Network Sampling to Estimate Distribution of Pavement Condition and Costs

Transcription

1 8 Transportation Research Record 148 Netork Sampling to Estimate Distribtion of Pavement Condition and Costs J. TEMPLETON, R. L. LYTTON, and A. GARCIA-DIAZ ABSTRACT Srveying the pavement condition of a highay netork by sampling is done to obtain crrent information that is accrate enogh for the prposes of planning and fnding needs estimates. Sampling is sed to redce the amont of time and manpoer that is reqired to collect this information to an irredcible minimm. In this paper are presented the reslts of a stdy of data from a 1982 srvey of the total mileage of three district netorks in Texas that as ndertaken to anser several qestions concerning the accracy that can be achieved ith different sample sies. It is fond that the sample sie reqired to estimate the distribtion of pavement scores to a given degree of accracy is smaller than the sample sie that mst be sed to estimate the average cost of maintenance and rehabilitation to the same level of accracy. The relative sies fo1md are presented in tables and figres and differ among classes of highay: Interstate, U.S., state, and farm-to-market. Several convenient relations ere fond among the mean pavement score and the variance of pavement scores, percentage of pavements needing no repairs, average costs per sqare yard, and percentage error in the estimated average costs. It is not srprising to find that the average costs are redced and the percentage error is increased as the mean pavement score increases. The mean score is easy to obtain accrately ith a small 5 percent sample. The reslts of this stdy give significant information that ill be sefl in planning ftre sampling srveys both in Texas and elsehere. An essential part of pavement management at the state and district netork level is to be able to estimate accrately the mean and the distribtion of pavement conditions and the cos ts to ma in ta in and rehabilitate the pavement netork. Becase of a limitation of fnds, time, eqipment, and manpoer this kind of information can be collected efficiently by sing sampling srveys of the condition of the netork. Reslts of the analysis of data collected on l on percent of the 2-mi-long pavement sections in three districts of the Texas State Department of Highays and Pblic Transportation, each of hich is responsible for beteen 2,5 and 3, mi of pavement of all fnctional classes, are reported. The objective of the stdy as to determine the ansers to several qestions abot the minimm sample sie and the method of sampling reqired to obtain estimates of costs and pavement condition that are accrate enogh for the prposes of planning and fnding needs estimates. The major qestions are 1. It is possible to estimate the cmlative probability distribtion of pavement scores accrately ith a 5 percent sample? 2. What kind of probability density fnction best fits the distribtion of pavement scores? 3. Ho accrately can the nmber of pavement sections ith pavement scores belo a minimm acceptable level be calclated sing a statistical distribtion derived from a 5 percent sample? 4. What sample sie is reqired to get an accrate estimate of the distribtion of pavement scores? 5. What sample sie is needed to accrately estimate the average cost of maintenance and rehabilitation? 6. Are there any overall relations beteen the mean pavement score, hich can be determined accrately ith a small sample, and the nmber of pavement sections that are not in need of any maintenance or rehabilitation? In short, the stdy attempted to find some basic information and rles of thmb that cold be spported by the data and cold assist in planning ftre condition srveys so as to minimie the effort spent and to maximie the accracy of the reslting information as mch as possible. PAVEMENT CONDITION SURVEYS In 1982 all highays ithin each Texas district ere divided into segments approximately 2 mi in length. Five percent of the total nmber of segments in each of 21 districts and on each of the for roaday systems (Interstate, farm-to-market, state, and U.S.) ere selected at random for sampling. A segment inclded all paved areas beteen to designated mileposts. Hence an Interstate highay segment inclded for roadays (to main roadays and to frontage roads). For the prpose of analysis of the srvey data, only the main roadays ere considered. One lane of each roaday ithin the selected segment as sampled and each of these observations as considered a sampling nit. Figre 1 (a) shos a divided highay segment ith main roadays only. The shaded area depicts the to observations associated ith the segment. Figre 1 (b) shos a to-lane highay segment. Only one observation is chosen from this segment. In the remaining three districts [Districts 8 (Abilene), 11 (Lfkin), and 15 (San Antonio)] a 1 percent sample in each roaday system as taken. Figre 2 shos the location of each of these districts.

2 Templeton et al. 9 T 2mi. L M s R L R T 2mi. t t Divided Hl9hay l ~ t To- Lane Hl9hay FIGURE 1 Observation from a 2-mi segment of a divided highay (a) and observation from a 2-mi segment of a to-lane highay (b). 2. One-inch asphaltic concrete pavement (ACP) overlay or seal pls level-p ($1.58/yd'); 3. To-and-one-half-inch ACP overlay ($3.41/ yd'); 4. For-inch ACP overlay ($6.5/yd'); or 5. Seven-and-one-half-inch ACP overlay ($11.93/ yd') or its eqivalent in reconstrction. FIGURE 2 Location of 1 percent sampled districts. For each observation, in both the 5 and the 1 percent samples, the serviceability index as determined ith the May's ridemeter and a visal defects rating as performed. In the visal rating, the folloing distress types ere recorded: rtting, raveling, flshing, alligator cracking, longitdinal cracking, and failres (or "potholes") For each, the rater noted the area covered (e.g., for alligator cracking--, 1 to 1, 11 to 25, or >25 percent). In Texas, the pavement evalation score is a nmber beteen O and 1 that represents a eighted condition of the riding qality rating and visal distress rating of the pavement. A detailed description of the scoring procedre is contained elsehere (!). The visal distress rating method is described by Epps et al. (_~) ESTIMATING COSTS OF MAJOR MAINTENANCE AND REHABILITATION In evalating the reslts of the stateide condition srvey, each pavement section ith a pavement score of less than 4 as considered to be in need of major maintenance or rehabilitation. Five fnding strategies ere considered in making the estimate of total costs, and one of these as selected for each pavement belo the specified minimm: 1. Seal coat, or fog seal, or extensive patching pls seal ($.36/yd 2 ); The selection of the appropriate strategy as made in the folloing ay. For each of the strategies, the estimated rehabilitated pavement score as compted and deterioration calclations ere made to determine life expectancy. This expected life as compared to a minimm alloable expected life of 3 to 5 years depending on the class of highay in order to determine hich of the five strategies had the smallest alloable expected life, and that one as chosen as the strategy to be sed. The selected strategy as assmed to be applied to the entice 2-mi section. Average costs ere determined for each roaday class in each of the 1 percent sampled districts. The nmber of pavement sections, the average costs, mean pavement scores, variance of pavement scores, and percentage of pavement sections ith no costs are tablated in Table 1. It is noted that District 11 has no Interstate highay mileage. When these reslts are plotted against the mean pavement score, clear trends emerge. For example, in Figre 3 the relations beteen the variance of pavement score and the mean pavement score for the different highay systems are shon. Knoing this relation and the probability density fnction for pavement scores, it is possible to constrct an accrate pavement score distribtion hen a good estimate of the mean score has been determined. In Figre 4 the relations beteen the mean pavement score and the percentage of pavements ith ero costs are shon. The trends are again qite clear, and it is significant that the farm-to-market system crve is above the state crve that, in trn, is above the U.S. and the Interstate highay crve. In Figre 5 the average costs per sqare yard are plotted against the mean pavement score to sho that even these costs can be estimated hen a good estimate of the mean pavement score is in hand. In this case, the average costs rise from the farm-to-market pard to the Interstate highay system. These reslts are encoraging becase they illstrate the clear trends that exist among the data that have been collected in the 1 percent sampled districts. If a 5 percent sample can be sed to obtain an accrate estimate of the mean pavement score, then it may be possible to se the relations in Figres 3-5 to make accrate estimates of the condition and the costs of rehabilitating pavement netorks.

3 1 Transportation Research Record 148 TABLE 1 Pavement Score Costs for the 1 Percent Sample Districts No. of Pavement Data Grop Distric.t Sections Percentage Average Variance of of Pavements Mean Cost Pavement Pavement ith Zero per Sqare Score Score Costs Yard($) Interstate U.S. highays State highays Farm-to-market roads , , , ::;: > BOO "- <1 > '-~~~'--~~---'~~~~~~~~~~~~ FIGURE 3 Relation of the variance of pavement score to the mean pavement score. 1 N 8 # ~ I!:::: -11" ;:!:: ::;: > 6 4 "- L'l < ~ " FIGURE 4 Relations beteen mean pavement score and the percentage of pavements ith ero costs. 1.2 c ~ 1. ::::> g.8 ~.6 L'l <1 ~.4 2 s '"~ FIGURE 5 Relations beteen mean pavement score and average cost per sqare yard. Ho large a sample is necessary to make accrate estimates of these qantities? Is a 5 percent sample large enogh? What is the best ay to organie a sample srvey to obtain the most accrate cost information for the effort? These are the qestions that ere asked at the beginning of this paper. They arise natrally from considering the information in Figres 3-5. The ansers are determined by simlating different sies of srveys sing the data from the 1 percent sampled districts. ACCURACY OF THE 5 PERCENT SAMPLE IN ESTIMATING NETWORK PAVEMENT SCORE DISTRIBUTIONS To determine the accracy reslting from a 5 percent sample in predicting pavement condition, the data from the 1 percent sampled districts ere first divided into 15 grops, and then a 5 percent random sample as taken from each grop. Observations ithin each of the three districts ere classified 1

4 ~.. Templeton et al. 11 by roaday system. Cmlative pavement score distribtions or histograms ranging from to 1 ere classified into 2 class intervals, each ith a idth of five. A comparison of the 1 percent sample histogram and the corresponding 5 percent histogram shos that the 5 percent histogram more closely resembles the 1 percent histogram in those data groped ith a larger nmber of observations. Table 2 gives the maximm absolte difference beteen the to histograms for each data grop along ith the nmber of observations in the 5 percent sample. The maximm difference ranges beteen.61 and TABLE 2 Maximm Absolte Difference Beteen 1 Percent and 5 Percent Sample Histograms Maximm Absolte Difference Beteen 1% and 5% Data Grop District Histograms Interstate ll U.S. highays State highays Farm-to-rarket 8.61 roads CONFIDENCE BANDS ON THE TRUE CUMULATIVE DISTRIBUTION No. of Observations in the 5% Sample To statistically compare the pavement score dis tr i btion based on the 5 percent sample ith the pavement score distribtion based on the 1 percent sample, a percentile of the Kolmogorov-Smirnov test statistic (}) is sed. This percentile along ith an empirical cmlative distribtion [Sn(X)] can be sed to form a 1 (1 - <>) percent confidence band for a tre cmlative distribtion [F (X) 1 As the percentage of confidence is increased, the band becomes ider. Figre 6 shos one of the better comparisons of the tre 1 percent histograms [F(X)] and the confidence bands generated s ing a 5 percent sample [Sn(X)]. The figre shos the comparison for farm-to-market roads in District Becase the Kolmogorov-Smirnov procedre reqires random sampling, it is assmed that the observations are independent. The actal departres from this assmption, hoever, are of only minor conseqence. According to Conover (}), if only discrete vales of the pavement scores are sed, the confidence band is conservative. That is, the tre bt nknon confidence coefficient is greater than 1 (1 - <>) percent. PROBABILITY DENSITY FUNCTION THAT FITS THE PAVEMENT SCORE DISTRIBUTION If a probability density fnction is fitted to the histogram of a 5 percent data sample, it is possible to estimate the nmber of pavement sections ith a pavement score belo 4. An investigation as made of the accracy of this procedre. The beta distribtion as chosen as the family of density fnctions to be sed. The probability density fnction is defined as F(X) = [l/b(a,b)] x l (I - X)b-l for a,b > ;< X< 1 In Eqation 1, B(a,b) is defined as 1 B(a,b) = J x I (1 - X)b-l dx Becase the random variable (X) mst be in the interval beteen ero and one, all of the pavement scores are divided by 1 to satisfy this condition. The parameters a and b are estimated by the folloing procedre. The mean of the 1 percent sample distribtion is set eqal to µ and the variance to cr'. The estimated vales of a and bare then given by and b= (a - iip.)/il (4) Table 3 gives the estimated vales of the parameters a and b for each of the data grops calclated from the 1 percent sample distribtion. According to Hogg and Craig (j_), a and b are consistent estimators of a and b. A Kolmogorov-Smirnov goodness of fit test (}) as sed to determine if the 1 percent samples do actally come from a beta distribtion ith the a and (!) (2) (3) 1 DO fl[) :>< 7 ::> "' a (ii) "' Upper limit.. ~ :'ill U (x) = Sn(x)+.167 > "' >:: <..., 1 ::> ::! '.lll ::> ~~I) ,--....' 1 r--,-- () 1 ---,--' - ' 1--' I ' -... I Loer limit L(x)= Sn(x)-.167 fl 1() PAVEMENT SCORE FIGURE 6 Confidence hand for cmlative distribtion fnction of pavement scores F(X), District 11, farm-to-market system.

5 12 Transportation Research Record 148 TABLE 3 Estimated Vales of the Parameters a and h of the Beta Distribtion Data Grop 1-8 FM-8 SH-8 US-8 FM-11 SH-11 US FM-15 SH-15 US-15 a , _ b parameters given in Table 3. The tests shoed that the samples did come from a beta distribtion. ACCURACY OF ESTIMATING THE PERCENTAGE OF PAVEMENTS BELOW A MINIMUM ACCEPTABLE SCORE To methods of predicting the percentage of pavements ith a pavement score belo 4 ere sed and then compared. In the first method the percentage of roads falling at or belo 4 as compted directly from randomly selected 5 percent samples. The second method made se of the cmlative beta distribtion. The vales of µ and cr 2 ere estimated from the sample. Making se of Eqations 3 and 4, a and b ere estimated from selected 5 percent samples. Using the International Mathematical and Statistical Libraries, Inc., (IMSL) sbrotine, MDBETA, the percentage of observations falling at or belo 4 as then estimated. Table 4 gives the mean error for each of the to methods sed for each data grop. As can be seen from the table, the mean error is less for every data grop hen the percentage is calclated sing the beta distribtion. This method is, therefore, recommended for se ith all sch sample srveys. sed to fill ot sample sies beteen 5 percent and 5 percent. A total of 3 rns ere made at each sample sie ith each data grop and the mean absolte maximm difference beteen the cmlative sample distribtion and the tre 1 percent sample cmlative distribtion as calclated. Figre 7 shos the reslts of the stdy for farm-to-market roads. The least accracy is fond for the Interstate sections, and the greatest accracy is for the farm-to-market roads. Figre 7 shos that there is a point of diminishing retrns that is reached by attempting to improve the accracy by increasing the sample sie. Althogh there is no prescribed ay of choosing the optimal sample sie from these data alone (5), it is apparent that this point occrs near -;here the crve begins to flatten, somehere beteen 2 and 35 percent, hich corresponds to mean maximm d i f f e r ences of arond 3 to 1 percent. USE OF SAMPLING TO ESTIMATE AVERAGE COSTS Simlation rns ere made for sample sies ranging from 5 to 5 percent to estimate average costs. This reqired the selection of a sampling method and a means of calclating the average cost., Simlation stdies ere sed to determine both of these and the detailed reslts are reported elsehere (. _). The most accrate sampling method proved to be a random sampling in each conty of each district. The most accrate method of calclating the average cost is to take an average of the costs per sqare yard on a sample-by-sample basis. Table 5 gives the reslts of the simlation stdy, in hich the percentage error in the average costs per sqare yard as calclated for different sample sies. It is orth noting that the better condition a highay netork is in, the more difficlt it is to make an accrate estimate of average costs per sqare yard. This is illstrated in Figre 8, hich shos that the mean percentage error in average costs per sqare yard for a 25 percent sample increases ith the mean pavement score. TABLE4 Mean Error in Predicting Percentage of Roads Belo a Pavement Score of 4 Mean Error Mean Error Calclated Calclated Directly from Beta Data Grop District from Sample Distribtion Interstate U.S. highays State highays Farm-to-market 8.36,31 roads SAMPLE SIZE FOR A SPECIFIED ACCURACY IN THE ESTIMATE OF PAVEMENT SCORE DISTRIBUTION Ho large a sample sie is needed to obtain consistent accracy in estimating the pavement score cmlative distribtion? This qestion as stdied sing a simlation program. Random selection of pavement sections ithin each conty of each distr i ct as CONCLUSIONS Throgh the se of simlation procedres, the folloing basic reslts ere obtained: It as fond that the beta distribtion fits the pavement score data in each of the 11 data grops. The percentage of roads ith a pavement score of 4 or less can best be estimated throgh the se of the cmlative beta distribtion. This procedre leads to a better estimate of the percentage of pavements ith a score belo 4 than does a direct estimate from the sample. The more variable the pavement scores are ithin a district, the greater is the mean maximm difference beteen the sample cmlative distribtion and the 1 percent sample cmlative distribtion. Pavement condition can be estimated accrately ith a smaller sample sie than is reqired to estimate the costs of maintenance and rehabilitation to the same degree of accracy. Simple random sampling on a conty-by-conty basis sally leads to the smallest relative errors in predicting costs. The better the condition of roads district, the more difficlt it is to estimate a mean cost per sqare yard. ithin a accrately

6 Templeton et al. 13 II Olslrlcl Olstrlcl II Dlslrlct Ill l Cl: 8 l ll ll.7. ::!'.6 :::i ::!' 5 x 4 ::!' 4. 4 t 3. ::!' 2 1 FH !) SllMPLE SIZE 1%) FIGURE 7 Comparison of district reslts of the mean maximm difference sing conty stratification Method A, farm-to-market system. TABLE 5 Percentage Error in Average Cost per Sqare Yard ith Conty Stratification System Sample Sie(%) FM SH s When relations have been established beteen mean pavement score and the variance of pavement scores, the percentage of pavements reqiring no costs and average costs per sqare yard, as illstrated in Figres 3-5, can be estimated if an accrate estimate of the mean score is knon. A 5 percent sample can give sch an estimate of the mean score. The se of a 5 percent sample to estimate the parameters of a beta distribtion can prodce reasonably accrate estimates of the percentage of pavements falling belo a minimm acceptable score. Althogh the details of the distress and riding qality of individal pavement sections can only be fond by observation of the sections themselves, estimates of average costs and pavement netork condition distribtions, hich are sed in netorklevel pavement management, may be achieved ith a 5 percent sample if the relations referred to previosly have been established. Althogh the data analyed here are for Texas conditions, costs, and pavement rating methods, the same general approach may be sed by any other state, ith the reslt that more efficient se can be made of state highay agency manpoer resorces in srveying the condition of the state highay netork. The level of accrate detail that is desired as a reslt of a stateide condition srvey ill dictate the percentage of the pavement sections that shold be sampled. From the observations made in this stdy, the percentage can range from 5 percent p to 4 or 5 percent, depending on the type of information desired and the level of accracy that is - VJ U;;j ~ ~ 4 <:Ii < VJ ~~ ~ :c 3 < > 2 Zct W:::i Uo VJ a.. ~ 1 < :::;; FM FIGURE 8 Relation beteen mean pavement score and mean percentage error in estimated average cost per sqare yard. reqired. This paper illstrates typical reslts that can be expected. REFERENCES 1. T. Scllion and R.L. Lytton. The Development of the RAMS-State Cost Estimating Program. Research Report 239-6F. Texas Transportation Institte, Texas A&M University System, College Station, Nov J.A. Epps, A.H. Meyer, I.E. Larrimore, and H.L. Jones. Roaday Maintenance Evalation User's Manal. Research Report Texas Transportation Institte, Texas A&M University System, College Station, Sept W.J. Conover. Practical Nonparametric Statistics. John Wiley and Sons, Inc., Ne York, R.V. Hogg and A.T. Craig. Introdction to Mathematical Statistics. MacMillan Pblishing co., Inc., Ne York, J.P. Mahoney and R.L. Lytton. Measrements of Pavement Performance Using Statistical Sampling

7 14 Transportation Research Record 148 Techniqes. Research Report Texas Transportation Institte, Texas A&M University System, College Station, C.J. Templeton and R.L. Lytton. Estimating Pavement Condition and Rehabilitation Costs Using Statistical Sampling Techniqes. Research Report Texas Transportation Institte, Texas A&M University System, College Station, Jne Pblication of this paper sponsored by Committee on Pavement Management Systems. Ontim~l - r Pavement M -. - ::in::iqem _, - ---ent - A - Using Roghness Measrements BENJAMIN COLUCCRIOS and KUMARES C. SINHA nnro::ich - - r r ABSTRACT Many state highay departments are placing major emphasis on the development of cost-effective procedres for maintaining their existing pavement netork. The state of Indiana is in need of a systematic procedre for allocating Interstate resrfacing fnds to its pavement netork. An optimiation procedre for establishing resrfacing priorities at the netork level, hich can be incorporated in a pavement management system for the state, is described. Roghness measrements, increase in roghness over time, and traffic are the primary factors considered in the optimiation scheme. Different types of resrfacing activities are considered in the model. A performance fnction model as developed to relate resrfacing strategies to the overall redction in pavement roghness present in the pavement section jst before resrfacing. Regression eqations based on roghness measrements ere also developed in this stdy for predicting ftre roghness levels. The optimiation model has the capability of considering deficient pavement sections at any point ithin the specified analysis period. In addition, it has the capability of analying the impact of different bdget scenarios. The model, in its present format, can predict hat pavement section and resrfacing strategy combination shold be adopted in order to achieve an optimal resrfacing program in Indiana dring the next 5-year period. 'l'he application of the optimiation model to the Indiana Interstate highay netork is discssed in the paper. The Indiana Department of Highays (!OOH) throgh the Research and Training Center (R&TC) has been collecting pavement roghness measrements on a contining basis for the entire highay system since These data are smmaried annally along ith other information inclding average daily traffic (ADT) in one direction, srface type and textre, contract nmber, length, and last time a major rehabilitation as performed. This information crrently forms the basis for most of the decisions abot major rehabilitation, primarily for the Interstate system. Althogh this information is sefl in identifying pavement sections that exceed the minimm acceptable vales established by the state in any given year, it is not sefl in the process of selecting those miles that have the greatest need given a constraint on the amont of money available for major rehabilitation. For an effective management approach, it is necessary to have a mathematical model that can anser qestions sch as (_!.,±_) 1. Which specific pavement contract sections as ell as ho many miles of roads shold be rehabilitated dring a given year or dring the time frame specified ith available bdget? 2. What type of resrfacing strategy shold be applied to the pavement contract section selected in order to se the total available bdget in the most cost-effective manner? 3. Ho many additional lane-miles can be improved if the bdget is increased by a certain percentage? 4. Ho mch additional bdget is reqired to pgrade the pavement condition of the entire netork (or a part of it) to a minimm acceptable level? In the folloing paragraphs a systematic procedre, hich ses a mathematical model for allocating maintenance and rehabilitation fnds to existing pavements ithin the state of Indiana, is described. The reslts of applying the mathematical model to