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The minimization of error contained within the residual D ( N1 ) j D ( N1 ) j
vector {rk} is analogous to the reduction of error employed j D ( N ) j D ( N1 )
in least squared error analysis. The squared error between 1
j +1
D ( N ) j - j
actual output and predicted output is calculated by using D ( N 2 ) j
j
2
a mathematical model to determine the sensitivity of j D ( N 2 ) j D ( N 2 )
the weighting parameters for allocating the squared error. j +1 - j
It is possible to adjust the model parameters until there is j
no squared error remaining; however, because of the pres- D ( N i ) j D ( N i ) j
ence of random error, the values in the residual matrix j D ( N i ) j D ( N i )
{rk} should not be forced to zero (31). The elements in the
residual vector {rk} which represents errors between the D ( N1 ) - D ( N1 )
actual and model outputs are determined based on model D ( N1 )
parameters pi assuming at each iteration in the process, that D( N ) - D( N )
they are known values. The sensitivity matrix [Fki], which 2 2
reflects the sensitivity of the output from mathematical = D( N2 ) (13)
model fk to the assumed parameters pi, is also a known
value. Therefore, the unknown change vector {i} presents
the relative changes of the model parameters and is the tar- D( Ni ) - D( Ni )
D( Ni )
get vector to be determined in the process. Equation 9 can
be rewritten as:
where
{ i } = [ Fki
T
Fki ] [ Fki ] {rk }
-1 T
(10)
D(Ni) = crack length at Ni, calculated using j and j, and
D(Ni) = measured crack length at Ni.
The change vector {i} obtained using an initial assumption
of parameters, can then be used to determine the following The parameters and in the model were determined by
new set of parameters. iteration. The convergence criterion was set to 1.0 percent
(i.e., the iteration was repeated until the elements in the
change vector were less than 0.01).
pij +1 = pij (1 + 0.6 i ) (11) The percent crack length at each of the pavement ages
was used to develop the model parameters and in the
Where j is the iteration count. reflection cracking model along with the system identifica-
Solutions for the parameters in the model are found by min- tion process. Table 18 presents an example of the percent
imizing the change vector {i}. Therefore, the iteration process of reflection cracking development of all severity levels of
using Equation 10 is continued until there is no squared error four LTPP sections, which were calculated based on reflec-
remaining or the desired convergence is reached. Thus, the tion crack data in Table 18. Table 19 shows the developed
iteration was repeated until the elements in the change vector model parameters, and Figures 27 through 30 present plots
{i} are less than 0.01. of the compiled and measured data for four LTPP sections.
The calibrated parameters for all asphalt overlay test sec-
tions in the LTPP, New York City, and Texas are listed in
Calibrating Reflection Cracking Model Appendix M.
of Test Sections The model parameters and for the three levels of
distress are the "field data" which will be calibrated to
Based on the system identification and the parameter
the number of days for a crack to propagate through the
adjustment algorithm, the reflection cracking models were
calibrated using the data obtained from LTPP, New York overlay computed with the reflection cracking model. The
City, and Texas asphalt overlay test sections. The process was coefficients by which the different modes of crack propa-
used to fit the predicted crack length to the measured crack gation relate to these field derived model parameters are
length by iteration. Thus, the parameter adjustment algo- the "calibration coefficients" which define a particular
rithm of Equation 9 can be expressed as follows: application (pavement structure, climatic zone, region) of
the reflection cracking model. The and calibration coef-
ficients for all of the overlaid sections are tabulated in
[ F ]{} = {r } (12) Appendix M.

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Table 18. Reflection cracking development of L+M+H for
LTPP test sections.
Number of Days
Section No. Overlay Type % Crack Length
after Overlay
0 0
1,200 0
1,492 0
2,283 9.86
2,640 8.28
340503 AC/AC OL
3,004 20.41
3,367 28.12
3,757 29.59
4,121 45.92
4,247 55.78
0 0
52 0
641 28.76
1,110 30.84
270506 AC/Mill/AC OL 1,803 46.60
2,595 64.55
3,183 67.02
3,600 66.46
3,992 66.07
0 0
1,226 20.60
1,799 65.76
2,590 70.47
240563 AC/FC/AC OL 2,990 64.76
3,375 81.89
3,802 88.59
4,012 92.06
4,395 93.30
0 0
114 3.91
358 37.32
55B901 JRC/AC OL 869 75.44
2,499 93.86
3,760 95.14
4,410 99.12
Table 19. Calibrated model parameters of LTPP sections.
Model Parameters (L+M+H)
LTPP Section No. Overlay Type
340503 AC/AC 2.365 3,617.12
270506 AC/Mill/AC 0.702 1,004.85
240563 AC/FC/AC 2.276 1461.25
55B901 JRC/AC 1.159 329.42

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100
90 Measured Data
80 Model
70
% Crack Length
60
50
40
30
20
10
0
0 1000 2000 3000 4000 5000
No. of Days
Figure 27. Computed and measured reflection crack for LTPP
section 340503.
100
90
80
70
% Crack Length
60
50
40
30
20 Measured Data
10 Model
0
0 1000 2000 3000 4000 5000
No. of Days
Figure 28. Computed and measured reflection crack for LTPP
section 270506.
100
90
80
70
% Crack Length
60
50
40
30
20 Measured Data
10 Model
0
0 1000 2000 3000 4000 5000
No. of Days
Figure 29. Computed and measured reflection crack for LTPP
section 240563.