Cc, , 0.93 1.045 36 0.98 52 Cc, WC, VST 0.98 0.178 38.6 0.1 54 Cc, WC, 0.93 1.043 36.2 0.98 58 Cc, , WC, VST 0.99 0.168 24.3 –0.5 60 Cc, , WC, 0.93 1.043 29.1 0.97 64 Cc, WC, VST, PF 0.98 0.164 25.8 –1.5 77 Cc, , 0.93 1.044 36.1 0.99 78 Cc, WC, VST 0.98 0.183 33.1 0.04 80 Cc, WC, 0.93 1.044 36.1 0.98 86 Cc, , WC, 0.93 1.044 29.1 0.96 88 Cc, , VST, 0.97 0.236 21 0.03 95 Cc, , WC, VST, 0.96 0.307 19 –11.3 101 Cc, Cu, PI 0.95 0.313 26.7 –0.44 104 Cc, Cu, VST 0.98 0.204 25.5 0.01 Table 50. Read the entire page →
From page 208... ... 208 Relationship Between Erodibility and Properties of Soils Figure 170. Number of data points in each of 135 combination groups for the JET/Global data set: critical shear stress. Read the entire page →
From page 209... ... Development of Correlation Equations 209 Figure 173. R2 results for the power models in JET/Global data set: critical shear stress. Read the entire page →
From page 210... ... 210 Relationship Between Erodibility and Properties of Soils After passing through Filters 1 and 2 (R2 and MSE) , the linear models associated with Groups 49, 76, 109, 112, and 113 in the JET/Global data set were selected for further analysis. Read the entire page →
From page 211... ... Development of Correlation Equations 211 HET Data Set. A similar approach was taken to select the best correlation equation for critical shear stress in the HET database. Read the entire page →
From page 212... ... 212 Relationship Between Erodibility and Properties of Soils Figure 178. Number of data points in each of 105 combination groups for the HET/Coarse data set: critical shear stress. Read the entire page →
From page 213... ... Development of Correlation Equations 213 behind the poor relationships for critical shear stress in the HET database was that the method used to calculate the erodibility parameters in the HET includes many crude judgements. More on the issues affecting the HET is discussed in Chapter 8. Read the entire page →
From page 214... ... 214 Relationship Between Erodibility and Properties of Soils Table 53. Selected linear models for critical shear stress in the HET/Global data set. Read the entire page →
From page 215... ... Development of Correlation Equations 215 7.3.4.2 Critical Velocity The same four-filter process discussed in Section 7.3.3 was implemented to obtain the best models for critical velocity. The first observation was that of the three erosion tests studied in this chapter (EFA, JET, HET) Read the entire page →
From page 216... ... 216 Relationship Between Erodibility and Properties of Soils Figure 185. R2 results for the linear models in EFA/Fine data set: critical velocity. Read the entire page →
From page 217... ... Development of Correlation Equations 217 After passing through Filters 1 and 2 (R2 and MSE) , the linear models associated with Groups 114, 124, 128, and 132 in the EFA/Fine data set were selected for further analysis. Read the entire page →
From page 218... ... 218 Relationship Between Erodibility and Properties of Soils Table 56. Selected power models for critical velocity in the EFA/Fine data set. Read the entire page →
From page 219... ... Development of Correlation Equations 219 Figure 190. Number of data points in each of 105 combination groups for the EFA/Coarse data set: critical velocity. Read the entire page →
From page 220... ... 220 Relationship Between Erodibility and Properties of Soils Both the linear and power models showed a few good groups in terms of R2 values; however, as shown in Figure 190, the number of data points in most groups was lower as compared with the EFA/Fine data set. Figure 193 and Figure 194 show the values of MSE for the linear and power models, respectively. Read the entire page →
From page 221... ... Development of Correlation Equations 221 Table 57. Selected linear models for critical velocity in the EFA/Coarse data set. Read the entire page →
From page 222... ... 222 Relationship Between Erodibility and Properties of Soils 7.3.4.3 Initial Slope of Erosion Rate–Shear Stress The same four-filter process discussed in Section 7.3.3 is implemented in this section to obtain the best models for the initial slope of the erosion rate–shear stress curve, Et. Regression analysis for Et was performed for the EFA, HET, and JET separately. Read the entire page →
From page 223... ... Development of Correlation Equations 223 Figure 198. R2 results for the power models in EFA/Fine data set: Es. Read the entire page →
From page 224... ... 224 Relationship Between Erodibility and Properties of Soils selected power models after the requirements of the first three filters (R2, MSE, and F-value/ F-stat) were met. Read the entire page →
From page 225... ... Development of Correlation Equations 225 Figure 202. R2 results for the linear models in EFA/Coarse data set: Es. Read the entire page →
From page 226... ... 226 Relationship Between Erodibility and Properties of Soils After passing through Filters 1 and 2 (R2 and MSE) , linear models associated with Groups 38, 42, 43, and 47 were selected for further analysis. Read the entire page →
From page 227... ... Development of Correlation Equations 227 Eτ Table 61. Selected power models for shear stress slope in the EFA/Coarse data set. Read the entire page →
From page 228... ... 228 Relationship Between Erodibility and Properties of Soils Figure 207. Number of data points in each of 135 combination groups for the JET/Fine data set: Es. Read the entire page →
From page 229... ... Development of Correlation Equations 229 The R2 values of the power models were generally higher than those of the linear models. Figure 210 and Figure 211 show the MSE results for each of the 135 combination groups in the EFA/Fine data set for the linear and power models, respectively. Read the entire page →
From page 230... ... 230 Relationship Between Erodibility and Properties of Soils Table 62. Selected linear models for shear stress slope in the JET/Fine data set. Read the entire page →
From page 231... ... Development of Correlation Equations 231 groups in the EFA/Coarse data set. Figure 214 and Figure 215 show the results for R2 for each combination group for the linear and power models, respectively. Read the entire page →
From page 232... ... 232 Relationship Between Erodibility and Properties of Soils Figure 214. R2 results for the linear models in JET/Coarse data set: Es. Read the entire page →
From page 233... ... Development of Correlation Equations 233 form was selected as the most promising equation. Figure 216 shows the plot of POO versus q for this model. Read the entire page →
From page 234... ... 234 Relationship Between Erodibility and Properties of Soils Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 66 shows the results of the selected power models after the requirements of the first three filters (R2, MSE, and F-value/F-stat) Read the entire page →
From page 235... ... Development of Correlation Equations 235 Table 66. Selected power models for shear stress slope in the HET/Coarse data set. Read the entire page →
From page 236... ... 236 Relationship Between Erodibility and Properties of Soils Figure 222. R2 results for the linear models in HET/Global data set: Es. Read the entire page →
From page 237... ... Development of Correlation Equations 237 Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 67 shows the results of the selected linear models after the requirements of the first three filters (R2, MSE, and F-value/F-stat) Read the entire page →
From page 238... ... 238 Relationship Between Erodibility and Properties of Soils Table 67. (Continued) Read the entire page →
From page 239... ... Development of Correlation Equations 239 7.3.4.4 Initial Slope of Erosion Rate–Velocity The same four-filter process discussed in Section 7.3.3 is implemented in this section to obtain the best models for Ev. Similar to the case of critical velocity, vc, among the three erosion tests studied in this chapter (EFA, JET, HET) Read the entire page →
From page 240... ... 240 Relationship Between Erodibility and Properties of Soils Results show that power models in general perform better than the linear models. R2 values of Groups 109 to 135 are generally higher than the rest for both power models. Read the entire page →
From page 241... ... Development of Correlation Equations 241 Figure 230. MSE results for the linear models in EFA/Fine data set: Ev. Read the entire page →
From page 242... ... 242 Relationship Between Erodibility and Properties of Soils Figure 232. Plot of POO versus correction factor for Group 126 (power) Read the entire page →
From page 243... ... Development of Correlation Equations 243 Both the linear and power models showed some good groups in terms of R2 values; however, as shown in Figure 233, the number of data points in most groups was very low. After passing through Filters 1 and 2 (R2 and MSE) Read the entire page →
From page 244... ... 244 Relationship Between Erodibility and Properties of Soils Table 71. Selected power models for velocity slope in the EFA/Coarse data set. Read the entire page →
From page 245... ... Development of Correlation Equations 245 EFA Database. It was observed that dividing the EFA/Global data set into the EFA/Fine and the EFA/Coarse data sets would improve the regression results. Read the entire page →
From page 246... ... 246 Relationship Between Erodibility and Properties of Soils Figure 239. R2 results for the power models in EFA/Fine data set: EC. Read the entire page →
From page 247... ... Development of Correlation Equations 247 Table 72. Selected linear models for erosion category in the EFA/Fine data set. Read the entire page →
From page 248... ... 248 Relationship Between Erodibility and Properties of Soils versus q for this model. The vertical axis in Figure 242 shows the probability that, when the selected model is used, the predicted EC will be smaller than the actual EC, in percentage. Read the entire page →
From page 249... ... Development of Correlation Equations 249 Figure 245. R2 results for the power models in EFA/Coarse data set: EC. Read the entire page →
From page 250... ... 250 Relationship Between Erodibility and Properties of Soils requirements of the first three filters were met. The Group 91 correlation equation in power form was selected as the most promising equation. Read the entire page →
From page 251... ... Development of Correlation Equations 251 JET Data Set. A similar approach was taken to select the best correlation equation for EC in the JET database. Read the entire page →
From page 252... ... 252 Relationship Between Erodibility and Properties of Soils Filter 3, F-value/F-stat, was determined for each group mentioned above. Table 76 shows the results of the selected linear models after the requirements of the first three filters (R2, MSE, and F-value/F-stat) Read the entire page →
From page 253... ... Development of Correlation Equations 253 linear form was selected as the most promising equation. Figure 250 shows the plot of POU versus q for this model. Read the entire page →
From page 254... ... 254 Relationship Between Erodibility and Properties of Soils Figure 251. Number of data points in each of 135 combination groups for the HET/Fine data set: EC. Read the entire page →
From page 255... ... Development of Correlation Equations 255 After passing through Filters 1 and 2 (R2 and MSE) , the linear models associated with Groups 48, 53, 74, 79, 100, 109, 112, 114, and 129 were selected for further analysis. Read the entire page →
From page 256... ... 256 Relationship Between Erodibility and Properties of Soils Table 78. Selected linear models for erosion category in the HET/Fine data set. Read the entire page →
From page 257... ... Development of Correlation Equations 257 Figure 256. Plot of POU versus correction factor for Group 12 (power) Read the entire page →
From page 258... ... 258 Relationship Between Erodibility and Properties of Soils Both the linear and power models showed some good groups in terms of R2 values; however, as shown in Figure 257, the number of data points was very low in most combination groups. In fact, close to half of the combination groups did not have any data points. Read the entire page →
From page 259... ... Development of Correlation Equations 259 Table 80. Selected linear models for erosion category in the HET/Coarse data set. Read the entire page →
From page 260... ... 260 Relationship Between Erodibility and Properties of Soils Table 81. Selected power models for erosion category in the HET/Coarse data set. Read the entire page →
From page 261... ... Development of Correlation Equations 261 characterization of all possible solutions of the model parameters and their relative probabilities while simultaneously providing a systematic and transparent approach to assess the performance of the proposed regression model. The major difference between Bayesian analysis and deterministic frequentist regression is in how each approach interprets the observed data and model parameters. Read the entire page →
From page 262... ... 262 Relationship Between Erodibility and Properties of Soils Understanding this varying evidence condition motivates a specialized calibration that points to systematic and transparent assessment of prediction confidence based on all available evidence. 7.4.2 Hypotheses The foregoing regression analysis selected several groups of experimental observations and corresponding statistical models that showed satisfying model/observation comparisons. Read the entire page →
From page 263... ... Development of Correlation Equations 263 where q̂ denotes mean of parameters and Δq represents the uncertainty component. Other than the deterministic methodology, the probabilistic calibration allows for an exhaustive exploration of all potential combinations of the model parameters that best resemble the experimental observations. Read the entire page →
From page 264... ... 264 Relationship Between Erodibility and Properties of Soils an = number of data points. bParameter values given by deterministic regression. Read the entire page →
From page 265... ... Development of Correlation Equations 265 goodness of fit for each erodibility variable (i.e., critical shear stress, tc; critical velocity, vc; erosion category, EC; velocity slope, Ev; and shear stress slope, Et, respectively) Read the entire page →
From page 266... ... 266 Relationship Between Erodibility and Properties of Soils (a) Read the entire page →
From page 267... ... Development of Correlation Equations 267 distribution, which indicates also the unbiased character of the proposed model. However, the change of bin size and starting position of the histogram may result in a variation to its shape. Read the entire page →
From page 268... ... 268 Relationship Between Erodibility and Properties of Soils (a) Read the entire page →
From page 269... ... Development of Correlation Equations 269 (a) Read the entire page →
From page 270... ... 270 Relationship Between Erodibility and Properties of Soils (a) Read the entire page →
From page 271... ... Figure 266. Joint relative frequency histogram of model parameters, two at a time. Read the entire page →
From page 272... ... 272 Relationship Between Erodibility and Properties of Soils To achieve this goal, composition of a multidimensional mesh grid retrieved across all independent variables is required, enabling the assessment of the model prediction along each domain of all independent variables versus the dependent variable or erodibility parameter. Each domain range is decided by minimum and maximum values retrieved from experimental observations; additionally, 10 uniform discretization steps were chosen for each variable to ensure computational efficiency and sufficiency of the inferences at the same time. Read the entire page →
From page 273... ... Development of Correlation Equations 273 equal aspect ratio context, where results lying along the 45° line indicate a perfect fit. Similar to the 1,000 posterior ensemble of realizations, this figure shows 1,000 model predictions at the same location of the available experimental EC observations. Read the entire page →
From page 274... ... 274 Relationship Between Erodibility and Properties of Soils (c) Read the entire page →
From page 275... ... Development of Correlation Equations 275 are presented for this model, as convergence of the MCMC posterior sample was also achieved. The matrix of plots presenting the posterior's joint relative frequency histograms of this model's parameters is introduced in Figure 271. Read the entire page →
From page 276... ... Figure 271. Joint relative frequency histogram of model parameters, two at a time. Read the entire page →
From page 277... ... Group No. Independent Variables Data Seta Model Expression (parameter values given by deterministic regression) Read the entire page →
From page 278... ... 278 Relationship Between Erodibility and Properties of Soils (c) Read the entire page →
From page 279... ... Development of Correlation Equations 279 (a) Read the entire page →

From page 168...

... 168 The main goal of this research was to develop equations correlating soil erosion parameters defined in previous chapters (i.e., EC, Ev, Et, vc, and tc) with the common soil engineering properties.