Sensitivity analysis produced a prioritized list of recommendations for improving the pavement distress/smoothness forecasting models used by the PHT analysis tool. This tasks objective of was to implement the improvements to enhance the PHT forecasting models to make them compatible with MEPDG version 1 and the AASHTO MEPDG manual of practice.

Along with updating the forecasting models, this task objective also included developing a reliability index that can be applied to the forecast RSL estimate to enable policy makers and engineers to model various uncertainties into the analysis. Pavement condition forecasting tools depend on many major assumptions such as climate, traffic growth, etc. Practically every input associated with forecasting the future pavement condition is variable in nature. The combined effect of variability in key inputs used for forecasting future pavement condition is variability and uncertainty in forecasted pavement condition and remaining service life. It is common to incorporate reliability into pavement condition forecasting tools for pavement design and management so as to consider these uncertainties and variations.

Since the completion of version 0.8 of the MEPDG software in 2004, the NCHRP has initiated and conducted several research projects to review the product, recommend improvements, and implement the recommended improvements. This led to considerable change in the MEPDG. Examples of key changes are as follows:

- Improvement in pavement analysis algorithms (errors and deficiencies found in the original computational algorithms have been identified and corrected).
- Improvement to climate models used to model temperature and moisture profiles within the pavement structure (the original climate models have been enhanced based on work conducted at Arizona State University (ASU)).
- Recalibration of all MEPDG models in 2007 with update LTPP project information (additional materials test data, up to 8 years of additional performance and traffic data).Identification and correction of systematic error in PCC CTE data and recalibration of all rigid pavement performance prediction models in 2011.

The flexible, rigid, and composite simplified pavement performance prediction models developed for the Pavement Health Track (PHT) Analysis Tool were done using MEPDG version 0.8 software (i.e., the models were calibrated using predicted distress/IRI from the MEPDG software). Since then significant changes has been done to the MEPDG. Under this FHWA contract, Battelle/ARA investigated the reasonableness of the PHT Analysis Tool pavement models (Task 2) and found the following:

- Observed trends in PHT Analysis Tool distress/IRI predictions are similar to trends in version 0.8 of the MEPDG. Thus inherent anomalies in version 0.8 of the MEPDG are apparent in PHT Analysis Tool distress/IRI predictions.
- Goodness of fit for the PHT Analysis Tool distress/IRI pavement performance prediction models was mostly inadequate.
- PHT Analysis Tool estimates of RSL without internal calibration were mostly biased.
- Internal PHT Analysis Tool calibration of models in general does reduce bias.

The general consensus from all of the observations and findings presented is that although the version 0.8 of the MEPDG and thus the PHT Analysis Tool pavement performance prediction models are a vast improvement on current pavement technology there was the need for further improvement in order to make it a practical and useable pavement condition forecasting tool.

Also, as pavement condition forecasting tools depend on many major assumptions such as climate, traffic growth, etc. and inputs for the PHT Tools were mostly guesstimates, there need for incorporating reliability into the PHT pavement condition forecasting methodology to account for this high level of reliability. Incorporating reliability allows policy makers and engineers to model various kinds of uncertainty into remaining service life estimates and analysis.

Calibration of the existing pavement models comprised of the following steps:

- Select pavement types of interest.
- Identify input data (source and data items).
- Assemble data and establish project database.
- Review assembled data for completeness and accuracy.
- Develop algorithms and parameters required for calibration.
- Calibrate models by maximizing goodness of fit and minimizing error between measured and predicted distress.
- Perform sensitivity analysis to determine calibrated models reasonableness. Modify model parameters as needed.
- Finalize new calibrated models.

All the four pavement types considered by the PHT tool were considered of interest and selected for models calibration.

- Bituminous Pavement.
- Jointed Plain Concrete Pavement (JPCP).
- Asphalt Concrete (AC) Overlay on Existing AC Pavement.
- Asphalt Concrete Overlay on Existing JPCP Pavement.

Data items of interest for developing the project calibration database are the HPMS2010 data items currently imported by the PHT analysis tool. No additional fields were required.

A total of 504 LTPP projects were assembled for establishing a project database for the calibration analysis. A description of the projects and key pavement features are described in the following paragraphs

The selected projects are well distributed within the continental U.S. The good geographical distribution implies that data assembled form these projects will collectively represent site, design, and construction practices throughout the U.S. given the calibrated models a national character.

A breakdown of the distribution of pavement type for the selected projects is listed below, which shows that each pavement type and base type of interest is well represented.

- Bituminous pavement:
- Conventional AC over granular base: 32
- Full-depth AC over asphalt treated base: 112
- Semi-rigid AC over cement treated base: 25

- Jointed plain concrete pavement: 155
- Asphalt concrete overlay over existing AC pavement: 104
- Asphalt concrete overlay over existing jointed concrete pavement:18

Highway functional class distribution is predominantly rural principal arterials. This is typical of pavements on the national highway system (NHS) and thus represents the type of pavements typical found in HPMS and State highway databases used for policy and asset management decision making.

New construction or AC overlay placement base year two-way average annual daily truck traffic and future volumes were used to characterize traffic for pavement forecasting. Historical truck AADT estimates was obtained from the LTPP and used to determine base year truck volumes and growth rates. The base year and growth rates were used to determine future truck AADT for 20 years after original construction or AC overlay placement. The truck AADT growth rate was determined by fitting a linear curve to the historical truck AADT data.

The selected projects have an adequate distribution among the four LTPP climate regions. The good climate distribution implies that data assembled from these projects will collectively represent climate conditions across the U.S.

- Dry, Freeze: 67
- Dry, Non-Freeze: 80
- Wet, Freeze: 201
- Wet, Non-Freeze: 135

The selected projects have an adequate distribution of projects among the two soil types. The good distribution implies that data assembled from these projects will collectively represent soil conditions across the U.S.

- Fine: 200
- Granular: 294

The design features of interest are the AC overlay thickness, existing surface layer thickness, and the base type. The selected projects have the following design features, illustrated in Figure 10.

- PCC thickness represents typical U.S. practice ranging from 7 to 13 inches
- AC overlays were thicker for existing AC pavements when compared to existing PCC.
- All typical base types were represented.

Measured distress (cracking, rutting, and faulting) along with smoothness (IRI) data was assembled for all the 504 selected projects.

The assembled data was reviewed for completeness and accuracy. Data review was done by computing the mean, standard deviation, and range statistics to identify outliers and developing distress/IRI versus age plots to determine reasonableness of trends. Key issues identified and resolved were outliers in data, inconsistencies with AC overlay thickness for new JPCP pavements, and atypical trends. The anomalies were resolved as needed.

The assembled data was used to compute the input variables and clusters required for forecasting pavement condition. The input variables and clusters were different for each pavement type and distress/IRI. A detailed description of the models input variables and clusters are summarized PHTv1.1 Forecasting Models Technical Information document.

Calibration comprised of the following steps:

- For all the projects, using the appropriate model inputs, execute models and equations and predict distress and smoothness. Distress/IRI predictions were done for a 40-year analysis period.
- Extracting relevant outputs, including inputs, clusters, predicted distress and IRI for each selected project.
- Reviewing the extracted data for accuracy and reasonableness.
- Matching the extracted predicted distress values with field-measured values and comparing the predicted distress with the measured values.
- Determine reasonableness of goodness of fit and bias.
- For models found to be inadequate, recalibrate prediction models as necessary to produce unbiased predictions.

Note that due to the nature of inputs, goodness of fit is not expected to be a good as the AASHTO MEPDG models. The goodness of it is expected to be as low as that reported for pavement management models (typically ranges from 10 to 30 percent). The goal thus was to meet this threshold of goodness of fit and eliminate bias between measured and predicted distress/IRI values.

The recalibrated pavement forecasting models were validated by performing a comprehensive sensitivity analyses. Sensitivity analysis comprised of the following steps:

- Develop typical "baseline " pavement sections for the four pavement types of interest.
- Determine key models inputs variables and the range of their typical values (e.g., PCC thickness ranges from 7 to 13 in).
- Predict distress/IRI for the baseline pavement sections and vary key inputs as needed within the range of typical values.
- Determine the impact of change in input variable values on predicted distress/IRI.
- Are changes in distress/IRI in accordance with engineering expectations (e.g., thicker PCC implies less distress).
- Is the magnitude of change reasonable (10 percent change in PCC thickness results in 5 to 15 percent change in distress/IRI within a 25 year analysis period).

- If the sensitivity analysis outcome is reasonable then the new recalibrated models are deemed as adequate. Otherwise, the models are modified as needed until an adequate outcome is obtained.

The final new improved recalibrated PHT Tool models was documented and incorporated into the existing PHT analysis software. This required modifications to the analysis engine dynamic link library (DLL) file as well as the PHT graphical user interface (GUI) application to support the improved models.

Practically every input associated with forecasting the future pavement condition is variable in nature. Perhaps the most obviously uncertain of all is future levels of truck traffic, material properties, and climate. Furthermore, pavements have been known to exhibit significant variation in condition along their length. The combined effect of variability in key inputs used for forecasting future pavement condition is variability and uncertainty in forecasted pavement condition/life as shown in Figure 11. Thus, it is common to incorporate reliability into pavement condition forecasting tools for pavement design and management so as to consider the uncertainties and variations in inputs when forecasting future pavement condition/life.

For pavement analysis, reliability has been described in many ways over the years.

- The probability that a pavement design will perform satisfactorily under prescribed traffic and environmental conditions over anticipated design period.
- The probability that a pavement system will perform its intended function over its design life (or time) and under the conditions (or environment) encountered during operation
- The probability that serviceability will be maintained at adequate levels from a user's point of view, throughout the design life of the facility

In the strictest sense reliability is defined as one minus the probability of failure:

*R* = 1 - *Pfailure* (1)

Traditionally pavement failure has been defined using serviceability loss (a subjective measure of pavement performance). The 1993 AASHTO Pavement Design Guide define reliability mathematically in terms of the number of predicted equivalent single axle loads to terminal serviceability (N) being less than the number of equivalent single axle loads actually applied (n) to the pavement.

The 1993 AASHTO Guide approach produced results that indicated that thicker pavements always increased design reliability. This assumption, however, is not always be true as several design features other than thickness (e.g., HMAC mixture design, dowels for jointed plain concrete pavements, and subgrade improvement for all pavement types) do influence reliability.

Thus for AASHTO's MEPDG, reliability was incorporated in a consistent and uniform fashion for all pavement types, allowing users to select a desired level of reliability for each distress type and smoothness. Design reliability was defined as the probability that each of the key distress types and smoothness will be less than a selected critical level over the analysis period (see equation 3).

*R = P*[Distress at Give Time during Design Period < Critical Distress Level] (3)

The diagram in Figure 12 illustrates the AASHTO MEPDG approach using a probability distribution for IRI. This diagram shows that the probability, R, that IRI is greater than its associated user-defined failure criteria can be computed over the entire analysis period.

Reliability was incorporated into MEPDG predicted distress/IRI as follows:

- Calibrate distress/IRI model using field measured distress/IRI data. Typically each distress/IRI model was calibrated using LTPP and other field performance data.
- Plot predicted distress/IRI (horizontal axis) versus residual error of prediction (i.e., difference of predicted distress/IRI and measured distress/IRI results for all sections used in calibration) (vertical axis). The residual error characterizes how the prediction model fails to properly explain the observed distress/IRI.
- Divide predicted distress/IRI into reasonably spaced increments and assume a distribution of residual error for each distresss/IRI. Typically a normal distribution is assumed.
- For each increment, estimate mean predicted distress/IRI and mean standard error of estimate for measured distress/IRI.
- Develop a mathematical relationship to predict distress/IRI standard error from mean predicted distress/IRI. The standard error is determined as a function of the predicted distress/IRI.
- An illustration for JPCP slab cracking and JPCP slab cracking at various reliability levels is shown in Figure 13 and Figure 14 respectively. Estimate cracking at the desired relability level using the following relationship:

*Distress / IRI = Distress / IRI _ mean + STDmeas × Zp* (4)

WHERE

*Distress/IRI _P* = Distress/IRI level corresponing to the reliability level p

*Distress/IRI _mean* = Distress/IRI predicted using the deterministic model with mean inputs (corresponing to 50 percent reliabality)

*STDmeas* = Standard deviation of distress/IRI corresponding to distress/IRI predicted using the deterministic model with mean inputs

*Zp* = Standardized normal deviate (mean 0 and standard deviation 1) corresponding to reliability level p.

The procedure described above was used for incorporating reliability/risk into PHT Tool pavement condition forecasting. This enables users to forecast pavement condition at a given reliability level/index and then estimate RSL based on that reliability level. The reliability index incorporated was a decimal value between 50 and 100 percent that describes the reliability percentage of the RSL forecast as reported by the PHT tool. Note that the existing PHT determines RSL at 50 percent reliability, mean value.

Measured alligator cracking data at different ages was available for most of the sections. The PHT Analysis Tool computed parameters that are inputs to the alligator cracking model were extracted for ages corresponding to field alligator cracking data measurements. The predicted alligator cracking was compared against the measured field data to compute the residual error for each age.

Plots of measured/predicted alligator cracking versus computed PHT Tool estimated fatigue damage was prepared and examined. Outliers were further examined for erroneous inputs and when found they were rerun in the PHT Tool. The updated data were then used to develop revised calibration coefficients and model that resulted in unbiased alligator cracking prediction. Unbiased prediction means that the model does not on average over all of the data over predict or under predict the measured data.

The new PHT Tool alligator cracking model developed from the S-shaped curve model relating cracking to fatigue damage for new bituminous pavements is as presented below:

WHERE

*ACRK* = predicted alligator cracking, percent lane area

*FDAM* = fatigue damage

*C0* = calibration coefficients = 0.115

*C1* = calibration coefficients = -1.25

The C0 and C1 coefficients were determined to minimize the prediction error of the model and reduce bias. The model was developed with 1095 data points and has an *R ^{2}* of 17.14 percent and an RMSE of 8.7 percent. The new model goodness of fit statistics indicates the level of accuracy typical for pavement management type models. A plot of predicted and measured cracking versus fatigue damage is shown in Figure 15 for the entire dataset used for model calibration development. This data plot illustrates the S-Shaped curve typically used in modeling cracking versus fatigue damage.

The new alligator cracking model was further evaluated for bias. Bias was defined as the consistent under or over estimation of cracking. Bias was determined by performing a statistical paired t-test to determine if measured and predicted alligator cracking was similar:

- Develop null and alternative hypothesis:
- Null hypothesis H0: PHT Tool cracking = LTPP measured cracking.
- Alternate hypothesis HA: PHT Tool cracking ≠ LTPP measured cracking.

- Perform statistical analysis to determine and evaluate test p-value.
- A rejection of the null hypothesis (p-value < 0.05) would imply cracking from the PHT Tool and measured LTPP cracking are from different populations. This indicates bias in PHT Tool alligator cracking estimates for the range of typical inputs used in analysis.

Note that a significance level, α, of 0.05 or 5 percent was assumed for all hypothesis testing. The outcome of the paired t-test was a p-value of 0.8587. The p-value showed that there was not significant bias in predicted alligator cracking.

The new bituminous pavement alligator cracking model can be used to predict future alligator cracking in AC overlay of existing AC pavements. The reflection of existing AC cracking for such pavements was considered as the HPMS and state PMS databases do not provide information on existing pavement past/historical distress and the extent of repairs done to the existing pavement prior to AC overlay placement. The nationally calibrated MEPDG alligator cracking reflection model was recommended. Default existing alligator cracking post repairs and AC overlay placement was determine using historical data from the LTPP database as shown below in Table 6.

Pavement Age at AC Overlay Placement, years | Alligator Cracking Post Repairs (ICRK), percent lane area |
---|---|

0 to 5 | 5 |

5 to 10 | 10 |

10 to 15 | 15 |

> 15 | 20 |

Risk associated with alligator cracking prediction or the reliability of the alligator cracking is defined as the one-tail confidence interval at a predefined reliability level around a given alligator cracking prediction. Specifically, the one-tailed confidence interval is as defined in equation 4. For this study, confidence interval was determined as follows:

- Use new PHT Tool alligator cracking model to estimate the distress (over typical range of cracking, i.e., 0 to 100 percent).
- Divided the typical range of the distress into subsets (e.g., 0 to 10, 10 to 20, etc.).
- For each subset of predicted alligator cracking, estimate the standard error of the mean prediction (i.e., standard deviation of measured - predicted distress (i.e., std. error of the estimate, SEE) for all individual data points that falls within the subset).
- Develop a relationship between the SEE and predicted alligator cracking.

The relationship developed was used in the PHT Tool to determine SEE for any predicted alligator cracking. SEE will be used to estimate predicted alligator cracking at any desired reliability level as shown in Table 7. The design reliability procedure described above requires the estimation of variability in the form of standard deviation at any given level of predicted alligator cracking. Predicted alligator cracking standard deviation was determined as follows:

- Divide predicted alligator cracking to five or more intervals.
- For each interval, determine mean predicted alligator cracking and standard error (i.e., standard variation of predicted - measured alligator cracking for all the predicted alligator cracking that falls within the given interval).
- Develop a non linear model to fit mean predicted alligator cracking and standard error for each interval.

Reliability Level P (One Sided Confidence Interval), percent | Standardized Normal Deviate (Mean 0 and Standard Deviation 1) Corresponding To Reliability Level P |
---|---|

75 | 0.674 |

80 | 0.842 |

85 | 1.036 |

90 | 1.282 |

95 | 1.645 |

The resulting standard error prediction model developed for the PHT Tool is presented below:

*Stderr(ACRK)* = 7.24 + (0.65 × *MPACRK*^{0.417}) (6)

WHERE

*Stderr(ACRK)* = cracking standard error of the estimate, percent

*ACRK* = predicted alligator cracking, percent lane area

*MPACRK* = mean predicted alligator cracking

The diagram in Figure 16 presents a plot of standard deviation versus predicted alligator cracking developed using the data presented in Table 8 which was obtained through analysis of predicted alligator cracking data. The region of predicted cracking that triggers maintenance and rehabilitation is in the range of 5 to 30 percent, and reported predicted alligator cracking SEE for this range was found to be reasonable.

Mean Cracking, percent | Standard Deviation of Predicted Cracking, percent |
---|---|

0 | 7.2 |

10 | 8.9 |

20 | 9.5 |

30 | 9.9 |

40 | 10.3 |

50 | 10.6 |

60 | 10.8 |

70 | 11.1 |

80 | 11.3 |

90 | 11.5 |

100 | 11.7 |

Measured rutting data for a wide range of ages and truck traffic application was available for most of the sections. PHT Tool computed parameters that are inputs to the rutting model. The rutting model input parameters were extracted for ages corresponding to field rutting measurements. The PHT predicted rutting was compared against the measured field rutting data to compute the residual error for each age.

Plots of field measured and PHT Tool predicted rutting versus age was prepared and examined. Outliers were further examined for erroneous inputs. Errors in inputs were corrected as needed and the PHT Tool was rerun for those sections to obtain new corrected predictions of rutting. The updated measured/predicted rutting dataset was then used to revise the existing PHT Tool rutting model algorithm and calibration coefficients as needed to increase goodness of fit, minimize error in measured and predicted rutting, and minimize bias.

The new PHT Tool rutting model developed for new bituminous pavements is as presented below:

TRUT = ACRUT + BASERUT + SUBGRUT (7)

ACRUT = *C*0 × *MAAT*^{0.792} × (ε_{HMA-V} × *CESALS*^{0.527285}) (8)

BASERUT = *C*1 × *BASETHK* × (ε_{BASE-V} × *CESALS*^{0.1307}) (9)

WHERE

*TRUT* = predicted total rutting in all layers

*ACRUT* = predicted rutting in the AC layer

*BASERUT* = predicted rutting in the base layer

*SUBGRUT* = predicted rutting in the sub-grade layer

*MAAT* = mean annual air temperature, °F

*CESAL* = cumulative 18-kip ESALs since last improvement or original construction

*PRECIP* = mean annual precipitation/rainfall, in

*C0* = 0 .01038

*C1* = 0 .112531

*C2* = 0 .000476

*C3* = 0 .0000221

The revised rutting models coefficients were determined to minimize total rutting prediction error and reduce bias. The model was developed with 592 data points and has an *R ^{2}* of 20.0 percent and an RMSE of 0.104 in. The new model goodness of fit statistics is typical for pavement management type models. The diagram in Figure 17 present plot of predicted versus measured rutting for the entire dataset used for model development.

The new rutting model was further evaluated for bias. Bias was defined as the consistent under- or over-estimation of rutting. Bias was determined by performing a statistical paired t-test to determine is measured and predicted rutting was similar (i.e., essentially from the same population). The paired t-test was performed as follows:

- Develop null and alternative hypothesis:
- Null hypothesis H0: PHT Tool rutting = LTPP measured rutting.
- Alternate hypothesis HA: PHT Tool rutting ≠ LTPP measured rutting.

- Perform statistical analysis to determine and evaluate test p-value.
- A rejection of the null hypothesis (p-value < 0.05) would imply rutting from the PHT Tool and measured LTPP rutting are from different populations. This indicates bias in PHT Tool rutting estimates for the range of typical inputs used in analysis.

Note that a significance level, α, of 0.05 or 5 percent was assumed for all hypothesis testing. The outcome of the paired t-test was a p-value of 0.0509. The p-value showed that there was not significant bias in predicted rutting.

Risk associated with rutting prediction or the reliability of the rutting is defined as the one-tail confidence interval at a predefined reliability level around a given rutting prediction. For this study, confidence interval was determined as follows:

- Use new PHT Tool rutting model to estimate the distress over typical range of rutting from 0.0 to 1.0 inches.
- Divided the typical range of the distress into subsets (e.g., 0 to 0.10, 0.10 to 0.20, etc.).
- For each subset of predicted rutting, estimate the standard error of the mean prediction (i.e., standard deviation of measured - predicted distress (i.e., std. error of the estimate, SEE) for all individual data points that falls within the subset).
- Develop a relationship between the SEE and predicted rutting.

The relationship developed was used in the PHT Analysis Tool to determine SEE for any predicted rutting. SEE will be used to estimate predicted rutting at any desired reliability level as shown in Table 7. The design reliability procedure described above requires the estimation of variability in the form of standard deviation at any given level of predicted rutting. Predicted rutting standard deviation was determined as follows:

- Divide predicted rutting to five or more intervals.
- For each interval, determine mean predicted rutting and standard error (i.e., standard variation of predicted - measured rutting for all the predicted rutting that falls within the given interval).
- Develop a non linear model to fit mean predicted rutting and standard error for each interval.

The resulting standard error prediction model developed for the PHT Tool is presented below:

*Stderr*(*TRUT*) = 0.0186 + (0.0729 × *MPRUT*^{0.1}) (11)

WHERE

*Stderr*(*TRUT*) = rutting standard error of the estimate, inches

*TRUT* = predicted rutting, inches

*MPRUT* = mean predicted rutting

The diagram in Figure 18 presents a plot of standard deviation versus predicted rutting developed using the data presented in Table 9 which was obtained through analysis of predicted rutting data. The region of predicted rutting that triggers maintenance and rehabilitation is 0.4 to 0.90 inches, and reported predicted rutting SEE for this range was found to be reasonable.

Mean Rutting, inches | Standard Deviation of Predicted Rutting, inches |
---|---|

0 | 0.018 |

0.1 | 0.076 |

0.2 | 0.080 |

0.3 | 0.083 |

0.4 | 0.085 |

0.5 | 0.086 |

0.6 | 0.087 |

0.7 | 0.088 |

0.8 | 0.089 |

0.9 | 0.090 |

1 | 0.091 |

Measured transverse cracking data was available for most of the sections for a wide range of pavement ages. PHT Tool was used to compute parameters that are inputs to the transverse cracking model. For each pavement section and for the ages for which measured transverse cracking data was available, relevant input computed parameters and corresponding field measured transverse cracking data was extracted and used to develop a project database for model evaluation and calibration. The input PHT Tool and output computed parameters and predicted transverse cracking was evaluated to identify errors and outlines in the input database. The outcome of this examination was to correct anomalies and errors. The PHT Tool was rerun using the corrected input database.

Plots of measured versus predicted transverse cracking was prepared and evaluated. Evaluation comprised of comparing measured field and PHT Tool predicted transverse cracking to assess goodness of fit and bias. The outcome of this evaluation was a determination that current PHT Tool transverse cracking model produced biased predictions. Thus there was a need for recalibration to improve goodness of fit and minimize bias. Unbiased prediction means that the model does not on average over all of the data over predict or under predict the measured data.

The new PHT Tool transverse cracking model developed from the S-shaped curve model relating a pseudo damage parameter to transverse cracking. The new bituminous pavement transverse cracking model is as presented below:

*FACTOR* = *AGE* / (62.5 + 14.9986**HMATHK* - 409967*loglog(η)- 6.9433**VA* - 0.4584**PCT34* - 3.3029**FTCYC*) (13)

WHERE

*TCRK* = predicted transverse cracking, feet/mile

*AGE* = pavement age, years

*HMATHK* = HMA thickness, inches

*VA* = as-constructed HMA mix air void content, percent

*PCT34* = cumulative percent retained on the ¾ in sieve for the HMA

*FTCYC* = mean annual air freeze-thaw cycles

*C0* = WF: 4.61, WNF: 1053, Dry: 223.6

*C1* = WF: -3.327, WNF: -4.5, Dry: -4.5

The model was developed with 700 data points and has an R^{2} of 53.5 percent and an RMSE of 502 ft/mi. The new model goodness of fit statistics indicates the level of accuracy typical for pavement management type models. The diagram in Figure 19 presents the plot of predicted versus measured transverse cracking for the entire dataset used for model development. The diagram in Figure 20 presents a plot of predicted transverse cracking versus FACTOR. This plot shows considerably higher predictions of transverse cracking for Freeze and Dry regions compared to Wet/No-freeze.

The new transverse cracking model was further evaluated for bias. Bias was defined as the consistent under or over estimation of cracking. Bias was determined by performing a statistical paired t-test to determine is measured and predicted transverse cracking was similar. The paired t-test was performed as follows:

- Develop null and alternative hypothesis:
- Null hypothesis H0: PHT Tool cracking = LTPP measured cracking.
- Alternate hypothesis HA: PHT Tool cracking ≠ LTPP measured cracking.

- Perform statistical analysis to determine and evaluate test p-value.
- A rejection of the null hypothesis (p-value < 0.05) would imply cracking from the PHT Tool and measured LTPP cracking are from different populations. This indicates bias in PHT Tool transverse cracking estimates for the range of typical inputs used in analysis.

Note that a significance level, α, of 0.05 or 5 percent was assumed for all hypothesis testing. The outcome of the paired t-test was a p-value of 0.7902. The p-value showed that there was not significant bias in predicted transverse cracking.

Risk associated with transverse cracking prediction or the reliability of the transverse cracking prediction was defined as the one-tail confidence interval at a predefined reliability level around a given transverse cracking prediction. The confidence interval was determined as follows:

- Use new PHT Tool transverse cracking model to estimate the distress over typical range of cracking of 0 to 5000 ft/mi.
- Divided the typical range of the distress into subsets (e.g., 0 to 1000, 1000 to 2000, etc.).
- For each subset of predicted transverse cracking, estimate the standard error of the mean prediction (i.e., standard deviation of measured - predicted distress (i.e., std. error of the estimate, SEE) for all individual data points that falls within the subset).
- Develop a relationship between the SEE and predicted transverse cracking.

The relationship developed was used in the PHT Analysis Tool to determine SEE for any predicted transverse cracking. SEE will be used to estimate predicted transverse cracking at any desired reliability level as shown in Table 7. The design reliability procedure described above requires the estimation of variability in the form of standard deviation at any given level of predicted transverse cracking. Predicted transverse cracking standard deviation was determined as follows:

- Divide predicted transverse cracking to five or more intervals.
- For each interval, determine mean predicted transverse cracking and standard error (i.e., standard variation of predicted - measured transverse cracking for all the predicted transverse cracking that falls within the given interval).
- Develop a non linear model to fit mean predicted transverse cracking and standard error for each interval.

The resulting standard error prediction model developed the PHT Tool is presented below:

*Stderr*(*TCRK*) = 1.0 + (59.23 × *MPTCRK*^{0.3953}) (14)

WHERE

*Stderr*(*TCRK*) = cracking standard error of the estimate, feet/mile

*TCRK* = predicted transverse cracking, feet/mile

*MPTCRK* = mean predicted transverse cracking

The diagram in Figure 21 presents a plot of standard deviation versus predicted transverse cracking developed using the data presented in Table 10 which was obtained through analysis of predicted transverse cracking data. The region of predicted cracking that triggers maintenance and rehabilitation is 1000 to 3000 ft/mi, and reported predicted transverse cracking SEE for this range was found to be reasonable.

Mean Transverse Cracking, ft/mi | Standard Deviation of Predicted Transverse Cracking, ft/mi |
---|---|

0 | 1 |

500 | 691.83 |

1000 | 909.59 |

1500 | 1067.54 |

2000 | 1195.99 |

2500 | 1306.19 |

3000 | 1403.73 |

3500 | 1491.87 |

4000 | 1572.67 |

4500 | 1647.58 |

5000 | 1717.61 |

Measured smoothness data was available for most of the sections for a wide range of pavement ages. PHT Tool was used to compute parameters that are inputs to the smoothness model such as alligator cracking, rutting, transverse cracking, and site factors. The initial IRI is a key smoothness model input and was estimated using historical field measured IRI available in the LTPP database.

For each LTPP section and for the ages for which measured smoothness data was available, the required smoothness inputs were estimated and used along with measured IRI to develop a project database for PHT Tool IRI model evaluation and calibration.

Current model evaluation began by reviewing the IRI calibration database for reasonableness and to identify errors and outliers. The outcome of this examination was to correct identified anomalies and errors. The PHT Tool was rerun using the corrected input database to develop a final IRI calibration database.

Next, plots of measured versus predicted smoothness was prepared and evaluated. Evaluation comprised of comparing measured field and PHT Tool predicted smoothness king to assess goodness of fit and bias. The outcome of this evaluation was a determination that current PHT Tool smoothness model produced biased IRI predictions. Thus there was a need for recalibration to improve goodness of fit and minimize bias. Unbiased prediction means that the model does not on average over all of the data over predict or under predict the measured data.

The new PHT Tool smoothness model was thus developed which was essentially recalibration of the existing IRI model to obtain new model coefficients that produce a better fit of measured and predicted IRI. The new bituminous pavement smoothness model is as presented below:

*IRI* = *IRI*0 + (*C*0 × *TCRK*) + (*C*1 × *TRUT*) + (*C*2 × *ACRK*) + *C*3 × *FACTOR*) (15)

*FACTOR* = *FROSTH* + *SWELLP* × *AGE*^{1.5} (16)

*FROSTH* = *LN*((*PRECIP* + 1)×*FINES*×(*FI* + 1)) (17)

*SWELLP* = *LN*((*PRECIP* + 1)×*CLAY*×(*PI* + 1)) (18)

WHERE

*IRI* = predicted IRI value

*IRI0* = initial IRI value

*TCRK* = predicted transverse cracking, feet/mile

*TRUT* = predicted rutting, inches

*ACRK* = predicted alligator cracking, percent

*AGE* = pavement age, years

*PRECIP* = mean annual precipitation, inches

*FINES* = amount of fine sand and silt particles in sub-grade, percent

*CLAY* = amount of clay particles in sub-grade, percent

*FI* = mean annual freezing index

*PI* = sub-grade soil plasticity index

*C0* = 0.000592

*C1* = 8.5571

*C2* = 0.8676

*C3* = 0.0175

The model was developed with 1507 data points and has an R^{2} of 72.7 percent and an RMSE of 7.54 in/mi. The new model goodness of fit statistics indicates the level of accuracy typical for pavement management type models. The diagram in Figure 22 presents the plot of predicted versus measured IRI for the entire dataset used for model development.

The new IRI model was further evaluated for bias. Bias was defined as the consistent under or over estimation of IRI. Bias was determined by performing a statistical paired t-test to determine is measured and predicted IRI was similar. The paired t-test was performed as follows:

- Develop null and alternative hypothesis:
- Null hypothesis H0: PHT Tool mean IRI = LTPP measured IRI.
- Alternate hypothesis HA: PHT Tool mean IRI ≠ LTPP measured IRI.

- Perform statistical analysis to determine and evaluate test p-value.
- A rejection of the null hypothesis (p-value < 0.05) would imply IRI from the PHT Tool and measured LTPP IRI are from different populations. This indicates bias in PHT Tool IRI estimates for the range of typical inputs used in analysis.

Note that a significance level, α, of 0.05 or 5 percent was assumed for all hypothesis testing. The outcome of the paired t-test was a p-value of 0.0780. The p-value showed that there was not significant bias in predicted IRI.

Risk associated with IRI prediction or the reliability of the IRI prediction was defined as the one-tail confidence interval at a predefined reliability level around a given IRI prediction. Specifically, the one-tailed confidence interval is as defined in equation 4. For this study, confidence interval was determined as follows:

- Use new PHT Tool IRI model to estimate the distress over typical range of IRI ranging from 30 to 300 inches/mile.
- Divide the typical range of the distress into subsets (e.g., 30 to 60, 60 to 90, etc.).
- For each subset of predicted IRI, estimate the standard error of the mean prediction (i.e., standard deviation of measured - predicted distress (i.e., std. error of the estimate, SEE) for all individual data points that falls within the subset).
- Develop a relationship between the SEE and predicted IRI.

The relationship developed was used in the PHT Analysis Tool to determine SEE for any predicted IRI. SEE will be used to estimate predicted IRI at any desired reliability level as shown in Table 7. The design reliability procedure described above requires the estimation of variability in the form of standard deviation at any given level of predicted IRI. Predicted IRI standard deviation was determined as follows:

- Divide predicted IRI to five or more intervals.
- For each interval, determine mean predicted IRI and standard error (i.e., standard variation of predicted - measured IRI for all the predicted IRI that falls within the given interval).
- Develop a non linear model to fit mean predicted IRI and standard error for each interval.

*Stderr*(*IRI*) = 0.001 + (1.5827 × *MPIRI*^{0.3809}) (19)

WHERE

*Stderr(IRI)* = IRI standard error of the estimate, inch/mile

*IRI* = predicted IRI, inch/mile

*MPIRI* = mean predicted IRI

The diagram in Figure 23 presents a plot of standard deviation versus predicted IRI developed using the data presented in Table 11 which was obtained through analysis of predicted IRI data. The region of predicted IRI that triggers maintenance and rehabilitation is 150 to 250 in/mi, and reported predicted IRI SEE for this range was found to be reasonable.

Mean IRI, in/mi | Standard Deviation of Predicted IRI, in/mi |
---|---|

30 | 5.8 |

60 | 7.5 |

90 | 8.8 |

120 | 9.8 |

150 | 10.7 |

180 | 11.4 |

210 | 12.1 |

240 | 12.8 |

270 | 13.4 |

300 | 13.9 |

Measured transverse cracking data was available for most of the sections for a wide range of pavement ages. PHT Tool was used to compute parameters that are inputs to the transverse cracking model such as edge support, climate, PCC compressive strength, and PCC elastic modulus. For each pavement section and for the ages for which measured slab cracking data was available, all required inputs along with field measured slab cracking data was assembled into a project database for model evaluation and calibration. The assembled data was reviewed to identify errors and outliers. The outcome of this examination was to correct identified anomalies and errors. The PHT Tool was rerun using the corrected input database to develop the final project database.

Plots of measured versus predicted slab cracking was prepared and evaluated. Evaluation comprised of comparing measured field and PHT Tool predicted slab cracking to assess goodness of fit and bias. The outcome of this evaluation was a determination that current PHT Tool slab cracking model produced biased predictions. Thus there was a need for recalibration to improve goodness of fit and minimize bias. Unbiased prediction means that the model does not on average over all of the data over predict or under predict the measured data.

The new PHT Tool slab cracking model developed from the S-shaped curve model relating a pseudo damage parameter to slab cracking. The new JPCP slab cracking model is below:

*FACTOR* = C0**EDGSUP* + C1**EPCC* + C2**CTB* + C3**ATB* + C4**PCC_COMP* + C5**PCCTHK* + C6**SUBGCOAR* + C7**CLIMWF* + C8**CLIMWNF* + C9**CLIMDNF* (21)

WHERE

*TCRK* = predicted transverse cracking, feet/mile

*AGE* = pavement age, years

*CESALS* = mean annual precipitation, inches

Coefficient | Value | Description |
---|---|---|

C0 | 200 | EDGSUP = (1 if a tied PCC shoulder or widened slab, otherwise 0) |

C1 | -0.0039 | EPCC = 28-day PCC slab elastic modulus in psi |

C2 | -20 | CTB = (1 if base type is cement treated material, otherwise 0) |

C3 | 752.4 | ATB = (1 if base type is asphalt treated material, otherwise 0) |

C4 | 1.9799 | PCCCOMP = 28-day PCC compressive strength in psi |

C5 | 730 | PCCTHK = PCC slab thickness in inches |

C6 | -315 | SUBGCOAR = (1 if sub-grade soil type is coarse grained, otherwise 0) |

C7 | 1000 | CLIMWF = (1 if pavement is located in a wet-freeze climate, otherwise 0) |

C8 | 100 | CLIMWNF = (1 if pavement is located in a wet-no-freeze climate, otherwise 0) |

C9 | 100 | CLIMDNF = (1 if pavement is located in a dry-no-freeze climate, otherwise 0) |

The model was developed with 618 data points and has an R^{2} of 67.8 percent and an RMSE of 6.8 percent. The new model goodness of fit statistics indicates the level of accuracy typical for pavement management type models. The diagram in Figure 24 present plot of predicted versus measured transverse cracking for the entire dataset used for model development.

The new slab cracking model was further evaluated for bias. Bias was defined as the consistent under or over estimation of slab cracking. Bias was determined by performing a statistical paired t-test to determine is measured and predicted slab cracking was. The paired t-test was performed as follows:

- Develop null and alternative hypothesis:
- Null hypothesis H0: PHT Tool slab cracking = LTPP measured cracking.
- Alternate hypothesis HA: PHT Tool slab cracking ≠ LTPP measured cracking.

- Perform statistical analysis to determine and evaluate test p-value.
- A rejection of the null hypothesis (p-value < 0.05) would imply cracking from the PHT Tool and measured LTPP cracking are from different populations. This indicates bias in PHT Tool slab cracking estimates for the range of typical inputs used in analysis.

Note that a significance level, α, of 0.05 or 5 percent was assumed for all hypothesis testing. The outcome of the paired t-test was a p-value of 0.0575. The p-value showed that there was not significant bias in predicted slab cracking.

Risk associated with slab cracking prediction or the reliability of the slab cracking prediction was defined as the one-tail confidence interval at a predefined reliability level around a given slab cracking prediction. For this study, confidence interval was determined as follows:

- Use new PHT Tool slab cracking model to estimate the distress over typical range of slab cracking of 0 to 100 percent.
- Divided the typical range of the distress into subsets (e.g., 0 to 10, 10 to 20, etc.).
- For each subset of predicted slab cracking, estimate the standard error of the mean prediction (i.e., standard deviation of measured - predicted distress (i.e., std. error of the estimate, SEE) for all individual data points that falls within the subset).
- Develop a relationship between the SEE and predicted slab cracking.

The relationship developed was used in the PHT Analysis Tool to determine SEE for any predicted slab cracking. SEE was used to estimate predicted slab cracking at any desired reliability level as shown in Table 7. The design reliability procedure described above requires the estimation of variability in the form of standard deviation at any given level of predicted slab cracking. Predicted slab cracking standard deviation was determined as follows:

- Divide predicted slab cracking to four or more intervals.
- For each interval, determine mean predicted slab cracking and standard error (i.e., standard variation of predicted - measured slab cracking for all the predicted slab cracking that falls within the given interval).
- Develop a non linear model to fit mean predicted slab cracking and standard error for each interval.

The resulting standard error prediction model developed for the PHT Tool is presented below:

*Stderr*(*TCRK*) = 0.2227 + (4.0127 × *MPTCRK*^{0.3691}) (22)

WHERE

*Stderr*(*TCRK*) = slab cracking standard error of the estimate, percent

*TCRK* = predicted slab cracking, percent

*MPTCRK* = mean predicted slab cracking

The diagram in Figure 25 presents a plot of standard deviation versus predicted slab cracking. The region of predicted cracking that triggers maintenance and rehabilitation is 10 to 30 percent, and reported predicted slab cracking SEE for this range was found to be reasonable.

Measured transverse joint faulting data was available for most of the sections for a wide range of pavement ages. The PHT Tool was used to compute parameters that are inputs to the joint faulting model such as edge support, climate, and joint spacing. For each pavement section and for the ages for which measured transverse joint faulting data was available, all required inputs along with field measured joint faulting data was assembled into a project database for model evaluation and calibration. The assembled data was reviewed to identify errors and outliers. The outcome of this examination was to correct identified anomalies and errors. The PHT Tool was rerun using the corrected input database to develop the final project database.

Plots of measured versus predicted transverse joint faulting was prepared and evaluated. Evaluation comprised of comparing measured field and PHT Tool predicted transverse joint faulting to assess goodness of fit and bias. The outcome of this evaluation was a determination that current PHT Tool transverse joint faulting model produced biased predictions. Thus there was a need for recalibration to improve goodness of fit and minimize bias. Unbiased prediction means that the model does not on average over all of the data over predict or under predict the measured data.

The new PHT Tool transverse joint faulting model developed from the S-shaped curve model relating a pseudo damage parameter to joint faulting. The new JPCP transverse joint faulting model is as presented below:

*FACTOR* = C0**DOWDIA* + C1**ATB* + C2**CTB* + C3 **EDGESUP* + C4 + C5**WET* + C6**PCCTHK* + C7**SUBGCOAR* (24)

WHERE

*TJFLT* = predicted transverse joint faulting, inches

*AGE* = pavement age, years

Coefficient | Value | Description |
---|---|---|

C0 | 652.9 | DOWDIA = dowel diameter, 0 for non doweled pavements and PCC thickness or 8 for doweled pavements |

C1 | 122.6 | ATB = 1 if base type is asphalt treated or permeable asphalt treated |

C2 | 441.7 | CTB = 1 if base type is cement treated or lean concrete |

C3 | 760.7 | EDGESUP (Edge support), 1 if a tied PCC shoulder (HPMS Shoulder_Type = 3) or widened slab (lane width > 12 ft) is used, otherwise 0 |

C4 | 703.3 | Site factor constant |

C5 | -501.8 | WET = 1 if mean annual precipitation > 20 in., else 0 |

C6 | -20.9 | PCCTHK = PCC slab thickness in inches |

C7 | -290.8 | SUBGCOAR = 1 if sub-grade soil type is coarse grained, otherwise 0 |

The model was developed with 527 data points and has an R^{2} of 66.3 percent and an RMSE of 0.028 inches. The new model goodness of fit statistics indicates the level of accuracy typical for pavement management type models. The diagram in Figure 26 presents the plot of predicted versus measured transverse joint faulting for the entire dataset used for model development.

The new transverse joint faulting model was further evaluated for bias. Bias was defined as the consistent under or over estimation of joint faulting. Bias was determined by performing a statistical paired t-test to determine is measured and predicted transverse joint faulting was similar. The paired t-test was performed as follows:

- Develop null and alternative hypothesis:
- Null hypothesis H0: PHT Tool joint faulting = LTPP measured faulting.
- Alternate hypothesis HA: PHT Tool joint faulting ≠ LTPP measured faulting.

- Perform statistical analysis to determine and evaluate test p-value.
- A rejection of the null hypothesis (p-value < 0.05) would imply faulting from the PHT Tool and measured LTPP faulting are from different populations. This indicates bias in PHT Tool transverse joint faulting estimates for the range of typical inputs used in analysis.

Note that a significance level, α, of 0.05 or 5 percent was assumed for all hypothesis testing. The outcome of the paired t-test was a p-value of 0.4558. The p-value showed that there was not significant bias in predicted transverse joint faulting.

Risk associated with transverse joint faulting prediction or the reliability of the transverse joint faulting prediction was defined as the one-tail confidence interval at a predefined reliability level around a given transverse joint faulting prediction. For this study, confidence interval was determined as follows:

- Use new PHT Tool transverse joint faulting model to estimate the distress over typical range of joint faulting of 0 to 0.5 inches.
- Divided the typical range of the distress into subsets (e.g., 0 to 0.10, 0.10 to 0.20, etc.).
- For each subset of predicted transverse joint faulting, estimate the standard error of the mean prediction (i.e., standard deviation of measured - predicted distress (i.e., std. error of the estimate, SEE) for all individual data points that falls within the subset).
- Develop a relationship between the SEE and predicted transverse joint faulting.

The relationship developed was used in the PHT Analysis Tool to determine SEE for any predicted transverse joint faulting. SEE was used to estimate predicted transverse joint faulting at any desired reliability level as shown in Table 7. The design reliability procedure described above requires the estimation of variability in the form of standard deviation at any given level of predicted transverse joint faulting. Predicted transverse joint faulting standard deviation was determined as follows:

- Divide predicted transverse joint faulting into four or more intervals.
- For each interval, determine mean predicted transverse joint faulting and standard error (i.e., standard variation of predicted - measured transverse joint faulting for all the predicted transverse joint faulting that falls within the given interval).
- Develop a non linear model to fit mean predicted transverse joint faulting and standard error for each interval.

The resulting standard error prediction model developed for the PHT Tool is presented below:

*Stderr*(*TJFLT*) = 0.0042 + (0.1363 × *MPTJFLT*^{0.5}) (25)

WHERE

*Stderr*(*TJFLT*) = joint faulting standard error of the estimate, inches

*TJFLT* = predicted transverse joint faulting, inches

*MPTJFLT* = mean predicted transverse joint faulting

The diagram in Figure 27 presents a plot of standard deviation versus predicted transverse joint faulting. The region of predicted transverse joint faulting that triggers maintenance and rehabilitation is 0.1 to 0.3 inches, and reported predicted joint faulting SEE for this range was found to be reasonable.

Measured smoothness data was available for most of the sections for a wide range of pavement ages. PHT Tool was used to compute parameters that are inputs to the smoothness model including slab cracking, transverse joint faulting, spalling, and site factors. The initial IRI is a key smoothness model input and was estimated using historical field measured IRI available in the LTPP database. For each LTPP section and for the ages for which measured smoothness data was available, the required smoothness inputs were estimated and used along with measured IRI to develop a project database for PHT Tool IRI model evaluation and calibration.

Current model evaluation began by reviewing the IRI calibration database for reasonableness and to identify errors and outliers. The outcome of this examination was to correct identified anomalies and errors. The PHT Tool was rerun using the corrected input database to develop a final IRI calibration database.

Next, plots of measured versus predicted smoothness was prepared and evaluated. Evaluation comprised of comparing measured field and PHT Tool predicted smoothness to assess goodness of fit and bias. The outcome of this evaluation was a determination that current PHT Tool smoothness model produced biased IRI predictions. Thus there was a need for recalibration to improve goodness of fit and minimize bias. Unbiased prediction means that the model does not on average over all of the data over predict or under predict the measured data.

The new PHT Tool smoothness model was thus developed which was essentially recalibration of the existing IRI model to obtain new model coefficients that produce a better fit of measured and predicted IRI. The new JPCP smoothness model is as presented below.

*IRI* = *IRI*_{0} + (*C*0 × *TCRK*) + (*C*1 ×* TJFLT*) + (*C*2 × *TJSPALL*) + (*C*3 × *FACTOR*) (26)

*FACTOR* = *AGE*×(1 + 0.5556×*FI*)×(1 + *P*_{200})×10^{-6} (27)

WHERE

*IRI* = predicted IRI value

*IRI0* = initial IRI value

*TCRK* = predicted slab cracking, percent

*TJFLT* = predicted transverse joint faulting, inches

*TJSPALL* = predicted transverse joint spalling, percent

*AGE* = pavement age, years

*FI* = mean annual freezing index

*C0* = 0.4

*C1* = 21.2

*C2* = 1.52

*C3* = 18.16

The model was developed with 777 data points and has an R^{2} of 73.35 percent and an RMSE of 15.02 in/mi. The new model goodness of fit statistics indicates the level of accuracy typical for pavement management type models. The diagram in Figure 28 presents the plot of predicted versus measured IRI for the entire dataset used for model development.

The new JPCP IRI model was further evaluated for bias. Bias was defined as the consistent under or over estimation of cracking. Bias was determined by performing a statistical paired t-test to determine is measured and predicted IRI was similar. The paired t-test was performed as follows:

- Develop null and alternative hypothesis:
- Null hypothesis H0: PHT Tool IRI = LTPP measured IRI.
- Alternate hypothesis HA: PHT Tool IRI ≠ LTPP measured IRI.

- Perform statistical analysis to determine and evaluate test p-value.
- A rejection of the null hypothesis (p-value < 0.05) would imply IRI from the PHT Tool and measured LTPP IRI are from different populations. This indicates bias in PHT Tool IRI estimates for the range of typical inputs used in analysis.

Note that a significance level, α, of 0.05 or 5 percent was assumed for all hypothesis testing. The outcome of the paired t-test was a p-value of 0.1108. The p-value showed that there was not significant bias in predicted IRI.

Risk associated with IRI prediction or the reliability of the IRI prediction was defined as the one-tail confidence interval at a predefined reliability level around a given IRI prediction. For this study, confidence interval was determined as follows:

- Use new PHT Analysis Tool IRI model to estimate the distress over the typical range of IRI of 30 to 300 in/mi.
- Divide the typical range of the distress into subsets (e.g., 30 to 60, 60 to 90, etc.).
- For each subset of predicted IRI, estimate the standard error of the mean prediction (i.e., standard deviation of measured - predicted distress (i.e., std. error of the estimate, SEE) for all individual data points that falls within the subset).
- Develop a relationship between the SEE and predicted IRI.

The relationship developed was used in the PHT Analysis Tool to determine SEE for any predicted smoothness IRI. SEE was used to estimate predicted IRI at any desired reliability level as shown in Table 7. The design reliability procedure described above requires the estimation of variability in the form of standard deviation at any given level of predicted IRI. Predicted IRI standard deviation was determined as follows:

- Divide predicted IRI to five or more intervals.
- For each interval, determine mean predicted IRI and standard error (i.e., standard variation of predicted - measured IRI for all the predicted IRI that falls within the given interval).
- Develop a non linear model to fit mean predicted IRI and standard error for each interval.

The resulting standard error prediction model developed for the PHT Tool is presented below:

*Stderr*(*IRI*) = 0.001 + (3.793 × *MPIRI*^{0.2952}) (28)

WHERE

*Stderr*(*IRI*) = IRI standard error of the estimate, inches/mile

*IRI* = predicted IRI value, inches/mile

*MPIRI* = mean predicted IRI

The diagram in Figure 29 presents a plot of standard deviation versus predicted IRI. The region of predicted IRI that triggers maintenance and rehabilitation is 150 to 250 in/mi, and reported predicted IRI SEE for this range was found to be reasonable.

Measured transverse cracking data was available for most of the sections for a wide range of pavement ages. PHT Tool was used to compute parameters that are inputs to the transverse cracking model. For each pavement section and for the ages for which measured transverse cracking data was available, relevant input computed parameters and corresponding field measured transverse cracking data was extracted and used to develop a project database for model evaluation and calibration. The input PHT Tool and output computed parameters and predicted transverse cracking was evaluated to identify errors and outlines in the input database. The outcome of this examination was to correct anomalies and errors. The PHT Tool was rerun using the corrected input database.

Plots of measured versus predicted reflection cracking was prepared and evaluated. Evaluation comprised of comparing measured field and PHT Tool predicted reflection cracking to assess goodness of fit and bias. The outcome of this evaluation was a determination that current PHT Tool reflection cracking model produced biased predictions. Thus there was a need for recalibration to improve goodness of fit and minimize bias. Unbiased prediction means that the model does not on average over all of the data over predict or under predict the measured data.

The new PHT Tool reflection cracking model developed from the S-shaped curve model relating age since overlay placement to reflection cracking. The AC overlaid JPCP reflection cracking model is as presented below:

*RCRK* = C0 * EXTCRK * LWIDTH / 1 + 2.718^{C1(a)+C2(AGE)(b)} (29)

WHERE

*RCRK* = predicted reflection cracking, feet/mile

*EXTCRK* = number of pre-overlay transverse joints and cracks

*LWIDTH* = underlying slab or land width, feet

*AGE* = pavement age, years

*C0* = 9.9639

*C1* = 0.3896

*C2* = 0.2826

The coefficients listed above were determined to minimize the prediction error of the model and reduce bias. The model was developed with 200 data points and has an R^{2} of 54.0 percent and an RMSE of 862 ft/mi. The new model goodness of fit statistics indicates the level of accuracy typical for pavement management type models. The diagram in Figure 30 presents the plot of predicted versus measured reflection cracking for the entire dataset used for model development.

The new reflection cracking model was further evaluated for bias. Bias was defined as the consistent under or over estimation of cracking. Bias was determined by performing a statistical paired t-test to determine is measured and predicted transverse cracking was similar. The paired t-test was performed as follows:

- Develop null and alternative hypothesis:
- Null hypothesis H0: PHT Tool cracking = LTPP measured cracking.
- Alternate hypothesis HA: PHT Tool cracking ≠ LTPP measured cracking.

- Perform statistical analysis to determine and evaluate test p-value.
- A rejection of the null hypothesis (p-value < 0.05) would imply cracking from the PHT Tool and measured LTPP cracking are from different populations. This indicates bias in PHT Tool reflection cracking estimates for the range of typical inputs used in analysis.

Note that a significance level, α, of 0.05 or 5 percent was assumed for all hypothesis testing. The outcome of the paired t-test was a p-value of 0.8706. The p-value showed that there was not significant bias in predicted reflection cracking.

Risk associated with transverse reflection cracking prediction or the reliability of the reflection cracking prediction was defined as the one-tail confidence interval at a predefined reliability level around a given reflection cracking prediction. For this study, confidence interval was determined as follows:

- Use new PHT Tool reflection cracking model to estimate the distress over typical range of cracking from 0 to 5000 ft/mi.
- Divided the typical range of the distress into subsets (e.g., 0 to 1000, 1000 to 2000, etc.).
- For each subset of predicted reflection cracking, estimate the standard error of the mean prediction (i.e., standard deviation of measured - predicted distress (i.e., std. error of the estimate, SEE) for all individual data points that falls within the subset).
- Develop a relationship between the SEE and predicted reflection cracking.

The relationship developed was used in the PHT Analysis Tool to determine SEE for any predicted reflection cracking. SEE was used to estimate predicted reflection cracking at any desired reliability level as shown in Table 7. The design reliability procedure described above requires the estimation of variability in the form of standard deviation at any given level of predicted reflection cracking. Predicted transverse reflection cracking standard deviation was determined as follows:

- Divide predicted reflection cracking to five or more intervals.
- For each interval, determine mean predicted cracking and standard error (i.e., standard variation of predicted - measured cracking for all the predicted reflection cracking that falls within the given interval).
- Develop a non linear model to fit mean predicted reflection cracking and standard error for each interval.

The resulting standard error prediction model developed for the PHT Tool is presented below.

*Stderr*(*RCRK*) = 1.0 + (134 × *MPRCRK*^{0.2964}) (30)

WHERE

*Stderr*(*RCRK*) = reflection cracking standard error of the estimate, feet/mile

*RCRK* = predicted reflection cracking, feet/mile

*MPRCRK* = mean predicted reflection cracking

The diagram in Figure 31 presents a plot of standard deviation versus predicted reflection cracking. The region of predicted reflection cracking that triggers maintenance and rehabilitation is 1000 to 3000 ft/mi, and reported predicted reflection cracking SEE for this range was found to be reasonable.

Measured smoothness data was available for most of the sections for a wide range of pavement ages. PHT Tool was used to compute parameters that are inputs to the smoothness model such as initial IRI and transverse cracking. The initial IRI is a key smoothness model input and was estimated using historical field measured IRI available in the LTPP database. For AC overlaid existing JPCP, the effect of site factors on future IRI was deemed negligible while future rutting and alligator cracking was also minimal. Thus, they were not included in this model.

For each LTPP section and for the ages for which measured smoothness data was available, the required smoothness inputs were estimated and used along with measured IRI to develop a project database for PHT Tool IRI model evaluation and calibration. Current model evaluation began by reviewing the IRI calibration database for reasonableness and to identify errors and outliers. The outcome of this examination was to correct identified anomalies and errors. The PHT Tool was rerun using the corrected input database to develop a final IRI calibration database.

Next, plots of measured versus predicted smoothness was prepared and evaluated. Evaluation comprised of comparing measured field and PHT Tool predicted smoothness king to assess goodness of fit and bias. The outcome of this evaluation was a determination that current PHT Tool smoothness model produced biased IRI predictions. Thus there was a need for recalibration to improve goodness of fit and minimize bias. Unbiased prediction means that the model does not on average over all of the data over predict or under predict the measured data.

The new PHT Tool smoothness model was thus developed which was essentially recalibration of the existing IRI model to obtain new model coefficients that produce a better fit of measured and predicted IRI. The new composite pavement smoothness model is as presented below.

*IRI* = *IRI*0 + (*C*0×*RCRK*) (31)

WHERE

*IRI* = predicted IRI smoothness, in/mile

*IRI0* = initial IRI smoothness, in/mile

*RCRK* = predicted reflection cracking, feet/mile

*C0* = 0.00401

The coefficient listed above was determined to minimize the prediction error of the model and reduce bias. The model was developed with 264 data points and has an R^{2} of 32.97 percent and an RMSE of 7.27 in/mi. The new model goodness of fit statistics indicates the level of accuracy typical for pavement management type models. The diagram in Figure 32 presents the plot of predicted versus measured IRI for the entire dataset used for model development.

The new IRI model was further evaluated for bias. Bias was defined as the consistent under or over estimation of IRI. Bias was determined by performing a statistical paired t-test to determine is measured and predicted IRI was similar. The paired t-test was performed as follows:

- Develop null and alternative hypothesis:
- Null hypothesis H0: PHT Tool IRI = LTPP measured IRI.
- Alternate hypothesis HA: PHT Tool IRI ≠ LTPP measured IRI.

- Perform statistical analysis to determine and evaluate test p-value.
- A rejection of the null hypothesis (p-value < 0.05) would imply IRI from the PHT Tool and measured LTPP IRI are from different populations. This indicates bias in PHT Tool IRI estimates for the range of typical inputs used in analysis.

Note that a significance level, α, of 0.05 or 5 percent was assumed for all hypothesis testing. The outcome of the paired t-test was a p-value of 0.2688. The p-value showed that there was not significant bias in predicted IRI.

Risk associated with IRI prediction or the reliability of the IRI prediction was defined as the one-tail confidence interval at a predefined reliability level around a given IRI prediction. For this study, confidence interval was determined as follows:

- Use new PHT Tool IRI model to estimate the distress over typical range of IRI ranging from 30 to 300 in/mi.
- Divide the typical range of the distress into subsets (e.g., 30 to 60, 60 to 90, etc.).
- For each subset of predicted IRI, estimate the standard error of the mean prediction (i.e., standard deviation of measured - predicted distress (i.e., std. error of the estimate, SEE) for all individual data points that falls within the subset).
- Develop a relationship between the SEE and predicted IRI.

The relationship developed was used in the PHT Analysis Tool to determine SEE for any predicted smoothness IRI. SEE was used to estimate predicted IRI at any desired reliability level as shown in Table 7. The design reliability procedure described above requires the estimation of variability in the form of standard deviation at any given level of predicted IRI. Predicted IRI standard deviation was determined as follows:

- Divide predicted IRI to five or more intervals.
- For each interval, determine mean predicted IRI and standard error (i.e., standard variation of predicted - measured IRI for all the predicted IRI that falls within the given interval).
- Develop a non linear model to fit mean predicted IRI and standard error for each interval.

The resulting standard error prediction model developed for the PHT Tool is presented below:

*Stderr*(*IRI*) = 6.43 + (0.56 × MPIRI^{0.5}) (32)

WHERE

*Stderr*(*IRI*) = IRI standard error of the estimate, inches/mile

*IRI* = predicted IRI value, inches/mile

*MPIRI* = mean predicted IRI

The diagram in Figure 33 presents a plot of standard deviation versus IRI. The region of predicted smoothness IRI that triggers maintenance and rehabilitation is 150 to 250 in/mi, and reported predicted IRI SEE for this range was found to be reasonable.

Updated: 02/19/2014