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Publication Number:  FHWA-HRT-15-036    Date:  December 2015
Publication Number: FHWA-HRT-15-036
Date: December 2015


Long-Term Pavement Performance Program Determination of In-Place Elastic Layer Modulus: Backcalculation Methodology and Procedures

Chapter 2. Backcalculation Methodology and Packages

This chapter summarizes the activities completed in phase I of this project.(13) It describes how specific backcalculation packages were selected as candidates to determine the elastic layer modulus values for all test sections included in the LTPP program.

Types of Backcalculation Methods

There are many programs and methods that can be used to estimate the in-place elastic modulus values of pavement structural layers from deflection basins. Hou developed one of the first solutions for backcalculation of elastic layer modulus values of more than two layers.(13) Hou’s approach is to search for the set of layer modulus values that minimize the sum of the squared differences between the calculated and measured deflections. Many backcalculation software programs use this general approach, matching measured deflections to deflections calculated with multilayer elastic theory.

The types of backcalculation methods can be grouped into five generalized categories, which are summarized in the following sections.

Iterative Search Methods

The iterative search method is based on the backcalculation program making repeated calls to an elastic layer subroutine to match the measured to calculated deflections for program-selected or derived layer moduli. The iteration process stops when the measured and calculated deflections are within a tolerance level or when the maximum number of iterations is reached. The user sets the tolerance and maximum number of iterations.

Lytton and Anderson provided a detailed description of two algorithms included in this category of programs.(15,16) The iterative search method involves selecting an initial set of layer modulus values. These modulus values are used to compute surface deflections, which are compared to the measured deflections. The assumed modulus values are adjusted, and the process is repeated until the calculated deflections match the measured deflections within a specified tolerance. Many backcalculation packages use this method, especially those that are used for production and research purposes.

EVERCALC© and MODCOMP© are two programs that fall within this category.(7,9) The MODCOMP© software package was used in the first round of backcalculation for all pavements.(4) Both EVERCALC© and MODCOMP© were selected for this study to backcalculate the elastic layer modulus for all of the LTPP sections.

Database Search Methods

Backcalculation by database search involves searching a database of calculated deflection basins generated for varying modulus values for specific pavement structure to find the calculated deflection basin that best matches the measured deflection basin. Database backcalculation programs work by generating a database of deflection basins for a matrix of elastic layer modulus values and either fixed layer thicknesses or a matrix of thicknesses and then searching the database for the deflection basin that most closely matches the measured basin.

The database search-based programs include MODULUS, COMDEF, and DBCONPAS, which were all developed about the same time. (See references 16–21 and 8.)

Equivalent Thickness Methods

Odemark presented an equivalent thickness method for analyzing deflections.(22) The equivalent thickness approach reduces a multilayer elastic system to an equivalent system with fewer layers (generally three or less) for which a solution is easily obtainable. Equivalent thickness-based methods use either the iterative or database search methods in finding a calculated deflection basin for a set of layer moduli that best match the measured basin.

Examples of equivalent thickness-based backcalculation programs include those developed by Ullidtz and by Lytton and Michalak.(23–25) Ullidtz’s program, Evaluation of Layer Moduli and Overlay Design (ELMOD©), is widely used since it is distributed by Dynatest® as part of a software package offered to purchasers of the Dynatest FWDs. ELMOD© is a proprietary software package.

Forward Calculation Methods or Closed-Form Solutions

Some methods use specific points of the measured deflection basin to directly calculate the modulus of limited layers. These methods are referred to as “forward” calculation methods and provide a unique solution for each deflection basin. The forward calculation method or closed-form solution was used during the first round of modulus determination for LTPP.(3)

Most of the closed-form solution methods are limited to three or fewer layers (including the subgrade or foundation)—a major disadvantage of these methods. This limitation makes individual layers with anomalies or defects difficult to identify because it requires use of the equivalent stiffness concept. More importantly, the error can be large between the measured and calculated deflections for the sensors excluded from the calculation process.

The Hogg model is an example of closed-form solutions for flexible pavements.(26) The area and best fit methods are a deviation from the elastic layer theory approach commonly used for flexible pavements. The area-based procedure for backcalculating the k-value from deflections for rigid pavement analyses was developed in the 1980s and enhanced in the 1990s. (See references 27–32.) The best fit method was used in the first round of backcalculation for rigid pavement structures and was also selected for use for this study.

Other Methods

Other methods that have been recently developed include the use of artificial neural networks (ANNs), genetic algorithms, and dynamic backcalculation methods.


DIPLOBACK is an example of one backcalculation package that uses ANN.(33) Software packages that use ANNs to complete the backcalculation in terms of pattern recognition similar to the procedure established by Lytton and Michalak through the SEARCH program.(25) Some researchers have developed neural network models using a large synthetic database generated from routine finite element analysis software programs, such as ILLI-SLAB for rigid pavements and ILLI-PAVE for flexible pavements. (See references 34–40.) Most ANN methods are confined to a small infra-space of pavements, and site features and are not used routinely by practitioners.

Genetic Algorithms

Multiple programs have been developed based on genetic algorithms. Zhang et al. developed a modified genetic algorithm called MGABPLM in 1998, which is based on the Homotopy method.(41) This category of backcalculation has been used very infrequently and only for research purposes.

Dynamic Analysis Methods

Dynamic analysis methods require the use of FWD deflection-time histories using frequency and time-domain solutions. DYNABACK-F is a program based on dynamic analyses.(42) This category of backcalculation packages is complex, requires a lot of data, and as such, is only used by a small group within the industry for research purposes.


Practitioners commonly use iterative and database search backcalculation packages for research, pavement evaluation, and rehabilitation design. Unfortunately, none of the programs that fall in these categories result in a unique set of modulus values for a specific pavement structure. Multiple combinations of elastic layer modulus values with similar errors can exist. This non-uniqueness of solutions is the major reason for debate on which method is the better one and which set of layer modulus values is the correct one. The closed form solution package ELMOD© is widely used, but it is a proprietary program. Moreover, a closed form solution (unique set of layer modulus values) does not necessarily imply a correct or accurate solution.

Candidate Backcalculation Programs

Many published documents have included a literature review and comparison of backcalculation methods. This section includes a brief review of the findings from other studies related to the selection and use of candidate backcalculation programs. It also identifies factors that can have a significant impact on the results and how those items can be handled as part of executing large batch files for production backcalculation.

Comparison of Backcalculation Programs

Rada et al. published a comparison of the features of several computer programs for backcalculation.(43) More recently, a European study tabulated the features of 20 different computer programs used for backcalculation.(44) Most backcalculation programs rely on
an elastic layer program to calculate surface deflections and use the iterative or database
search methods.

Von Quintus and Killingsworth used multiple software packages to backcalculate layer modulus from the same deflection basin data for selected LTPP test sections.(45–47) The following programs were included in this study:

These programs were evaluated on the basis of technical merit, functionality, and data processing compatibility using 18 test sections located in the LTPP southern region. Von Quintus and Killingsworth concluded that one software package should be used because of the large differences found between many of the programs.

Von Quintus and Simpson followed up with a similar study for selecting a package to backcalculate elastic layer modulus values for the deflection basins measured on the approach and leave ends of each LTPP test section.(4) The approach and leave ends were used because many cores were extracted from these locations for layer thickness determination after the first round of deflection basin measurements. Von Quintus and Simpson compared multiple packages using the following five evaluation factors:

They recommended the use of MODCOMP4.0© for backcalculating elastic layer moduli.

Generally, nearly all previous studies recommended one method over others, recommended multiple methods, or determined that all methods compared produced similar results. The following list summarizes the results from a few other projects that compared backcalculation programs:

The first backcalculation study sponsored by FHWA for flexible pavements used the software package MODCOMP4.0© for consistency of results between linear and nonlinear solutions.(4) MODCOMP4.0© was also used for rigid pavements along with a forward calculation procedure developed by Khazanovich.(3) MODCOMP4.0© was selected over EVERCALC©, MODULUS, and other programs because of its capability and versatility to consider different constitutive equations for unbound layers and soils. This same finding was reached in backcalculating layer elastic modulus values for multiple local calibration and commercial projects.(51)

Many agencies have backcalculated layer modulus values for the LTPP test sections located within their agency. States in which layer modulus values have been backcalculated from LTPP FWD deflection data include Arizona, Colorado, Michigan, Mississippi, Missouri, Montana, Ohio, Texas, Utah, Washington, Wisconsin, and Wyoming. EVERCALC©, MODULUS, and ELMOD© have been used most extensively for backcalculating in-place elastic layer modulus values used in the local calibration studies. However, MODCOMP© and MICHBACK© have also been used by selected agencies.(10) The MODCOMP© program was used to generate in-place layer modulus values for the global calibration effort of the MEPDG.(52)

FHWA sponsored two more recent studies related to the use of FWD deflection data and backcalculation of layer modulus data (Smith et al. and Rada et al.).(53,5) These two studies and reports are referred to throughout this report as the Smith and Rada studies.(53,5) Both studies include a review of FWD deflection data and backcalculation software packages for estimating in-place modulus values. The Smith study includes a more extensive review of backcalculation methods for both rigid and flexible pavements, while the Rada study was prepared using a complete set of deflection data from the LTPP SPS-1 experiment. These literature reviews also documented the programs that are being used by different agencies for rehabilitation, evaluation, and forensic investigations. Knowing the programs that agencies are using is important so that the results will be readily useable to as many agencies as possible.

While both the Smith and Rada studies provide a comparison of backcalculation programs, the Smith study includes a more detailed comparison of selected backcalculation packages.(53,5) Both of these studies used case studies from the LTPP program to evaluate and compare the results from different software packages. The summaries included in the Smith study and the earlier backcalculation projects provide an excellent comparison of the different software packages. The Rada study uses the MODULUS and MODCOMP© software packages and found that the MODULUS program resulted in many more acceptable solutions.(5) The Smith study documented the use of the MODCMOP, MICHBACK©, and EVERCALC© programs and found varying results.(53) The EVERCALC© program, however, consistently resulted in lower error terms.

Results from the two literature reviews confirmed the summary of available backcalculation programs that have already been summarized in the existing literature. Hall also completed a similar unpublished literature review of deflection data and backcalculation methods. Hall’s review included international projects and software packages that are available from outside the United States. These summaries and others list and compare the following:

In summary, many of the backcalculation procedures are similar, but the results can be different due to assumptions, iteration technique, backcalculation, or forward calculation schemes used within the programs. Some of these methods, such as MODCOMP© and EVERCALC©, also have the capability to estimate the elastic layer modulus values and coefficients of nonlinear constitutive relationships (stress-sensitivity properties). Specifically, these methods can be grouped in those with different constitutive models for describing the behavior between load and deflection or stress and strain: linear, quasi-nonlinear, and nonlinear backcalculated modulus values. Regardless, results from the backcalculation packages have been debated and questioned since their development in the mid-1970s. There is no consensus on the best procedure or the one providing the most reliable and accurate results.

Ground Truth Regarding Backcalculated Modulus Values

The deflection-based elastic moduli represent composite values and do not result in a unique set of elastic layer moduli. It is challenging to determine which set of elastic layer moduli represents the real values and which backcalculation package consistently results in those values. The following list contains some points that must be considered in explaining any discrepancy between the modulus values determined through different techniques and in defining an acceptable set of elastic layer moduli:

As an example, five asphalt concrete layers or lifts are shown in figure 1. These five individual layers result in different dynamic moduli. The upper three layers are less than 2 inches each, which are too thin for backcalculating the elastic moduli from deflection basins. The backcalculation process would combine all layers into one hot mix asphalt (HMA) layer for which an equivalent or composite elastic modulus would be determined from the deflection basins. One main issue is that these structures, which are typical, increase the variance between the laboratory measured and backcalculated modulus values. Some procedures combine the laboratory measured values into an equivalent modulus using the equivalent modulus concept.

It is difficult, if not impossible, to determine the ground truth values because of the simplifying assumptions used in all backcalculation programs. More importantly, there is significant debate within the pavement engineering community of what constitutes ground truth. Thus, the software packages that result in values consistent with the perceived realistic values and results between multiple programs were used in selecting the candidate packages to be used in the case studies as well as for the production runs.

Figure 1. Photo. Core recovered from an LTPP test section used to equate laboratory-measured modulus values to backcalculated elastic modulus values (Texas SPS-5 section). This photo shows a core recovered from a Long-Term Pavement Performance (LTPP) Texas Specific Pavement Studies (SPS)-5 test section. Five arrows point to the individual asphalt concrete layers of the core.

Figure 1. Photo. Core recovered from an LTPP test section used to equate laboratory-measured modulus values to backcalculated elastic modulus values (Texas SPS-5 section).

Figure 2. Photo. Cross-section of the pavement layers exposed during a forensic investigation to measure the rutting within individual pavement layers (Arizona SPS-1 section). This photo shows the sides of a trench through an asphalt concrete surface pavement from one of the Arizona Specific Pavement Studies (SPS)-1 sections. String lines are included that show the individual layers of the pavement. Arrows are also included with descriptors to designate the following individual layers (from the bottom to the top): coarse-grained subgrade soil layer with higher amount of fines and higher water content, 12- to 15-inch coarse-grained embankment layer, 4-inch aggregate base layer, 3- to 4-inch asphalt-stabilized base mixture, 2-inch dense-graded hot mix asphalt (HMA) binder layer, and 2-inch dense-graded HMA wearing surface.

Figure 2. Photo. Cross-section of the pavement layers exposed during a forensic investigation to measure the rutting within individual pavement layers (Arizona SPS-1 section).

Factors Considered in Selecting Candidate Programs

The following factors were considered in selecting candidate programs for use in the case studies:

Six programs were selected for use in the case studies for calculating layer stiffness properties from deflection basins on a production basis. The selection of these six programs was based on the results and recommendations from published documents and, to a minor degree, on the experience of the authors. Table 1 summarizes the features of the six packages selected for use in the case studies.

Table 1. Backcalculation software packages used in the case studies.
Feature of Packagea Backcalculation Software Packages/Methods
Backcalculation method Iterative Iterative (optimization) Raphson-Newton Iterative Database (optimization) Closed form solution
Convergence scheme Sum of squares of absolute error Sum of absolute error Least squares Relative deflection error at sensors Sum of relative squared error N/A
Non-linear analysis capability No Yes, limited No Yes No No
Visual image of basin convergence Yes Yes No No No N/A
Pavement type specific Rigid and flexible Rigid, flexible, and composite Flexible and composite Rigid, flexible, and composite Flexible Rigid and composite
Forward response calculation method Multilayer elastic theory Multilayer elastic theory Multilayer elastic theory Multilayer elastic theory Multilayer elastic theory Closed form solution
Ability to fix modulus Yes Yes Yes Yes Yes No
Maximum number of Layers 5 5 3 5 4 2
Layer interface analysisc Variable Fixed Fixed Fixed Fixed Fixed

N/A = Not applicable.
aForward calculation methods used to determine seed values for the backcalculation procedures are not listed in this table.
bThe best fit method was evaluated simultaneously, and the differences in the computed values as well as the feasibility to automate the procedure were evaluated. In addition, the forward calculation spreadsheets prepared using the Hogg model were used.(26)
cThe layer interface condition was selected to be consistent with the field distress development.

Standardization of Backcalculation Process

The first version of ASTMD 5858-96(2015), Standard Guide for Calculating In Situ Equivalent Elastic Moduli of Pavement Materials Using Layered Elastic Theory, was published in 1996 and last updated in 2008.(54) This guide presents the concepts for backcalculating pavement layer elastic modulus values from measured deflections using elastic layer theory. The guide does not address adjustments for load level, frequency, temperature, or seasonal variation. Since the backcalculation guidance provided in ASTM D5858-96 is based on elastic layer theory, it is applicable to flexible pavements and only to a limited extent to rigid pavements (i.e., interior loading and slab size/stiffness ratios less than 8). Neither ASTM International nor the American Association of State Highway and Transportation Officials (AASHTO) has approved further standards or guidance. As such, ASTM D5858-96 was followed in backcalculating elastic layer moduli.




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