# Verification of LTPP Virtual Weather Stations Phase I Report: Accuracy and Reliability of Virtual Weather Stations

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**Figure 1. Diagram. Schema of VWS concept.** The figure is a schematic chart showing the virtual weather station (VWS) concept. For each general pavement studies (GPS) test section and specific pavement studies (SPS) site, the climatic conditions (temperature, precipitation, freezing index, etc.) are estimated using the weather data from up to five nearby weather stations. The diagram resembles a wheel with spokes, with the center, or hub, being a VWS box containing data from: automated weather stations (AWS) installed at SPS–1, SP–2, and SP–8 sites; seasonal monitoring program (SMP) sites; and operating weather stations (OWS). Small circles surrounding the center box represent sites and weather stations. The spokes connecting the circles to the center are the distances (R).

**Figure 2. Equation. Estimating climatic data.** Uppercase V subscript lowercase M equals the quotient of two summations. The numerator is the summation, from lowercase I equal 1 to lowercase I equal lowercase K, of the quotient of uppercase V, subscript lowercase MI, divided by the square of uppercase R subscript lowercase I. The denominator is the summation, from lowercase I equal 1 to lowercase I equal lowercase K, of the quotient of 1 divided by the square of uppercase R subscript lowercase I.

**Figure 3. Graph. Estimated versus measured maximum temperature for section 331001.** The figure is a comparison plot of the VWS-estimated (Y-axis) and site-measured (X-axis) maximum temperatures in degrees Celsius for section number 331001. The data points concentrate on a 45-degree line going upwards from the origin.

**Figure 4. Graph. Estimated versus measured maximum temperature for site 010100.** The figure is a comparison plot of the VWS-estimated (Y-axis) and site-measured (X-axis) maximum temperatures in degrees Celsius for section number 010100. The data points scatter around a 45-degree line going upwards from the origin.

**Figure 5. Graph. Estimated versus measured maximum temperature for site 040100.** The figure is a comparison plot of the VWS-estimated (Y-axis) and site-measured (X-axis) maximum temperatures in degrees Celsius for section number 040100. The data points scatter around a virtual line slightly above and parallel to a 45-degree line going upwards from the origin.

**Figure 6. Bar chart. Error distribution of maximum temperature estimates for AWS sites.** The figure is a bar chart showing the frequency distribution of the errors between the VWS-estimated and site-measured maximum temperatures in degrees Celsius for AWS sites. The Y-axis is the frequency, while the X-axis is the maximum temperature error in degrees Celsius. The plot approximates a normal distribution with a mean close to 0 and a wide spread.

**Figure 7. Bar chart. Error distribution of minimum temperature estimates for AWS sites.** The figure is a bar chart showing the frequency distribution of the errors between the VWS-estimated and site-measured minimum temperatures in degrees Celsius for AWS sites. The Y-axis is the frequency, while the X-axis is the minimum temperature error in degrees Celsius. The plot approximates a normal distribution with a mean close to 0 and a narrower spread compared to figure 6.

**Figure 8. Bar chart. Error distribution of maximum temperature estimates for SMP sites.** The figure is a bar chart showing the frequency distribution of the errors between the VWS-estimated and site-measured maximum temperatures in degrees Celsius for the SMP sites. The Y-axis is the frequency, while the X-axis is the maximum temperature error in degrees Celsius. The plot approximates a normal distribution with a mean close to 0 and a wide spread.

**Figure 9. Bar Chart. Error distribution of minimum temperature estimates for SMP sites.** The figure is a bar chart showing the frequency distribution of the errors between the VWS-estimated and site-measured minimum temperatures in degrees Celsius for the SMP sites. The Y-axis is the frequency, while the X-axis is the minimum temperature error in degrees Celsius. The plot approximates a normal distribution with a mean close to 0 and a wide spread.

**Figure 10. Graph. AWS versus SMP measured maximum temperature for site 010100.** The figure is a comparison plot of the AWS (Y-axis) and SMP site-measured (X-axis) maximum temperatures in degrees Celsius for section number 010100. The data points concentrate on a 45-degree line going upwards from the origin.

**Figure 11. Graph. AWS versus SMP measured maximum temperature for site 040100.** The figure is a comparison plot of the AWS (Y-axis) and SMP site-measured (X-axis) maximum temperatures in degrees Celsius for section number 040100. The data points scatter around a 45-degree line going upwards from the origin.

**Figure 12. Graph. Temperature difference between site 460800 and nearby weather stations versus distance.** The graph is a plot of the maximum temperature difference at the site and the nearby weather stations (Y-axis) versus distance (X-axis) for section number 460800. Five vertical strings of data points intersect, at distances of 11, 23, 32, 38, and 41 kilometers, with a horizontal reference line at 0 on the Y-axis.

**Figure 13. Graph. Mean maximum temperature difference versus distance for AWS and SMP sites.** The graph contains the data points of the mean maximum temperature difference at a site and the nearby weather stations (Y-axis) versus distance (X-axis) for the AWS and SMP sites. The data points are mostly within a band of plus or minus 2 degrees Celsius about a horizontal reference line at 0 on the Y-axis and to a distance on the X-axis of 60 kilometers.

**Figure 14. Graph. Mean minimum temperature difference versus distance for AWS and SMP sites.** The graph contains the data points of the mean minimum temperature difference at a site and the nearby weather stations (Y-axis) versus distance (X-axis) for the AWS and SMP sites. The data points are mostly within a band of plus or minus 3 degrees Celsius about a horizontal reference line at 0 on the Y-axis and to a distance on the X-axis of 60 kilometers.

**Figure 15. Graph. Standard deviation of maximum temperature difference versus distance for AWS and SMP sites.** The graph contains the data points of the standard deviation of the maximum temperature difference at a site and the nearby weather stations (Y-axis) versus distance (X-axis) for the AWS and SMP sites. The data points are evenly scattered in a rectangular area enclosed by approximately 1 and 6 degrees Celsius on the Y-axis and 0 and 60 kilometers on the X-axis.

**Figure 16. Graph. Standard deviation of minimum temperature difference versus distance for AWS and SMP sites. **The graph contains the data points of the standard deviation of the minimum temperature difference at a site and the nearby weather stations (Y-axis) versus distance (X-axis) for the AWS and SMP sites. The data points are evenly scattered in a rectangular area enclosed by 1 and 4 degrees Celsius on the Y-axis and 0 and 60 kilometers on the X-axis.

**Figure 17. Graph. Mean maximum temperature difference versus distance for NCDC sites.** The graph contains the data points of the mean maximum temperature difference at a site and the nearby weather stations (Y-axis) versus distance (X-axis) for the National Climatic Data Center (NCDC) sites. The data points are mostly within a band of plus or minus 10 degrees Celsius about a horizontal reference line at 0 on the Y-axis to a distance on the X-axis of 100 kilometers.

**Figure 18. Graph. Mean minimum temperature difference versus distance for NCDC sites.** The graph contains the data points of the mean minimum temperature difference at a site and the nearby weather stations (Y-axis) versus distance (X-axis) for the NCDC sites. The data points are mostly within a band of plus or minus 10 degrees Celsius about a horizontal reference line at 0 on the Y-axis to a distance on the X-axis of 100 kilometers.

**Figure 19. Graph. Standard deviation of maximum temperature difference versus distance for NCDC sites.** The graph contains the data points of the standard deviation of the maximum temperature difference at a site and the nearby weather stations (Y-axis) versus distance (X-axis) for the NCDC sites. The data points are scattered in a rectangular area enclosed by approximately 1 and 7 degrees Celsius on the Y-axis and 0 to 100 kilometers on the X-axis.

**Figure 20. Graph. Standard deviation of minimum temperature difference versus distance for NCDC sites.** The graph contains the data points of the standard deviation of the minimum temperature difference at a site and the nearby weather stations (Y-axis) versus distance (X-axis) for the NCDC sites. The data points are scattered in a rectangular area enclosed by approximately 1 and 7 degrees Celsius on the Y-axis and 0 to 100 kilometers on the X-axis.

**Figure 21. Graph. Average maximum temperature difference versus elevation difference for AWS and SMP sites.** The graph contains the data points of the mean maximum temperature difference at a site and the nearby weather stations (Y-axis) versus the elevation difference in meters (X-axis) for the AWS and SMP sites. The data points are mostly within a circular area bounded on the Y-axis by plus or minus 2 degrees Celsius from a horizontal reference line at 0 and on the X-axis by plus or minus 150 meters. Outliers exist primarily in the upper left quadrant and the lower right quadrant.

**Figure 22. Graph. Average minimum temperature difference versus elevation difference for AWS and SMP sites.** The graph contains the data points of the mean minimum temperature difference at a site and the nearby weather stations (Y-axis) versus the elevation difference in meters (X-axis) for the AWS and SMP sites. The data points are mostly within a circular area bounded on the Y-axis by plus or minus 3.5 degrees Celsius from a horizontal reference line at 0 and on the X-axis by plus or minus 200 meters. Outliers exist primarily in the upper left quadrant and the lower right quadrant.

**Figure 23. Graph. Standard deviation of maximum temperature difference versus elevation difference for AWS and SMP sites.** The graph contains the data points of the standard deviation of the maximum temperature difference at a site and the nearby weather stations (Y-axis) versus distance (X-axis) for the AWS and SMP sites. The data points are scattered in a roughly rectangular area enclosed by approximately 0.5 and 6.2 degrees Celsius on the Y-axis and plus or minus 200 meters on the X-axis.

**Figure 24. Graph. Standard deviation of minimum temperature difference versus elevation difference for AWS and SMP sites.** The graph contains the data points of the standard deviation of the minimum temperature difference at a site and the nearby weather stations (Y-axis) versus distance (X-axis) for the AWS and SMP sites. The data points are scattered in a roughly rectangular area enclosed by approximately 0.6 and 4.0 degrees Celsius on the Y-axis and plus or minus 200 meters on the X-axis.

**Figure 25. Graph. Mean maximum temperature difference versus elevation difference for NCDC sites.** The graph contains the data points of the mean maximum temperature difference at a site and the nearby weather stations (Y-axis) versus the elevation difference in meters (X-axis) for the NCDC sites. The data points are scattered in a roughly rectangular area beginning between 10 and 15 degrees Celsius on the Y-axis and minus 1900 meters on the X-axis and descending diagonally to the right to between minus 15 and minus 8 degrees Celsius on the Y-axis and 1900 meters on the X-axis. The data points are concentrated most heavily in the vicinity of the point located at 0 on the X-axis and 0 on the Y-axis.

**Figure 26. Graph. Mean minimum temperature difference versus elevation difference for NCDC sites. **The graph contains the data points of the mean minimum temperature difference at a site and the nearby weather stations (Y-axis) versus the elevation difference in meters (X-axis) for the NCDC sites. The data points are scattered in a roughly rectangular area beginning between 1 and 13 degrees Celsius on the Y-axis and minus 1800 meters on the X-axis and descending diagonally to the right to between minus 15 and minus 3 degrees Celsius on the Y-axis and 1800 meters on the X-axis. The data points are concentrated most heavily in the vicinity of the point located at 0 on the X-axis and 0 on the Y-axis.

**Figure 27. Graph. Mean maximum temperature difference versus north-south distance for AWS and SMP sites.** The graph contains the data points of the mean maximum temperature difference at a site and the nearby weather stations (Y-axis) versus the north-south distance in kilometers (X-axis) for the AWS and SMP sites. The data points are mostly scattered in a roughly rectangular area enclosed by approximately minus 1 and plus 3 degrees Celsius on the Y-axis and plus or minus 50 kilometers on the X-axis.

**Figure 28. Graph. Mean minimum temperature difference versus north-south distance for AWS and SMP sites.** The graph contains the data points of the mean minimum temperature difference at a site and the nearby weather stations (Y-axis) versus the north-south distance in kilometers (X-axis) for the AWS and SMP sites. The data points are mostly scattered in a roughly rectangular area enclosed by approximately minus 3 and plus 3 degrees Celsius on the Y-axis and plus or minus 50 kilometers on the X-axis.

**Figure 29. Graph. Standard deviation of maximum temperature difference versus north-south distance for AWS and SMP sites.** The graph contains the data points of the standard deviation of the maximum temperature difference at a site and the nearby weather stations (Y-axis) versus the north-south distance in kilometers (X-axis) for the AWS and SMP sites. The data points are widely scattered between 0.5 and 6.5 degrees Celsius on the Y-axis and plus or minus 60 kilometers on the X-axis.

**Figure 30. Graph. Standard deviation of minimum temperature difference versus north-south distance for AWS and SMP sites.** The graph contains the data points of the standard deviation of the minimum temperature difference at a site and the nearby weather stations (Y-axis) versus the north-south distance in kilometers (X-axis) for the AWS and SMP sites. Most of the data points are in a rectangular area between 0.8 and 4.4 degrees Celsius on the Y-axis and plus or minus 60 kilometers on the X-axis.

**Figure 31. Equation. Average temperature calculation.** Uppercase V subscript lowercase M equals the quotient of a summation divided by lowercase K. The summation is from lowercase I equal 1 to lowercase I equal lowercase K and is of uppercase V, subscript lowercase MI.

**Figure 32. Bar chart. Percent contribution of closest weather stations to VWS using the 1 divided by R squared rule.** The figure is a bar chart showing, for all 880 Long-Term Pavement Performance sites, the frequency (Y-axis) of the percent contribution (X-axis) of the closest weather station to a VWS estimate using the inverse distance squared weight, which is 1 divided by the distance, which is R, squared. The distribution is very roughly uniform. Percents 40, 60, 70, 80, and 90 each have a frequency of approximately 110. Percent 20 has a frequency of approximately 2; percent 30 has a frequency of approximately 80; percent 50 has a frequency of approximately 150; and percent 100 has a frequency of approximately 90.

**Figure 33. Bar chart. Percent contribution of closest weather stations to VWS using the 1 divided by R rule.** The figure is a bar chart showing, for all 880 Long-Term Pavement Performance sites, the frequency (Y-axis) of the percent contribution (X-axis) of the closest weather station to a VWS estimate using the inverse distance weight, which is 1 divided by the distance, which is R. The distribution is skewed to the right. Percent 20 has a frequency of approximately 20; percent 30 has a frequency of approximately 240; percent 40 has a frequency of approximately 230; percent 50 has a frequency of approximately 150; percent 60 has a frequency of approximately 100; percent 70 has a frequency of approximately 70; percent 80 has a frequency of approximately 40; percent 90 has a frequency of approximately 20; and percent 100 has a frequency of approximately 10.

**Figure 34. Bar chart. Distribution of mean error for maximum temperature (NCDC data).** The figure is a bar chart showing the frequency distribution of the mean error between estimated and measured maximum temperatures at NCDC sites. The Y-axis is the frequency, and the X-axis is the average maximum temperature error in degrees Celsius. The distribution is approximately normal, with a peak frequency close to 1300 at a mean error of 0 degrees Celsius.

**Figure 35. Bar chart. Distribution of mean error for minimum temperature (NCDC data).** The figure is a bar chart showing the frequency distribution of the mean error between estimated and measured minimum temperatures at NCDC sites. The Y-axis is the frequency, and the X-axis is the average minimum temperature error in degrees Celsius. The distribution is not quite normal, with a minor amount of skewness to the right. The peak frequency is close to 900 at a mean error of 0.5 degrees Celsius.

**Figure 36. Bar chart. Distribution of standard deviation of error for maximum temperature (NCDC data).** The figure is a bar chart showing the frequency distribution of the standard deviation of the error between estimated and measured maximum temperature at NCDC sites. The distribution is skewed to the right, with a peak frequency close to 1400 at a mean error of 2.0 degrees Celsius.

**Figure 37. Bar chart. Distribution of standard deviation of error for minimum temperature (NCDC data). **The figure is a bar chart showing the frequency distribution of the standard deviation of the error between estimated and measured maximum temperature at NCDC sites. The distribution is slightly skewed to the right, with a peak frequency close to 1900 at a mean error of 2.0 degrees Celsius.

**Figure 38. Graph. Maximum air temperature for SMP sections 010101 versus 010102.** The figure is a plot of the maximum air temperature at SMP section 010102 (Y-axis) versus the maximum air temperature at SMP section 010101 (X-axis) in degrees Celsius, both sections being within the same site. The data points are scattered around a 45-degree line extending from the origin (0 on both the X- and Y-axes) upwards to the right. The most pronounced scattering occurs between the 5 and 30 degrees Celsius readings on both axes.

**Figure 39. Graph. Minimum air temperature for SMP sections 010101 versus 010102.** The figure is a plot of the minimum air temperature at SMP section 010102 (Y-axis) versus the minimum air temperature at SMP section 010101 (X-axis) in degrees Celsius, both sections being within the same site. The data points are scattered around a 45-degree line extending from the origin (0 on both the X- and Y-axes) both upwards to the right and downward to the left. The most pronounced scattering occurs between the minus 10 and plus 20 degrees Celsius readings on both axes.