Temporal and spatial variation in allocating annual traffic activity across an urban region and implications for air quality assessments

Abstract

Patterns of traffic activity, including changes in the volume and speed of vehicles, vary over time and across urban areas and can substantially affect vehicle emissions of air pollutants. Time-resolved activity at the street scale typically is derived using temporal allocation factors (TAFs) that allow the development of emissions inventories needed to predict concentrations of traffic-related air pollutants. This study examines the spatial and temporal variation of TAFs, and characterizes prediction errors resulting from their use. Methods are presented to estimate TAFs and their spatial and temporal variability and used to analyze total, commercial and non-commercial traffic in the Detroit, Michigan, U.S. metropolitan area. The variability of total volume estimates, quantified by the coefficient of variation (COV) representing the percentage departure from expected hourly volume, was 21, 33, 24 and 33% for weekdays, Saturdays, Sundays and holidays, respectively. Prediction errors mostly resulted from hour-to-hour variability on weekdays and Saturdays, and from day-to-day variability on Sundays and holidays. Spatial variability was limited across the study roads, most of which were large freeways. Commercial traffic had different temporal patterns and greater variability than noncommercial vehicle traffic, e.g., the weekday variability of hourly commercial volume was 28%. The results indicate that TAFs for a metropolitan region can provide reasonably accurate estimates of hourly vehicle volume on major roads. While vehicle volume is only one of many factors that govern on-road emission rates, air quality analyses would be strengthened by incorporating information regarding the uncertainty and variability of traffic activity.

1. Introduction

Traffic activity encompasses the number, mix, speed and acceleration of vehicles on roads. The spatial and temporal pattern of traffic activity depends commuting and working schedules, less frequent events such as non-routine congestion, construction, accidents and weather, and other factors. Traffic activity is a key determinant of vehicle-related or “on-road” mobile source emissions and the resulting concentrations of traffic-related air pollutants. While modern vehicles have significantly reduced exhaust emissions, the transport sector remains the largest emitter of nitrogen oxides (NO_x) and carbon monoxide (CO) in the US and elsewhere, and mobile sources are major sources of particulate matter (PM_2.5), volatile organic compounds (VOCs), and other pollutants (Baldauf, Thoma et al. 2008, Batterman 2013, European Environment Agency 2013). Traffic activity must be characterized to estimate vehicle emissions and model urban and local-scale pollutants, especially in near-road environments and traffic corridors (Baldauf, Thoma et al. 2008, Gokhale 2011, Batterman 2013). Understanding traffic-related emissions and impacts at project-level, urban and national scales is increasingly important for exposure, risk, epidemiologic, regulatory and accountability studies (Health Effects Institute 2010). Knowledge of traffic flows and related information is also critical for road design, signal optimization, pedestrian crossings, and other purposes.

Well-known and regular patterns of traffic activity include weekday morning and afternoon “rush hour” peaks due to individuals commuting between home and work yielding a bimodal diurnal pattern; low traffic on Saturday and especially Sunday mornings; and high truck traffic during the workday. In addition to temporal changes, traffic activity varies spatially, reflecting differences among roads (e.g., freeways versus local arterials), regions (e.g., city, suburban and rural areas), and transit types (“urban” versus “through” traffic), among other factors. While road-specific measurements at continuous counting sites (CCSs) using induction loops or other technologies can provide accurate quantification, such data are comparatively rare. Measurements are often limited to short-term counts that provide limited coverage (e.g., only a few days are monitored, and weekends are usually excluded). In addition, monitoring may not classify activity by vehicle type, an important omission for air quality analyses since in terms of exhaust emissions, a single heavy duty vehicle can represent many passenger car equivalents (PCEs) (Lindhjem, Pollack et al. 2012). For example, each large truck is estimated to represent 12 PCEs of NO_x and 50 PCEs of PM_2.5, based on the emission factor model MOVES 2010a and 2010 scenarios for Detroit, Michigan (Batterman, Cook et al. 2015).

Emissions from on-road sources are most commonly calculated using city-wide estimates of vehicles-km-traveled (VKT). To provide spatial and temporal allocations, VKT may be allocated to the link-level (roadway segment) using travel demand models or empirical extrapolations, such as the U.S. Highway Performance Monitoring System that consolidates CCS data (North American Research Strategy for Tropospheric Ozone 2005). While one of the most important sectors of emissions inventories, on-road vehicle emissions are difficult to quantify, in part due to the variability of activity data, which is used in conjunction with emissions factors (e.g., grams emitted per km driven) to estimate emission rates (Parrish 2006).

1.1 Allocating traffic to the hour

Traffic volume undergoes regular changes that can be modeled using temporal adjustment factors (TAFs) or “temporal profiles” that reflect the monthly, day-of-week, and hour-of-the-day patterns. Hourly volume on a road can be approximated as the product of the TAFs and estimates or measurements of annual average daily traffic (AADT) or commercial annual average daily traffic (CAADT) (Cook, Isakov et al. 2008). Hourly class-specific volume is estimated as:

$${\mathrm{V}}_{\mathrm{i},\mathrm{k},\mathrm{t}}={\text{AADT}}_{\mathrm{i}}{\text{FM}}_{\mathrm{i},\mathrm{k}}{\text{MAF}}_{\mathrm{i},\text{MON}(\mathrm{t})}{\text{DAF}}_{\mathrm{i},\mathrm{k},\text{DTYPE}(\mathrm{t})}{\text{HAF}}_{\mathrm{i},\text{DTYPE}(\mathrm{t})}$$ (1)

where V_i,k,t (vehicles h⁻¹) = number of vehicles for road link i, vehicle class k, and hour t; AADT_i = annual average daily traffic (vehicles h⁻¹); FM_i,k = fleet mix allocation factor, the fraction of vehicles in vehicle class k on link i; MAF_i,MON(t), DAF_i,k,DAY(t) and HAF_i,t respectively are monthly, daily and hourly temporal allocation factors (dimensionless); and DTYPE(t) = day type for hour t, e.g., day of week, weekdays and weekends. Rather than having fleet mix and temporal factors specific to each road segment or link, TAFs usually are based on road type and location, thus greatly reducing the number of parameters needed. For example, in the U.S., interstates and freeways are designated as Functional Classification Codes 1 and 2, respectively; a subdivision of urban and rural also can be useful given different traffic patterns.

We recently have shown that separate sets of TAFs are needed for total and commercial vehicles, and for weekdays, Saturdays, Sundays and observed holidays, and that site-specific or urban-wide TAFs can explain provide accurate predictions with a few exceptions, e.g., low volume due to adverse weather (Batterman, Cook et al. 2015). The U.S. Environmental Protection Agency has compiled numerous TAFs in the SMOKE database, which is designed to assist the spatial and temporal allocation of emissions for air quality modeling purposes (http://www.cmascenter.org/smoke/). This database provides month-of-year, day-of-week, and hour-of-day TAFs for mobile sources in 156 unique Source Classification Codes that separate region, road and vehicle type. The current database includes thousands of profiles, which are provided mainly by the states. However, this database has important gaps: many profiles are identical; few Saturday and Sunday profiles are available; only a subset of states and regions have local data; smaller roads are under-represented; some profiles are old; and the generalizability of national and local profiles has not been evaluated. Default SMOKE profiles provided substantially poorer agreement, especially for commercial vehicles, with historical data than TAFs developed for the local area (Batterman, Cook et al. 2015).

1.2 Uncertainty in traffic volume estimates and dispersion modeling

TAF-based estimates of traffic volume given by eq. (1) are deterministic and do not account for uncertainty (i.e., the lack of knowledge regarding true values) and variation (fluctuations of a quantity over time, space or subgroups such as vehicle types). Uncertainty and variation are associated with both intrinsic fluctuations of traffic, called “inherent variability,” as well as modeling imperfections, called “modeling” or “prediction error.” The latter results from errors in estimating parameters, e.g., a result of a limited number of observations used to estimate TAFs, as well as errors in the model structure, e.g., possible violation of the assumption of independence between monthly, daily and hourly TAFs implicit in eq. (1). Prediction error can include both systematic and random components. TAFs estimated using local data should match means and thus minimize systematic errors.

It is important to characterize and quantify uncertainty and variability in emission inventories to prevent erroneous inferences in air quality modeling and exposure assessment, which may lead to major environmental policy implications (Frey and Zhao 2004). Despite calls for explicit quantification of uncertainties in mobile source emission factors (National Research Council 2000), including the temporal trend of NO_x emissions (North American Research Strategy for Tropospheric Ozone 2005), few air quality analyses have used reliability- or probability–based approaches, particularly for traffic-related air pollutants. A general approach to estimate uncertainties in mobile source emission inventories using Monte Carlo methods has been reported, however, few parameters were presented (Ho, Clappier et al. 2014). A probabilistic emissions inventory compiled for Houston found wide confidence ranges for the four pollutants investigated, including two predominantly from mobile sources (benzene, formaldehyde), largely due to uncertainty in emission factors (Frey and Zhao 2004). Estimates of on-road mobile emissions in the Pearl River Delta region of China also showed large uncertainties due to emission factors, as well as a lack of knowledge regarding fleet characteristics and vehicle classifications (Zheng, Zheng et al. 2009). Uncertainties ascribed to the distribution of vehicle activity among vehicle types (e.g., trucks, motorcycles, mopeds) was limiting for a mobile source inventory developed for Shanghai, China (Wang, Chen et al. 2008). Other earlier studies examining the uncertainty of mobile source inventories are cited elsewhere (Frey and Zhao 2004). While only one of several governing factors, the spatial and temporal allocation of traffic activity can significantly affect emissions, concentrations and exposures of traffic-related air pollutants (Lindhjem, Pollack et al. 2012). Hourly allocations, especially, can affect predictions of primary pollutants, e.g., CO, NO, PM_2.5, while daily allocations, especially weekday/weekend differences, may significantly affect secondary pollutants such as O₃.

1.3 Study objectives

This study seeks to understand the variability of vehicle activity and, specifically, to examine the spatial and temporal variability at monthly, daily and hourly levels. It extends our previous analysis that focused on understanding the TAFs that apply to the Detroit area and differences from default TAFs used in SMOKE (Batterman, Cook et al. 2015) by presenting methods to estimate TAFs and their spatial and temporal variability. Other measures of vehicle activity, including speed, vehicle age distributions, and vehicle mix, also contribute to uncertainties in emissions inventories, but are beyond the present scope. Long term (multiyear) trends and the variability of emission factors, also beyond the scope, have been discussed elsewhere (Parrish 2006).

2. Materials and methods

2.1 Calculating TAFs

This section presents an efficient way to calculate monthly, daily and hourly TAFs from traffic volume measurements. Quality-assured and reasonably complete hourly traffic measurements are assumed to be available over an extended period, specifically, multiple years. TAFs are estimated that match mean (or expected) traffic volumes for month-of-year, day-of-week, and hour-of-day periods. The method does not provide maximum likelihood estimates or address issues related to missing data, outliers, and other statistical issues. (While not issues in the present dataset, outlier issues could be addressed using nonparametric statistics, e.g., substituting medians for means, or by using censored (trimmed) datasets.)

First, V_i,k is calculated as the long term (multiyear) average daily traffic volume for road link i and vehicle class k. Second, V_i,k,DAY(t) is defined as the daily (24-hr) average traffic volume and calculated for each day t that has at least 75% valid hourly observations (18 or more valid hourly observations per day). Next, V_i,k,MON(t) is calculated as the average monthly volume, calculated from the daily averages, again requiring at least 75% valid observations (23 or more valid daily observations per month). Then, from eq. (1), monthly TAFs are determined as

$${\text{MAF}}_{\mathrm{i},\mathrm{k},\text{MON}(\mathrm{m})}=1/(\mathrm{t}{\mathrm{V}}_{\mathrm{i},\mathrm{k}}){\sum }_{\mathrm{t}}{\mathrm{V}}_{\mathrm{i},\mathrm{k},\text{MON}(\mathrm{m},\mathrm{t})}$$ (2)

where t = number of observations of month m available, e.g., the number of Januarys available in the record, which in most cases will be the number of years of data available. Eq. (2) is simply the average volume in month m divided by the long term annual average. The average monthly allocation factor should equal 1.0 when weighted by the frequency of occurrence of each month (which may vary due to missing data and an unequal number of months available). Monthly TAFs are calculated by link i and vehicle type k.

Next, daily TAFs are calculated as

$${\text{DAF}}_{\mathrm{i},\mathrm{k},\text{DTYPE}(\mathrm{d})}=1/(\mathrm{t}{\mathrm{V}}_{\mathrm{i},\mathrm{k}}){\sum }_{\mathrm{t}}[{\mathrm{V}}_{\mathrm{i},\mathrm{k},\text{DAY}(\mathrm{d},\mathrm{t})}/{\text{MAF}}_{\mathrm{i},\mathrm{k},\text{MON}(\mathrm{t})}]$$ (3)

where t = number of observations of day type d (e.g., Mondays) available. This equation represents the average volume on a particular day type divided by the long term annual average, accounting for month-to-month variation. The average daily allocation factor (e.g., across days of the week) should equal 1.0 when weighted by the frequency of occurrence of each day type (i.e., holidays are comparatively rare). Daily TAFs are calculated by link and vehicle type.

Finally, hourly TAFs are determined as

$${\text{HAF}}_{\mathrm{i},\mathrm{k},\text{HOUR}(\mathrm{h})}=1/(\mathrm{t}{\mathrm{V}}_{\mathrm{i},\mathrm{k}}){\sum }_{\mathrm{t}}[{\mathrm{V}}_{\mathrm{i},\mathrm{k},\mathrm{t}}/({\text{MAF}}_{\mathrm{i},\mathrm{k},\text{MON}(\mathrm{t})}\times {\text{DAF}}_{\mathrm{i},\mathrm{k},\text{DTYPE}(\mathrm{t})})]$$ (4)

where t = number of observations of hour h (e.g., 1 am) available. This equation represents the average volume on a particular hour of a particular day type, e.g., 1 am on weekdays, divided by the long term annual average, after accounting for monthly and day-of-week variation. In this case, the sum of the hourly allocation factors (for each hour of the day) should equal 1.0.

To account for interactions between day-of-week and hour-of-day, hourly TAFs are calculated for different day types, i.e., weekdays, Saturdays, Sundays, and observed holidays. In addition, TAFs can be calculated by vehicle type, e.g., total vehicles, passenger vehicles, and trucks.

2.2 Variability and uncertainty of TAF-based models

The variability or uncertainty of prediction errors can be quantified as the difference between observations and predictions, which is the mean or expected value. Because errors were generally proportional to expected volume, and because volumes differed significantly over time and between roads, percentage differences between an observation and the TAF-based prediction are used to quantify errors:

$${\mathrm{D}}_{\mathrm{i},\mathrm{k},\mathrm{t}}=100\%\times ({\mathrm{O}}_{\mathrm{i},\mathrm{k},\mathrm{t}}-{\mathrm{V}}_{\mathrm{i},\mathrm{k},\mathrm{t}})/{\mathrm{V}}_{\mathrm{i},\mathrm{k},\mathrm{t}}$$ (5)

where D_i,k,t = percentage difference for road link i, vehicle type k and time (hour) t, O_i,k,t = measured volume, and V_i,k,t = predicted volume obtained from eq. 1. Dividing by V_i,k,t is advantageous as compared to O_i,k,t since observed values can sometimes be outliers, e.g., very low flows due to accidents or construction, which could greatly elevate D_i,k,t. Percentage differences are used to examine both the spatial and temporal variability of prediction errors, using measures that include the coefficient of variation (COV) and differences at various percentiles.

2.3 Distributional analyses

Distributions to the percentage differences using bootstrap analyses (1000 draws), K-S and A-D tests, and a wide range of distributions (e.g., Erf, Erlang, exponential, extreme value, Gamma, inverse Gauss, logistic, loglogistic, lognormal, lognormal2, normal, Pareto, Pareto2, Pearson5, PERT, Rayleigh, triangular, uniform, Weibull, extreme value minimum, Laplace, Levy). Based on Aitken information criteria, logistic, Gaussian, triangular, and other distributions often had the “best” fits, however, very few of the cases (e.g., differences for a particular day type) fitted any of these distributions at a 95% confidence level. Many distributions appeared asymmetric and had “fat” tails. However, histograms and probability plots often showed that normal distributions often provided “reasonable” fits.

2.4 Spatial variability

Monthly, daily and hourly TAFs are estimated at each CCS using eqs. (2) to (4). The variation of the TAFs across CCSs is a measure of spatial variability. This variation is displayed graphically, calculated as percentage differences from the expected volume (for each CCS) using eq. (5), and summarized as a coefficient of variation (COV) by month, day and hour (e.g., for monthly TAFs, the COV for the January allocation is the standard deviation of January TAFs across the sites, divided by the mean January TAF). As COVs tend to be similar for a specific averaging period (e.g., COVs for each monthly TAF are similar), average COV across the time periods were calculated as an overall measure of spatial variability. Thus, the average monthly COV is the average of monthly COVs over the 12 months, and the average daily COV is the average of the daily COVs over all day types (day of the week and holidays). COVs for hourly TAFs are further broken down by day type (weekdays, Saturdays, Sundays and holidays.) For example, the hourly COV for weekdays at 9 am is the COV of the 9 am hourly TAF (across the 13 sites) on weekdays; the average hourly COV for weekdays is the average of the 24 hourly COVs for weekdays. Low traffic periods (e.g., very early morning) often had greater variation. Since these periods will have low emissions and are of less interest for air pollution purposes, the traffic-weighted average COV is calculated. With the assumptions that prediction errors in the monthly, daily and hourly TAFs are independent and Gaussian distributed, the “total” spatial variability was computed by propagating errors:

$${\text{COV}}_{\mathrm{S},\text{TOT}}={[{({\text{COV}}_{\mathrm{S},\text{MON}})}^2+{({\text{COV}}_{\mathrm{S},\text{DAY}})}^2+{({\text{COV}}_{\mathrm{S},\text{HOUR}})}^2]}^{1/2}$$ (6)

where COV_S,MON = spatial COV of month-of-year TAFs, COV_S,DAY = spatial COV of day-of-week TAFs, and COV_S,HOUR = spatial COV of hour-of-day TAFs. For each of COV, the average was calculated over all periods (e.g., all hours of the day for COV_S,HOUR).

2.4 Temporal variability

The temporal variability of TAFs is the departure from expected traffic volumes at a given road link, a result of (non-routine) changes in weather, commuting, shopping, construction and other factors. The temporal variability is quantified as the difference between observed and predicted volumes (based on eq. 1), again quantified using percentage differences. Temporal variation is defined at monthly, daily, and hourly levels. Equivalently and perhaps more simply, temporal variation is the uncertainty in monthly, daily and hourly TAFs. The following derivation follows from eqs. (2) to (4) used to calculate monthly, daily and hourly TAFs. The temporal COV for monthly TAFs is

$${\text{COV}}_{\text{MAFi},\mathrm{k},\text{MON}(\mathrm{m})}=100\%\times {\sigma }_{\text{MAFi},\mathrm{k},\text{MON}(\mathrm{m})}/{\text{MAF}}_{\mathrm{i},\mathrm{k},\text{MON}(\mathrm{m})}$$ (7)

where σ_{MAFi,k,MON(m)} = standard deviation of the monthly TAFs at a site, which is divided by its monthly TAF (from eq. 2). At the daily level, the temporal COV is estimated as

$${\text{COV}}_{\text{DAFi},\mathrm{k},\text{DTYPE}(\mathrm{d})}={[{(100\%\times {\sigma }_{\text{DAFi},\mathrm{k},\text{DTYPE}(\mathrm{d})}/{\text{DAF}}_{\mathrm{i},\mathrm{k},\text{DTYPE}(\mathrm{d})})}^2-{({\text{COV}}_{\text{MAFi},\mathrm{k},\text{MON}(\mathrm{m})})}^2]}^{1/2}$$ (8)

where σ_{DAFi,k,DTYPE(d)} = standard deviation sample of the daily TAF, which is divided by the daily TAF from eq. (3). The temporal variability at the monthly level is removed from this estimate. Finally, the temporal COV for hourly TAFs is:

$${\text{COV}}_{\text{HAFi},\mathrm{k},\text{HOUR}(\mathrm{h})}={[{(100\%\times {{\sigma }^2}_{\text{HAFi},\mathrm{k},\text{HOUR}(\mathrm{h})}/{\text{HAF}}_{\mathrm{i},\mathrm{k},\text{HOUR}(\mathrm{h})})}^2-{({\text{COV}}_{\text{DAFi},\mathrm{k},\text{DTYPE}(\mathrm{d})})}^2-{({\text{COV}}_{\text{MAFi},\mathrm{k},\text{MON}(\mathrm{m})})}^2]}^{1/2}$$ (9)

where σ_{HAFi,k,HOUR(h)} = sample estimate of the variance of the hourly TAF, which is divided by the hourly TAF from eq. (4). The temporal variability at the monthly and daily levels is removed from this estimate.

Because of potential differences between day types, daily and hourly COVs were calculated separately for weekdays, Saturdays, Sundays and holidays. Monthly, daily and hourly COVs were computed using eqs. (7) to (9) for each time period (e.g., monthly COVs were determined for January, February, etc.) and for each CCS. The range of TAFs across the CCSs were plotted. With the same assumptions used for spatial variability, namely, that the variability and uncertainty of monthly, daily, and hourly TAFs are independent and Gaussian distributed, the “total” temporal variability is

$${\text{COV}}_{\mathrm{T},\text{TOT}}={[{({\text{COV}}_{\mathrm{T},\text{MON}})}^2+{({\text{COV}}_{\mathrm{T},\text{DAY}})}^2+{({\text{COV}}_{\mathrm{T},\text{HOUR}})}^2]}^{1/2}$$ (10)

where COV_T,MON = temporal COV of month-of-year TAFs (average COV over 12 months of the year), COV_T,MON = temporal COV of day-of-week TAFs (average COV over all day-types), and COV_T,HOUR = temporal COV of hour-of-day TAFs (average COVs over 24 hours of the day, by day type). As before, the traffic-weighted average COV is calculated as the most representative measure.

2.5 Total variability

Eqs. (6) and (10) give COVs representing the spatial and temporal variability for each set of TAFs. To estimate the “total” variability, spatial and temporal variation is assumed to be uncorrelated and normally distributed, and again Gaussian quadrature is used:

$${\text{COV}}_{\text{TOT}}={[{({\text{COV}}_{\mathrm{S},\text{TOT}})}^2+{({\text{COV}}_{\mathrm{T},\text{TOT}})}^2]}^{1/2}$$ (11)

This analysis was conducted for each hour-of-the-day (24 hours), each month-of-the-year (12 months), the four day types (weekdays, Saturdays, Sundays, holidays), each road link (13 links), and the four year study period. The fraction of variance attributable to spatial and temporal factors at the monthly, daily and hourly levels was calculated. The full set of analyses was performed for the total number of vehicles, as well non-commercial vehicles (motorcycles, cars, light trucks; sum of FHWA classes 1 to 3, see below), and commercial vehicles (buses, trucks; sum of FHWA classes 4 to 13).

2.6 Data sources

The analysis used hourly data from CCSs in southeast Michigan in the Detroit area for 2009 through 2012, provided by the Michigan Department of Transportation (MDOT). The CCS sites are mapped in Figure 1. The CCS data represent actual vehicle volumes (unweighted by vehicle type), and were reviewed and quality checked by MDOT staff. All sites but one were located on urban interstates or freeways. Nine sites used induction loops to measure total vehicle counts, and four sites (on I-94, I-75, M-24 and I-275) used weigh-in-motion (WIM) sensors which derived counts in the 13 FHWA vehicle classes (1 = motor cycles; 2 = passenger cars; 3 = other two-axles, four tire single unit vehicles; 4 = buses; 5 = two axles, six tire, single unit trucks; 6 = three axles, single unit trucks; 7 = four or more axles, single unit trucks; 8 = three or four axles, single trailer trucks; 9 = five axles, single trailer trucks; 11 = five or less axles, multi trailer trucks; 12 = six axles, multi trailer trucks; 13 = seven or more axles, multi trailer trucks).

Figure 1.

Map showing locations of continuous counting sites (CCS) in the Detroit, Michigan area with and without weigh-in-motion measurements. Map shows 50 x 35 km area. Axes show UTM scales (in meters).

The CCSs were located on seven different roads that had from 6 to 8 traffic lanes. All road segments except the US-24 CCS were limited access highways (classified as FCC 1 and 2). Based on short-term count and classification data, AADT ranged from 74,000 to 161,000 vehicles/day at the CCS sites. Three of the interstate highways (I-75, I-94, I-275) had considerable (7–11%) commercial traffic; the fourth (I-96) had less commercial traffic (4%). The other roads (M-10, M-39, US-24) had small fractions of commercial traffic (1–4%) and primarily serve commuters. Overall, the 13 sites represent a range of major roads and vehicle mixes across an urban area.

3. Results

3.1 Total volume – TAFs and spatial variability

Month-of-year TAFs for total vehicle volume are listed in Table 1 and plotted in Figure 2 (top left panel). The TAF gives the fraction of annual average volume for each month averaged across the sites, e.g., the January TAF of 0.909 in Table 1 means that the (total) January volume is 8.9% below the annual average. Across the 13 CCSs, month-to-month variation in traffic volume was modest, e.g., volume increased by 5 to 6% in summer months and decreased by 4 to 9% in winter, compared to the annual average. The spatial COVs indicate the site-to-site variation for each month, e.g., the January spatial COV of 6.4% indicates that a ±1 standard deviation range of the volume reduction is 8.9 ± 6.4%, or 2.5 to 15.3%, across the sites. The spatial COVs were fairly constant over the year, e.g., monthly COVs averaged 5.5% (range from 4.1% in August and September to 7.9% in November. The interquartile and extreme range of TAFs across the CCSs is shown in Figure 2; the highest and lowest values varied by about 10% from the mean. Like the daily and hourly TAFs discussed later, monthly TAFs occasionally fell to low values, likely due to extended closures or lane restrictions from construction that reduced traffic, e.g., monthly TAFs fell below 0.8 at a few CCSs on several months. However, most TAFs remained near average values (as shown by the narrow interquartile range in Figure 2), indicating that month-to-month changes in volume across the study roads were very similar, and thus only a small degree of spatial variation was present. (The temporal and total COVs listed in Tables 1–3 is discussed later.)

Table 1.

Month-of-year TAFs for total volume averaged across 13 sites and spatial, temporal and total variability expressed as coefficient of variation. COVs use average across 13 sites. Weighted average adjusts by volume on each month. Total COV based on Gaussian quadrature.

Month	Site-Ave. TAF	Coefficient of Variation (%)
		Spatial	Temporal	Total
Jan	0.909	6.4	10.4	12.2
Feb	0.980	5.0	5.3	7.3
Mar	1.033	4.2	6.4	7.6
Apr	1.011	4.4	11.4	12.2
May	1.013	7.6	10.2	12.7
June	1.057	5.3	11.6	12.7
July	1.011	6.2	10.9	12.6
Aug	1.054	4.1	8.0	9.0
Sept	1.015	4.1	11.8	12.5
Oct	1.009	5.2	8.8	10.2
Nov	0.968	7.9	9.2	12.1
Dec	0.941	5.6	6.7	8.7
Average	1.000	5.5	9.2	10.7
Weighted Average		5.5	9.2	10.7

Figure 2.

Monthly, daily and hourly TAFs for total volume showing spatial variation across 13 sites. Plots show median, 25^th and 75^th percentiles (as error bars), and minimum and maximum TAFs across 13 sites.

Table 3.

Hour-of-day TAFs for total volume averaged across 13 sites and spatial, temporal and total variability expressed as coefficient of variation. Spatial COVs use average COVs across 13 sites. Temporal COVs use average COV across 13 sites, adjusted for monthly and daily temporal variation by Gaussian quadrature. Weighted average adjusts by volume on each month. Total COV based on Gaussian quadrature. Weekdays are average of Mondays, Tuesdays, Wednesdays, Thursdays and Fridays.

Hour	Weekdays				Saturdays				Sundays				Holidays
	Site-Ave TAF	COV (%)			Site-Ave TAF	COV (%)			Site-Ave TAF	COV (%)			Site-Ave TAF	COV (%)
		Spatial	Temp.	Total		Spatial	Temp	Total		Spatial	Temporal	Total		Spatial	Temporal	Total
0	0.013	17.7	17.7	25.0	0.025	18.0	9.5	20.4	0.030	21.8	15.4	26.7	0.026	20.0	5.0	20.7
1	0.008	18.6	15.3	24.1	0.018	21.7	9.4	23.6	0.022	25.4	14.6	29.3	0.019	24.2	31.7	39.9
2	0.007	19.8	14.6	24.6	0.016	27.9	8.4	29.2	0.020	33.9	13.7	36.6	0.015	28.4	31.0	42.0
3	0.007	25.5	13.8	29.0	0.011	21.8	9.6	23.8	0.013	28.5	15.7	32.6	0.011	21.4	34.0	40.2
4	0.011	41.5	15.8	44.4	0.011	22.7	12.2	25.7	0.010	17.7	15.6	23.6	0.011	17.7	34.5	38.8
5	0.027	30.0	15.1	33.6	0.016	25.5	14.0	29.1	0.012	20.1	16.0	25.7	0.014	21.3	42.0	47.2
6	0.049	15.7	13.7	20.8	0.023	19.8	12.1	23.2	0.018	14.6	12.3	19.1	0.023	14.0	47.9	49.9
7	0.068	8.7	13.5	16.0	0.030	13.6	11.6	17.9	0.022	11.4	10.2	15.3	0.028	8.6	57.4	58.0
8	0.063	11.7	12.4	17.0	0.037	10.1	11.5	15.3	0.027	13.3	10.2	16.8	0.030	8.2	49.2	49.8
9	0.050	5.8	8.8	10.5	0.044	7.2	9.5	11.9	0.038	7.9	8.8	11.8	0.036	8.6	32.6	33.7
10	0.046	5.2	6.5	8.3	0.050	6.5	7.9	10.2	0.050	5.7	8.8	10.5	0.044	9.9	23.2	25.2
11	0.048	5.1	6.5	8.3	0.055	6.0	6.3	8.7	0.056	6.7	8.8	11.1	0.054	8.4	17.0	18.9
12	0.051	4.8	6.4	7.9	0.061	5.5	6.1	8.2	0.060	9.9	5.7	11.4	0.061	8.2	10.3	13.1
13	0.054	4.8	6.9	8.4	0.063	5.6	6.9	8.9	0.068	6.2	3.7	7.2	0.064	8.2	0.0	8.2
14	0.062	3.9	7.6	8.6	0.066	4.2	7.9	9.0	0.072	5.8	5.7	8.1	0.068	5.9	4.0	7.1
15	0.071	5.0	8.6	9.9	0.068	3.7	8.4	9.1	0.072	4.9	5.5	7.4	0.071	3.1	8.9	9.4
16	0.074	9.5	8.7	12.9	0.066	4.6	8.9	10.0	0.071	5.7	9.7	11.2	0.069	3.2	9.6	10.1
17	0.072	11.7	9.6	15.1	0.063	4.7	9.1	10.2	0.068	5.4	8.9	10.4	0.065	4.3	7.5	8.7
18	0.058	7.6	9.2	11.9	0.059	4.4	8.2	9.3	0.063	3.5	8.3	9.0	0.061	3.9	0.0	3.9
19	0.044	7.4	8.6	11.3	0.052	5.2	8.4	9.9	0.056	2.8	10.7	11.0	0.056	3.6	0.0	3.6
20	0.036	7.8	9.4	12.2	0.045	5.0	8.0	9.5	0.049	4.5	13.3	14.1	0.052	5.5	0.0	5.5
21	0.032	9.2	12.3	15.4	0.043	8.3	9.5	12.6	0.042	7.4	15.9	17.6	0.047	8.9	0.0	8.9
22	0.028	12.7	18.0	22.0	0.041	14.7	11.1	18.4	0.035	11.9	16.5	20.4	0.041	13.8	4.7	14.6
23	0.021	15.3	20.1	25.3	0.035	18.5	13.1	22.6	0.027	16.0	19.3	25.1	0.033	17.5	14.7	22.9
Average	-	12.7	11.6	17.2	-	11.9	9.5	15.2	-	12.1	11.4	16.6	-	11.5	19.4	22.6
Weighted Ave.		9.5	10.2	14.0	-	8.6	8.9	12.4	-	8.8	9.8	13.2	-	8.6	13.2	15.8

We also computed monthly TAFs by day-type. These had the same trend as shown in Figure 2 (e.g., highest in summer, lowest in winter), but had slightly greater seasonal variation (e.g., greater reductions in winter). Because TAFs for nearly all months were within a few percent of each other, however, a breakdown by day-type did not significantly alter performance of the TAF model. Also, given the limited number of Saturdays and Sundays available in each month, day-type-specific monthly TAFs may be less reliable. Thus, while monthly TAFs showed some dependence on day-type, we opted to utilize monthly TAF that apply to all days in the month.

Day-of-week TAFs for total vehicle volume are listed in Table 2 and plotted in Figure 2 (top right panel). Volumes increased slightly through the workweek, reaching a maximum on Friday (10% higher than on Monday). Compared to the weekday average, volumes were reduced by 21% on Saturdays, 33% on Sundays, and 32% on observed holidays (New Year's Day, Memorial Day, Independence Day, Labor Day, Thanksgiving, Christmas). Changes on the four other federal holidays were generally negligible; these holidays are not widely observed (Batterman, Cook et al. 2015). The spatial COVs show that the site-to-site variation was small, e.g., the COV averaged only 2% on weekends, and 4 to 5% on Saturdays, Sundays and observed holidays (Table 2).

Table 2.

Day-of-week TAFs for total volume averaged across 13 sites and spatial, temporal and total variability expressed as coefficient of variation. Spatial COVs use average COVs across 13 sites. Temporal COVs use average COV across 13 sites, adjusted for monthly temporal variation by Gaussian quadrature. Weighted average adjusts by volume on each month. Total COV based on Gaussian quadrature. Weekdays are average of Mondays, Tuesdays, Wednesdays, Thursdays and Fridays.

Day	Site-Ave. TAF	Coefficient of Variation (%)
		Spatial	Temporal	Total
Sun	0.735	5.5	14.0	15.0
Mon	1.042	1.9	10.2	10.4
Tue	1.075	1.5	9.2	9.4
Wed	1.084	1.8	10.6	10.8
Thur	1.105	1.5	0.0	1.5
Fri	1.148	1.3	10.9	11.0
Sat	0.860	4.1	12.4	13.1
Holiday	0.738	5.5	18.6	19.4
Weekdays	1.091	1.6	8.2	8.3
Average	0.973	2.9	10.8	11.1
Weighted Average		2.6	10.2	10.5

Hour-of-day TAFs are listed in Table 3 and plotted for four day-types in Figure 2 (center and lower panels). Weekday TAFs show the expected bi-modal shape, reflecting morning and afternoon rush hour commuting periods (highest volumes between 7:00 and 9:00 am, and between 3:00 to 6:00 pm). Profiles for other day types are unimodal, with the highest volume between 3:00 and 4:00 pm, and the lowest between 3:00 and 5:00 am. Site-to-site variability was 9 to 10% (COV averaged across all hours). The greatest variation among sites occurred during early morning hours, particularly on weekdays. This largely resulted from higher traffic on interstate highways (I-75, I-94 and I-275), which have considerable long-distance and through traffic (including a high fraction of trucks), compared to the other roads (M-10, US-24, M-39), which are dominated by passenger vehicles and commuters, and which have a low fraction of trucks.

Table 4 summarizes the analysis of spatial variability and propagates monthly, daily and hourly contributions to estimate the total spatial variability. On weekdays, for example, the spatial COVs for monthly, daily and hourly TAFs (5.5, 2.0 and 9.5% respectively) give a total spatial variability of 11.1%. Results for the other day-types are comparable. Overall, monthly, daily and hourly TAFs across the Detroit area CCSs are comparable, thus, the typical or expected level of volume changes in uniform manner across the study roads. This may reflect the similarity of the study roads, which were mostly large limited-access urban freeways. Greater spatial variation would be anticipated for a larger cross-section of roads (including freeways, surface arterials, collectors, rural and urban freeways, etc.) Roads that serve primarily serve “through” traffic would be expected to differ from roads that serve primarily local or urban traffic, but no such distinctions were found, suggesting that both types of traffic were carried on most roads. These results apply to the total volume of vehicles, which is dominated by light-duty passenger vehicles. (Later we show that spatial differences are greater for commercial traffic, where the distinction between urban and through traffic may be more meaningful.) The conclusion of limited spatial variability and the statistics in Table 4 apply to typical cases. More extreme variation can occur, e.g., Figure 2 shows that the extreme range of monthly and hourly TAFs (but not daily) could vary by 20%.

Table 4.

Summary of variability analysis for total traffic at 13 sites. Shows spatial, temporal and total (by propagation) COVs (in percent) and contributions from monthly, daily, and hourly averaging periods for four day types (weekdays, Saturdays, Sundays and holidays). Fraction of total variation (in percent) shown in parentheses.

Period	COV Type	Monthly	Daily	Hourly	Total (1)	Total (2)
Weekdays	Spatial	5.5 (7.9)	1.6 (0.7)	9.5 (23.7)	11.1 (32.3)	11.1 (29.3)
	Temporal	9.2 (22.5)	8.2 (17.6)	10.2 (27.6)	16.0 (67.7)	17.2 (70.7)
	Total	10.7 (30.4)	8.3 (18.3)	14.0 (51.3)	19.5 (100.0)	20.5 (100.0)
Saturdays	Spatial	5.5 (6.8)	4.1 (3.8)	8.6 (16.8)	11.0 (27.5)	11.0 (26.6)
	Temporal	9.2 (19.4)	12.4 (35.1)	8.9 (18.0)	17.9 (72.5)	18.2 (73.4)
	Total	10.7 (26.3)	13.1 (38.9)	12.4 (34.8)	21.0 (100.0)	21.3 (100.0)
Sundays	Spatial	5.5 (5.8)	5.5 (5.9)	8.8 (15.1)	11.8 (26.8)	11.8 (24.6)
	Temporal	9.2 (16.6)	14.0 (38.0)	9.8 (18.7)	19.4 (73.2)	20.6 (75.4)
	Total	10.7 (22.4)	15.0 (43.8)	13.2 (33.8)	22.7 (100.0)	23.7 (100.0)
Holidays	Spatial	5.5 (4.1)	5.5 (4.0)	8.6 (10.0)	11.6 (18.0)	11.6 (12.2)
	Temporal	9.2 (11.5)	18.6 (46.9)	13.2 (23.6)	24.6 (82.0)	31.0 (87.8)
	Total	10.7 (15.6)	19.4 (50.9)	15.8 (33.5)	27.2 (100.0)	33.0 (100.0)

3.2 Total volume - Temporal variation

The temporal variability in TAFs is more complex than spatial variability. Figure 3 illustrates results for a segment on I-94. The left panel shows the variability of monthly TAFs on this segment as COVs over the four year study period. Minimum and maximum observations ranged from −9 to +11% of the expected (long-term) value; the interquartile range was −6 to 6%; and the volume-weighted monthly COV across all 12 months was 6.1%. This means that the fluctuation in January volumes, for example, was typically about 6% of the long term average. These results are based on only 4 observations per month, which may be too short to obtain robust statistics for individual months. Temporal variation increases at the daily level, particularly on Saturdays, Sundays and holidays (center panel, Figure 3). Considering extremes, daily volumes increased as much as 26% over the expected volume on one Sunday and by 71% on one holiday (probably a holiday that was not observed one year); flows decreased by as much as 28 to 70%, depending on the day (probably due to adverse weather). The interquartile range was −13% to +5%. The volume-weighted COVs were 7 to 8% on weekdays, 10% on Saturdays, 13% on Sundays, and 18% on holidays; the volume-weighted average COV was 9.4%. Importantly, variation on weekdays was similar, supporting the grouping of weekdays. These results are based on nearly 200 observations of each day type at this site (27 for holidays), and thus are robust. The right panel of Figure 3 shows temporal variation at the hourly level for weekdays. Observations ranged from −87 to +149% of the expected volume; the interquartile range was −12% to +10%; and the weekday volume-weighted average COV was 10.7%. This is based on over 950 observations for each hour. Variability increased on Saturdays (13.7%), Sundays (15.3%) and holidays (22.8%; all average volume-weighted COVs). Overall, temporal variability is dominated by hourly variation, followed by daily and then monthly variation, and results are asymmetrical with a possibility of much lower than average volume, while much higher than average volume is unlikely. Other CCSs showed very similar patterns to the results for the I-94 CCS.

Figure 3.

Example of temporal variation for example road (I-94, CCS=9489). Panels show relative differences at monthly (left), daily (center) and hourly (right) averaging periods. Each plot shows minimum, 25^th, 50^th, 75^th percentile, and maximum differences. Day types as follows: 1 = Sundays, 2=Mondays, 3=Tuesdays, 4=Wednesdays, 5=Thursdays, 6=Fridays, 7=Saturdays, 8=holidays.

Results of the temporal analysis are presented in Tables 2 to 4, which list COVs showing temporal variation at monthly, daily and hourly levels across the 13 CCS sites, and in Figure 4, which plots monthly, daily and hourly COVs, including the median, interquartile range, minimum and maximum COVs across the sites. The plots also show the average and median COV across the time periods as dashed lines. Monthly COVs varied considerably, ranging up to 50% at some sites and some months (Figure 4 top left), thus the average COV considerably exceeds the median COV. There were no strong trends by site or month, although variation was slightly higher in the summer, likely reflecting the summer construction season. On I-94 and M-39, construction may have restricted volume for several months, increasing COVs beyond 40%. The average monthly temporal variability, 9.2%, may be slightly inflated due to outliers.

Figure 4.

Coefficient of variation (COV) for monthly, daily and hourly TAFs for total volume showing temporal variation. Plots show median, 25^th and 75^th percentiles (as error bars), and minimum and maximum COVs (as points) across 13 sites. Hourly TAFs show four day types.

At the daily level, COVs averaged 14.2%. As noticed earlier, COVs on weekends and especially holidays were higher (Figure 4, right). At the hourly level, weekday, Saturday and Sunday results were similar (COVs averaged 16.2, 19.7, and 17.9%, respectively). Hourly COVs tended to increase during low traffic periods (early morning and late evening). On holidays, hourly COVs averaged 26.8% and the variation over the day was notable, a contrast to the other day-types. With this exception, however, hourly COVs were reasonably consistent over the day.

Hourly traffic volumes were highly autocorrelated, e.g., the lag 1 autocorrelation for total volume on Detroit area freeways is typically from 0.92 to 0.95. This high correlation results from the smooth and regular changes in volumes (as demonstrated by the hourly TAFs). More important for the temporal analysis, however, is that the residuals from the TAF models (observed - predicted hourly volumes) also were highly autocorrelated, meaning that an unusually high (or low) volume on one hour will likely continue for the next few hours. The present analysis provides correct results of temporal variability for monthly, daily and hourly averaging periods; other periods require a separate analyses to account for autocorrelation.

3.3 Total volume - spatial and temporal variability

Table 4 consolidates the spatial and temporal analyses, listing the volume-weighted average COVs and propagated uncertainties at monthly, daily and hourly levels. A separate analysis is performed for each day-type. (Each day-type uses the same monthly COVs.) The volume-weighted averages are intended to make the analysis more relevant to emission estimates. Two estimates of temporal variability are provided: COVs using quadrature from monthly, daily and hourly components (listed as “Total 1”), and actual COVs (“Total 2”) that cannot be further apportioned. These two estimates are very similar, supporting the error propagation approach. Since Total 2 fully accounts for any correlations or dependencies that may be present, this estimate is emphasized.

The total (spatial and temporal) variability of total volume estimates, derived using TAF models and represented by COVs, was 21, 33, 24 and 33% for weekdays, Saturdays, Sundays and holidays, respectively (Table 4). These values include effects of monthly, daily and hourly fluctuations. Temporal variability had the largest share of the variability (71 to 89%, depending on day type). On weekdays and Saturdays, temporal variability largely arises from hour-to-hour variability (51 to 64% of total variability), followed by daily and monthly variability. In comparison, most of the temporal variability on Sundays and holidays arises from daily variability (44 to 51%), reflecting the less predictable or repeatable schedules on Sundays and holidays compared to the more regular weekday and Saturday patterns. The contribution of spatial variability to the total variability was under 29%, and the portions attributed to monthly and especially daily variability were small (<8% of the total for monthly, <6% for daily). As noted, the average temporal traffic activity patterns across the study roads were similar.

Several examples help interpret the results. The total weekday COV of 20.5% means that the 75^th, 90^th and 98^th percentile volumes would exceed mean volumes by 14, 34 and 42%, respectively. The higher COVs on Saturdays, Sundays and holidays further increase these values (by 4, 16 and 61%, respectively). Expressed differently, on weekdays, there is a 22% probability that the volumes will deviate from the mean by more than 25% (either higher or lower), and there is a 3% probability of a 50% or greater difference. The emissions inventory guidance from the Intergovernmental Panel on Climate Change suggests using a 95% confidence interval (Intergovernmental Panel on Climate Change 1996). This translates to ±40% interval about the mean. While these examples illustrate the variations from mean values, the overall variability is considered relatively modest. In particular, a 25% accuracy criterion, which often is applied in quality assurance activities and air quality modeling applications, is not frequently exceeded. This conclusion is limited to total vehicle volume (not emission rates), and as discussed below, has several restrictions.

The variability estimates may be under-estimated for several reasons. We assumed independence of temporal and spatial factors, used Gaussian distributions, and present results averaged across the 13 road segments. In addition, the analysis was based mostly on urban freeways in a single region, and the distributions of percentage differences had “fat” tails. However, tests using percentage differences (eq. 3) and TAF models provided comparable estimates, supporting the approach, and the largest differences during high traffic periods will be skewed downward, a result of capacity limitations on the roads. Given the parametric measures used, results may be sensitive to outliers, however, results were not altered substantially using trimmed data sets, e.g., 10^th to 90^th and 2^nd to 98^th percentile values. Lastly, results differ by vehicle type. The variability of commercial and heavy duty vehicle volumes, which are responsible for a disproportionate share of on-road emissions, is examined next.

3.4 Spatial and temporal variability by vehicle type

Analyses performed for the four road links (I-94, US-24, I-75, I-275) that used weigh-in-motion vehicle sensors allowed a breakdown by vehicle type. Figures S2 and S3 compare TAFs and the spatial variability across the four roads for noncommercial and commercial vehicles, respectively; the corresponding TAFs and COVs are listed in Tables S1 to S6. Spatial and temporal variability for non-commercial and commercial traffic activity is summarized in Tables 5 and 6, respectively.

Table 5.

Summary of variability analysis for passenger (non-commercial) traffic at 4 sites. Shows spatial, temporal and total (by propagation) COVs (in percent) and contributions from monthly, daily, and hourly averaging periods for four day types (weekdays, Saturdays, Sundays and holidays). Fraction of total variation (in percent) shown in parentheses.

Period	COV Type	Monthly	Daily	Hourly	Total (1)	Total (2)
Weekdays	Spatial	2.6 (1.5)	2.2 (1.1)	10.6 (24.7)	11.2 (27.2)	11.2 (23.7)
	Temporal	6.9 (10.3)	12.7 (35.3)	11.2 (27.1)	18.3 (72.8)	20.0 (76.3)
	Total	7.3 (11.8)	12.9 (36.4)	15.4 (51.8)	21.4 (100.0)	22.9 (100.0)
Saturdays	Spatial	2.6 (2.0)	3.7 (4.2)	7.6 (17.6)	8.8 (23.8)	8.8 (21.4)
	Temporal	6.9 (14.5)	11.0 (37.0)	9.0 (24.7)	15.8 (76.2)	16.9 (78.6)
	Total	7.3 (16.5)	11.6 (41.2)	11.8 (42.3)	18.1 (100.0)	19.1 (100.0)
Sundays	Spatial	2.6 (0.8)	9.0 (9.4)	8.2 (7.8)	12.5 (17.9)	12.5 (16.0)
	Temporal	6.9 (5.4)	24.3 (67.9)	8.7 (8.7)	26.7 (82.1)	28.7 (84.0)
	Total	7.3 (6.2)	25.9 (77.3)	12.0 (16.5)	29.5 (100.0)	31.3 (100.0)
Holidays	Spatial	2.6 (1.2)	8.3 (12.5)	7.3 (9.6)	11.4 (23.3)	11.4 (15.1)
	Temporal	6.9 (8.5)	15.2 (41.2)	12.3 (27.0)	20.7 (76.7)	27.1 (84.9)
	Total	7.3 (9.7)	17.3 (53.7)	14.3 (36.7)	23.6 (100.0)	29.4 (100.0)

Table 6.

Summary of variability analysis for commercial traffic at 4 sites. Shows spatial, temporal and total (by propagation) COVs (in percent) and contributions from monthly, daily, and hourly averaging periods for four day types (weekdays, Saturdays, Sundays and holidays). Fraction of total variation (in percent) shown in parentheses. (1) Total estimate based on quadrature from spatial and temporal components. (2) Total based on quadrature of spatial components and total variability of temporal component.

Period	COV Type	Monthly	Daily	Hourly	Total (1)	Total (2)
Weekdays	Spatial	3.6 (1.8)	2.7 (1.0)	14.8 (30.8)	15.5 (33.7)	15.5 (29.6)
	Temporal	8.5 (10.2)	14.7 (30.4)	13.5 (25.7)	21.7 (66.3)	23.9 (70.4)
	Total	9.2 (12.0)	15.0 (31.4)	20.1 (56.6)	26.7 (100.0)	28.4 (100.0)
Saturdays	Spatial	3.6 (0.8)	12.9 (10.8)	10.8 (7.5)	17.2 (19.2)	17.2 (15.9)
	Temporal	8.5 (4.7)	20.6 (27.6)	27.3 (48.4)	35.2 (80.8)	39.5 (84.1)
	Total	9.2 (5.6)	24.3 (38.5)	29.3 (56.0)	39.2 (100.0)	43.0 (100.0)
Sundays	Spatial	3.6 (0.5)	26.8 (28.6)	14.2 (8.0)	30.6 (37.1)	30.6 (35.7)
	Temporal	8.5 (2.9)	27.9 (30.9)	27.1 (29.1)	39.8 (62.9)	41.0 (64.3)
	Total	9.2 (3.4)	38.7 (59.5)	30.6 (37.1)	50.2 (100.0)	51.1 (100.0)
Holidays	Spatial	3.6 (0.3)	10.8 (2.3)	14.7 (4.3)	18.5 (6.9)	18.5 (7.3)
	Temporal	8.5 (1.5)	60.2 (73.3)	30.0 (18.3)	67.8 (93.1)	66.2 (92.7)
	Total	9.2 (1.7)	61.2 (75.7)	33.4 (22.6)	70.3 (100.0)	68.7 (100.0)

TAFs for non-commercial vehicles are similar to those discussed previously for total traffic with one exception: weekday hourly TAFs have more pronounced and sharper rush hour peaks, a result of omitting commercial traffic that tends to follows a different diurnal pattern (discussed below). Although data from only four roads were included, the spatial, temporal and total variation at monthly, daily and hourly levels, as well as the total variability, were very similar to that shown earlier (Tables 4 and 5), suggesting that these four roads are representative of the larger set.

TAFs for commercial vehicles differ from those discussed earlier, as noted in earlier work (Batterman, Cook et al. 2015). At the monthly level, the overall trend remains very similar, although the one month decline in July was more pronounced. At the daily level, volumes were very significantly reduced on Saturdays, Sundays and holidays. At the hourly level, a single broad peak captured the trend for each day type; on weekdays, the peak was relatively flat over typical working hours (8 am to 3 pm); on Sundays, volume was relatively constant from 11 am to the end of the day (although volumes are low: 79% below weekday averages). These trends carry over to the variability analysis (Table 6). For commercial vehicles, total variability on weekdays, Saturdays, Sundays and holidays was 28, 43, 51 and 69%, all significantly higher than seen for non-commercial (or total) vehicles. The large increase on holidays reflects that commercial vehicle volumes depend on the holiday and which day of the week it falls. While only the six federal holidays typically observed in Michigan were considered, the variability remains large for this subset. Considering weekdays that have much higher commercial traffic than other day types, 38% of volume estimates fall outside the ±25% accuracy criterion, and 8% would be off by 50% or more. A portion of the increase is attributable to temporal variation on US-24, a road with little truck traffic (390 vehicles per day). Excluding this road reduced the total variability to 25, 28, 44, and 64% for weekdays, Saturdays, Sundays and holidays, respectively.

Overall, commercial traffic has greater variability than total or and non-commercial vehicle traffic. On an hourly and weekday basis, variability of 25 to 28% is expected. Variability is higher on weekends and holidays, but truck volume during these periods is much lower. While subject to the same limitations expressed earlier, as well as a more restrictive sample size, the TAF-based models using region-specific data performed well, providing accurate estimates of the number and variability of both commercial and non-commercial vehicles.

4. Discussion

Traffic volume, speed, acceleration and vehicle mix fluctuate due to commuting and work schedules, construction activities, weather, and many other factors that vary over time and location. The relatively simple TAF-based models could represent much of the historical variability of traffic volume at the urban scale. The models give the expected or average volume based on historical data, and prediction errors are typically on the order of 25% for weekdays and higher (especially for commercial vehicles) on Sundays and holidays. These errors mostly resulted from hour-to-hour variability on weekdays and Sundays, and from day-to-day variability on Saturdays and holidays. The spatially variability was limited in the study region, thus, a single set of TAFs could portray traffic volume on all study roads. These results may represent a “best-case” analysis since region-specific TAFs were derived using local data, which minimized systematic biases, and only larger roads were considered, which likely reduced spatial variation. Errors may increase for a larger region (e.g., state-wide), other types of roads (e.g., secondary arterial, rural highways, through versus local traffic), specific vehicle types (e.g., heavy-duty trucks), and with less comprehensive data. Still, our results may be applicable to traffic activity on larger roads in many mid-to-large size U.S. cities. These roads include interstates, other freeways and major arterials that collectively account for much or most of VKT and on-road emissions. Continuous traffic monitoring on smaller roads (e.g., local, collector, and minor arterial roads) is uncommon, and no evaluation of uncertainty or variability was attempted for such roads.

Traffic activity might be modeled in other ways. The arrival of vehicles at a location (CCS) might be modeled as a Poisson random process, which is characterized by an average arrival rate λ and other parameters. Agent-based or “microsimulation” models can be used to represent behaviors of individual vehicles. Alternatives to such “microscopic” level approaches include various types of time series statistical models that account for the periodicity and growth of traffic volume. Instead, a “macroscopic” or “engineering” approach was used in which TAFs were derived from hourly, daily and monthly averages and variability using COVs. This extends the prevailing approach for allocating annual estimates of traffic activity to the hourly level using TAFs, which forms the basis of temporally resolved link-based emissions inventories. In turn, these inventories provide inputs to dispersion models used to estimate near-road concentrations and exposures of air pollutants.

Our grouping of day types, namely, weekdays, Saturdays, Sundays and holidays, was based on repeatable patterns shown for both total and commercial vehicle volumes. However, other groupings of day type, vehicle type, month and possibly other variables (e.g., road type, urban vs. rural) may reduce uncertainties, e.g., Mondays and Fridays may differ from midweek days (Tuesday, Wednesday, Thursday). Approaches to grouping variables, e.g., cluster analysis, might be useful in future studies to define such subgroups.

Vehicle volume is only one of many factors governing on-road emission rates. Other aspects of traffic activity that undergo spatial and temporal variation include vehicle speed, acceleration, age, and fleet composition. Given the “U” shaped profile of vehicle emission rates (g/km-vehicle) versus speed or engine power, moderate congestion on freeways may decrease speed (and average engine power) thus lowering emissions; extreme congestion may lead to stop-and-go traffic, transient acceleration, and higher emissions. Such effects are important if peak volumes approach road capacity and lead to congested conditions. Conversely, at low levels of congestion, representing the majority of observations in the study, emissions will be proportional to volume (assuming vehicle mix and other factors are constant). Thus, the 25% variability estimated for vehicle volume at the hourly level may apply to emission rates. However, emission rates can have much higher uncertainties since they incorporate the uncertainty of emission factors, which can be large, especially for toxic pollutants like benzene and formaldehyde (Frey and Zhao 2004).

Volume errors on the order of 25% using TAF-based models may apply if the TAFs use local data, separate each day type (weekdays, Saturdays, Sundays, holidays), and distinguish major vehicles classes (e.g., passenger and commercial vehicles). Importantly, many or most TAFs in the literature are deficient: they are old and may no longer reflect traffic patterns; vehicle classes are not adequately distinguished (e.g., a single fixed vehicle mix is assumed); and day types are not separated appropriately. As a result, errors can be large, especially for commercial vehicles that account for a disproportionately large share of pollutant emissions. We previously investigated the use of literature TAFs for Detroit and found volume errors that averaged 35 to 40%; errors were larger (sometimes over a factor of two) for commercial vehicles since literature TAFs did not represent their unimodal pattern on weekdays and the large decreases on weekends (compare Supplemental Figures S2 and S3) (Batterman, Cook et al. 2015).

Air quality analyses would be strengthened by incorporating uncertainty and variability information. For example, compliance with short-term air quality standards for NO₂ or CO demonstrated using a 90 or 95^th percentile value for traffic volume would provide a high degree of confidence that standards would be attained in “hot spot” and other applications. Such analyses require the average traffic volume, which can be derived using AADT and the TAFs, and upper bound volumes, which can be calculated using the COVs in Tables 4 to 6 and the inverse normal distribution with the desired statistical confidence level. For example, for a maximum weekday hourly traffic volume of 1000 vehicles/h, a COV of 20.5% (Table 4), and 95% confidence (z-score of 1.645), the upper bound volume is 1337 vehicle/h (1000 vehicles/h × 1.205 × 1.645), i.e., volumes would not be expected to exceed this level more than 5% of the time. Ideally, such analyses would be performed by vehicle class since COVs and emission factors differ by vehicle class.

Other air quality applications would benefit by considering the variability of traffic volume data. For example, the uncertainty of emission rates could be incorporated in the evaluation of air quality model performance. Rather than stating the number of predictions that fall within a factor or two (or other bound) of observations, an arbitrary and not altogether satisfying criterion, a probabilistic analysis could account for the uncertainty in predictions and the likelihood that predictions match observations (Rao 2005). A second example is the reconciliation of dispersion modeling predictions with the (inevitably limited) monitoring data available, and the potential use of Bayesian melding and other techniques to improve predictions and reduce uncertainties. Third, to guide research aimed at further improving models, sources of uncertainty could be grouped into categories (due to emissions, dispersion, background estimates, etc.), apportioned, and prioritized. Unfortunately, these approaches are not routinely used. Alternatives to the use of probabilistic approaches have deficiencies. For example, analyses using the maximum-possible emission rate can overestimate concentrations and be unrealistic, while analyses using expected or average emission rates do not represent peak emission and concentration events.

5. Conclusions

This analysis has shown that link-based and temporally-resolved traffic activity measures can be developed with reasonable accuracy using annual estimates of traffic activity and appropriate temporal allocation factors. These are necessary inputs for the development of spatially- and temporally-resolved on-road emissions inventories, which in turn can be used to predict concentrations and exposures of traffic-related air pollutants. These results apply to large roads in U.S. cities that carry a major fraction of vehicle traffic and that are responsible for a large portion of emissions in urban areas. While vehicle volume is only one of many factors governing emission rates of on-road vehicles, air quality analyses could be strengthened by incorporating information regarding the uncertainty and variability of traffic activity.

Supplementary Material

supplement

Figure S1. Relative differences in total volume by hourly, road and day type. Shows minimum, 5^th, 25^th, 50^th, 75^th, 95^th percentile and maximum differences.

Figure S2. TAFs for non-commercial vehicles at 4 sites showing spatial variability (range and median).

Figure S3. TAFs for commercial vehicles at 4 sites showing spatial variation (range and median at 4 sites).

Table S1. Month-of-year TAFs for passenger vehicle volume averaged across 4 sites and spatial, temporal and total variability expressed as coefficient of variation. COVs use average across 4 sites. Weighted average adjusts by volume on each month. Total COV based on Gaussian quadrature.

Table S2. Day-of-week TAFs for passenger vehicle volume averaged across 4 sites and spatial, temporal and total variability expressed as coefficient of variation. COVs use average across 4 sites. Weighted average adjusts by volume on each month. Total COV based on Gaussian quadrature. Weekdays are average of Mondays, Tuesdays, Wednesdays, Thursdays and Fridays.

Table S3. Hour-of-day TAFs for non-commercial volume averaged across 4 sites and spatial, temporal and total variability expressed as coefficient of variation. COVs use average across 4 sites. Weighted average adjusts by volume on each month. Total COV based on Gaussian quadrature. Weekdays are average of Mondays, Tuesdays, Wednesdays, Thursdays and Fridays.

Table S4. Month-of-year TAFs for commercial vehicle volume averaged across 4 sites and spatial, temporal and total variability expressed as coefficient of variation. COVs use average across 4 sites. Weighted average adjusts by volume on each month. Total COV based on Gaussian quadrature.

Table S5. Day-of-week TAFs for commercial vehicle volume averaged across 4 sites and spatial, temporal and total variability expressed as coefficient of variation. COVs use average across 4 sites. Weighted average adjusts by volume on each month. Total COV based on Gaussian quadrature. Weekdays are average of Mondays, Tuesdays, Wednesdays, Thursdays and Fridays.

Table S6. Hour-of-day TAFs for commercial volume averaged across 4 sites and spatial, temporal and total variability expressed as coefficient of variation. COVs use average across 4 sites. Weighted average adjusts by volume on each month. Total COV based on Gaussian quadrature. Weekdays are average of Mondays, Tuesdays, Wednesdays, Thursdays and Fridays.

Highlights.

• Temporally resolved emissions are derived using temporal allocation factors or TAFs.

• TAF-based models can accurately predict expected hourly traffic flows.

• Schedules of passenger and commercial vehicles must be distinguished.

• Spatial and temporal variability strongly affects traffic volume and emissions.

• Temporal variability at especially daily and hourly levels is significant.