# Estimation of Regional Economic Development Indicator from Transportation
Network Analytics111This is a pre-print of an article submitted to Scientific
Reports. The final authenticated version is available online at:
https://doi.org/10.1038/s41598-020-59505-2

Bin Li  School of Automobile, Chang’an University, Xi’an, Shaanxi, 710064,
China  Song Gao  Geospatial Data Science Lab, Department of Geography,
University of Wisconsin-Madison, WI, 53706, USA  Corresponding author email:
<EMAIL_ADDRESS>Yunlei Liang  Geospatial Data Science Lab, Department of
Geography, University of Wisconsin-Madison, WI, 53706, USA  Yuhao Kang
Geospatial Data Science Lab, Department of Geography, University of Wisconsin-
Madison, WI, 53706, USA  Timothy Prestby  Geospatial Data Science Lab,
Department of Geography, University of Wisconsin-Madison, WI, 53706, USA  Yuqi
Gao  Geospatial Data Science Lab, Department of Geography, University of
Wisconsin-Madison, WI, 53706, USA  Runmou Xiao  School of Automobile, Chang’an
University, Xi’an, Shaanxi, 710064, China

###### Abstract

With the booming economy in China, many researches have pointed out that the
improvement of regional transportation infrastructure among other factors had
an important effect on economic growth. Utilizing a large-scale dataset which
includes 3.5 billion entry and exit records of vehicles along highways
generated from toll collection systems, we attempt to establish the relevance
of mid-distance land transport patterns to regional economic status through
transportation network analyses. We apply standard measurements of complex
networks to analyze the highway transportation networks. A set of traffic flow
features are computed and correlated to the regional economic development
indicator. The multi-linear regression models explain about 89$\%$ to 96$\%$
of the variation of cities’ GDP across three provinces in China. We then fit
gravity models using annual traffic volumes of cars, buses, and freight trucks
between pairs of cities for each province separately as well as for the whole
dataset. We find the temporal changes of distance-decay effects on spatial
interactions between cities in transportation networks, which link to the
economic development patterns of each province. We conclude that
transportation big data reveal the status of regional economic development and
contain valuable information of human mobility, production linkages, and
logistics for regional management and planning. Our research offers insights
into the investigation of regional economic development status using highway
transportation big data.

## Introduction

With the booming economy in China, many researches have pointed out that the
improvement of regional transportation infrastructure, the mobility of labor
and capital, and industry reform along with other socioeconomic factors play
an important role on economic growth. Timely estimation of social and economic
status of cities and regions has important implications for enterprise
investment and government policy making. Traditional inference approaches to
economic status mainly rely on official reports and census surveys, which
usually take a long period and are labor intensive. With the rapid development
of information, communication and technology (ICT), new data sources of human
activities [1] and vehicle movement flow [2, 3, 4], air transport flow [5, 6],
financial flow [7], information flow [8], communication flow [9, 10, 11, 12] ,
and others [13] have become available for better understanding and monitoring
the status of our socioeconomic environments [14, 15]. Liu et al. [1] found
that online social activity could reflect the macro economic status of 282
prefecture-level cities in China. Recently, Gao et al. [16] conducted a
comprehensive review on data resources, computational tools, analytical
methods, theoretical models, and applications in computational socioeconomics.

In the past decades, a wealth of works have been dedicated to studying the
pattern of human mobility involving passenger transportation [17, 18, 19, 20,
21, 22, 23]. A comparatively smaller literature has been dedicated to the
pattern of transportation activity embedded in goods movement [24, 25]. The
scarcity of reliable data sources on freight transportation appears to be one
of the challenges [26]. Early studies rely on traditional freight traffic
surveys, which are typically enterprise questionnaire surveys to obtain
information such as the traffic volume and speed in specific road sections.
For example, Ogunsanya [27] examined the field survey data collected from
several freight terminals using questionnaires and freight delivery records of
major warehouses in Lagos, Nigeria. With floating car technology being
increasingly applied in transportation, large-scale global positioning system
(GPS) tracking data of trucks have been used by researchers in exploring the
spatial interaction patterns. In addition, several applications have been
created in transportation planning through the prediction of road travel time,
logistic demand, and analysis of vehicle operation characteristics. Comendador
et al. [28] presented urban freight distributions in two Spanish cities using
GPS data, and compared the general mobility patterns of five groups of freight
according to the type of goods including frequency and average distance. Fu
and Shi [26] analyzed the spatiotemporal features of freight truck trips using
GPS data that record the truck operation in real time. Zanjani et al. [29]
estimated origin-destination truck flows based on truck GPS data, and the
results indicate the value of GPS data in freight travel demand modeling.
Mrazovic et al. [30] explored the spatial and temporal patterns of trucks
based on data on urban freight deliveries. Boarnet et al. [31] explored the
freight flows in Los Angeles and provided evidence on employment being an
important driver of freight activity. With the burgeon of new data sources on
transportation of goods, De Montis et al. [32] obtained a weighted network
representation in which the vertices correspond to the Sardinian
municipalities in Italy and the weighted edges correspond to the amount of
commuting traffic among them. They characterized the structure of human
traffic (at the intercity level) and investigated its relation to the
topological structure of cities (defined by the connectivity pattern among
them). Zhao et al. [4] used logistics data on origin-destination flows of
goods in Hong Kong and analyzed the intra-urban freight movement based on the
gravity model with a specific interest in uncovering potential trends in the
distance decay effect, reflected by the parameter $\beta$, through multiple
years of observation. They took the population of sub-districts in the area of
study as the values for nodal attractions and estimated the distance decay
parameter $\beta$ by selecting among the candidates the parameter value that
yielded the best goodness of fit. With the obtained estimates for $\beta$,
they demonstrated that the estimated interaction flows coincide reasonably
well with observed interaction flows, which argues for the effectiveness of
the gravity model in determining spatial interactions of freight movements.
Recently, Ding et al. [33] reviewed the applications of complex network theory
in urban traffic studies. Our study differs from theirs in terms of the
activity space (i.e. intra-urban vs. inter-urban). Accordingly, our study
investigates regional connections and includes both intracity and intercity
transportation of people and goods through highways.

Apart from the studies using data on coach (vehicle) services, the analysis of
airline services [34, 5, 35] and railway services [36] have been catching the
attention of researchers on the studies of spatial interactions. Barrat et al.
[37] provided an early study of the worldwide airports network including the
traffic flow and its correlation with the topological structure. A closely
related field of study on transportation is dedicated to investigating the
impact of transportation infrastructure on socioeconomic status such as
economic growth or population growth which Beyzatlar et al. [38], Iacono and
Levinson [39] investigated. The rapid development of high-speed rails plays an
important role in regional economic development. Zheng and Kahnconnect [40]
demonstrated the effects of China’s bullet trains in facilitating labor market
integration and mitigating the housing price and living costs in megacities.
Jia et al. [41] showed that the high-speed rail construction has a positive
effect on regional economic growth in China. Cheng et al. [42] and Chen $\&$
Vickerman [43] investigated how the new development of high-speed rail
infrastructure impacts on the economy structure of cities and regions in
Europe. However, the endogenous problem of inverse causality of development
potential on infrastructure investment has been a hurdle in producing a
convincing conclusion on causality. Although the new development of high-speed
rails reduces the transportation costs of people between large cities, there
is a significant reduction in GDP after the high-speed rail upgrade in the
counties located along the affected railway lines. The reduction was largely
driven by the concurrent drop in fixed asset investments of those bypassed
counties [44]. Gao et al. [45] introduced the concept of inter-industry and
inter-regional learnings of regional economic development. Using 25 years of
economic data in China between 1990 and 2015, they addressed the endogeneity
concerns by using the difference-in-differences analysis. Results showed that
the high-speed rail development increased the industrial similarity of
connected pairs of neighboring provinces [45].

In this research, we examined the regional economic development indicator
measured by the gross domestic product (GDP) values of cities and the
relationship between GDP and transportation activities of human and goods in
three provinces of China. Over 3.5 billion records of vehicle entry and exit
data were collected in highway toll stations (287 in Liaoning, 421 in Jiangsu,
and 335 in Shaanxi, respectively) using toll collection systems and surveys in
these three provinces between years 2014 and 2017. The raw station-level data
were then aggregated and summarized to the city level as our main focus is to
investigate the relationship between regional economic development and the
transportation networks. More detailed data, feature construction, and method
descriptions can be found in the "Methods" section. We conducted the multiple
linear regression analysis using a set of traffic flow features with/without
regularization techniques, and the modeling results explained about 89$\%$ to
96$\%$ of the variation of cities’ GDP across three provinces. To further
investigate the regional spatial interaction characteristics, we then
constructed the highway transportation networks to fit gravity models using
annual traffic volumes of cars, buses, and trucks between pairs of cities for
each province separately as well as together for the whole dataset. We found
the temporal changes of distance-decay effects on spatial interactions in
transportation networks, which link to the regional economic development
patterns in each province. In summary, the major contributions of this
research are three-fold: (1) we present an analytical workflow to investigate
the regional economic development status using highway transportation big
data; (2) the analyses of highway transportation networks using the gravity
model and the principle component analysis provide a good interpretation of
the spatial structure of regional highway transportation development and the
temporal economic changes; (3) the weighted network measures using the traffic
flows correlate better with regional economy than that using the physical
distance-based ones.

## Results

### Estimation of GDP from traffic flows

The relationship between the fitted city GDP values using the multiple linear
regression (MLR) models of transportation features and the actual city GDP in
three provinces (i.e., Liaoning, Jiangsu, and Shaanxi) are summarized in Fig.
1. It shows that simple transportation flow features (i.e., intra-city and
inter-city flows of cars, buses, and trucks) extracted from the transportation
networks of cities (in Equation 2) can explain the variation of the economic
development indicator (i.e., GDP) among cities very well, with the goodness of
fit: R-squared of 0.934 (in Liaoning province), 0.892 (in Jiangsu province),
and 0.967 (in Shaanxi province). In addition, the R-squared further increased
a margin (+0.025 in Liaoning, +0.021 in Jiangsu, and +0.014 in Shaanxi) by
including the volume of passengers in cars $\&$ buses and the freight truck
weights in the MLR model. After transforming the dependent variable (GDP) with
the natural log form (Ln), the generalized linear model had goodness of fit
R-squared of 0.849 (in Liaoning), 0.727 (in Jiangsu), and 0.914 (in Shaanxi),
respectively. The results are not as good as the original MLR approach. The
prediction root-mean-square error (RMSE) of city GDP using original MLR model
in three provinces are 53.5 (Liaoning), 119.8 (Jiangsu), and 30.11 (Shaanxi)
billion CNY, respectively. As shown in the residual plots Figs. S2, S3, S4,
and S5, the residuals center on zero and are not correlated with any
predictors, which indicate that these models’ predictions have a relatively
constant variance and show homoscedasticity and normality. In addition, by
applying two regularized regression methods: the Ridge regression [46] and the
least absolute shrinkage and selection operator (LASSO) regression [47, 48],
the smallest RMSE values were 54.2 (Liaoning), 121.0 (Jiangsu), and 30.66
(Shaanxi) billion CNY, respectively. Moreover, these two methods only select a
few predictors while reducing the coefficients of other highly correlated
predictors to zero. It helps mitigate the problem of multicollinearity in MLR
based on the ordinary least squares estimation, and the model still keeps a
high goodness of fit for each province’ data (See the ’Discussion and
Conclusion’ section in detail). However, the selected three provinces had
different economic development patterns in the study period. Accordingly, the
goodness of fit (R-squared = 0.587) of a universal MLR model which combines
all the GDP values and traffic flow data across three provinces is not as good
as that in each province. The corresponding prediction RMSE increased to about
212 billion CNY using MLR. This demonstrates the complexity and heterogeneity
of the economic development in different cities and provinces. The findings
correspond to existing research on the regional economic complexity in China
using non-monetary metrics [49, 50]. As for the impact of predictors, we found
both similarities and variations of their standardized regression coefficients
in different provinces (as shown in Table 1). Specifically, the intracity flow
of cars and buses has the largest positive impact to the city GDP with a
standardized coefficient (0.362) in Jiangsu province, followed by the
intercity truck in-flow (0.263) and the intercity car/bus in-flow (0.262),
while in the case of Liaoning province, the intercity car/bus in-flow has the
largest positive standardized coefficient (7.371) followed by the intracity
flow of cars and buses (1.134). In Shaanxi province, the intercity out-flow of
cars and buses has the largest positive impact to the city GDP with a
standardized coefficient (2.527), followed by the intracity flow of trucks
(0.476) and the intracity flow of cars and buses (0.401). With the Ridge and
LASSO regressions, we found very similar results (in Table 2) to the non-
regularized MLR regarding the predictor selection. Accordingly, car/bus
intracity flow and intercity in-flow are both selected to explain the GDP
variation among cities in Jiangsu and Liaoning provinces. One different
feature selection result is that the intercity out-flow of trucks selected in
the Ridge and LASSO regression models for determining cities’ GDP in Jiangsu
province had a smaller standardized coefficient than that of the intercity in-
flow of trucks in the non-regularized MLR. As for the Shaanxi province, six
out of eight transport predictors are selected and the intracity flows of cars
$\&$ buses as well as trucks play an important role in predicting the city GDP
as reported in Table 2.

In addition, Fig. 2 and supplementary figures S6 and S7 show that the temporal
changes of traffic volumes of cars & buses and trucks over the consecutive
four-year period match the city GDP overall changes well. Moreover, our
experiments based on the total traffic volume (including all flows of cars &
buses and trucks) of each city over four years and city GDP data using a
Bayesian structural time-series model [51] verify that transportation
infrastructure can boost economic growth (with p-value 0.002), since it has
significant impact on transport activities including both human movements and
freight flows.

Standardized Coefficients | Liaoning | Jiangsu | Shaanxi
---|---|---|---
$I_{C}$ | 7.371 | 0.262 | -2.183
$O_{C}$ | -7.273 | -0.070 | 2.527
$N_{C}$ | 1.134 | 0.362 | 0.401
$R_{C}$ | -0.173 | 0.117 | 0.139
$I_{K}$ | -0.310 | 0.263 | 0.072
$O_{K}$ | 0.248 | 0.113 | -0.226
$N_{K}$ | -0.222 | 0.063 | 0.476
$R_{K}$ | 0.026 | -0.183 | -0.044
Table 1: The standardized multi-linear regression coefficients using the
ordinary least squares technique. Note: intercity incoming flow of cars $\&$
buses ($I_{C}$), intercity outgoing flow of cars $\&$ buses ($O_{C}$),
intracity flow of cars $\&$ buses ($N_{C}$), and the ratio of
incoming/outgoing intercity flow for cars $\&$ buses ($R_{C}$), intercity
incoming flow of trucks ($I_{K}$), intercity outgoing flow of trucks
($O_{K}$), intracity flow of trucks ($N_{K}$), and the ratio of
incoming/outgoing intercity flow for trucks ($R_{K}$).

| Ridge $\lambda$ | Ridge R-squared | Ridge RMSE |  | Ridge Selected Features
---
and Coefficients
LASSO $\lambda$ | R-squared | RMSE |  | LASSO Selected Features
---
and Coefficients
Liaoning | 21.62 | 0.887 | 69.9 |  | Intercept: 4.97e+01
---
$I_{C}$: 1.18e-05
$N_{C}$:1.85e-05
0.06 | 0.932 | 54.2 |  | Intercept: 6.64e+02
---
$I_{C}$: 4.4e-04
$O_{C}$: -4.2e-04
$N_{C}$: 3.30e-05
$R_{C}$: -6.5e+02
$I_{K}$: -2.45e-06
$O_{K}$: -2.0e-06
$N_{K}$: -3.2e-05
$R_{K}$: 5.26
Jiangsu | 52.38 | 0.888 | 121.0 |  | Intercept: 2.9e+02
---
$I_{C}$: 3.2e-07
$O_{C}$: 6.8e-07
$N_{C}$: 1.0e-05
$O_{K}$: 3.4e-05
18.39 | 0.882 | 124.2 |  | Intercept: 2.32e+02
---
$I_{C}$: 5.8e-06
$N_{C}$: 1.0e-05
$O_{K}$:3.8e-05
Shaanxi | 16.89 | 0.961 | 32.59 |  | Intercept: 7.92e+01
---
$I_{C}$: 6.61e-08
$O_{C}$: 1.29e-06
$N_{C}$: 2.82e-06
$N_{K}$: 1.27e-05
0.70 | 0.965 | 30.66 |  | Intercept: -2.02e+02
---
$I_{C}$: 1.15e-06
$O_{C}$: 7.55e-07
$N_{C}$: 2.83e-06
$R_{C}$: 2.94e+02
$N_{K}$: 1.47e-05
$R_{K}$: -3.05e+01
All | 25.23 | 0.570 | 213.4 |  | Intercept: 1.31e+02
---
$I_{K}$: 3.53e-05
$O_{K}$: 5.05e-06
$N_{K}$: 2.31e-05
6.70 | 0.569 | 213.5 |  | Intercept: 3.15e+02
---
$I_{K}$: 4.28e-05
$N_{K}$: 2.62e-05
$R_{K}$: -1.99e+02

Table 2: Ridge and LASSO regression analysis results. Note: intercity incoming
flow of cars $\&$ buses ($I_{C}$), intercity outgoing flow of cars $\&$ buses
($O_{C}$), intracity flow of cars $\&$ buses ($N_{C}$), and the ratio of
incoming/outgoing intercity flow for cars $\&$ buses ($R_{C}$), intercity
incoming flow of trucks ($I_{K}$), intercity outgoing flow of trucks
($O_{K}$), intracity flow of trucks ($N_{K}$), and the ratio of
incoming/outgoing intercity flow for trucks ($R_{K}$).  Figure 1: The
relationships between the estimated and actual GDP values of cities in three
provinces using multi-linear regression. (a) Liaoning province; (b) Jiangsu
province; (c) Shaanxi province; (d) combine all cities together. (CI:
confidence interval; PI: prediction interval.) Figure 2: (a) The temporal
changes of cities’ GDP; (b) the correlation between city GDP and traffic
volumes; (c) the temporal changes of traffic volumes of cars and buses; (d)
the temporal changes of traffic volumes of trucks in Shaanxi province.

### City attractiveness and distance-decay effects

Next, we utilize the gravity model to estimate the city nodal attractiveness
and the distance-decay effect on spatial interactions of cities based on the
traffic-flow-weighted networks derived from cars $\&$ buses and freight
trucks’ entry and exit records along highways. Figure 3 shows the spatial
distributions of the traffic volumes of cars and buses between cities in
Shaanxi province. The magnitude of the traffic volumes of cars and buses
within cities are larger than their pairwise inter-city traffic volumes for
each category of vehicles (i.e., cars $\&$ buses, and trucks). The maps
visualizing the traffic volumes of freight trucks in Shaanxi over four years
can be found in Fig. S8. Similar maps of the other two provinces can be found
in Supplement Figs. S9, S10, S11, and S12. The spatiotemporal distributions of
traffic volumes in each province could reflect the highway construction and
transportation trends in each province [52, 53]. Figure 4 displays the
relationships between the estimated and observed flow volumes of freight
trucks, cars $\&$ buses in three provinces (LN: Liaoning; JS: Jiangsu; SX:
Shaanxi) over four consecutive years using the gravity models. The goodness of
fit R2 of those models are all over 0.99 using the linear regression approach.
The results indicate that the gravity model fits well in the regional land
transportation patterns for cars $\&$ buses and freight trucks in all three
provinces over the study period.

In addition, we estimated the distance-decay coefficient $\beta$ in the
gravity models for both categories of transportation networks using three
different approaches: linear regression, linear programming (MINIMAX), and the
Null model. Figure 5 shows the temporal changes of the distance-decay
coefficient $\beta$ in three provinces using different parameter estimation
approaches. Although we got different estimated absolute values of $\beta$
using varying parameter estimation strategies, the relative overall trend
remains. Specifically, as the regional economy grows, we would expect more
spatial interactions of people and goods (and other types of flows [7]) among
cities across different ranges of distances. Thus, it results in a smaller
$\beta$ and vice versa. Furthermore, the rapid development of transportation
infrastructures boosts the long-distance travels in a province. We found that
the distance-decay effects decreased significantly in Shaanxi province and
aligned well with its fast economic growth trend in the study period (see
Figure 2). In contrast, the regional economy in Northeast China grew slowly in
recent years and even decreased in several cities. Liaoning province was
selected as a representative province to illustrate this trend. From year 2015
to 2016, most cities’ GDP decreased which may limit some long-distance travels
of passengers and goods. Thus, the distance-decay coefficient $\beta$
increased from 1.03 to 1.11 for traffic flows of cars and buses using the null
model estimation approach, and increased from 0.85 to 0.94 for flows of
freight trucks respectively. Besides, the distance decay effects of traffic
flows in Jiangsu province didn’t change much while it kept a stable economic
growth across all cities over years. The spatial concentration of economic
activities and interactions within the southern part of Jiangsu but fewer
transportation interactions between the southern and the northern parts may
contribute to the high distance-decay effect (as shown in Figure S9).
Moreover, we found that the distance-decay coefficients $\beta$ for the truck
flow networks are smaller than those of the cars $\&$ buses flow networks in
Shaanxi and Liaoning provinces, where long-distance travels of freight trucks
are important components of regional economic activities.

(a) 2014

(b) 2015

(c) 2016

(d) 2017

Figure 3: Mapping the annual traffic volumes of cars and buses among cities in
Shaanxi province from 2014 to 2017. Note: The maps were generated using ArcMap
version 10.6 and Adobe Illustrator CC version 20. Figure 4: Relationships
between the estimated and observed flow volumes of freight trucks, cars and
buses in three provinces (LN: Liaoning; JS: Jiangsu; SX: Shaanxi) over four
years using the gravity model.  Figure 5: The temporal changes of distance
decay effects on traffic-flow-based spatial interactions among cities in three
provinces using three different parameter estimation approaches to the decay
coefficient $\beta$. (a) using linear regression approach; (b) using MINIMAX
linear programming approach; (c) using the null model.

### Network measures and sub-network structure

Network science methods are useful in uncovering inherent characteristics of
transportation networks and the spatial structure of cities and regions [10,
7]. A variety of network indicators such as Centrality, PageRank, LeaderRank
and many others have been proposed for identifying influential nodes in
complex networks [54, 55, 56]. We constructed three spatial interaction
networks of cities based on different weight choices: (1) the physical
distances between city centers $G_{D}<V,E,W_{D}>$; (2) the inter-city flow
volumes of cars and buses $G_{C}<V,E,W_{C}>$; and (3) the inter-city flow
volumes of freight trucks $G_{K}<V,E,W_{K}>$. Then, the betweenness and
closeness centrality measures were computed in the network $G_{D}$, which help
understand the spatial structure of cities in the physical space. The
betweenness quantifies node importance based on how often shortest paths pass
through a specific node, but it needs to incorporate human population
distribution and the distance decay function to better explain traffic flow
distribution [2]. As for the flow-weighted networks $G_{C}$ and $G_{K}$, the
closeness centrality and the weighted PageRank [57, 58, 59] were computed to
measure the node importance. Then, we computed the Pearson’s correlation
coefficient between city GDP and each of these node importance measures. As
shown in Figure 6, the city PageRank measures in the networks of cars $\&$
buses $PageRank(C)$ and trucks $PageRank(K)$ strongly positively correlate
with city GDP in all three provinces. The flow-weighted closeness centrality
$Closeness(C)$ and $Closeness(K)$ are positively correlated with city GDP
whereas neither $Closeness(D)$ nor $Betw(D)$ centrality in physical space had
a significant correlation with city GDP. In sum, the weighted network measures
using the traffic flows correlate better with regional economy than that using
the physical distance-based ones.

Figure 6: The Pearson’s correlation coefficients between city GDP value,
betweenness, closeness centrality measures and the PageRank index in transport
flow networks of cars $\&$ buses (C) and trucks (K) in three provinces. The
Betw (D) and Closeness (D) measures are calculated using the spatial
interaction networks of cities with the inter-city distances as edge weights.

In addition, by applying principle component analysis (PCA) on the spatial
interaction network matrices, we extract the subsystem of flows with a large
portion of the total variance of interactions among cities. The first few
components represent a substantial part of the total variance, which could
reveal prominent regional transportation connection patterns. Taking the
inter-city spatial interaction network of Shaanxi as an example, using the
observed flow volumes of cars and buses in year 2014 as the weights, the PCA
results in Figure 7(a) show that first two principle components already
account for over 80$\%$ (PC1: 63.1$\%$ and PC2: 17.0$\%$) of the total
variance of the inter-city flows. Xi’an, as the provincial capital city of
Shaanxi province, has the largest standardized component score for the first
principle component. By linking each group of city destinations (filtered by
the factor loadings) to a common set of strongly connected origins (filtered
by component scores) using the Goddard’s approach [60], a dominant nodal sub-
system is extracted. As shown in Figure 7(b) and Figure 3, this sub-system
links several important cities in the central and southern parts of Shaanxi
province through the highway transportation system, including Xi’an, Xianyang,
Weinan, Baoji, Shangluo, Ankang, and Hanzhong. Interestingly, the same
dominant nodal sub-system is also extracted using the freight truck traffic
volumes as the weights for the inter-city spatial interaction networks as
shown in Figure S8. This finding confirms the strong regional spatial
connections of people and goods to support economic development in this area.
Similarly, the PCA analysis results for extracting the dominant nodal sub-
systems of Jiangsu province and Liaoning province are shown in Figure S13 and
S14. The sub-network structure reflects the complexity and the spatial
connection characteristics of the regional economy.

Figure 7: The PCA analysis results of spatial interaction networks of buses
and cars in Shaanxi province. (a) the city coordinates using the first two
principle components PC1 and PC2; the axes in red represent original
coordinate space. (b) the graph structure of first principle component PC1
with factor loading > 0.3 and component score > 1.0. Note: The figures were
generated using RStudio version 1.2.

## Discussion and Conclusion

When estimating the city GDP from traffic flow variables, the existence of
multicollinearity among independent variables may change the coefficient
estimates of the MLR model erratically in response to small changes in the
data. We used the variance inflation factors (VIF), which is a measure of how
much the variance of the estimated regression coefficient $b_{k}$ (where $k$
is the index of a predictor) is "inflated" by the existence of correlation
among the predictor variables in the model.

$VIF_{k}=\frac{1}{1-R^{2}_{k}}$ (1)

Where $R^{2}_{k}$ is the R-squared obtained by regressing the $k_{th}$
predictor on the remaining predictors.

The VIF scores for the predictor variables incoming flow ($I_{C}$), outgoing
flow ($O_{C}$) of cars and buses are very high (> 2000) across all three
provinces. As the original multi-linear regression model includes the ratio of
incoming/outgoing flow ($R_{C}$), such high multicollinearity is expected.
After removing the two variables with largest VIF scores in each regression
model for predicting the city GDP, the R-squared values slightly reduced: from
0.934 to 0. 909 (Liaoning), from 0.892 to to 0.889 (Jiangsu), and from 0.967
to 0.955 (Shaanxi), respectively.

Moreover, by calibrating the model penalty term of L1-norm and L2-norm using
empirical data in each province, we derived the regularized regression results
using the Ridge and LASSO approaches. As shown in Table 2, both Ridge and
LASSO methods got a high goodness of fit in the predictive modeling of city
GDP: R-squared of 0.887 (Ridge) and 0.932 (LASSO) in Liaoning province, 0.888
(Ridge) and 0.882 (LASSO) in Jiangsu province, and 0.961 (Ridge) and 0.965
(LASSO) in Shaanxi province. The R-squared results are almost as good as the
non-regularized MLR results using all features, but the regularized regression
models are more stable in the cross-validation experiments although with some
additional cost of bias. Regarding the feature/predictor selection, each
province has its own unique combination of transport flow predictors to its
city economy. The models also identify the prominent predictor (i.e., the
intracity flow of cars and buses $N_{C}$) that has a large positive impact on
their GDP values but with different magnitudes of coefficients across three
provinces. The detailed model coefficient results can be found in Table 2.
Interestingly, if we combine all the cities’ data from three provinces into
one linear model, the top two selected predictors are the intercity incoming
flow of trucks ($I_{K}$) and the intracity flow of trucks ($N_{K}$) for both
Ridge and LASSO models, which have distinctive patterns in each province and
thus help explain the city GDP variance across three provinces.

In addition, it is worth noting that if we took the province information as a
dummy variable in the non-regularized MLR model of predicting GDP values using
aforementioned traffic variables for all cities, the goodness of fit
(R-squared) in the MLR model increased from 0.587 to 0.818, and the
coefficients of this dummy variable had p-values close to 0. These values
indicated that each province has its own structure of variability and such
effect is statistically significant when predicting the GDP value from the
transportation flow volumes of cars, buses, and trucks.

In sum, this study demonstrates that highway transportation big data could
reveal the status of regional economic development and contain valuable
information of human mobility, production linkages, and logistics for regional
management and planning.

## Methods

### Data

Annual transportation data between years 2014 and 2017 for three provinces,
namely Shaanxi (10 cities), Jiangsu (13 cities) and Liaoning (14 cities), in
China were collected and summarized in the city level. All vehicles are
required to pay for their trips within the highway in China (Except in Hainan
province). Using manual and electronic toll collection systems including
radio-frequency identification (RFID) and automatic plate recognition
technology as well as surveys in each station, detailed data including the
entrance and exit stations, and the type of each vehicle (two categories: cars
$\&$ buses, and freight trucks) were recorded. In addition, for cars and
buses, the average number of passengers of each vehicle was estimated based on
the type of cars and buses and the survey per vehicle type. Whereas for
trucks, the total weight of each vehicle was also recorded and the loading
weight was estimated based on the type of truck. More details about the data
collection in highway toll stations in China can be found in previous works
[52, 61]. City level data were aggregated and summarized from the raw station-
level data as our main focus was to investigate the relationship between
regional economy and transportation networks. We eliminated the traffic flows
between different provinces and only kept the traffic flows within one
province. In sum, the total number of vehicles, the sum of passengers (for
cars and buses) and the weights (for trucks) as well as the distance (km)
between paired cities were calculated respectively (see Table 3). In addition,
the gross domestic product (GDP) data for each city in the same study period
(2014-2017) were collected from the City Statistics Yearbook in each province,
which measured the market value of all the final goods and services produced
by all resident and institutional units engaged in a city.

Province | Number of Entry and Exit Stations | Year | Vehicle Type | Number of Vehicles (million) | Number of Passengers or Volume of Weights (million)
---|---|---|---|---|---
Jiangsu | 421 | 2014 | Cars & Buses | 267.9 | 929.0
Trucks | 116.3 | 1150\. 9
2015 | Cars & Buses | 259.8 | 639\. 2
Trucks | 100.5 | 940,603.3
2016 | Cars & Buses | 422\. 9 | 1289.4
Trucks | 122.7 | 1,189,972.7
2017 | Cars & Buses | 486.5 | 1429.6
Trucks | 141.3 | 1,421,325.6
Liaoning | 287 | 2014 | Cars & Buses | 107.0 | 398.1
Trucks | 41.1 | 429,189.2
2015 | Cars & Buses | 111.4 | 397.9
Trucks | 38.9 | 391,497.2
2016 | Cars & Buses | 121.0 | 424.9
Trucks | 43.4 | 445,433.7
2017 | Cars & Buses | 131.4 | 459.9
Trucks | 48.6 | 516,205.0
Shaanxi | 335 | 2014 | Cars & Buses | 146.9 | 525.9
Trucks | 56.7 | 839,483.4
2015 | Cars & Buses | 165.3 | 566.1
Trucks | 57.3 | 787,819.5
2016 | Cars & Buses | 221.5 | 730.2
Trucks | 65.6 | 826,738.1
2017 | Cars & Buses | 233.3 | 757.2
Trucks | 68.3 | 824,578.3

Table 3: Summarized highway transportation data between 2014 and 2017 in three
provinces: Jiangsu, Liaoning, and Shaanxi.

### Estimating GDP based on the traffic flow data

The multiple linear regression model (MLR) with ordinary least squares (OLS)
parameter estimation is used to discover the statistical relationship between
the transportation flow data and the economic development indicator (i.e.
GDP). The GDP of a given city $i$ is considered as a dependent variable
$GDP_{i}$ predicted by eight independent variables (also known as features or
predictors) related to its transport flows of people and goods: intercity
incoming flow of cars $\&$ buses ($I_{C}$), intercity outgoing flow of cars
$\&$ buses ($O_{C}$), intracity flow of cars $\&$ buses ($N_{C}$), and the
ratio of incoming/outgoing intercity flow for cars $\&$ buses ($R_{C}$),
intercity incoming flow of trucks ($I_{K}$), intercity outgoing flow of trucks
($O_{K}$), intracity flow of trucks ($N_{K}$), and the ratio of
incoming/outgoing intercity flow for trucks ($R_{K}$). The formula is as
follows:

$GDP_{i}=b_{0}+b_{1}I_{Ci}+b_{2}O_{Ci}+b_{3}N_{Ci}+b_{4}R_{Ci}+b_{5}I_{Ki}+b_{6}O_{Ki}+b_{7}N_{Ki}+b_{8}R_{Ki}+e_{i}$
(2)

The intercity incoming flow ($I$) is the traffic flow with all origins outside
the city and all destinations inside the city. The intercity outgoing flow
($O$) is the flow that originates inside the city and goes outside of the
city. The intracity flow ($N$) is the flow with all origins and destinations
inside the same city. The index ($R$) is the ratio of incoming and outgoing
flows for a city, which may be able to indicate the role of a city compared
with other cities among a transportation network. To demonstrate, when the
ratio is larger than 1, it means there are more transportation flows coming to
the city than leaving it, and the city may work as a ‘sink’ in the network. It
may be a commercial center attracting people coming to find jobs or an
industry center consuming a large amount of goods. When the ratio is less than
1, it means that the city works more like a ‘source’ which sends people or
provides materials to other places. In addition, this study takes two types of
traffic flows into consideration: the cars $\&$ buses flow and the truck flow.
These two kinds of flows can reflect different aspects of the city economy,
from the perspectives of human movement and goods movement. There are in total
eight independent variables together with the constant ($b_{0}$) and the error
term ($e$) included in the multi-linear regression model. By solving this
linear model using the OLS technique, the relationship between GDP and the
traffic flow of each city in each year can be discovered. The model is first
used for each province separately, in attempt to distinguish the pattern of
economic development in each province. Then the data of three provinces are
merged together to identify the general relationship of GDP and traffic flow
across provinces.

In addition, the standardized coefficient $b_{j}^{\prime}$ of an independent
variable from the multi-linear regression model is calculated as follows. It
can be interpreted as the change of dependent variable in standard deviations,
per standard deviation change in the predictors.

$b_{j}^{\prime}=b_{j}*\frac{S_{X_{j}}}{S_{y}}$ (3)

Where $b_{j}$ is a regression coefficient. $S_{y}$ is the standard deviation
of the dependent variable and $S_{X_{j}}$ is the standard deviation of
independent variable $X_{j}$.

Besides, we also investigated a generalized linear model (GLM) with the
natural log transformation of each city GDP given its log-normal or gamma
distribution characteristics (in Supplement Fig. S1).

$Ln(GDP_{i})=b_{0}+b_{1}I_{Ci}+b_{2}O_{Ci}+b_{3}N_{Ci}+b_{4}R_{Ci}+b_{5}I_{Ki}+b_{6}O_{Ki}+b_{7}N_{Ki}+b_{8}R_{Ki}+e_{i}$
(4)

### Linear models with regularization

When some independent variables are highly correlated in the MLR, the OLS
coefficient estimations will have a large variance. The introduced
regularization techniques in statistics and machine learning help reduce
variance at the cost of introducing some bias in a model and avoid the
overfitting issue [62, 63]. There are two types of regularization techniques
with two different cost functions regarding the model complexity: the $L1$
norm (least absolute deviations) and the $L2$ norm (least squares).

The Ridge regression applies the $L2$ penalty term to control the coefficient
of each independent variable in a linear regression model [46]. The objective
of Ridge approach is to minimize the following cost function:

$\sum_{i=1}^{N}(y_{i}-b_{0}-\sum_{j=1}^{p}x_{ij}b_{j})^{2}+\lambda\sum_{j=1}^{p}b^{2}_{j}$
(5)

Where $y_{i}$ is the observed value of the dependent variable for each sample
$i$ (i.e., the city GDP); $N$ is the total number of sample observations;
$b_{0}$ is the constant in MLR; $x_{ij}$ is the value of the independent
variable $j$ for each sample $i$; $b_{j}$ is the estimated coefficient for the
independent variable $j$ (abovementioned transportation flow variables); $p$
is the total number of independent variables (predictors), and $\lambda>=0$ is
a regularization penalty parameter. If $\lambda$ is 0, the cost function backs
to the OLS estimation. However, if $\lambda$ is very large then it will add
too much penalty to the model complexity and may lead to the model
underfitting. The parameter calibration is performed to find the best
$\lambda$ that fits the data well and achieves the bias-variance balance.

The LASSO regression has a similar cost function to minimize but it applies
the $L1$ penalty term rather than the $L2$ norm to regularize the coefficient
of each independent variable [47]:

$\sum_{i=1}^{N}(y_{i}-b_{0}-\sum_{j=1}^{p}x_{ij}b_{j})^{2}+\lambda\sum_{j=1}^{p}|b_{j}|$
(6)

By shrinking some coefficients, the Ridge regression and the LASSO regression
are able to control the multicollinearity in the model. It tends to pick one
predictor from a few very correlated predictors and set the coefficients of
the others to zero [47, 48]. Therefore, the regularization techniques are very
helpful when there are many intercorrelated features and feature selection is
necessary [64].

### The gravity model fitting with linear regression and linear programming

To further estimate the relative attractions of cities and identify the
important cities in each province, the gravity model is applied to the study
area. This widely used model assumes that the flows between nodes are
generated by some attraction force from the nodes[65]. With known flow values
between cities, the attraction value of each city in this model can be
estimated through a reverse calibration process [65].

The traffic flow ${G_{ij}}$ is proposed to be computed by the following
equation:

$G_{ij}=k(P_{i}P_{j})/(d_{ij})^{\beta}\;for\;all\;i\neq j$ (7)

where k is a constant and $d_{ij}$ is the distance between two cities. The
attraction of each city($P_{i}$ and $P_{j}$) and the exponent on distance
$\beta$ (i.e., the coefficient of distance decay effect) are unknown and need
to be estimated. The equation can then be transferred by taking a natural
logarithm.

$lnG_{ij}=lnP_{i}+lnP_{j}-\beta(lnd_{ij})+lnk$ (8)

By representing the formula with simpler symbols, we got the following
equation:

$b_{ij}+D_{ij}=X_{i}+X_{j}-a_{ij}\beta$ (9)

where $X_{i}=lnP_{i}$, $X_{j}=lnP_{j}$, $b_{ij}=lnG_{ij}-lnk$,
$a_{ij}=lnd_{ij}$, $D_{ij}$ is the deviation between the estimation and the
observation (i.e., the error term).

Here the $b_{ij}$ can be considered as a known dependent variable and $X_{i}$
and $X_{j}$ are the coefficients of a series of dummy variables showing the
relationship between cities [65]. For each given flow, the dummy variable will
be 1 for the origin city and the destination city, and 0 for other cities. By
forming such a linear regression relationship, it is possible to find the best
$X_{i}$, $X_{j}$, and $\beta$ that are closest to the real situation.

Besides the linear regression approach, a linear programming approach is also
conducted as a comparison. It solves the problem starting from the deviation.

$D_{ij}=X_{i}+X_{j}-a_{ij}\beta-b_{ij}\;for\;all\;i\neq j$ (10)

and $D_{ij}$ can be represented as

$D_{ij}=D_{ij}^{1}-D_{ij}^{2}$ (11)

$D_{ij}^{1}$ and $D_{ij}^{2}$ is always greater than or equal to 0. When
$D_{ij}$ >0, $D_{ij}^{1}$ = $D_{ij}$ and $D_{ij}^{2}$ = 0; When $D_{ij}$ <0,
$D_{ij}^{1}$ = 0 and $D_{ij}^{2}$ = $-D_{ij}$. When $D_{ij}$ = 0, both
$D_{ij}^{1}$ and $D_{ij}^{2}$ is 0 [66]. The goal of the linear programming is
to minimize the maximum absolute error (MINIMAX). Here the $M$ represents the
maximal absolute deviation and the optimization model is listed as:

$\displaystyle Minimize:M$ $\displaystyle Subject\;to:$
$D_{ij}^{1}-D_{ij}^{2}=X_{i}+X_{j}-a_{ij}\beta-b_{ij}$ (12)
$M-D_{ij}^{1}-D_{ij}^{2}>=0$ (13)
$D_{ij}^{1},D_{ij}^{2},X_{i},X_{j},\beta,M>=0$ (14)

By solving this model, the values of $X_{i}$ for all cities and the distance-
decay coefficient $\beta$ can be estimated. The attraction of each city and
the $\beta$ values of four years calculated from the MINIMAX approach and the
linear regression approached are then compared.

### The null model

A null model is also used to examine the effect of distance decay (i.e. the
value of $\beta$). The null model first constructs an interactive network
without considering the distance among nodes[67, 68]. Then by comparing the
estimated flow in the null model with the observed flow in reality, the
difference can reflect the effect of distance. The total flow of a specific
node can be denoted as:

$W_{i}=\sum_{j}G_{ij}$ (15)

Letting $F=\sum_{i}\sum_{j}G_{ij}$ and $N=\sum_{i}\sum_{j}W_{i}W_{j}$, and the
estimated flow between node $i$ and $j$ is:

$G_{ij}^{null}=W_{i}W_{j}F/N$ (16)

The ratio $R_{ij}=G_{ij}/G_{ij}^{null}$ is used to measure the effect of
distance decay and since it should be related to the distance itself. A linear
regression model between $R_{ij}$ and $d_{ij}$ is constructed. Then the best
fitted slope is taken as the value of the distance-decay coefficient $\beta$.

### Network structure analyses

The centrality and PageRank measures are often used in transportation network
analysis to quantify the importance of a node in road networks [55, 2, 69].
Given a network $G<V,E,W>$ of nodes (V) and edges (E) with weights (W), The
betweenness and closeness centrality measures are defined as follows. The
shortest paths can be calculated based on different weights such as physical
distance and traffic volume, as discussed in the section "Network measures and
sub-network structure".

The betweenness centrality of a node $Betw(i)$ quantifies how often the
shortest paths passes through a node $i$ in a network.

$Betw(i)=\sum_{j\neq k\neq i}\frac{n_{jk}(i)}{n_{jk}}$ (17)

Where $n_{jk}$ is the number of shortest paths between nodes $j$ and $k$, and
$n_{jk}(i)$ is the number of shortest paths between nodes $j$ and $k$ that
contain node $i$.

The closeness centrality of a node $Clos(i)$ is the reciprocal of the sum
length of the shortest paths between the node and all other nodes in the
network.

$Clos(i)=\frac{1}{\sum^{n}_{j}d_{ij}}$ (18)

Where $n$ is the total number of nodes; $d_{ij}$ is the length of the shortest
path between nodes $i$ and $j$.

The PageRank $PR(i)$ quantifying the node importance in networks is computed
as follows [57, 59].

$PR(i)=\frac{1-d}{n}+d*\sum_{j\in M(i)}PR(j)*W_{ij}/L(j)$ (19)

Where $d$ is a damping factor between 0 and 1 and is usually set as 0.85 [57];
$n$ is the total number of nodes; $j$ is the index of a set of nodes $M(i)$
that link to $i$, $L(j)$ is the number of outbound links on node $j$, and
$PR(j)$ is the PageRank of node $j$. As for the weighted networks, the
PageRank algorithm interprets the normalized edge weight $W_{ij}$ as
connection strengths and an edge with a larger weight is more likely to be
selected for connection.

In addition, the Principle Component Analysis (PCA) is utilized to identify
the structure of the network and discover potential sub-networks inside a
province [70]. By applying a standard PCA to the transport flow matrix
$[F_{ij}]$, every principal component (PC) can be related to a sub-system of
flows and the first few components represent stronger sub-systems [70, 60].
The factor loadings and component scores are computed for each PC and they are
used to identify the dominant sub-networks. Given a dimension of $n*m$ flow
data set $F_{ij}$, the first principal component of a set of destination
features $X_{1}$,$X_{2}$, …, $X_{j}$, $X_{m}$, is the normalized linear
combination of the features that has the largest variance. As shown below, the
elements $\phi_{11},…,\phi_{m1}$ as the factor loadings of the first principal
component. The loading vector defines a direction in feature space along which
the data varies most. Similarly, we can derive other at most $min(n-1,m)$
principle components.

$\begin{bmatrix}F_{11}&F_{12}&...&F_{1j}&...F_{1m}\\\
F_{21}&F_{22}&...&F_{2j}&...F_{2m}\\\ ...\\\
F_{i1}&F_{i2}&...&F_{ij}&...F_{im}\\\ ...\\\
F_{n1}&F_{n2}&...&F_{nj}&...F_{nm}\\\ \end{bmatrix}$

$Z_{1}=\phi_{11}X_{1}+\phi_{21}X_{2}+...+\phi_{j1}X_{j}+...+\phi_{m1}X_{m}$
(20)

$X_{j}=(F_{1j},F_{2j},...,F_{ij},...,F_{nj})$ (21)

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## Acknowledgments

Bin Li and Runmou Xiao would like to thank the funding support of this
research by the Fundamental Research Funds for the Central Universities (Grant
No. 300102229108), CHD; Transportation Strategic Planning Policy Project Plan
2018 (Grant No. 2018-22-3), Ministry of Transport of the People’s Republic of
China. Song Gao would like to thank the research support by the University of
Wisconsin - Madison Office of the Vice Chancellor for Research and Graduate
Education with funding from the Wisconsin Alumni Research Foundation.

## Author contributions statement

B.L., S.G., and R.M.X. designed this research; B.L., S.G., Y.L.L. and Y.H.K.
performed experiments, analyzed data and wrote the paper; T.P. created the
maps and data visualization; Y.Q.G wrote the paper. All authors reviewed and
edited the manuscript.

## Additional information

Supplementary information accompanies this paper at
http://www.nature.com/scientificreports Competing financial interests: The
authors declare no competing financial interests. How to cite this
article:.XXX

## Appendix

### Modeling the GDP distribution

The empirical histogram distribution of the economic development indicator
(GDP) values across all cities in the study period is a positively skewed
distribution with a long-tail end (Fig. S1a). By fitting the data observations
into four theoretical models (i.e., normal, gamma, log-normal, and Weibull
distributions) using the maximum likelihood estimation [71], the results show
that the log-normal model is the best with the maximum log-likelihood of
-994.39 and the minimum Akaike information criterion (AIC) value of 1992.78.
AIC is used to determine which model performs the best regarding the goodness
of fit and the simplicity of a model using information theory [72]. Meanwhile,
the log-likelihood and AIC values for other models are gamma distribution
(log-likelihood=-1002.82 $\&$ AIC=2009.64), Weibull distribution (log-
likelihood=-1004.23 $\&$ AIC=2012.45), and normal distribution (log-
likelihood=-1066.17 $\&$ AIC=2136.35) respectively. Fig. S1b and Fig. S1d show
the curves of empirical and theoretical cumulative distribution functions of
these models and their quantile-quantile plots. The skewness-kurtosis plot
(i.e., the Cullen and Frey graph [73]) is used to examine the characteristics
of the skewness and the degree of tailedness (i.e., kurtosis) of the GDP data
with uncertainty compared with the theoretical models. As shown in the S1c,
the bootstrapping values fit better with the log-normal, gamma and Weibull
models rather than the normal, uniform and logistic models. The consistent
positive skewness and kurtosis values verify the "heavy-tailed on the right"
characteristic of the city GDP distribution in the study areas.

Figure S1: The empirical and theoretical distributions of all city GDP values.
(a) the histogram of city GDPs; (b) the observed empirical cumulative
distribution function (CDF) in black and four theoretical CDFs; (c) the Cullen
and Frey graph shows the probability distribution fittings; (d) the Q-Q plot
compares the empirical and theoretical quantile distributions. Figure S2: The
residual plots of predicting city GDP values in Liaoning province using the
multiple linear regression. (a) the scatter plot of residuals and fitted
values; (b) the Normal Quantile-Quantile plot for standardized residuals; (c)
the Scale-Location plot to check the residual spread; (d) the Residuals vs
Leverage plot to find influential samples if any. Figure S3: The residual
plots of predicting city GDP values in Jiangsu province using the multiple
linear regression. (a) the scatter plot of residuals and fitted values; (b)
the Normal Quantile-Quantile plot for standardized residuals; (c) the Scale-
Location plot to check the residual spread; (d) the Residuals vs Leverage plot
to find influential samples if any. Figure S4: The residual plots of
predicting city GDP values in Shaanxi province using the multiple linear
regression. (a) the scatter plot of residuals and fitted values; (b) the
Normal Quantile-Quantile plot for standardized residuals; (c) the Scale-
Location plot to check the residual spread; (d) the Residuals vs Leverage plot
to find influential samples if any. Figure S5: The visualization of
correlation matrix of predictors and residuals in each province.  Figure S6:
(a) The temporal changes of cities’ GDP values; (b) the correlation between
city GDP and traffic volumes; (c) the temporal changes of traffic volumes of
cars and buses; (d) the temporal changes of traffic volumes of trucks in
Liaoning province. Figure S7: (a) The temporal changes of cities’ GDP values;
(b) the correlation between city GDP and traffic volumes; (c) the temporal
changes of traffic volumes of cars and buses; (d) the temporal changes of
traffic volumes of trucks in Jiangsu province.

(a) 2014

(b) 2015

(c) 2016

(d) 2017

Figure S8: Traffic Volume of Freight Trucks in Shaanxi. Note: The maps were
generated using ArcMap version 10.6 and Adobe Illustrator CC version 20.

(a) 2014

(b) 2015

(c) 2016

(d) 2017

Figure S9: Traffic Volume of Cars and Buses in Jiangsu. Note: The maps were
generated using ArcMap version 10.6 and Adobe Illustrator CC version 20.

(a) 2014

(b) 2015

(c) 2016

(d) 2017

Figure S10: Traffic Volume of Freight Trucks in Jiangsu. Note: The maps were
generated using ArcMap version 10.6 and Adobe Illustrator CC version 20.

(a) 2014

(b) 2015

(c) 2016

(d) 2017

Figure S11: Traffic Volume of Cars and Buses in Liaoning. Note: The maps were
generated using ArcMap version 10.6 and Adobe Illustrator CC version 20.

(a) 2014

(b) 2015

(c) 2016

(d) 2017

Figure S12: Traffic Volume of Freight Trucks in Liaoning. Note: The maps were
generated using ArcMap version 10.6 and Adobe Illustrator CC version 20.

Figure S13: The PCA analysis results of spatial interaction networks of buses
and cars in Jiangsu. Note: The figures were generated using RStudio version
1.2.

Figure S14: The PCA analysis results of spatial interaction networks of buses
and cars in Liaoning. Note: The figures were generated using RStudio version
1.2.


# Derivation of capture cross section from quasielastic excitation function

V.V.Sargsyan1,2, G.G.Adamian1, N.V.Antonenko1, and P.R.S.Gomes3 1Joint
Institute for Nuclear Research, 141980 Dubna, Russia
2International Center for Advanced Studies, Yerevan State University, 0025
Yerevan, Armenia
3Instituto de Fisica, Universidade Federal Fluminense, Av. Litorânea, s/n,
Niterói, R.J. 24210-340, Brazil

###### Abstract

The relationship between the quasielastic excitation function and the capture
cross section is derived. The quasielastic data is shown to be a useful tool
to extract the capture cross sections and the angular momenta of the captured
systems for the reactions 16O+144,154Sm,208Pb, 20Ne+208Pb, and 32S+90,96Zr at
near and above the Coulomb barrier energies.

###### pacs:

25.70.Jj, 24.10.-i, 24.60.-k
Key words: capture cross section, quasielastic excitation function, cold
fusion reactions

##  I Introduction

The partial capture cross section is one of the important ingredients to
calculate and predict the production cross sections of exotic and superheavy
nuclei in the cold, hot, and sub-barrier astrophysical fusion reactions.
Therefore, more experimental and theoretical studies of the capture process
are required. There is a relationship between the capture and the quasielastic
scattering processes because of the conservation of the reaction flux
PRSGomes1 ; PRSGomes3 . Any loss from the quasielastic channel directly
contributes to the capture and vise versa. The quasielastic measurements are
usually not as complex as the direct capture (fusion) measurements. Thus, the
quasielastic data are suited for the extraction of the capture probabilities
and of the capture cross sections.

The paper is organized in the following way. In Sec. II we derive the formulas
for the extraction of the capture cross section and of the angular momentum of
the captured system by employing the experimental quasielastic excitation
function. In Sec. III, using these formulas, we extract the capture cross
sections and the angular momenta of the captured systems and compare with
those of direct measurements. Using the available experimental quasielasic
data, we predict the capture cross sections for the cold fusion reactions. In
Sec. IV the paper is summarized.

##  II Relationship between capture and quasielastic scattering

The expression

$P_{qe}(E_{\rm c.m.},J)+P_{cap}(E_{\rm c.m.},J)=1$ (1)

connecting the quasielastic (reflection) $P_{qe}$ and the capture
(transmission) $P_{cap}$ probabilities follows from the conservation of the
reaction flux PRSGomes1 ; PRSGomes3 . Thus, one can extract the capture
probability $P_{cap}(E_{\rm c.m.},J=0)$ at $J=0$ from the experimental
quasielastic probability $P_{qe}(E_{\rm c.m.},J=0)$:

$P_{cap}(E_{\rm c.m.},J=0)=1-P_{qe}(E_{\rm c.m.},J=0)=1-d\sigma_{qe}(E_{\rm
c.m.})/d\sigma_{Ru}(E_{\rm c.m.}).$ (2)

Here, the quasielastic probability PRSGomes1 ; Timmers ; Timmers2 ; Zhang

$\displaystyle P_{qe}(E_{\rm c.m.},J=0)=d\sigma_{qe}/d\sigma_{Ru}$ (3)

for angular momentum $J=0$ is given by the ratio of the quasielastic
differential cross section and Rutherford differential cross section at 180
degrees. Further, one can approximate the $J$ dependence of the capture
probability $P_{cap}(E_{\rm c.m.},J)$ at a given energy $E_{\rm c.m.}$ by
shifting the energy Bala :

$\displaystyle P_{cap}(E_{\rm c.m.},J)\approx P_{cap}(E_{\rm
c.m.}-\frac{\hbar^{2}\Lambda}{2\mu R_{b}^{2}},J=0)=1-P_{qe}(E_{\rm
c.m.}-\frac{\hbar^{2}\Lambda}{2\mu R_{b}^{2}},J=0),$ (4)

where $\Lambda=J(J+1)$, $R_{b}=R_{b}(J=0)$ is the position of the Coulomb
barrier at $J=0$. Then, we extract the capture cross section
$\sigma_{cap}(E_{\rm c.m.})$ from the experimental quasielastic probabilities
$P_{qe}$:

$\displaystyle\sigma_{cap}(E_{\rm c.m.})=\sum_{J=0}^{J_{cr}}\sigma_{\rm
cap}(E_{\rm
c.m.},J)=\pi\lambdabar^{2}\sum_{J=0}^{J_{cr}}(2J+1)[1-P_{qe}(E_{\rm
c.m.}-\frac{\hbar^{2}\Lambda}{2\mu R_{b}^{2}},J=0)],$ (5)

where $\lambdabar^{2}=\hbar^{2}/(2\mu E_{\rm c.m.})$ is the reduced de Broglie
wavelength, $\mu=m_{0}A_{1}A_{2}/(A_{1}+A_{2})$ is the reduced mass ($m_{0}$
is the nucleon mass), and at given bombarding energy $E_{\rm c.m.}$ the
summation is over the possible values of angular momentum $J$ from $J=0$ to
the critical angular momentum $J=J_{cr}$. For values $J$ greater than
$J_{cr}$, the potential pocket in the nucleus-nucleus interaction potential
vanishes and the capture is not occur. To calculate the critical angular
momentum $J_{cr}$ and the position $R_{b}$ of the Coulomb barrier, we use the
nucleus-nucleus interaction potential $V(R,J)$ of Ref. Pot . For the nuclear
part of the nucleus-nucleus potential, the double-folding formalism with the
Skyrme-type density-dependent effective nucleon-nucleon interaction is
employed Pot .

If one sets $R_{b}(J)\approx R_{b}$ in Eq. (5) for approximating the $J$-wave
penetrability by the $s$-wave penetrability at a shifted energy, one obtains
only the leading term in the series expansion in $\Lambda$. The next term in
this expansion can be easily calculated in the same way as in Ref. Bala
[$R_{b}(J)\approx R_{b}-\frac{\hbar^{2}\Lambda}{\mu\alpha R_{b}^{3}}$,
$V_{b}(J)\approx V_{b}+\frac{\hbar^{2}\Lambda}{2\mu
R_{b}^{2}}+\frac{\hbar^{4}\Lambda^{2}}{2\mu^{2}\alpha R_{b}^{6}}$,
$\alpha=-\partial^{2}V(R,J=0)/\partial R^{2}|_{R=R_{b}}=\mu\omega_{b}^{2}$,
$\omega_{b}=\omega_{b}(J=0)$ is the curvature of the $s$-wave potential
barrier with the height $V_{b}=V_{b}(J=0)=V(R=R_{b},J=0)$]:

$\displaystyle P_{cap}(E_{\rm c.m.},J)\approx P_{cap}(E_{\rm
c.m.}-\frac{\hbar^{2}\Lambda}{2\mu
R_{b}^{2}}-\frac{\hbar^{4}\Lambda^{2}}{2\mu^{2}\alpha R_{b}^{6}},J=0).$ (6)

With this improved expression for the $P_{cap}$, we obtain

$\displaystyle\sigma_{cap}(E_{\rm
c.m.})=\pi\lambdabar^{2}\sum_{J=0}^{J_{cr}}(2J+1)[1-P_{qe}(E_{\rm
c.m.}-\frac{\hbar^{2}\Lambda}{2\mu
R_{b}^{2}},J=0)][1-\frac{2\hbar^{2}\Lambda}{\mu^{2}\omega_{b}^{2}R_{b}^{4}}].$
(7)

Converting the sum over $J$ into an integral and changing variables to
$E=E_{\rm c.m.}-\frac{\hbar^{2}\Lambda}{2\mu R_{b}^{2}}$ in Eq. (7), we obtain
the following simple expression:

$\displaystyle\sigma_{cap}(E_{\rm c.m.})=\frac{\pi R_{b}^{2}}{E_{\rm
c.m.}}\int_{E_{\rm c.m.}-\frac{\hbar^{2}\Lambda_{cr}}{2\mu R_{b}^{2}}}^{E_{\rm
c.m.}}dE[1-d\sigma_{qe}(E)/d\sigma_{Ru}(E)][1-\frac{4(E_{\rm
c.m.}-E)}{\mu\omega_{b}^{2}R_{b}^{2}}],$ (8)

which relates the capture cross section with quasielastic excitation function.
Note that $\Lambda$ is not a small parameter, there is a natural cutoff
$\Lambda_{cr}=J_{cr}(J_{cr}+1)$ in this parameter. Because of this cutoff, the
second term $\frac{\hbar^{2}\Lambda}{2\mu R_{b}^{2}}$ in Eq. (6) is always
larger than the third one $\frac{\hbar^{4}\Lambda^{2}}{2\mu^{2}\alpha
R_{b}^{6}}$ Bala . By using the experimental quasielastic probabilities
$P_{qe}(E_{\rm c.m.},J=0)$ and Eq. (8) one can obtain the capture cross
sections.

For the systems with $Z_{1}\times Z_{2}<2000$, the critical angular momentum
$J_{cr}$ is large enough and Eqs. (7) and (8) can be approximated with a good
accuracy as:

$\displaystyle\sigma_{cap}(E_{\rm
c.m.})\approx\pi\lambdabar^{2}\sum_{J=0}^{\infty}(2J+1)[1-P_{qe}(E_{\rm
c.m.}-\frac{\hbar^{2}\Lambda}{2\mu
R_{b}^{2}},J=0)][1-\frac{2\hbar^{2}\Lambda}{\mu^{2}\omega_{b}^{2}R_{b}^{4}}]$
(9)

and

$\displaystyle\sigma_{cap}(E_{\rm c.m.})\approx\frac{\pi R_{b}^{2}}{E_{\rm
c.m.}}\int_{0}^{E_{\rm
c.m.}}dE[1-d\sigma_{qe}(E)/d\sigma_{Ru}(E)][1-\frac{4(E_{\rm
c.m.}-E)}{\mu\omega_{b}^{2}R_{b}^{2}}].$ (10)

Following the procedure of Ref. Bala  and using the extracted $\sigma_{cap}$
and the experimental $P_{qe}$, one can find the average angular momentum

$\displaystyle<J>=\frac{\pi R_{b}^{2}}{E_{\rm c.m.}\sigma_{cap}(E_{\rm
c.m.})}\int_{E_{\rm c.m.}-\frac{\hbar^{2}\Lambda_{cr}}{2\mu
R_{b}^{2}}}^{E_{\rm c.m.}}$ $\displaystyle
dE[1-d\sigma_{qe}(E)/d\sigma_{Ru}(E)][1-\frac{5(E_{\rm
c.m.}-E)}{\mu\omega_{b}^{2}R_{b}^{2}}]$ (11) $\displaystyle\times[(\frac{2\mu
R_{b}^{2}}{\hbar^{2}}(E_{\rm c.m.}-E)+\frac{1}{4})^{1/2}-\frac{1}{2}]$

and the second moment of the angular momentum

$\displaystyle<J(J+1)>=\frac{2\pi\mu R_{b}^{4}}{\hbar^{2}E_{\rm
c.m.}\sigma_{cap}(E_{\rm c.m.})}\int_{E_{\rm
c.m.}-\frac{\hbar^{2}\Lambda_{cr}}{2\mu R_{b}^{2}}}^{E_{\rm c.m.}}$
$\displaystyle dE[1-d\sigma_{qe}(E)/d\sigma_{Ru}(E)][1-\frac{6(E_{\rm
c.m.}-E)}{\mu\omega_{b}^{2}R_{b}^{2}}]$ (12) $\displaystyle\times[E_{\rm
c.m.}-E]$

of the captured system.

##  III Results of calculations

For the verification of our method of the extraction of $\sigma_{cap}$,
firstly we compare the extracted capture cross sections with experimental one.
In Figs. 1 and 2 one can see a good agreement between the extracted and
directly measured capture cross sections for the reactions 16O + 120Sn, 18O +
124Sn, 16O + 208Pb, and 16O + 144Sm at energies above the Coulomb barrier. The
results on the sub-barrier energy region are discussed later on. To extract
the capture cross section, we use both Eq. (8) (solid lines) and Eq. (10)
(dotted lines). The used values of critical angular momentum are $J_{cr}$=54,
56, 57, and 62 for the reactions 16O + 120Sn, 18O + 124Sn, 16O + 144Sm, and
16O + 208Pb, respectively. The difference between the results of Eqs. (8) and
(10) is less than 5$\%$ at the highest energies. At low energies, Eqs. (8) and
(10) lead to the same values of $\sigma_{cap}$. The factor $1-\frac{4(E_{\rm
c.m.}-E)}{\mu\omega_{b}^{2}R_{b}^{2}}$ in Eqs. (8) and (10) very weakly
influences the results of the calculations for the systems and energies
considered. Hence, one can say that for the relatively light systems the
proposed method of extracting the capture cross section is model independent
(particular, independent on the potential used).

Figure 1:  The extracted capture cross sections for the reactions 16O + 120Sn
(a) and 18O + 124Sn (b) by employing Eq. (8) (solid line) and Eq. (10) (dotted
line). These lines are almost coincide. The used experimental quasielastic
data are from Ref. Sinha . The experimental capture (fusion) data (symbols)
are from Refs. Sinha ; JACOBS .  Figure 2: The same as in Fig. 1, but for the
reactions 16O + 208Pb(a),144Sm(b). The used experimental quasielastic data are
from Refs. Timmers2 ; Timmers . For the 16O + 208Pb reaction, the experimental
capture (fusion) data are from Refs. Pbcap  (open squares), Pbcap1  (open
circles), Pbcap2  (closed stars), and Pbcap3  (closed triangles). For the 16O
+ 144Sm reaction, the experimental capture (fusion) data are from Refs. SmCap1
(closed squares) and SmCap2  (open squares).

One can see that the used formulas are suitable not only for almost spherical
nuclei (Figs. 1 and 2), but also for the reactions with strongly deformed
target- or projectile-nucleus (Figs. 3 and 4). The deformation effect is
effectively contained in the experimental $P_{qe}$. $J_{cr}=58$, 68, 74, and
76 for the reactions 16O+154Sm, 32S+90Zr, 32S+96Zr, and 20Ne+208Pb,
respectively. The results obtained by employing the formula (10) are almost
the same and not presented in Figs. 3 and 4.

Figure 3:  The same as in Fig. 1, but for the reactions 20Ne + 208Pb and 16O +
154Sm. The used experimental quasielastic data are from Refs. Piasecki ;
Timmers . The experimental capture (fusion) data (symbols) are from Refs.
SmCap2 ; Piasecki . For the 16O + 154Sm reaction, the dashed line is obtained
from the shift of the solid line by 1.7 MeV higher energies.  Figure 4:  The
same as in Fig. 1, but for the reactions 32S + 90Zr (a) and 32S + 96Zr (b).
For the 32S+90Zr reaction, we show the extracted capture cross sections,
increasing the experimental $P_{qe}$ by 1% (dashed line), 2% (dotted line),
and 3% (dash-dotted line). The used experimental quasielastic data are from
Ref. Zhang3 . The experimental capture (fusion) data (symbols) are from Ref.
ZhangS32Zn9096 . For the 32S + 96Zr reaction, the energy scale for the
extracted capture cross sections is adjusted to that of direct measurements.

For the reactions 16O+154Sm and 32S+96Zr, the extracted capture cross sections
are shifted in energy by 1.7 and 1.9 MeV, respectively, with respect to the
measured capture data. This could be the result of different energy
calibrations in the experiments on the capture measurement and on the
quasielastic scattering. Because of the lack of systematics in these energy
shifts, their origin remains unclear and we adjust the Coulomb barriers in the
extracted capture cross sections to the values following the experiments.

Note that the extracted and experimental capture cross sections deviate from
each other in the reactions 16O+208Pb, 16O+144Sm, and 32S+90Zr at energies
below the Coulomb barrier. Probably this deviation is a reason for the large
discrepancies in the diffuseness parameter extracted from the analyses of the
quasielastic scattering and fusion (capture) at deep sub-barrier energies. One
of the possible reasons for the overestimation of the capture cross section
from the quasielastic data at sub-barrier energies is the underestimation of
the total reaction differential cross section taken as the Rutherford
differential cross section. Indeed, for the 32S+90Zr reaction, the increase of
$P_{qe}$ within 2–3% is in order to obtain the agreement between the extracted
and measured capture cross sections at the sub-barrier energies [Fig. 4(a)].

One can use Eq. (8) and available experimental quasielasic data Ikezoe  to
predict the capture cross sections for the reactions 48Ti,54Cr,56Fe,64Ni,70Zn
+ 208Pb, using $J_{cr}=78$, 74, 58, 51, 31, respectively. The extracted
capture cross sections $\sigma_{cap}(E_{\rm c.m.})$ as a function of $E_{\rm
c.m.}$ are presented in Fig. 5 (a).

Figure 5: (a) The extracted capture cross sections employing Eq. (8) (solid
line) and Eq. (10) (dotted line) for the reactions 48Ti,54Cr,56Fe,64Ni,70Zn +
208Pb. The used experimental quasielastic data are from Ref. Ikezoe . (b) The
extracted values of the maximal angular momenta vs. energy for the above
mentioned reactions. The solid and dotted lines show the results of
calculations of $J_{max}$ by using the extracted capture cross sections
calculated with Eqs. (8) and (10), respectively.

The formulas (8) and (10) give almost the same capture cross sections for
reactions 48Ti,54Cr + 208Pb at energies under consideration. Thus, for these
systems, the values of $J_{cr}$ are relatively large and the account of
$J_{cr}$ does not affect the results. However, for heavier systems with
smaller $J_{cr}$ (the smaller potential pockets in the nucleus-nucleus
interaction potentials), the deviation between the results obtained with Eqs.
(8) and (10) increases strongly with the factor $Z_{1}\times Z_{2}$. The
$\sigma_{cap}$, calculated with the finite value of critical angular momentum,
decreases with increasing Coulomb repulsion in the system. One can try to
check experimentally these predictions of $\sigma_{cap}(E_{\rm c.m.})$ by the
direct measurement of the capture cross sections. Note that the values of the
extracted capture cross sections for the 48Ti + 208Pb system are close to
those found in the experiments 50Ti + 208Pb Naik ; Clerc . However, for the
64Ni + 208Pb system, there are strong deviations in the energy between the
extracted and experimental Bock  capture cross sections.

By using the extracted $\sigma_{cap}(E_{\rm c.m.})$ and the sharp-cutoff
approximation, one can determine the maximal angular momentum $J_{max}$ in the
captured system as a function of the bombarding energies:

$\displaystyle J_{max}=[2\mu E_{\rm c.m.}\sigma_{cap}(E_{\rm
c.m.})/(\pi\hbar^{2})]^{1/2}-1.$ (13)

The extracted $J_{max}$ for the cold fusion reactions are shown in Fig. 5(b).
For the system 70Zn + 208Pb, the small depth of the potential pocket in the
nucleus-nucleus interaction potential leads to the decrease of $J_{max}$ by
the factor about of 2.4 at highest energy considered (about of 17 MeV above
the Coulomb barrier).

In the reactions with weakly bound nuclei one can extract the capture cross
section by employing the conservation of the reaction flux PRSGomes1 ; Nash ;
Be9Pb ; Li6Pb

$\displaystyle P_{cap}(E_{\rm c.m.},J=0)=1-[P_{qe}(E_{\rm
c.m.},J=0)+P_{BU}(E_{\rm c.m.},J=0)]$ (14)

and the measured probabilities of the quasielastic scattering ($P_{qe}(E_{\rm
c.m.},J=0)=d\sigma_{qe}/d\sigma_{Ru}$) and of the breakup ($P_{BU}(E_{\rm
c.m.},J=0)=d\sigma_{BU}/d\sigma_{Ru}$) which are defined as the differential
cross sections ratios between quasielastic scattering, breakup reaction and
the Rutherford scattering at backward angle. As seen in Fig. 6, the extracted
capture cross sections $\sigma_{cap}(E_{\rm c.m.})$ (solid line) for the
6Li+208Pb reaction are rather close to those found in the direct measurements
Li6Pbcap  at energies above the Coulomb barrier.

Figure 6:  (Colour online) The extracted capture cross sections
$\sigma_{cap}(E_{\rm c.m.})$ (solid line) and $\sigma^{noBU}_{cap}(E_{\rm
c.m.})$ (dotted line) for the 6Li+208Pb reaction. The used experimental
quasielastic and quasielastic plus breakup data are from Ref. Li6Pb . The
experimental capture cross sections (solid squares) are from Refs. Li6Pbcap .
The energy scale for the extracted capture cross sections is adjusted to that
of direct measurements.

It looks that at energies near and below the Coulomb barrier the extracted
$\sigma_{cap}(E_{\rm c.m.})$ deviates from the direct measurements. It is
similarly possible to calculate the capture excitation function

$\displaystyle\sigma^{noBU}_{cap}(E_{\rm c.m.})=\frac{\pi R_{b}^{2}}{E_{\rm
c.m.}}\int_{E_{\rm c.m.}-\frac{\hbar^{2}\Lambda_{cr}}{2\mu R_{b}^{2}}}^{E_{\rm
c.m.}}dEP^{nBU}_{cap}(E,J=0)[1-\frac{4(E_{\rm
c.m.}-E)}{\mu\omega_{b}^{2}R_{b}^{2}}]$ (15)

in the absence of the breakup process (Fig. 6, dotted line) by using the
following formula for the capture probability in this case Nash :

$\displaystyle P^{nBU}_{cap}(E_{\rm c.m.},J=0)=1-\frac{P_{qe}(E_{\rm
c.m.},J=0)}{1-P_{BU}(E_{\rm c.m.},J=0)}.$ (16)

By employing the measured excitation functions $P_{qe}$ and $P_{BU}$ at
backward angle Li6Pb , Eqs. (8), (15), and the formula

$\displaystyle<P_{BU}>(E_{\rm c.m.})=1-\frac{\sigma_{cap}(E_{\rm
c.m.})}{\sigma^{noBU}_{cap}(E_{\rm c.m.})},$ (17)

we extract the mean breakup probability $<P_{BU}>(E_{\rm c.m.})$ averaged over
all partial waves $J$ (Fig. 7).

Figure 7:  The extracted mean breakup probability $<P_{BU}>(E_{\rm c.m.})$
[Eq. (14)] as a function of bombarding energy $E_{\rm c.m.}$ for the 6Li+208Pb
reaction. The used experimental quasielastic and quasielastic plus breakup
data are from Ref. Li6Pb .

The value of $<P_{BU}>$ has a maximum at $E_{\rm c.m.}-V_{b}\approx 4$ MeV
($<P_{BU}>$=0.26) and slightly (sharply) decreases with increasing
(decreasing) $E_{\rm c.m.}$. The experimental breakup excitation function at
backward angle has the similar energy behavior Li6Pb . By comparing the
calculated capture cross sections in the absence of breakup and experimental
capture (complete fusion) data, the opposite energy trend is found in Ref.
Nash , where $<P_{BU}>$ has a minimum at $E_{\rm c.m.}-V_{b}\approx 2$ MeV
($<P_{BU}>$=0.34) and globally increases in both sides from this minimum. It
is also shown in Refs. Nash ; PRSGomes4  that there are no systematic trends
of breakup in the complete fusion reactions with the light projectiles 9Be,
6,7,9Li, and 6,8He at near-barrier energies. Thus, by employing the
experimental quasielastic backscattering, one can obtain the additional
information about the breakup process.

Figure 8:  The extracted $<J>$ and $<J^{2}>$ for the reactions 16O + 208Pb (a)
and 16O + 154Sm (b) by employing Eqs. (11) and (12). The used experimental
quasielastic data are from Ref. Timmers2 . The experimental data of $<J^{2}>$
and $<J>$ are from Refs. Vand  (open squares) and Gil ; Vand2  (open squares
and circles), respectively.  Figure 9:  The extracted $<J>$ for the reactions
32S + 96Zr (a) and 16O + 120Sn (b) by employing Eq. (11). The used
experimental quasielastic data are from Refs. Zhang3 ; Sinha .

By using the Eqs. (11) and (12) and experimental $P_{qe}$, we extract $<J>$
and $<J^{2}>$ of the captured system for the reactions 16O + 154Sm and 16O +
208Pb, respectively (Fig. 8). The agreements with the results of direct
measurements of the $\gamma-$multiplicities in the corresponding complete
fusion reactions are quite good. For the 16O + 208Pb reaction at sub-barrier
energies, the difference between the extracted and experimental angular
momenta is related with the deviation of the extracted capture excitation
function from the experimental one (see Fig. 2). In Fig. 9 we present the
predictions of $<J>$ for the reactions 16O + 120Sn and 32S + 96Zr.

##  IV Summary

We realized that the found relationship between the quasielastic excitation
function and capture cross sections is working well, and the quasielastic
technique could be an important and simple tool in the study of the capture
(fusion) research, especially, in the cold and hot fusion reactions and in the
breakup reactions at energies near and above the Coulomb barrier. Employing
the quasielastic data, one can also extract the moments of the angular
momentum of the captured system.

We thank S. Heinz, S. Hofmann and H.Q. Zhang for fruitful discussions and
suggestions. We are grateful to H. Ikezoe, C.J. Lin, E. Piasecki, and H.Q.
Zhang for providing us their experimental data. This work was supported by
DFG, NSFC, RFBR, and JINR grants. The IN2P3(France)-JINR(Dubna) and Polish -
JINR(Dubna) Cooperation Programmes are gratefully acknowledged. P.R.S.G.
acknowledges the partial financial support from CNPq and FAPERJ.

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