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Measuring solar reflectance of variegated flat roofing materials using quasi-Monte Carlo method
Energy and Buildings 114 (2016) 234–240
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Energy and Buildings journal homepage: www.elsevier.com/locate/enbuild
Measuring solar reflectance of variegated flat roofing materials using quasi-Monte Carlo method Hamid Reza Hooshangi, Hashem Akbari ∗ , Ali G. Touchaei Concordia University, Canada
a r t i c l e
i n f o
Article history: Received 10 March 2015 Received in revised form 29 June 2015 Accepted 30 June 2015 Available online 4 July 2015 Keywords: Solar reflectance measuring Asphalt shingle Quasi-Monte Carlo Solar spectrum reflectometer
a b s t r a c t The Cool Roof Rating Council recommends applying the Monte Carlo (MC) method to ASTM standard C1549 to estimate the mean solar reflectance (R) of a variegated roofing sample. For samples with high degree of variation in the solar reflectance, the MC approach is slow in convergence. Applying proper set point of low-discrepancy sequences on the variegated roof sample can increase the convergence rate for about 40%. We measure solar reflectance of a variegated roofing sample (gridded to 1 × 1 cells) with C1549 and calculate R by averaging the measured solar reflectance. Then, we estimate R using MC and quasi-Monte Carlo (QMC) technique as a function of the number of random spots on the measured sample. To further investigate the performance of QMC, we analyze the sensitivity of the standard error of the mean reflectance of selected sample spots for a variety of simulated samples, where the range and distribution of the solar reflectance is varied. Based on the simulated and experimental results, we recommend using QMC for measuring the solar reflectance of variegated surfaces and propose an equation to estimate the required number of random spots as a function of the mix and the range of solar reflectance of samples. Crown Copyright © 2015 Published by Elsevier B.V. All rights reserved.
1. Introduction There are several instruments and standards for measuring and labeling solar reflectance and thermal emittance of roofing materials. To measure solar reflectance, pyranometer, portable solar reflectometer and spectrophotometer are widely used. ASTM E903-12: Standard Test Method for solar absorptance, reflectance, and transmittance of materials using integrating spheres [6] provides a standard for measuring near-normal beam-hemispherical spectral reflectance using a spectrophotometer equipped with an integrating sphere. Levinson et al. [17] summarized several limitations related to measurements using the spectrophotometer. For instance, the commercially available spectrophotometers can only illuminate an area of about 10 mm2 of a sample. Also the sample could not be larger than 150 cm2 to properly fit in instrument’s port. ASTM E1918-06: Standard Test Method for measuring solar reflectance of horizontal and low-sloped surfaces in the field [4], uses a pyranometer for measuring global solar reflectance of flat
∗ Corresponding author. E-mail address: [email protected] (H. Akbari). http://dx.doi.org/10.1016/j.enbuild.2015.06.073 0378-7788/Crown Copyright © 2015 Published by Elsevier B.V. All rights reserved.
or rough horizontal and low sloped surfaces with area of at least 10 m2 . However, the need for clear sky and certain range of solar zenith angles (<45) limits the use of ASTM E1918. Solar reflectance of roofing products are also measured by portable solar reflectometer. Unlike pyranometer which the reflectance of entire sample will be measured, the reflectance of surfaces is estimated by the reflectance of a considerably small area of the surface that is measured by solar reflectometer. In addition, the light source of solar reflectometer is independent from sky condition [17,18]. In comparison with spectrophotometer, this laboratory instrument can measure an area of 6.5 cm2 in much shorter time. Measurements by solar reflectometer can be conducted based on ASTM C1549-09: Standard Test Method for determination of solar reflectance near ambient temperature using a portable solar reflectometer [5]. ASTM C1549-09 is applicable for measuring the reflectance of homogeneous samples. Solar reflectance of nonuniform or variegated surfaces cannot be estimated by measuring one spot. Hence, statistical methods are required to apply for estimating solar reflectance, or solar absorptance of variegated roofing materials [23]. Therefore, the reflectance of a sufficient area should be measured to estimate the reflectance of sample (R) based on the reflectance of a measured area (r). The acceptable measurement
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area depends on the level of variegation of a sample, the method to select sample spots, and reflectance estimation accuracy. CRRC has developed a method [3], to measure the solar reflectance of flat (or nearly-flat) variegated roofing materials using C1549-09. This method requires measuring solar reflectance of several non-overlapped random spots (Monte Carlo method) to estimate the mean solar reflectance of the sample (R). ANSI/CRRC S100 suggests selection of 30 random spots for this process; however, for some samples, this selected number of random points does not yield the required accuracy. The objective of this study is to investigate the application of Quasi-Monte Carlo (QMC) method on estimating the reflectance of variegated or binary samples. QMC samplings technique is based on the quasi-random or deterministic points. In other words, the deterministic version of MC method is called QMC method. The application of MC and QMC is tested on a sample of a fiberglass asphalt shingle. We further compare the convergence rate of MC and QMC methods for a variety of simulated samples with different levels of variegation. We provide an equation to estimate the required number of random spots as a function of the range and distribution of the solar reflectance of spots. 2. Theory 2.1. Monte Carlo methods Monte Carlo (MC) techniques are powerful methods to highdimensional numerical integration. Random sampling is the basis of Monte Carlo method. In MC method, random quantities are used to estimate the law of large numbers. Applying proper transformations, in many cases, make the integration domain as the s-dimensional unit cube (Is := [0, 1]s ). In addition to this assumption, we presume that “integrand f is square integrable over Is ” [20]. So according to Monte Carlo approximation, Eq. (1) is accepted for the integral.
1 f (xn ) N N
f (u)du ≈ Is
(1)
n=1
Is
1 f (xn ) N N
f (u)du=lim N→∞
(2)
n=1
Acworth et al. [1] prove that the square of the error in Eq. (1) is equal to 2 (f)N−1 ( 2 (f) is the variance of f). Hence, with overwhelming probability we have [20]:
Is
1 1 f (xn ) = 0(N − 2 ) N
N
f (u)du −
n=1
numbers are sufficiently large [16]. In this study, the random spot selected, using MATLAB rand function. Second issue regarding to application of MC is related to the probabilistic error bounds for numerical integration. The last, the convergence rate is too low in many applications. 2.1.1. Application of MC in measuring solar reflectance of variegated roofing materials ASTM C1549-09 is applicable to solar reflectance measurements of uniform surfaces. To measure the solar reflectance of variegated flat surfaces that have significantly higher variation of solar reflectance, [2] applied a statistical Monte Carlo (MC) method. The proposed technique estimates the mean solar reflectance of sample (R) by averaging the measured solar reflectance of several random non-overlapped spots (¯r ). Appendix A provides a summary of the MC method and resulted error of estimated mean solar reflectance. The method is used by Cool Roof Rating Council (CRRC) for measuring and labeling solar reflectance of flat or almostflat variegated roofing materials. This method is called ANSI/CRRC S100: Standard Test Methods for determining radiative properties of materials [10]. Based on ANSI/CRRC S100, at least 30 random non-overlapped spots of a sample should be selected and measurements should be performed in accordance with ASTM C1549-09. Then the mean standard error (ε) of the estimated average solar reflectance, r, is calculated by Eq. (4). ε≈ √ n
(4)
where represents the standard deviation of a set of spots and n is a number of random spots. According to current CRRC requirement of measuring the solar reflectance if ε is equal or less than 0.005 then r can represent R of sample with ±0.01 accuracy (with the probability of 95%). Otherwise, the number of random spots must be increased. This procedure should be repeated until the mean standard error of spots does not exceed 0.005. 2.2. Quasi-Monte Carlo method
where x1 ,. . ., xN represent the independent random spots selected from the uniform distribution on Is . Based on the law of large numbers, we have Eq. (2). It indicates that for “almost all” set of sample points, we converge almost surely by using Monte Carlo method.
235
(3)
√ Eq. (3) states that the error of MC is of order 1/ n. It can be con4 cluded that we need about 10 sample points to reach the error tolerance of 10−2 . Also, Eq. (3) asserts that convergence rate is not a function of dimension s. It is the main advantage of MC method for high-dimensional problems. On the other hand, there are several disadvantageous in using MC method. First of all, truly random selection is of crucial importance in MC method. Nevertheless generating perfect random numbers is difficult and cannot be generated through an artificial (computer) means [16]. Computer programs generate pseudorandom numbers; rather truly random numbers. However, they can satisfy standard statistical tests especially when the generated
Quasi-Monte Carlo (QMC) method was introduced to address the drawbacks of MC method. MC method implements the random sample points to estimate the law of large numbers. However, QMC samplings technique is based on the quasi-random or deterministic points. In other words, the deterministic version of MC method is called QMC method. We seek deterministic points with good distribution properties that can increase the convergence rate. A low-discrepancy point sets or sequences are correlated to provide greater uniformity [11]. An ideal form of quasi-random points are low-discrepancy sequences. Using low-discrepancy sequences can significantly increase the convergence rate compared to the convergence rate of MC (Eq. (3)). It is verified by [15] that the order of convergence is reduced to N−1 (logN)s−1 for s = 1 and s = 2. Therefore, in many cases of integrals, the QMC method will surpass the MC method. Moreover, answers produced by QMC typically are more accurate than MC approach [22]. Over the years, different low-discrepancy sequences have been derived; mostly linked to the van der Corput sequence. Various experimental studies have been performed to evaluate the performance of QMC sequences [8,25,9]. It was stated that the sample size should be sufficiently large to QMC outperforms the MC. The Holton and Sobol [12,24] sequences are well-known lowdiscrepancy sequences. The outstanding features of Halton and Sobol sequences distinguish them from other quasi-random set points. The Halton sequence can produce uniform distribution for lower number of dimensions (1–10). Also, increasing s (number of
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Fig. 1. Santa Fe variegated asphalt shingle.
dimensions) does not impact the uniformity of generated numbers by Sobol sequence [14]. In addition, there are several approaches to improve the uniformity of Halton and Sobol sequences. One is to permutate the radical inverse coefficients calculated by implementing a reverse-radix operation to all of the possible coefficient values [14]. This approach is also known as scrambled method. The scramble for Halton sequences is proposed by [8]. Further analysis regarding permutation were studied by Hellekalek [13]. Matouˇsek [19] investigated the quality of uniform distribution of low- discrepancy sequences. In addition, he recommends a fairly precise random linear scramble combined with a random digital shift to increase the uniformity of Sobol set points. In this study, we applied Sobol scrambled version introduced by Owen [21] and improved by Matouˇsek [19]. Our investigations indicate that the uniformity in low number generations (less than 1000) in dimensions less than 100 can be significantly improved by applying formula in [19].The other approach to increase the uniformity is based on eliminating some numbers from set points. We applied the Halton Sequence Leaped method proposed by Kocis and Whiten [14]. They investigate the effect of eliminating Lth Halton number from sequence. They prove that the best leaps values are 31, 61, 149, 409 and 1949, for s and N of 1–400 and 10–105 , respectively. The most satisfying leap for Sobol sequence is found to be L = 2m where m is a positive integer. Finally, Kocis and Whiten [14] concluded that applying the modifications on QMC sequences can reduce the maximum error of QMC to a bit worse than 1/N, for a large number of dimensions. In this study, we compare the application of MC method using random numbers with both low-discrepancy methods in selecting spots for measuring and estimating the overall solar reflectance of binary and variegated surfaces. 2.3. Calculation of standard error of the mean reflectance for simulated samples The standard deviation (Eq. (5)) shows the variation or dispersion of data from the mean, where n is the number of data points, r¯ n describes the mean solar reflectance of the samples, and ri is the reflectance of spot i. s represents the standard deviation of population.
s=
1 n (ri − r¯ n )2 n i=1
(5)
If the sample consists of only black and white spots then Eq. (5) is modified to:
s=
Nb (Rb − r¯ n )2 + Nw (Rw − r¯ n )2
r¯n = Nb (Rb ) + Nw (Rw )
(6) (7)
Nb + Nw = 1
(8)
Combining Eqs. (5)–(8) leads to:
(Rw − Rb ) (Nb Nw ) ε= √ n
(9)
3. Methodology We measure the solar reflectance of all 418 spots, of a commercial sample, according to ASTM C1549-09 using the solar reflectometer V.6. Then, we calculate the mean solar reflectance (R) by averaging the measured solar reflectance of all spots. Second, we apply the ANSI/CRRC S100 (i.e. MC method) to estimate mean solar reflectance of the sample. Third, we investigate different scenarios of selecting spots using QMC. Finally, we repeat our analysis for several simulated variegated binary samples (combination of spots with high and low solar reflectance). For application of MC, we initially select 30 random spots. We measure the average solar reflectance and the standard error of estimated mean solar reflectance. If the standard error of estimated mean solar reflectance is less than or equal to the selected criteria, we terminate the process and record the number of measurements that led to the convergence (initially 30). Otherwise, we add non-repeating random spot and continue the process. We increase the number of additional spots until convergence is achieved. We repeat the entire process for each approach 100 times. 3.1. Variegated asphalt shingle The solar reflectance of Santa Fe asphalt shingle (Fig. 1) manufactured by CertainTeed Corporation was measured based on ANSI/CRRC S100 in conjunction with C1549-09. The total area of Santa Fe asphalt shingle is 0.294 m2 (456 in.2 ). We neglect the measurements on the horizontal gap where the two courses are laminated. This gap causes a significant displacement of the probe on the sample; resulting in a significant error caused by light escaped from probe and improperly illumination of a sample. The total measured area (excluding the overlaps) is 0.270 m2 (418 in.2 ), which satisfies the minimum requirement of ANSI/CRRC S100 (360 in2 ). The sample is gridded to 1 × 1 cells (Fig. 2). 3.2. Simulated samples We simulate 35 variegated samples (each one with area of 5.16 m2 or 8000 in.2 ) consisting of black and white spots. We randomly distribute the white and black spots with solar reflectance of Rw and Rb , respectively, to resemble the binary variegated surfaces. The object of this study is to investigate selection methods to reach a faster convergence rate. So the effect of surface roughness
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Table 2 Statistics of solar reflectance measurements using SSR V.6 and selection G1. Mean (R) Minimum Maximum Median Standard deviation
0.354 0.250 0.459 0.350 0.041
Fig. 2. Portable solar spectrum reflectometer model SSR V.6.
Table 1 Characteristics of black and white simulated samples. Samples
Rb
Rw
Range
#1 to #5 #6 to #10 #11 to #15 #16 to #20 #21 to #25 #26 to #30 #31 to #35
0.425 0.40 0.55 0.60 0.70 0.80 0.90
0.475 0.50 0.35 0.30 0.20 0.10 0.04
0.05 0.10 0.20 0.30 0.50 0.70 0.86
(e.g., size of granules and texture; [7]) on solar reflectance is not considered. The samples are categorized into 5 ranges of solar reflectance as shown in Table 1. The range (RNG) defined as the difference between Rw and Rb . The black area to total area ratio (Nb ) of samples is specified as 10%, 20%, 30%, 40% and 50% for each range. Each row shows the samples as fraction of dark areas that it increases from 0.1 to 0.5.
Fig. 3. Solar reflectance distribution based on SSR V.6 measurements.
Fig. 4. Monte Carlo simulation for calculating R − r¯ and ε of variegated sample based on SSR V.6 measured data (the number of trials is 100).
4. Results 4.1. Measured solar reflectance of the fiberglass asphalt shingle
4.2. Monte Carlo estimate of solar reflectance of the fiberglass asphalt shingle
The statistics of measured solar reflectance for the fiberglass asphalt shingle is shown in Table 2. The calculated average solar reflectance is 0.353 according to SSR V.6 (selection G1). The range and standard deviation was calculated as 0.2 and 0.04, respectively. The distribution of measured solar reflectance is shown in Fig. 3.
By the central limit theorem, the mean solar reflectance of n measured spots (r) (Eq. (A1)) is fluctuates about a sample mean solar reflectance (R). If n is large enough we can assume r¯ is equal to R (Eq. (A2)). Fig. 4 illustrates how quickly the sample mean (¯r ) converges to the population mean (R), with the error tolerance of
Fig. 5. Monte Carlo simulation for ε calculation of variegated sample. The results are shown in an interval of five numbers. The number of trials is 100 for each ε calculation.
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Fig. 6. Sample points generated by (a, b) random function of MATLAB, (c) Halton Leaped (s: 400, L: 31), (d) Halton Leaped (s: 400, L: 61), (e) Sobol scrambled (s: 100), and (f) Sobol scrambled (s: 30).
10−2 . In addition, the convergence rate of the standard error of the mean (ε) is also shown. Fig. 4 elaborates that convergence rate of method implemented by ANSI/CRRC S100 (i.e., Internal Estimate or Confidence Interval approach) is significantly low compared to R−¯r . The main reason behind this fact is that the mean standard error is calculated based on the estimated values (r¯n ) (Eq. (A5)). Hence the confidence level of 95% is considered in ANSI/CRRC S100. The convergence rate of R − r¯ ≤ 0.01 is more likely to ε = 0.01 performance in Fig. 5. We also investigated the effect of increasing the standard error of the mean on convergence rate. Fig. 5 indicates that increasing ε from 0.005 to ε = 0.0075 and ε = 0.01 can significantly reduce the required numbers of spots. The average number of random spots for reaching ε = 0.005, ε = 0.0075 and ε = 0.01 based on SSR V.6 measurements is 65, 30 and 20, respectively. 4.3. Quasi-Monte Carlo estimate of solar reflectance of the fiberglass asphalt shingle We compare the application of MC with the Halton and Sobol low-discrepancy methods regarding to number of required random spots for estimating the mean solar reflectance of variegated asphalt shingle. We implement MATLAB Quasi-random generator (Halton and Sobol sequences) and rand function to produce the location of 80 spots in Fig. 6. We calculate the standard error of the mean reflectance by considering the solar reflectance of the spots determined by QMC methods. Fig. 7 illustrates that applying QMC sampling methods can decreases the number of required measured spots. The average number of spots to reach ε = 0.005 is reduced from nearly 65 (MC method) to 41 and 34 by applying Halton and Sobol sequences, respectively. The rate of convergence for MC is of order 1/N0.5 . Implementing Halton and Sobol sequences rather than random set points can increase the convergence rate about 35% and 45%, respectively. It is strongly recommended to implement formula provided by Matouˇsek [19] to generate Sobol sequence. The classic Sobol sequence [24] is not perform well in low number generations. The modified Halton sequence by Kocis & Whiten [14] significantly
Fig. 7. Quasi-Monte Carlo simulations for calculation of variegated sample, (a) Sobol sequences (Scrambled and Scrambled-Leaped), (b) Halton (ScrambledLeaped).
surpass the original construction of Halton [12]. It is worth to mention that our investigations verify that considering leaps of 31 and 61 performs much better than L = 149 and 409 for low number generation.
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Fig. 8. Minimum, 10%, 1st quartile, median, mean, 3rd quartile for different number of selected spots (N) (sample with range = 0.2 and Nb = Nw = 0.5). The dots show individual data for less that 10% and greater than 90%. The dashed lines show the acceptable standard error of ±0.005. The range of mean standard error, derived from Eq. (7), are shown in adjacent to box and whisker plots.
Fig. 9. Calculated mean standard error using QMC methods.
4.4. MC and QMC application for estimating number of required spots for simulated samples
Quasi-Monte Carlo (QMC) method can increase the convergence rate. We measured solar reflectance of a variegated roofing sample based on the C1549-09 (the sample is gridded to 1 × 1 cells and spots with diameter of 1 on cells). We calculated R by averaging the measured solar reflectance of all spots. Then, we estimated R using MC and QMC as a function of the number of random spots on our sample. We compared the convergence rate of MC and QMC. We analyzed the sensitivity of standard error of the mean reflectance of selected spots for a variety of simulated samples where the range and distribution of the solar reflectance is varied. QMC typically requires about 20% to 50% less random spots compared to MC to converge to the required standard error of estimate. For application of MC approach, we propose an equation to estimate the required number of random spots as a function of mix and range of solar reflectance of samples.
Acknowledgements We investigated the application of MC and QMC methods to estimate the convergence rate of simulated samples that was described in Section 3.2. Lower variation and range of solar reflectance increases the convergence rate of both MC and QMC. Fig. 8 shows the result of analysis for the simulated sample with equal areas covered in high and low reflectance spots (Nb = Nw = 0.5). For the MC approach about 400 random points are needed to achieve the required criteria. Note that Eq. (7) estimation of the number of required random points compared well with the results of the MC estimates. Knowing the product mix and the range of solar reflectance over the sample, Eq. (7) can be used to calculate n (the number of required random spots for a selected criteria, e.g., ε = 0.005). We applied the QMC based on Sobol and Halton sequences on the above simulated sample (RNG = 0.2 and Nb = Nw = 0.5) and the results are shown in Fig. 9. It shows that we can reach to the acceptable ε by using 283 spots rather than 400 needed spots by using MC. Further analysis is necessary to reach a conclusion on the performance of QMC. 5. Conclusions The Cool Roof Rating Council (CRRC) recommends using the Monte Carlo (MC) method to estimate the mean solar reflectance (R) of the variegated roofing samples. The MC approach is slow in converging to the solar reflectance of sample. For samples with high degree of variation in the solar reflectance, the convergence rate is slower than typical variegated materials and requires a large sample size to estimate R with ±0.01 required accuracy. Using
This research was in part funded by a grant from the Cool Roof Rating Council (CRRC) and a scholarship from the Huntsman Corporation. The authors appreciate their support throughout this project. The writing of this paper was supported by a grant from the Natural Resources and Engineering Council of Canada (NSERC).
Appendix A. By the central limit theorem, the mean solar reflectance of n measured spots (¯r ) (Eq. (A1)) is fluctuates about a sample mean solar reflectance (R). If n is large enough we can assume r¯ is equal to R (Eq. (A2)). The selection of spots is based on MC approach. The cells should not be overlapped or selected more than one time because cells have to have equal probability. Therefore the probability of each spot is the same as other spots. 1 ri N N
r≡
(A1)
i=1
R ≡ A−1
rdA
(A2)
A
where A represent the area of sample. ANSI/CRRC S100 employed the Internal Estimate or Confidence Interval approach as the statistical method to estimate the mean solar reflectance of sample (R) (Eq. (A3)). Also, the estimated solar
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reflectance (r) is calculated through averaging solar reflectance of selected spots based on the MC approach. R = r ± zε
(A3)
where ε is the standard error of mean solar reflectance (Eq. (A4)), and z is defined as random variable and derived from standard normal probability table. ε≈ √ n
(A4)
is standard deviation of population (Eq. (A5)). If n is higher than 30, we can assume the standard deviation of measured spots is well approximated the standard deviation of population.
n 1 s≡
(ri − r¯n )2 n−1
(A5)
i=1
The values of z is determined by how confident we want estimate R. In applied practice, such as ANSI/CRRC S100, the confidence level of 95% is typically stated. It means in 95% of the calculated r, the difference in values between R and r¯ is less than or equal to zε [26]. The z value for confidence level of 95% is 1.96 or approximately 2, as being employed by ANSI/CRRC S100 (Eq. (A6)). R = r ± 2ε
(A6)
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Measuring solar reflectance of variegated flat roofing materials using quasi-Monte Carlo method Hamid Reza Hooshangi, Hashem Akbari ∗ , Ali G. Touchaei Concordia University, Canada
a r t i c l e
i n f o
Article history: Received 10 March 2015 Received in revised form 29 June 2015 Accepted 30 June 2015 Available online 4 July 2015 Keywords: Solar reflectance measuring Asphalt shingle Quasi-Monte Carlo Solar spectrum reflectometer
a b s t r a c t The Cool Roof Rating Council recommends applying the Monte Carlo (MC) method to ASTM standard C1549 to estimate the mean solar reflectance (R) of a variegated roofing sample. For samples with high degree of variation in the solar reflectance, the MC approach is slow in convergence. Applying proper set point of low-discrepancy sequences on the variegated roof sample can increase the convergence rate for about 40%. We measure solar reflectance of a variegated roofing sample (gridded to 1 × 1 cells) with C1549 and calculate R by averaging the measured solar reflectance. Then, we estimate R using MC and quasi-Monte Carlo (QMC) technique as a function of the number of random spots on the measured sample. To further investigate the performance of QMC, we analyze the sensitivity of the standard error of the mean reflectance of selected sample spots for a variety of simulated samples, where the range and distribution of the solar reflectance is varied. Based on the simulated and experimental results, we recommend using QMC for measuring the solar reflectance of variegated surfaces and propose an equation to estimate the required number of random spots as a function of the mix and the range of solar reflectance of samples. Crown Copyright © 2015 Published by Elsevier B.V. All rights reserved.
1. Introduction There are several instruments and standards for measuring and labeling solar reflectance and thermal emittance of roofing materials. To measure solar reflectance, pyranometer, portable solar reflectometer and spectrophotometer are widely used. ASTM E903-12: Standard Test Method for solar absorptance, reflectance, and transmittance of materials using integrating spheres [6] provides a standard for measuring near-normal beam-hemispherical spectral reflectance using a spectrophotometer equipped with an integrating sphere. Levinson et al. [17] summarized several limitations related to measurements using the spectrophotometer. For instance, the commercially available spectrophotometers can only illuminate an area of about 10 mm2 of a sample. Also the sample could not be larger than 150 cm2 to properly fit in instrument’s port. ASTM E1918-06: Standard Test Method for measuring solar reflectance of horizontal and low-sloped surfaces in the field [4], uses a pyranometer for measuring global solar reflectance of flat
∗ Corresponding author. E-mail address: [email protected] (H. Akbari). http://dx.doi.org/10.1016/j.enbuild.2015.06.073 0378-7788/Crown Copyright © 2015 Published by Elsevier B.V. All rights reserved.
or rough horizontal and low sloped surfaces with area of at least 10 m2 . However, the need for clear sky and certain range of solar zenith angles (<45) limits the use of ASTM E1918. Solar reflectance of roofing products are also measured by portable solar reflectometer. Unlike pyranometer which the reflectance of entire sample will be measured, the reflectance of surfaces is estimated by the reflectance of a considerably small area of the surface that is measured by solar reflectometer. In addition, the light source of solar reflectometer is independent from sky condition [17,18]. In comparison with spectrophotometer, this laboratory instrument can measure an area of 6.5 cm2 in much shorter time. Measurements by solar reflectometer can be conducted based on ASTM C1549-09: Standard Test Method for determination of solar reflectance near ambient temperature using a portable solar reflectometer [5]. ASTM C1549-09 is applicable for measuring the reflectance of homogeneous samples. Solar reflectance of nonuniform or variegated surfaces cannot be estimated by measuring one spot. Hence, statistical methods are required to apply for estimating solar reflectance, or solar absorptance of variegated roofing materials [23]. Therefore, the reflectance of a sufficient area should be measured to estimate the reflectance of sample (R) based on the reflectance of a measured area (r). The acceptable measurement
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area depends on the level of variegation of a sample, the method to select sample spots, and reflectance estimation accuracy. CRRC has developed a method [3], to measure the solar reflectance of flat (or nearly-flat) variegated roofing materials using C1549-09. This method requires measuring solar reflectance of several non-overlapped random spots (Monte Carlo method) to estimate the mean solar reflectance of the sample (R). ANSI/CRRC S100 suggests selection of 30 random spots for this process; however, for some samples, this selected number of random points does not yield the required accuracy. The objective of this study is to investigate the application of Quasi-Monte Carlo (QMC) method on estimating the reflectance of variegated or binary samples. QMC samplings technique is based on the quasi-random or deterministic points. In other words, the deterministic version of MC method is called QMC method. The application of MC and QMC is tested on a sample of a fiberglass asphalt shingle. We further compare the convergence rate of MC and QMC methods for a variety of simulated samples with different levels of variegation. We provide an equation to estimate the required number of random spots as a function of the range and distribution of the solar reflectance of spots. 2. Theory 2.1. Monte Carlo methods Monte Carlo (MC) techniques are powerful methods to highdimensional numerical integration. Random sampling is the basis of Monte Carlo method. In MC method, random quantities are used to estimate the law of large numbers. Applying proper transformations, in many cases, make the integration domain as the s-dimensional unit cube (Is := [0, 1]s ). In addition to this assumption, we presume that “integrand f is square integrable over Is ” [20]. So according to Monte Carlo approximation, Eq. (1) is accepted for the integral.
1 f (xn ) N N
f (u)du ≈ Is
(1)
n=1
Is
1 f (xn ) N N
f (u)du=lim N→∞
(2)
n=1
Acworth et al. [1] prove that the square of the error in Eq. (1) is equal to 2 (f)N−1 ( 2 (f) is the variance of f). Hence, with overwhelming probability we have [20]:
Is
1 1 f (xn ) = 0(N − 2 ) N
N
f (u)du −
n=1
numbers are sufficiently large [16]. In this study, the random spot selected, using MATLAB rand function. Second issue regarding to application of MC is related to the probabilistic error bounds for numerical integration. The last, the convergence rate is too low in many applications. 2.1.1. Application of MC in measuring solar reflectance of variegated roofing materials ASTM C1549-09 is applicable to solar reflectance measurements of uniform surfaces. To measure the solar reflectance of variegated flat surfaces that have significantly higher variation of solar reflectance, [2] applied a statistical Monte Carlo (MC) method. The proposed technique estimates the mean solar reflectance of sample (R) by averaging the measured solar reflectance of several random non-overlapped spots (¯r ). Appendix A provides a summary of the MC method and resulted error of estimated mean solar reflectance. The method is used by Cool Roof Rating Council (CRRC) for measuring and labeling solar reflectance of flat or almostflat variegated roofing materials. This method is called ANSI/CRRC S100: Standard Test Methods for determining radiative properties of materials [10]. Based on ANSI/CRRC S100, at least 30 random non-overlapped spots of a sample should be selected and measurements should be performed in accordance with ASTM C1549-09. Then the mean standard error (ε) of the estimated average solar reflectance, r, is calculated by Eq. (4). ε≈ √ n
(4)
where represents the standard deviation of a set of spots and n is a number of random spots. According to current CRRC requirement of measuring the solar reflectance if ε is equal or less than 0.005 then r can represent R of sample with ±0.01 accuracy (with the probability of 95%). Otherwise, the number of random spots must be increased. This procedure should be repeated until the mean standard error of spots does not exceed 0.005. 2.2. Quasi-Monte Carlo method
where x1 ,. . ., xN represent the independent random spots selected from the uniform distribution on Is . Based on the law of large numbers, we have Eq. (2). It indicates that for “almost all” set of sample points, we converge almost surely by using Monte Carlo method.
235
(3)
√ Eq. (3) states that the error of MC is of order 1/ n. It can be con4 cluded that we need about 10 sample points to reach the error tolerance of 10−2 . Also, Eq. (3) asserts that convergence rate is not a function of dimension s. It is the main advantage of MC method for high-dimensional problems. On the other hand, there are several disadvantageous in using MC method. First of all, truly random selection is of crucial importance in MC method. Nevertheless generating perfect random numbers is difficult and cannot be generated through an artificial (computer) means [16]. Computer programs generate pseudorandom numbers; rather truly random numbers. However, they can satisfy standard statistical tests especially when the generated
Quasi-Monte Carlo (QMC) method was introduced to address the drawbacks of MC method. MC method implements the random sample points to estimate the law of large numbers. However, QMC samplings technique is based on the quasi-random or deterministic points. In other words, the deterministic version of MC method is called QMC method. We seek deterministic points with good distribution properties that can increase the convergence rate. A low-discrepancy point sets or sequences are correlated to provide greater uniformity [11]. An ideal form of quasi-random points are low-discrepancy sequences. Using low-discrepancy sequences can significantly increase the convergence rate compared to the convergence rate of MC (Eq. (3)). It is verified by [15] that the order of convergence is reduced to N−1 (logN)s−1 for s = 1 and s = 2. Therefore, in many cases of integrals, the QMC method will surpass the MC method. Moreover, answers produced by QMC typically are more accurate than MC approach [22]. Over the years, different low-discrepancy sequences have been derived; mostly linked to the van der Corput sequence. Various experimental studies have been performed to evaluate the performance of QMC sequences [8,25,9]. It was stated that the sample size should be sufficiently large to QMC outperforms the MC. The Holton and Sobol [12,24] sequences are well-known lowdiscrepancy sequences. The outstanding features of Halton and Sobol sequences distinguish them from other quasi-random set points. The Halton sequence can produce uniform distribution for lower number of dimensions (1–10). Also, increasing s (number of
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Fig. 1. Santa Fe variegated asphalt shingle.
dimensions) does not impact the uniformity of generated numbers by Sobol sequence [14]. In addition, there are several approaches to improve the uniformity of Halton and Sobol sequences. One is to permutate the radical inverse coefficients calculated by implementing a reverse-radix operation to all of the possible coefficient values [14]. This approach is also known as scrambled method. The scramble for Halton sequences is proposed by [8]. Further analysis regarding permutation were studied by Hellekalek [13]. Matouˇsek [19] investigated the quality of uniform distribution of low- discrepancy sequences. In addition, he recommends a fairly precise random linear scramble combined with a random digital shift to increase the uniformity of Sobol set points. In this study, we applied Sobol scrambled version introduced by Owen [21] and improved by Matouˇsek [19]. Our investigations indicate that the uniformity in low number generations (less than 1000) in dimensions less than 100 can be significantly improved by applying formula in [19].The other approach to increase the uniformity is based on eliminating some numbers from set points. We applied the Halton Sequence Leaped method proposed by Kocis and Whiten [14]. They investigate the effect of eliminating Lth Halton number from sequence. They prove that the best leaps values are 31, 61, 149, 409 and 1949, for s and N of 1–400 and 10–105 , respectively. The most satisfying leap for Sobol sequence is found to be L = 2m where m is a positive integer. Finally, Kocis and Whiten [14] concluded that applying the modifications on QMC sequences can reduce the maximum error of QMC to a bit worse than 1/N, for a large number of dimensions. In this study, we compare the application of MC method using random numbers with both low-discrepancy methods in selecting spots for measuring and estimating the overall solar reflectance of binary and variegated surfaces. 2.3. Calculation of standard error of the mean reflectance for simulated samples The standard deviation (Eq. (5)) shows the variation or dispersion of data from the mean, where n is the number of data points, r¯ n describes the mean solar reflectance of the samples, and ri is the reflectance of spot i. s represents the standard deviation of population.
s=
1 n (ri − r¯ n )2 n i=1
(5)
If the sample consists of only black and white spots then Eq. (5) is modified to:
s=
Nb (Rb − r¯ n )2 + Nw (Rw − r¯ n )2
r¯n = Nb (Rb ) + Nw (Rw )
(6) (7)
Nb + Nw = 1
(8)
Combining Eqs. (5)–(8) leads to:
(Rw − Rb ) (Nb Nw ) ε= √ n
(9)
3. Methodology We measure the solar reflectance of all 418 spots, of a commercial sample, according to ASTM C1549-09 using the solar reflectometer V.6. Then, we calculate the mean solar reflectance (R) by averaging the measured solar reflectance of all spots. Second, we apply the ANSI/CRRC S100 (i.e. MC method) to estimate mean solar reflectance of the sample. Third, we investigate different scenarios of selecting spots using QMC. Finally, we repeat our analysis for several simulated variegated binary samples (combination of spots with high and low solar reflectance). For application of MC, we initially select 30 random spots. We measure the average solar reflectance and the standard error of estimated mean solar reflectance. If the standard error of estimated mean solar reflectance is less than or equal to the selected criteria, we terminate the process and record the number of measurements that led to the convergence (initially 30). Otherwise, we add non-repeating random spot and continue the process. We increase the number of additional spots until convergence is achieved. We repeat the entire process for each approach 100 times. 3.1. Variegated asphalt shingle The solar reflectance of Santa Fe asphalt shingle (Fig. 1) manufactured by CertainTeed Corporation was measured based on ANSI/CRRC S100 in conjunction with C1549-09. The total area of Santa Fe asphalt shingle is 0.294 m2 (456 in.2 ). We neglect the measurements on the horizontal gap where the two courses are laminated. This gap causes a significant displacement of the probe on the sample; resulting in a significant error caused by light escaped from probe and improperly illumination of a sample. The total measured area (excluding the overlaps) is 0.270 m2 (418 in.2 ), which satisfies the minimum requirement of ANSI/CRRC S100 (360 in2 ). The sample is gridded to 1 × 1 cells (Fig. 2). 3.2. Simulated samples We simulate 35 variegated samples (each one with area of 5.16 m2 or 8000 in.2 ) consisting of black and white spots. We randomly distribute the white and black spots with solar reflectance of Rw and Rb , respectively, to resemble the binary variegated surfaces. The object of this study is to investigate selection methods to reach a faster convergence rate. So the effect of surface roughness
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Table 2 Statistics of solar reflectance measurements using SSR V.6 and selection G1. Mean (R) Minimum Maximum Median Standard deviation
0.354 0.250 0.459 0.350 0.041
Fig. 2. Portable solar spectrum reflectometer model SSR V.6.
Table 1 Characteristics of black and white simulated samples. Samples
Rb
Rw
Range
#1 to #5 #6 to #10 #11 to #15 #16 to #20 #21 to #25 #26 to #30 #31 to #35
0.425 0.40 0.55 0.60 0.70 0.80 0.90
0.475 0.50 0.35 0.30 0.20 0.10 0.04
0.05 0.10 0.20 0.30 0.50 0.70 0.86
(e.g., size of granules and texture; [7]) on solar reflectance is not considered. The samples are categorized into 5 ranges of solar reflectance as shown in Table 1. The range (RNG) defined as the difference between Rw and Rb . The black area to total area ratio (Nb ) of samples is specified as 10%, 20%, 30%, 40% and 50% for each range. Each row shows the samples as fraction of dark areas that it increases from 0.1 to 0.5.
Fig. 3. Solar reflectance distribution based on SSR V.6 measurements.
Fig. 4. Monte Carlo simulation for calculating R − r¯ and ε of variegated sample based on SSR V.6 measured data (the number of trials is 100).
4. Results 4.1. Measured solar reflectance of the fiberglass asphalt shingle
4.2. Monte Carlo estimate of solar reflectance of the fiberglass asphalt shingle
The statistics of measured solar reflectance for the fiberglass asphalt shingle is shown in Table 2. The calculated average solar reflectance is 0.353 according to SSR V.6 (selection G1). The range and standard deviation was calculated as 0.2 and 0.04, respectively. The distribution of measured solar reflectance is shown in Fig. 3.
By the central limit theorem, the mean solar reflectance of n measured spots (r) (Eq. (A1)) is fluctuates about a sample mean solar reflectance (R). If n is large enough we can assume r¯ is equal to R (Eq. (A2)). Fig. 4 illustrates how quickly the sample mean (¯r ) converges to the population mean (R), with the error tolerance of
Fig. 5. Monte Carlo simulation for ε calculation of variegated sample. The results are shown in an interval of five numbers. The number of trials is 100 for each ε calculation.
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Fig. 6. Sample points generated by (a, b) random function of MATLAB, (c) Halton Leaped (s: 400, L: 31), (d) Halton Leaped (s: 400, L: 61), (e) Sobol scrambled (s: 100), and (f) Sobol scrambled (s: 30).
10−2 . In addition, the convergence rate of the standard error of the mean (ε) is also shown. Fig. 4 elaborates that convergence rate of method implemented by ANSI/CRRC S100 (i.e., Internal Estimate or Confidence Interval approach) is significantly low compared to R−¯r . The main reason behind this fact is that the mean standard error is calculated based on the estimated values (r¯n ) (Eq. (A5)). Hence the confidence level of 95% is considered in ANSI/CRRC S100. The convergence rate of R − r¯ ≤ 0.01 is more likely to ε = 0.01 performance in Fig. 5. We also investigated the effect of increasing the standard error of the mean on convergence rate. Fig. 5 indicates that increasing ε from 0.005 to ε = 0.0075 and ε = 0.01 can significantly reduce the required numbers of spots. The average number of random spots for reaching ε = 0.005, ε = 0.0075 and ε = 0.01 based on SSR V.6 measurements is 65, 30 and 20, respectively. 4.3. Quasi-Monte Carlo estimate of solar reflectance of the fiberglass asphalt shingle We compare the application of MC with the Halton and Sobol low-discrepancy methods regarding to number of required random spots for estimating the mean solar reflectance of variegated asphalt shingle. We implement MATLAB Quasi-random generator (Halton and Sobol sequences) and rand function to produce the location of 80 spots in Fig. 6. We calculate the standard error of the mean reflectance by considering the solar reflectance of the spots determined by QMC methods. Fig. 7 illustrates that applying QMC sampling methods can decreases the number of required measured spots. The average number of spots to reach ε = 0.005 is reduced from nearly 65 (MC method) to 41 and 34 by applying Halton and Sobol sequences, respectively. The rate of convergence for MC is of order 1/N0.5 . Implementing Halton and Sobol sequences rather than random set points can increase the convergence rate about 35% and 45%, respectively. It is strongly recommended to implement formula provided by Matouˇsek [19] to generate Sobol sequence. The classic Sobol sequence [24] is not perform well in low number generations. The modified Halton sequence by Kocis & Whiten [14] significantly
Fig. 7. Quasi-Monte Carlo simulations for calculation of variegated sample, (a) Sobol sequences (Scrambled and Scrambled-Leaped), (b) Halton (ScrambledLeaped).
surpass the original construction of Halton [12]. It is worth to mention that our investigations verify that considering leaps of 31 and 61 performs much better than L = 149 and 409 for low number generation.
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Fig. 8. Minimum, 10%, 1st quartile, median, mean, 3rd quartile for different number of selected spots (N) (sample with range = 0.2 and Nb = Nw = 0.5). The dots show individual data for less that 10% and greater than 90%. The dashed lines show the acceptable standard error of ±0.005. The range of mean standard error, derived from Eq. (7), are shown in adjacent to box and whisker plots.
Fig. 9. Calculated mean standard error using QMC methods.
4.4. MC and QMC application for estimating number of required spots for simulated samples
Quasi-Monte Carlo (QMC) method can increase the convergence rate. We measured solar reflectance of a variegated roofing sample based on the C1549-09 (the sample is gridded to 1 × 1 cells and spots with diameter of 1 on cells). We calculated R by averaging the measured solar reflectance of all spots. Then, we estimated R using MC and QMC as a function of the number of random spots on our sample. We compared the convergence rate of MC and QMC. We analyzed the sensitivity of standard error of the mean reflectance of selected spots for a variety of simulated samples where the range and distribution of the solar reflectance is varied. QMC typically requires about 20% to 50% less random spots compared to MC to converge to the required standard error of estimate. For application of MC approach, we propose an equation to estimate the required number of random spots as a function of mix and range of solar reflectance of samples.
Acknowledgements We investigated the application of MC and QMC methods to estimate the convergence rate of simulated samples that was described in Section 3.2. Lower variation and range of solar reflectance increases the convergence rate of both MC and QMC. Fig. 8 shows the result of analysis for the simulated sample with equal areas covered in high and low reflectance spots (Nb = Nw = 0.5). For the MC approach about 400 random points are needed to achieve the required criteria. Note that Eq. (7) estimation of the number of required random points compared well with the results of the MC estimates. Knowing the product mix and the range of solar reflectance over the sample, Eq. (7) can be used to calculate n (the number of required random spots for a selected criteria, e.g., ε = 0.005). We applied the QMC based on Sobol and Halton sequences on the above simulated sample (RNG = 0.2 and Nb = Nw = 0.5) and the results are shown in Fig. 9. It shows that we can reach to the acceptable ε by using 283 spots rather than 400 needed spots by using MC. Further analysis is necessary to reach a conclusion on the performance of QMC. 5. Conclusions The Cool Roof Rating Council (CRRC) recommends using the Monte Carlo (MC) method to estimate the mean solar reflectance (R) of the variegated roofing samples. The MC approach is slow in converging to the solar reflectance of sample. For samples with high degree of variation in the solar reflectance, the convergence rate is slower than typical variegated materials and requires a large sample size to estimate R with ±0.01 required accuracy. Using
This research was in part funded by a grant from the Cool Roof Rating Council (CRRC) and a scholarship from the Huntsman Corporation. The authors appreciate their support throughout this project. The writing of this paper was supported by a grant from the Natural Resources and Engineering Council of Canada (NSERC).
Appendix A. By the central limit theorem, the mean solar reflectance of n measured spots (¯r ) (Eq. (A1)) is fluctuates about a sample mean solar reflectance (R). If n is large enough we can assume r¯ is equal to R (Eq. (A2)). The selection of spots is based on MC approach. The cells should not be overlapped or selected more than one time because cells have to have equal probability. Therefore the probability of each spot is the same as other spots. 1 ri N N
r≡
(A1)
i=1
R ≡ A−1
rdA
(A2)
A
where A represent the area of sample. ANSI/CRRC S100 employed the Internal Estimate or Confidence Interval approach as the statistical method to estimate the mean solar reflectance of sample (R) (Eq. (A3)). Also, the estimated solar
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reflectance (r) is calculated through averaging solar reflectance of selected spots based on the MC approach. R = r ± zε
(A3)
where ε is the standard error of mean solar reflectance (Eq. (A4)), and z is defined as random variable and derived from standard normal probability table. ε≈ √ n
(A4)
is standard deviation of population (Eq. (A5)). If n is higher than 30, we can assume the standard deviation of measured spots is well approximated the standard deviation of population.
n 1 s≡
(ri − r¯n )2 n−1
(A5)
i=1
The values of z is determined by how confident we want estimate R. In applied practice, such as ANSI/CRRC S100, the confidence level of 95% is typically stated. It means in 95% of the calculated r, the difference in values between R and r¯ is less than or equal to zε [26]. The z value for confidence level of 95% is 1.96 or approximately 2, as being employed by ANSI/CRRC S100 (Eq. (A6)). R = r ± 2ε
(A6)
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