Introduction

The implications of human mobility can be found not only in transportation sciences and spatial economics, but also in social sciences and politics due to the manifold effects it has on the individual forms of mobility, infrastructure and urban planning, but most notably in respect to the environment, energy and climate (IEA, 2019; OECD, 2020). Such implications are also strongly related to the individual travel behaviour, where the individual traveller experiences the everyday-related mobility behaviour and is therefore confined in his or her perception and conditions of the personal mobility options. The individual discernments of mobility issues may therefore differ to the actual issues and objective causes of mobility behaviour and their implications on the social and environmental level.

In relation to this complexity, mobility studies should provide explanations to such problems and the most important variables characterising and explaining mobility behaviour are generally assumed to be the entities of generalised cost, i.e. travel time and—as part of the socio-economic variables—direct travel costs, car-ownership or income, culminating in the socio-economic approaches of utility theory (Barbosa et al., 2018; Ben-Akiva and Lerman, 1985; McFadden, 2000; Mokhtarian and Chen, 2004; Small, 2012). Despite the basic nature of these measures and their “extreme importance for mobility modelling” (Barbosa et al., 2018), their functional descriptions and respective predictions are acknowledged to be ambiguous and have not been combined in a consistent model (Brathwaite and Walker, 2018). Also, approaches explaining the extent of daily travel time itself, have stated a law of constant travel time or a universal constant across space and time at around 1–1.3 h per day (Ahmed and Stopher, 2014; Schafer, 1998). Although this phenomenon of the so-called travel time budget (TTB) has been recognised since the early 1960s, it is generally given as average statistics without decisive reasons for its stability or conclusive underlying functional relationships (Zahavi et al., 1981; Ahmed and Stopher, 2014; Mokhtarian and Chen, 2004).

Other concepts of travel behaviour modelling have often been made in analogies to the original physical concepts and especially Newton’s mechanics, which have been used in two respects: Firstly, in terms of Newton’s laws of motion as, e.g. in pedestrian modelling (Helbing et al., 2001), and, secondly, in terms of the gravity models (GM), where the frequency of trips is proportional to the “masses of origin and destination” and the relative distance (function), which was evaluated in the early stages by ticket prices of train, bus and aeroplane (Barbosa et al., 2018; Lill, 1891; Wilson, 2010, 1967; Zipf, 1946). Further developments evaluated variants of the model structure (Wilson, 1967, 2010; Yan et al., 2013), with respect to commuting or migration (Masucci et al., 2013; Simini et al., 2012), opportunity and radiation modelling (Ruiter, 1967; Simini et al., 2012; Stouffer, 1940; Yang et al., 2014), comparisons of gravity vs. scaling, or maximum entropy (Anas, 1983; Brockmann et al., 2006; Bazzani et al., 2010; Song et al., 2010; Wilson, 2010; Bettencourt, 2013; Chen, 2015; Curiel et al., 2018).

Many of these studies have been developed based on motorised means of transport, foremost on car travel, and in combination with one specific travel purpose, i.e. commuting. However, if we envisage a general model, then all modes of transport, including the active modes with walking and cycling, and all purposes have to be equally accounted for, irrespectively of their current level of the modal split. This may be one main reason why such models are often not accepted in the wider community as generally applicable mobility models (Brathwaite and Walker, 2018).

In this paper, a physical/physiological mobility model (PHM) is developed based on a physical methodology and physiological energy effort, which consistently connects travel time, travel distance, and the extent of daily travelling for all modes of mobility. The basic hypothesis assumes the existence of specific probability density functions for spending physiological energy for one-, two- or multi-modal travel. The model, described in its most basic and fundamental form, is termed ‘Grundmodell’, focusing on the methodological consistency and the verification using real data. As such, it can be used to explain the extent of travel behaviour for all modes and modal combinations at a macroscopic level. Furthermore, the TTB model is explained as a natural consequence of the PHM; but the PHM concept has a wider range of validity and additional explanatory power. It is shown that, without any loss of generality (Kölbl and Helbing, 2003), the PHM can be systematically refined with further disaggregation, such as trip purpose or income groups, or adjusted for given cities or regions with a detailed knowledge of the transport infrastructure. However, it should be noted that, the comparison with the socio-economic variables is not an objective, since it would go far beyond the scope of this paper. The focus on the work described here is on the applicability to all forms of human mobility, i.e. in mono- and multi-modality. This has not been shown before in such a stringent and consistent methodology and on such a comprehensive data set.

Material and data methods

Data and design of the study

The data collection has been chosen, because these data are considered as the official travel survey data, used by official government departments, and have similar timespans of consecutive surveying. They still constitute the “gold-standard” of travel surveying and are the longest, publicly and electronically available surveys.

The selected countries for our data verification are—in alphabetic order—Germany with the survey of KONTIV (1976–2000) & MID (2000–2017) & the Mobility Panel (1995–2017) (Bundesministerium für Verkehr und digitale Infrastruktur, 2019), Switzerland (1974–2015) (Bundesamt für Statistik, 2015), the UK National Travel Survey (1972–2016) (Department for Transport, 2019) and the US National Household Travel Survey (1977–2017) (U.S. Department of Transportation, 2019).

The sources for all data have been referenced under the reference section. All data are officially and publicly available, either free of charge or for a data service charge. The changes in survey methodologies within each national survey over the years have been ignored as secondary as can be seen in the Germany data, where two parallel surveying methods yielded matching results. All data have been used without correction, since the statistical background for weighting is not always clearly stated and these have been ignored. Furthermore, all trips without any distinction in trip purposes are considered as for a daily trip making analysis, such a distinction cancels out.

Data pre-processing

The study data base was set up using the following fields: household-identity number (id), person-id, day-id, trip-id, year of travel, overall travel time and distance of a trip and mode of transport (MoT), which were standardised with overall door-to-door travel and main mode of transport.

The definition of the main mode of travel is the general standard of travel surveying, since for example, walking is nearly part of every trip. The following definition is generally used: “The main mode of a trip is that used for the longest stage of the trip.” “With stages of equal length the mode of the latest stage is used” (Department for Transport, 2019). However, this level of trip stage detail is not present in all the surveys, especially those the earlier ones. An analysis of surveys with such a detailed distinction of trips into coded modal trip stages revealed that the average number of modal stages per trip is around 1.1, limiting the extent in the modal definition of main mode of transport. It is to be noted that this distinction falls in the same methodological category as multi-modal trip making and therefore this does not change any assumptions of the PHM. However, most importantly, the travel surveys specify all modes and contain all modes alike, including the active modes with walking and cycling. All the surveys assume include walking trips except the UK NTS, which makes a distinction regarding long walks (>1 mile) and short walks (>50 yards). This distinction has been considered in the above analysis, so only the days, which included short walks have been used.

Correctness of records

Since there are incomplete data sets in the data and to have a criteria for the correctness of records, two conditions for the inclusion of an individual data set have been defined. Firstly, the daily record according to all trips-ids has to be complete with data for time, distance and mode of transport. Secondly, the given record is within the physically possible limits, for example, average trip speeds for walking (<33 km/h), cycle (<70 km/h), car driver (<150 km/h), bus (<100 km/h), rail (<250 km/h), which resulted in a data usage of more than 80% of all data. This issue of the upper limits is also relevant for the distribution tail. But the limits have been retained in order to show the full range of validity of PHM.

Data analysis

The data analysis has been undertaken using MATLAB. The code was generated with the standard functions of the software package, including those for estimating the parameters of the distribution function.

The following steps were undertaken to produce a consistent data base:

  • Travel distance is given in kilometre and miles, where all entries have been converted to kilometres.

  • Travel time is used as given “overall travel time”.

  • Mode of transport is given different forms. The early data bases provide only the “main mode of transport” as the mostly used mode throughout a trip. This has been cross-checked with detailed stage information, when provided. The definition of mode of transport is also combined with other variables. In the US-data, for example, a distinction between car-driver and car-passenger is given through a combination with other variables. Hence, the combination of “mostly used” and other related variables leads to an overall standard definition of (main) mode of transport.

  • Travel speed is used as a measure for correctness of the single trips according to the physically possible criteria of the MoT used as described above.

  • Completeness and consistency of the single travel day was checked where all the daily trip values have to stay within the defined physical limits. This assumption is actually a criteria for the carefulness of the surveyed person, i.e. that he or she took the surveying seriously and therefore provided correct values.

Over 80% of all the data were able to be used for the data analysis following the application of this standardisation process.

Table 1 provides the number of observations regarding survey, days and trips:

Table 1 Number of days and trips per country survey and year, respectively.
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Theory and method

The PHM-Model

We consider a region, which is partitioned as a grid structure of locations and where people travel between locations, from \(\overrightarrow {r_i}\) to \(\overrightarrow {r_j}\), using certain modes of transportation m (e.g. walking, bike or velo, car, train, etc.) over a respective day (Fig. 1). At a microscopic level, the number of people travelling can be modelled by

$$N\left( {\overrightarrow {r_i} ,\,\overrightarrow {r_j} ,\,m} \right) = n_{\rm {t}}\left( {\overrightarrow {r_i} } \right)n_{d}\left( {\overrightarrow {r_j} } \right)W\left( {\overrightarrow {r_i} ,\overrightarrow {r_j} ,\,m} \right)$$
(1)

where \(n_{t}\left( {\overrightarrow {r_i} } \right)\) and \(n_{d}\left( {\overrightarrow {r_j} } \right)\) contain all details about the total number of travellers and available destinations in different areas. Apart from travel time and distance, the mean human energy \(E_{ij}^{m}\) consumed during a trip is given with

$$E_{ij}^{m} = P_{m}\left| {\overrightarrow {r_i} - \overrightarrow {r_j} } \right|/v_{m}$$
(2)

where Pm is the mean power and vm the mean velocity of a respective mode travelled. The power Pm is the human physiological energy effort of an activity (Ainsworth et al., 2011; Bouchard et al., 1983; Dowd et al., 2018; Spitzer et al., 1982; WHO, 1985) given in kJ/min, since it is the human traveller per se, who starts and stops walking, driving, riding a bus, etc.

Fig. 1: Conception of study area.
figure 1

Schematic concept of a region described as a grid structure of locations, origin \(\overrightarrow {{{{\boldsymbol{r}}}}_{{{\boldsymbol{i}}}}}\) and destination \(\overrightarrow {{{{\boldsymbol{r}}}}_{{{\boldsymbol{j}}}}}\) with a visualisation of possible destinations at the same travel distance (circles).

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In the following we consider a travel behavioural distribution \(W\left( {\overrightarrow {r_i} ,\,\overrightarrow {r_j} ,\,m} \right)\) for a given mode m

$$W\left( {\overrightarrow {r_i} ,\,\overrightarrow {r_j} ,\,m} \right)\sim e^{ - E/E_{0,m}}$$
(3)

where E0,m is a single global scaling factor and E is the associated physiological energy consumed along a specific path.

This Grundmodell is based on the assumption that the amount of human movement or travel is constrained primarily by the physiological energy consumed, where energy usage is a hallmark for every human activity (Dowd et al., 2018; Spitzer et al., 1982). In the literature, physical activities are quantitatively given in different units such as the metabolic equivalent of task (MET), where 1 MET describes 3.5 mL of oxygen per minute per kilogram of an adult, often characterised as the metabolic cost of resting quietly. A respective activity is then assigned an intensity unit on the basis of their rate of energy expenditure expressed as multiples of 1 MET (Ainsworth et al., 2011). Although there are approximate conversions with corrections factors of MET to kJ/min, the tables with unconverted values of kJ/min (Spitzer et al., 1982) are used for the developed approach, because they can be directly applied to time and distance travelled.

In order to provide the most general form of a region, i.e. without any human location-based accumulations or settlements, a simple uniform distribution for both \(n_{t}\left( {\overrightarrow {r_i} } \right)\) and \(n_{d}\left( {\overrightarrow {r_j} } \right)\) is assumed. Specifically, we set

$$n_{d}\left( {\overrightarrow {r_j} } \right) = n_{d} = \rho _{d}\left( {{{{\mathrm{{\Delta}}}}}x} \right)^2$$
(4)

where ρd is the homogenous density of destinations. Considering the travel movement on a path we introduce the density of path length ℓ with ρ(ℓ) = 2πℓρd, where the number of available destinations scales linearly with the circumference of a circle around the origin as the path length ℓ increases (Fig. 1).

With Eq. (2) we obtain the physiological energy spent for a trip with the path length ℓ as

$$E = P_m\ell /v_m$$
(5)

Since Eq. (5) implies that

$$E\sim \ell$$

an additional energy term E should exist in the Grundmodell. Thus, the corresponding probability density function is given by

$$\widetilde w_m\left( E \right) = E_{0,m}^{ - 2}E{e}^{ - E/E_{0,m}}$$
(6)

where the probability of a travel path with an energy effort E1 < E < E2 is given by \(\widetilde W_m\left( {E_1\, < \,E\, \,< E_2} \right) = {\int}_{E_1}^{E_2} {\widetilde w_m\left( E \right){d}E}\). The additional energy term E leads also to a natural reduction of probabilities for low E, guaranteeing that \(\widetilde w_m \to 0\) for E → 0 without any additional ad hoc regularisations. The average energy expenditure 〈Em〉 from the frequency distribution can now be calculated from Eq. (6) where

$$\langle E_m \rangle = {\int}_0^\infty {E\,\widetilde w_m\left( E \right){d}E = 2E_{0,m}}$$
(7)

Multi-modal mobility and modal split

The travel survey data are based on daily travel behaviour, where a household with one or all persons is observed over a course of one or more days. The modal mobility behaviour is thus defined in relation to the number of modal trips made per person and day, i.e. the daily trip making by using one, two or more modes of transport (MoT).

The daily travel effort of a person \(E_d^{nm}\) can therefore be calculated by Eq. (2) as

$$E_d^{nm} = \mathop {\sum}\nolimits_n {\mathop {\sum}\nolimits_m {E_{ij}^m\left( n \right)} }$$
(8)

where n is the number of trips per day d, done by a person with n-MoTs m, again satisfying the assumption of independence of averaging. The probability density for a trip comprising multi-modal travel is given by

$$\widetilde w_{nm}\left( E \right) = E_{0,nm}^{ - 2}E{e}^{ - E/E_{0,nm}}$$
(9)

where the only parameter E0,nm is again proportional to the average energy effort for a specific combination of multi-modal travel.

Results

Physiological travel distribution functions

Out of more than 870 daily modal combinations, only four have been selected. These are common, quantitatively very different and still hold a common combination, i.e. walking (wk), car-driver (cd), rail train (rt), and their combination (wkcdrt). For such distributions, the general definition of daily mobility is used, where a person uses only one (main) mode of transport, i.e. walking, driving the car or the train for the whole day (d1), or in the modal combination, or using all three modes (d3).

An impression of the raw data variations and the modal distribution functions on a linear scale is seen in Fig. 2. The raw data show large variations, which could be reduced with larger bins for smaller variations. With a focus on the Grundmodell and without any loss of generality, a uniform distribution of all travellers over the travelling area is assumed as already given with Eqs. (6) and (9), leading to only one distribution function for all travellers over the whole area.

Fig. 2: Travel time and distribution function.
figure 2

A comparison of daily modal travel time distributions per capita between the relative frequencies of the raw data (individual markers) and the exemplary distribution function (continuous lines) of the proposed mobility model.

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Although a separate modal distribution function could be depicted for each country, for reasons of modal differences and clarity, only one modal function for all countries has been plotted in Fig. 2. For the parameter analysis, fitting was done using the maximum-likelihood method and for each modal behavioural set, year and survey.

The main frequencies of travel lie in the region below 400 min and above, i.e. to the right, the values approach zero. The selected d1-modes with their respective colour are walk (wk), car driver (cd), rail (rt); the d3-mode combination, (where the data are not a combination of three former) is with car-driver & rail & walk (cdrtwk). The markers show the different countries: Germany kontiv & MID ▷, Germany mobility panel ▽, Switzerland ◯, UK □ and US ⬨. The distribution functions give a representative course of the data with w(0) = 0 and w(∞) → 0. The data frequencies are averages of the respective survey countries, the distribution functions represent the overall averages.

The model distributions show a correct asymptotic behaviour by design, as the relative frequencies for t → 0 and for t → ∞ are exactly zero. The model also captures the region of small travel times, irrespective of the large variations in data. The main area of the distributions is in the range between 0 to 360 min, or statistically speaking, up to the 0.975-quantile or about five scaled means of car travel.

Additionally, the overall average daily travel time per capita and at their relative frequency of each surveyed year is depicted in Fig. 2, representing the TTB-model (and marked by the greyish points of d0). They are in full agreement with the values between 60 and 120 min of the TTB-literature (Ahmed and Stopher, 2014). This comparison of different modal distribution functions shows the compliance with the theoretical distribution function, its agreement with different modal travel behaviour and the methodological relationship and consistency with previous TTB-research.

To obtain a relational understanding of the distribution functions with regard to the different physical entities, the daily mobility frequency distributions of the different MoTs are plotted in a logarithmic scale in Fig. 3. The functions over time in minutes (being the equivalent plot to Fig. 2) are shown in Fig. 3a; in Fig. 3b the distributions are depicted over distance in km, and in Fig. 3c over travel energy in kJ. All three subfigures agree well with the PHM-model (solid lines).

Fig. 3: Travel distribution functions.
figure 3

Data and distribution functions of daily travel time a, distance b and energy c in a double logarithmic representation. Note that the x-axis can now depict all values.

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The three 1-modal modes are walk (wk), car-driver (cd), rail (rt); the 3-modal mode is the combination of the three, i.e. (cdrtwk). Figure 3a is exactly the same as Fig. 2 and presents already a clearer course of the different data wk is within the vicinity to cd and rt is close to cdrtwk. In Fig. 3b wk is far off cd; cd, and particular rt and cdrtwk are relatively (very) close. In Fig. 3c all 1-modes collapse nearly to one curve and the 3-modal curve is shifted, showing the amount of the different energy effort. All data points <0.975 quantile are depicted. The other values would over-proportionally distort the plot in terms of horizontally flattening out, which is only due to the binning of a single frequency observation in this area.

The microscopic relationship of Eqs. (2) and (8) and the macroscopic distribution of Eqs. (6) and (9) enable, that travel time (Fig. 3a) and travel distance (Fig. 3b) can be directly derived from travel energy with the respective modal powers (Fig. 3c). These functional relationships show the microscopic and macroscopic agreement between all three plots and, thus, the methodological consistency of the PHM.

In terms of goodness-of-fit of the 870 modal combinations, an R2-values has been calculated for each modal distribution function and for each survey year and country. These are then classified according to the modal usage and depicted in boxplots (Figs. 4–6). To capture nearly all modes such as air and many multi-modal combinations, the minimum data size is set to >8. Furthermore, to attain daily mobility in all combinations, the following denotations apply: m = d1 means, that only 1 MoT is used throughout the day; similarly m = d2 denotes the daily usage of 2 MoTs, and m = d3, where 3 or more MoTs per day are used. For example, a d1-behaviour is one, where a person only walks or only takes a car, or only a bus or a train for the whole daily trip making; a d2-behaviour is where a person takes the bus plus the car, or the bike plus the train, and so on. Without a modal distinction of daily travel behaviour, which is the general measure in all standard travel statistics, the notation is m = d0.

Fig. 4: Boxplot of R2-value-distribution of daily travel time according the respective modes of transport.
figure 4

The abbreviations of the modes of transport are: d0, d1, d2, d3, as defined above, wk—walking, vo—velo or bike, bs—stage bus, cd—car driver, cp—car passenger, rt—rail, and the respective combinations.

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Fig. 5: Boxplot of R2-distribution of daily travelled distance according the respective modes of transport.
figure 5

The abbreviations of the modes of transport are: d0, d1, d2, d3, as defined above, wk—walking, vo—velo or bike, bs—stage bus, cd—car driver, cp—car passenger, rt—rail, and the respective combinations.

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Fig. 6: Boxplot of R2-distribution of daily travel energy according the respective modes of transport.
figure 6

The abbreviations of the modes of transport are: d0, d1, d2, d3, as defined above, wk—walking, vo—velo or bike, bs—stage bus, cd—car driver, cp—car passenger, rt—rail, and the respective combinations.

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The boxplots are given for the specific modes and modal combinations and the three main indicators, i.e. daily travel time (Fig. 4), daily travel distance (Fig. 5) and daily travel energy (Fig. 6), providing the statistical accuracy to Fig. 3 for the three entities. It should be noted that travel time and distance has been recorded separately and are treated in that way. Travel time appears to be well approximated by the distribution function with a general median of R2 of around 0.9. By comparison, travel distance and travel energy yield median values of around 0.8. The odd-one-out appears to be car-passenger (cp), which shows greater differences in relation to travel distance. This may be due to the estimation of distance travelled or to the definition of the main mode of transport, where walking stages may have a greater influence. Such influences of modal combinations by stages may also be a reason for the greater spread of values of d0 and d1, or this may indicate that travelled distance is more sensitive than travelled time.

Estimation of the physiological modal powers

The stability over time of daily travel by mode can be seen in Fig. 7. On an overall level, denoted by d0, i.e., the stability of daily travel behaviour without modal distinctions is well known from the TTB-approaches (Ahmed and Stopher, 2014; Schafer, 1998). The data are based on the raw data and the markers show the different countries: Germany kontiv ▷, Germany mobility panel ▽, Switzerland ◯, UK □ and US ⬨. Whereas travel time lies between 50 and 200 min, i.e. a 4 fold scale, travel distance lies between 3 km and 100 km, a 30-fold scale. Both entities, however, show a stability of the years, irrespectively of the countries. The differences in country values seem to be relatively consistent, which could be traced back to the surveying methods. A similar stability shown in Fig. 7, can also be observed with daily travel behaviour of walking (wk), car-driver (cd) or rail train (rt). The daily behaviour of the multi-modal combination (cdrtwk) are fairly consistent, where the variations may be mainly linked to the result of a lower numbers of observations.

Fig. 7: A comparison of daily travel time and travel distance.
figure 7

Average daily travel time a and distance b per capita over years on a semi-logarithmic scale of walking (wk), car driver (cd), rail train (rt), the modal combination of car-driver, walking and rail, and the standard average daily travel time, without modal distinctions (d0), representing the travel time budget approaches.

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The plot related to distance (Fig. 7b) indicates similarly stable behaviour, with the modal values further apart, especially walking. The TTB values shown in Fig. 7 are higher than for car-driver, they are smaller with regard to distance.

To estimate the mobility effort we, firstly, make use of Eq. (5) and Fig. 7 with respect to stable travel times and physiological measurements of walking and cycling. That is, we make a rough estimate of an average energy expenditure \(\overline E _{d1}\) of a daily single modal behaviour, i.e. \(\overline E _{d1} = \overline P _m\left( t \right)t_m\) with m = wk, vo and accounts for roughly around 800 kJ. By assuming this average \(\overline E _{d1}\) for all daily single modal behaviour, we can estimate all other average power values with \(\overline P _m\left( t \right) = \overline E _{d1}/t_m\) for the other d1-behaviours, such as car-driver, bus, train, etc.

These average modal powers \(\overline P _m\left( t \right)\) can, according to Eq. (2), be applied to all related individual trips, yielding an energy expenditure \({\it{\epsilon }}_i\) of an individual trip with \({\it{\epsilon }}_i = \overline P _m\left( t \right)t_{i,m}\), where the ti,m are the raw data values. With that, all individual daily travel energy expenditures \({\it{\epsilon }}_d = \sum {\it{\epsilon }}_i\) are then calculated for all modal combinations, i.e. again d1- and also d2- and d3-modal travel effort. This approach ensures, that the microscopic variations of travel effort of each trip and each person with different travel times is retained through all surveys and years. Thus, the estimated 800 kJ is not a fixed constant, but only a macroscopic average.

This method can further be substantiated by the following considerations: \(\overline E _{d1}\) is different to 〈Em〉, i.e. \(\overline E _{d1}\) is an assumed average value only for d1-modal behaviours, 〈Em〉 is a calculated average from the distribution function of any modal combinations; (only in idealised, theoretical terms, they are quantitatively equal, and only for d1-values). Hence, the values in Fig. 3c still vary. Also, from a theoretical point of view, the assumption of a modal average power value is evaluated through the constant elasticity measure E0 of the exponential distribution function, where the distribution function retains its validity over a scaling range of around 5.

Furthermore, since the speed vi of an individual trip i is vi = li/ti = Pi(t)/Pi(l), the distance-related powers can be calculated with Pi(l) = Pi(t)/vi. Including the modal classification, these power values yield averages of 〈Pm(l)〉. In turn, these have to comply with the distance-related distribution functions of Eq. (6) and distance-related physiological measurements and constitutes a further verification of Fig. 3c. The intrinsic nature of the physical relationships shows that the methodology is also consistent with outside measurements. Thus, the estimate of 800 kJ for \(\overline E _{d1}\) is only a requirement for missing measurements in terms of power or daily expenditure. It does not constitute a necessary pre-requisite, and the relative ratios to the d2- or d3-modal behaviour will remain proportional. However, and most importantly, the methodology and the estimation approach allow replacement with real measurements at each point of the procedure, without any changes to the overall derivation.

Modal physiological powers and measurements

The estimations of the modal physiological power of time and distance according to speed can be seen in Fig. 8, providing a time–distance–energy space of the PHM and the innate human mobility behaviour, which occurs on an unconscious level. Figure 8 gives a 3-dimensional depiction with the common horizontal axis of speed in km/h and split into two dimensions. The upper plots show distance-specific modal powers in kJ/km, while the corresponding lower plots show time-specific modal powers in kJ/min. The left column uses a linear scaling while the right is doubly-logarithmic, covering a larger range of values. Each point denotes a modal power average of the respective survey years, using the same country markers as in previous figures. The abbreviations of the selected MoT are: ar—air, bs—stage bus, cd—car driver, rt—rail, vo—velo or bike, wk—walking and the respective combinations of 2- and 3-modal day travel (which are based on different data). The markers show again the different countries: Germany kontiv ▷, Germany mobility panel ▽, Switzerland ◯, UK □ and US ⬦. For reasons of clarity, the values >50 km/h are omitted in the linear plot Fig. 8a. The logarithmic plot Fig. 8b with travel speeds scale up to 103 can include power values of air travel. In general, the time values have an around 10-times larger variability than the distance values. The logarithmic plots indicate that the data can be approximated by a diagonal line, where all MoTs and also future modes should align to.

Fig. 8: Time-distance-energy-space.
figure 8

Power of distance and time over speed of selected modes of transport in linear a and double logarithmic b representation. The figure is actually a 3-dimensional plot in horizontal (Pm(t)—bottom) and vertical (Pm(l)—top) projection over travel speed, showing modal human behaviour in a time–distance–energy–space.

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The range of specific power values is visualised in Fig. 8. Table 2 provides examples of calculated averages of the whole modal hierarchy with a comparison to some power values of physiological measurements (Ainsworth et al., 2011; Bouchard et al., 1983; Spitzer et al., 1982; WHO, 1985) marked with *. This comparison verifies the consistency and complementarity of the physical methodology between modal travel behaviour and physiological measurements as meaningful and realistic quantities. The values for the proposed modes are given as mean values over all data. The overall modes (d0, d1, d2, d3, as defined above) show similar power values, which can explain the differences in time and distance travelled. With increasing level of specifications, the values of single MoT-s vary greatly between time, distance and modal powers. The measured \(P_m^ \ast\) are slightly lower, because they have been measured on an even path and do not include stop & go or up & downs. Car driver (cd) varies from driving on a country road (5.9 kJ/min) to driving the city under congestion (12.6 kJ/min). The high ride comfort of air travel is reflected by a \(P_m^ \ast \left( t \right)\) of 1.5 kJ/min which is approximately equivalent to “sitting on a chair”-measurements (Spitzer et al., 1982).

Table 2 Comparison of ground truth, with overall averages of time, distance and speed, calculated and measured power values of selected modes of transport (MoT), nominated as in Fig. 4, including air travel (ar).
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The daily travel time of the overall daily modal behaviour (d0) lies in the range of the TTB-approaches (Ahmed and Stopher, 2014; Schafer, 1998). The time values of the d1-mode per day, up to d3-modes per day rise steadily as the time-specific power values do not vary. As the MoTs are further specified, values become more and more diverse and the common functional relationships of Fig. 8 can hardly be envisaged.

A comparison of walking (wk) and cycling (vo) shows that the travel time and Pm(t) of wk should slightly be higher as shown by the physiological measurements, which is most likely due to the non-recorded short walking trips. Also, these physiological measurements have been made on “even paths” (Bouchard et al., 1983; Spitzer et al., 1982). This is not the case under real mobility conditions, where a certain amount of stop & go or up & down is involved. Taking the motion of the means of transport further account, i.e. into for example of public transport, an effort for an additional balancing out of the motion should lead to higher physiological values, as it can be seen with Pm(t) of air, where “sitting on the chair” with 1.5 kJ/min (Spitzer et al., 1982) corresponds well to the calculated 2.6 kJ/min. It should be noted that physiological measurements of many different activities especially in terms of travel and mobility, are uncommon and more data studies are needed. However, the values may not have changed significantly due to the similar physiological human constitutions over time (Bouchard et al., 1983; Spitzer et al., 1982; WHO, 1985).

Overall, there is an agreement between the calculated values and the respective measurements, and their ratios, which are more or less fixed because of the independent measured travel time and distance related to the ground-truth. This underlines the validity of the distribution function and its applicability to time, distance and energy.

Classification of daily mobility

With the above classification, a different definition for the Modal Split, which is usually defined on the basis of trips made of d0, can be obtained without the above daily behavioural distinction. If the daily modal travel behaviour in terms of the number of modes used, i.e. a d1-, d2- and d3-behaviour, then in all surveys the relative frequencies of these three categories are roughly 70% for d1-, 25% for d2- and 5% for d3-behaviour. This has remained stable over the course of the 5 decades for all countries (Fig. 9a). It should be noted that Fig. 9a is based only on the raw data and the above definition of daily mobility, yielding therefore an independent account for the daily mobility combinations of the observed countries.

Fig. 9: Modal split and travel energy.
figure 9

Modal split over survey years (a) and corresponding daily energy effort per capita (b) of 1-, 2- & 3+ modal behaviour (marked by d1, d2, d3).

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In Fig. 9d1 means that 1 mode (e.g. bike, car driver or train), is used by a person throughout the travel day; d2 means a combination of 2 modes (e.g. walking plus bus) and so on. The d3 values contain the values for 3 and more modes per day. The markers show again the different countries: Germany kontiv & MID ▷, Germany mobility panel ▽, Switzerland ◯, UK □ and US ◊. The relative share of all daily travel, i.e. the modal split, is around 70/25/5 percent over all survey countries and years. The daily energy effort per capita, measured in kilo Joule, increases by a ratio of around 1:1.5:2 between d1:d2:d3, which means that for d3 a persons will spend twice as much physiological energy as for d1-behaviour. Figure 9b also contains d0-values, which is the daily travel behaviour per capita with no modal distinctions. These values correspond to the TTB approaches and comprehend all modal behaviour, therefore the modal split = 1.

From Eqs. (2) and (8) with the d1-modal behaviour, a further consequence can be obtained for the specific MoT in question. Since modal travel time and distance are independent observations, the multiplication with the (average) physiological power Pm with m = walking, velo or bike, car driver, bus, rail train, etc. results in the daily energy effort Ed of a person, Eq. (8). Such daily travel behaviour has only one specific average physiological power value for the daily energy effort, which should comply with physiological measurements.

Such modal powers are more or less the same for the other daily travelling behaviour, where two (d2), three or more modes (d3) are used throughout the day, since the topology of the areas is the same. Hence, for the Grundmodell of PHM, the same modal powers can and are used for the trips of the d2-, d3-efforts, which yields realistic effort values for all daily modal trip combinations. Based on the independently recorded trip time and trip distance, a cross-check can be made and the consistency of the methodology can be validated. From a methodological view point, this means that, the effort ratios between d1, d2 and d3-behaviour are preserved even without any specifications for the power values.

An estimation for the absolute amount of the daily mobility effort is shown in Fig. 9b, where the d2-effort is 1.5-fold of d1 and 2-fold from d1 to d3-effort. A two 2-dimensional plot of a 3-dimensional relation is shown in Fig. 9, where each single point over the years on the left Fig. 9a has a corresponding energy expenditure on the right in Fig. 9b. This increase in required effort for multi-modal mobility may be a reason for its small proportion, thus, a clear indication for the assumption of a least effort (Zipf, 1949).

The effort values without modal distinctions of d0-behaviour is also shown in Fig. 9b, i.e. the sum of d1, d2, d3 and a modal split of 1. These values correspond to the TTB-approaches and show the mobility effort with the stability of the assumed “constancy” of daily travel time per capita (Ahmed and Stopher, 2014; Schafer, 1998).

Discussion

In the following, some comparative considerations can be made with the commonly used transport model, i.e. the GM and its methods, to provide possible options for future research and applications.

Physical consistency of the PHM

From a theoretical point of view, the distribution function of the PHM is in principle not a chosen statistical distribution according to its goodness-of-fit (Chen and Fan, 2020; Li, 2019; Small, 2012). However, according to the innate functional derivation, the distribution function satisfies the fitting of the three dimensions, time, distance and energy, simultaneously. The PHM can be based on the maximum entropy principle, also the concept of the Grundmodell, with the minimum required information. Furthermore, the PHM determines the distribution function of Eqs. (6) and (9) and hence, for example, the average energy can directly be derived using Eq. (7). In contrast to the GM, the exponential function cannot be altered, in order to comply with the three time, distance and energy dimensions simultaneously and consistently. It provides the derivative rigour, required for theoretical stringency and methodological guidance. Beyond the Grundmodell, the distribution function allows further specifications, without any loss of generality (Kölbl and Helbing, 2003). These conditions ensure, that the methodological edifice retains its consistency, even when modal compositions and decompositions are made. This consistency can also be observed in the parameters, having a defined specification of units, which can be measured and verified micro- and macroscopically.

Accuracy, applicability and limitations

It can be seen from Fig. 2, that the quantitative accuracy depends on the scale of granularity, i.e. related to measurement definitions, binning or zone sizes. The PHM uses the R2-measures with regard to distance bins of 0.05–0.3 km, depending on the MoT and the minimum survey distance. Thus, the PHM also provides the sensitivity for non-motorised MoTs and applicability at a microscopic scale.

The accuracy of the goodness-fit statistics is foremost dependent on the distribution function, which goes to zero in the vicinity of the origin (Figs. 2 and 3). These values are often ignored in transportation (Barbosa et al., 2018), due to the exponential or power distributions usually applied, but is accurately captured by the PHM model. A further influence are the data of the (fat) tails, which become sparse as data move towards the frequency of a single observation, i.e. 1/nt . Hence, the data points in the distribution must flatten out horizontally, which may question their representativeness. This is not apparent in a linear depiction, but in a logarithmic one, where data frequencies do not decrease towards zero, as they would do if the number of observations would be very high or go to infinity.

The current level of accuracy is also limited to the available data, especially from a perspective of the physiological data, since power values depend on age, sex, speed of the action, body height and weight, level of fitness, the inclination of the surface and infrastructure. Whereas the first three items could be related to the dedicated data categories of the travel data, the others variables could be measured and integrated with modern observation gadgets such as physiological watches or accurate geo-referencing. But most importantly, all variables with a required categorisation have to be equivalently represented on both, on the areas of physiology as well as on those of the mobility surveys.

Physiological variables and diversity

Whilst the Grundmodell considers physiological behaviour in its simplest form, which is the prime focus of this paper, further specifications are possible. Additional levels and extensions for transport modelling may be added, with developments towards a new understanding of transport supply with modal split assessments of a city or region or in respect to transport & land-use (Bart, 2010; Barthélemy, 2011; Wegener and Fuerst, 2004; Wilson, 2010). From a microscopic perspective for a practical application, this approach supports the growing literature on mobility & health, where the positive effects of the physiologically active modes such as walking and cycling for a healthier and longer life have been shown with statistical significance (Ainsworth et al., 2011; Batista Ferrer et al., 2018; Cooper et al., 2003; Dowd et al., 2018; Gopinath et al., 2018; Kujala et al., 1998; Pyky et al., 2018). The individual physiology depends further on age, sex, the daily activities of work or leisure or stress (Cooper et al., 2003; Spitzer et al., 1982; WHO, 1985). Such group classifications can be directly assessed with the PHM which can be correlated with the body mass index, which raises issues such as the discussion on the lack of physiological activity or obesity. Furthermore, children, mobility impaired or elderly who have lower physiological performance limits in medical terms, can directly be addressed methodologically. Twofold: firstly, with their lower available daily energy budget, i.e. that they have a lower level of their daily trip making and therefore are tempted to be driven around to meet their daily schedule (which is then not only due to their time management); or secondly, from a power perspective, for example, uneven road surfaces, curbs, steps or stairs demand relative higher power values especially for mobility impaired for overcoming such simple barriers. Even more, Spitzer (Spitzer et al., 1982) shows the variance of power values of walking, for example on slope pavement surface or stair usage, which can be used for the infrastructure design, D-tour assessment, an evaluation of active or barrier friendly infrastructure or for (public) transport access. In addition, because of the physiological approach, these effort evaluations can be applied to new modes, such as e-mobility with e-scooter, and in each country. Hence, the PHM enables already a direct verification of the energy function through physiological measurements, on a mono- and multi-modal level.

Homo economicus vs. homo mobilis

From the onset of mobility research (Dupuit, 1844; Gossen, 1854; Lill, 1891; Zipf, 1946) human travel behaviour has been connected with economic behaviour. The related economic driven hypotheses of the homo economicus are still the main underlying principle of travel behaviour research (Barbosa et al., 2018; Ben-Akiva and Lerman, 1985; Lohse and Schnabel, 2011; McFadden, 1974; Ortúzar et al., 2011; Wilson, 2010, 1967; Yan et al., 2013). According to the fit quality or to the parameter values, either power or the exponential law or combinations are chosen (Barbosa et al., 2018; Barthélemy, 2011; Gallotti et al., 2016). The problem of monotonically decreasing fit functions has already been discussed by Lill (1891) in the derivation of his travel law given as a hyperbolic distribution, where, at x = 0, the number of trips goes to infinity, which could be termed as the zero-origin problem. He simply dismissed the argument due the (methodological) consistency with the functional monotony and such “0”-trips are from a practical perspective excluded. However, an exponential function implies, that trips with zero distances have the highest frequency which is clearly unrealistic. The problem has either been ignored or explained with the help of additional variables and model extensions. A similar pragmatic argument is used due to large scaling validity fit of \(5\,t/\overline t\)- units (Barthélemy, 2011; Gallotti et al., 2016).

By contrast, the PHM with E0 has a definite meaning, also in terms of measureable dimensions, defining the physical level of the homo mobilis for all modes of mobility alike. It resolves the zero-origin problem with the usage of Eq. (4) and the function of the possible density of destinations. This consideration has not been taken into account explicitly in any other models although it is a basic fact and a fundamental prerequisite, i.e. with zero trip length there cannot be any trip. Furthermore, the PHM retains substantially the modal methodology as well as the scaling validity.

Economic implications for mobility planning

The equal treatment and inclusion of all daily trips per person, i.e. the basis for the TTB-approaches (Ahmed and Stopher, 2014), showed that average daily travel time should be independent of the average GDP per capita on a global scale (Schafer, 1998). The extent of daily travel time with a detailed justification of the disaggregation into different modes used per person per day, based on the limited physiological energy effort for all modes of mobility alike, has been shown in this paper.

From this definition of mobility our results indicate that mobility remains stable and does not increase as it is often assumed, for example, in public white papers (European Commission, 2016). Only modal split and, as such, the modes of mobility have changed toward the motorised modes with a disproportional increase in travel time and distance. Therefore, methodological adaptations can be established for current mobility performance indicators such as modal spit assessment or transport capacity. This raises questions about rational or bounded rational behaviour (Hargreaves-Heap and Hollis, 1987; Mahmassani and Chang, 1987; Sun et al., 2018; Vuong, 2018). Rational behaviour in physical terms is clearly a choice of the homo mobilis towards minimising expenses or least travel effort (Zipf, 1949) (as it can be seen in the ratio of mono-modal vs. duo or multimodal behaviour of Fig. 9). In addition, short term gains through minimising the travel power with a development towards a motorised modal choice has led to an increase in absolute travel time in the long term with additional monetary travel expenditure of the homo economicus. A reassessment of the current methodology of cost-benefit analyses would be required to answer such a question, where macro-economic time savings play a major part in infrastructure or land-use planning (Hensher, 2011; Li, 2019; Metz, 2008; Vickerman, 2017).

Future work

Multi-modal travel can be considered in the same way as mono-modal travel in the PHM, where the modal trip-energy expenditures are totalled. This satisfies the assumption of independence of averaging, so that the (macroscopic) distribution function of Eqs. (6) and (9) are methodologically the same. In fact, the trip modes in a multi-modal travel do not seem to be independent, and the sum of all related energy efforts is constrained (Frodesen et al., 1979). Furthermore, other entities of trips or activity patterns, with variables such as starting or ending point with the trip purpose, have not been presented. These problems would be addressed in the daily modal choice modelling, which is not the focus of this paper. Similarly, verification regarding number of trips would also be a further step in the model development.

Conclusions

The main purpose of this paper is to provide a physical human mobility model (PHM), based on the daily physiological effort and the modal power consumptions, a component, which has not been taken into account explicitly in any other transport and mobility models, which are based foremost on socio-economic variables. This variable enables an application to all modes of transport alike and therefore for mono- as well as for multi-modal travel behaviour and in simultaneous relation to time and distance.

With survey data of four countries on two continents and over five decades, it is possible to describe, measure and explain the extent of human travel behaviour for all modes and modal combinations. The PHM presented is shown in the most basic form, i.e. the Grundmodell, where only the relationships of physiological energy effort related to power, time and distance are utilised in a consistent model of daily travel behaviour per capita.

The methodological pre-requisite is the classification in daily travel behaviour per person. This simple statistical analysis of the raw data reveals a 70/25/5 percentage share, where respectively only 1/2/3+ modes are used throughout the day and across the survey years and countries. This definition of modal split is different to the one generally used, which is based only on trips made. The daily modal split definition allows a more consistent and general methodology, especially for modal usage. For the mobility effort, only one parameter is assumed, which can be verified with respect to the surveyed ground-truth of travel time and distance. Since daily physiological measurements do not exist, an assumed average estimate of 800 kJ as the only parameter for 1-modal daily mobility has been made to quantify all modal behaviour, single and multimodal ones. The PHM then is able to describe a consistent time–distance–energy space, which yields comparable results to available physiological measurements. In addition, the PHM can therefore provide a fully measureable and theoretically verifiable ansatz based on physical methods, where microscopic and macroscopic behaviour are consistently and complementary integrated. The discussion on the physiological variables shows that the approach developed is only a first step for a novel mobility planning methodology, where further differentiations are possible without any loss in generality.

Due to the firm basis of the human physiology and its travel behaviour, it is now possible to provide an explanation to the well-known phenomenon of the TTB, which has been observed globally (Ahmed and Stopher, 2014; Schafer, 1998). Through the definition of daily modal behaviour, it is also possible to explain the variations according to time and distance, which has been termed in the literature and policy papers as an increase in mobility, which is actually related foremost to a modal shift from non-motorised to motorised modes of transport, i.e. the induced traffic. Some implications for transport and infrastructure planning have already been discussed in the literature (e.g. Metz, 2008), however, there are even more fundamental consequences, for example, regarding bounded rational behaviour of homo economicus vs. homo mobilis, and with that a redefinition of transport economics and its methods for appraisal such as travel time savings or cost-benefit analysis. Such methodological re-developments are required in order to meet the challenges of climate change and the required transition of mobility.