Travel Time Formulation Research Articles

Travel times in quasi-dynamic traffic assignment

ABSTRACT By extending static traffic assignment with explicit capacity constraints, quasi-dynamic traffic assignment yields more realistic results while avoiding many disadvantages of a dynamic assignment. We analyse the computation of travel times in quasi-dynamic assignment models. We formulate and check requirements for the correctness of resulting travel times, addressing both the calculation of travel times for individual routes and links itself, as well as the differences between travel times of different travel choices. We demonstrate that existing approaches for travel time computation in the literature fail to satisfy all requirements and derive a new link travel time formula from the vertical queuing theory that does meet all requirements. We discuss expected changes to assignment results and methodological advantages for pathfinding and model extensions, including horizontal queuing. The new link travel time formulation is finally applied to three example scenarios from literature.

Steady-state link travel time methods: Formulation, derivation, classification, and unification

Travel times are one of the most important outputs of transport planning models, especially in a strategic context. It is therefore paramount that the methods that underpin the construction of travel times are well understood. A plethora of methods exists to extract and/or construct travel times given some underlying network loading procedure, also known as the traffic flow propagation model. However, the relation between these different travel time methods and the consistency between such methods has received relatively little attention in the literature. This might in part be due to the many different traffic flow propagation models in existence, ranging from vehicle based (microscopic), to flow based (macroscopic), and models that explicitly account for the time varying nature of traffic flows (dynamic) to models that do not (static). In this work, we limit ourselves to flow based, i.e. macroscopic, traffic flow models. Within this modelling paradigm we consider dynamic, semi-dynamic, and static traffic flow propagation formulations used to construct link travel times. The semi-dynamic and static approaches are considered as more aggregate versions of the dynamic formulation. Within this context we formulate a unified (link) travel time formulation that is consistent across these three modelling paradigms under the assumption of steady-state flow conditions. The dynamic link travel time formulation is based on a recent state-of-the-art continuous time macroscopic dynamic network loading model. In the dynamic model we assume steady-state conditions to remain consistent with steady-state semi-dynamic and static approaches. This allows us to derive semi-dynamic link travel time formulations from our dynamic model, while the static formulation is derived from its semi-dynamic counterpart. Throughout this work we explore link travel times from three different perspectives; an experienced perspective, which actively tracks the tail of a physical queue, and two functional perspectives, both of which do not require explicit queue tracking. Based on the existing literature and proposed formulations, a classification framework is proposed allowing one to compare existing (and future) methods in the literature in an objective fashion. We provide a number of explicit derivations of existing model formulations that can be considered special cases of our unified approach. A numerical example across the different perspectives is included and a significant number of representative existing methods in the literature has been classified based on our proposed framework for the reader's convenience.

Thermodynamics in the hydrologic response: Travel time formulation and application to Alpine catchments

AbstractThis paper presents a spatially explicit model for hydrothermal response simulations of Alpine catchments, accounting for advective and nonadvective energy fluxes in stream networks characterized by arbitrary degrees of geomorphological complexity. The relevance of the work stems from the increasing scientific interest concerning the impacts of the warming climate on water resources management and temperature‐controlled ecological processes. The description of the advective energy fluxes is cast in a travel time formulation of water and energy transport, resulting in a closed form solution for water temperature evolution in the soil compartment. The application to Alpine catchments hinges on the boundary conditions provided by the fully distributed and physically based snow model Alpine3D. The performance of the simulations is illustrated by comparing modeled and measured hydrographs and thermographs at the outlet of the Dischma catchment (45 km2) in the Swiss Alps. The Monte Carlo calibration shows that the model is robust and that a simultaneous fitting of streamflow and stream temperature reduces the uncertainty in the hydrological parameters estimation. The calibrated model also provides a good fit to the measurements in the validation period, suggesting that it could be employed for predictive applications, both for hydrological and ecological purposes. The temperature of the subsurface flow, as described by the proposed travel time formulation, proves warmer than the stream temperature during winter and colder during summer. Finally, the spatially explicit results of the model during snowmelt show a notable hydrothermal spatial variability in the river network, owing to the small spatial correlation of infiltration and meteorological forcings in Alpine regions.

Catchment-scale herbicides transport: Theory and application

This paper proposes and tests a model which couples the description of hydrologic flow and transport of herbicides at catchment scales. The model accounts for streamflow components’ age to characterize short and long term fluctuations of herbicide flux concentrations in stream waters, whose peaks exceeding a toxic threshold are key to exposure risk of aquatic ecosystems. The model is based on a travel time formulation of transport embedding a source zone that describes near surface herbicide dynamics. To this aim we generalize a recently proposed scheme for the analytical derivation of travel time distributions to the case of solutes that can be partially taken up by transpiration and undergo chemical degradation. The framework developed is evaluated by comparing modeled hydrographs and atrazine chemographs with those measured in the Aabach agricultural catchment (Switzerland). The model proves reliable in defining complex transport features shaped by the interplay of long term processes, related to the persistence of solute components in soils, and short term dynamics related to storm inter-arrivals. The effects of stochasticity in rainfall patterns and application dates on concentrations and loads in runoff are assessed via Monte Carlo simulations, highlighting the crucial role played by the first rainfall event occurring after herbicide application. A probabilistic framework for critical determinants of exposure risk to aquatic communities is defined. Modeling of herbicides circulation at catchment scale thus emerges as essential tools for ecological risk assessment.

Derivation of Travel Time Based on Diffusive Wave Approximation for the Time-Area Hydrograph Simulation

Derivation of travel time based on Saint-Venant equations may substantially improve rainfall-runoff routing, particularly in the Clark approach and GIS-based hydrological routing models. Kinematic wave (KW) and diffusive wave (DW) approximations are often applied in lieu of the dynamic wave (DYW) solution to study overland flow regime. In this study, for the first time, travel-time relationships are presented based on DWapproximation. In terms of equilibrium time, the DWapproximation improved results from 5% up to 15% over the KW approximation. After obtaining the semianalytical solution of the DW travel time, discharge hydrographs over a rectangular plane through the well-known time-area method were derived. Considering the DYW solution as the benchmark, it was found that the time-area approach by adopting the proposed DW wave travel time formulation, provided satisfactory results for 2:5 0:1. Using the travel time based on KW approach, the time-area method was found fairly accurate for K0F 2 > 25 and F0 > 0:1. K0 and F0 are the well-known kinematic wave number and Froude number, respectively. Therefore, it is perceived that as K0F 2 decreases, the difference between KW and DW approximations becomes greater. DOI: 10.1061/(ASCE)HE.1943-5584.0000399. © 2012 American Society of Civil Engineers. CE Database subject headings: Hydrographs; Simulation; Travel time; Equilibrium; Kinematic wave theory. Author keywords: Diffusion wave; Travel time; Equilibrium time; Time-area method; Hydrograph; Kinematic wave; Isochrones.

Significance of higher moments for complete characterization of the travel time probability density function in heterogeneous porous media using the maximum entropy principle

The travel time formulation of advective transport in heterogeneous porous media is of interest both conceptually, e.g., for incorporating retention processes, and in applications where typically the travel time peak, early, and late arrivals of contaminants are of major concern in a regulatory or remediation context. Furthermore, the travel time moments are of interest for quantifying uncertainty in advective transport of tracers released from point sources in heterogeneous aquifers. In view of this interest, the travel time distribution has been studied in the literature; however, the link to the hydraulic conductivity statistics has been typically restricted to the first two moments. Here we investigate the influence of higher travel time moments on the travel time probability density function (pdf) in heterogeneous porous media combining Monte Carlo simulations with the maximum entropy principle. The Monte Carlo experimental pdf is obtained by the adaptive Fup Monte Carlo method (AFMCM) for advective transport characterized by a multi‐Gaussian structure with exponential covariance considering two injection modes (in‐flux and resident) and lnK variance up to 8. A maximum entropy (MaxEnt) algorithm based on Fup basis functions is used for the complete characterization of the travel time pdf. All travel time moments become linear with distance. Initial nonlinearity is found mainly for the resident injection mode, which exhibits a strong nonlinearity within first 5IY for high heterogeneity. For the resident injection mode, the form of variance and all higher moments changes from the familiar concave form predicted by the first‐order theory to a convex form; for the in‐flux mode, linearity is preserved even for high heterogeneity. The number of moments sufficient for a complete characterization of the travel time pdf mainly depends on the heterogeneity level. Mean and variance completely describe travel time pdf for low and mild heterogeneity, skewness is dominant for lnK variance around 4, while kurtosis and fifth moment are required for lnK variance higher than 4. Including skewness seems sufficient for describing the peak and late arrivals. Linearity of travel time moments enables the prediction of asymptotic behavior of the travel time pdf which in the limit converges to a symmetric distribution and Fickian transport. However, higher‐order travel time moments may be important for most practical purposes and in particular for advective transport in highly heterogeneous porous media for a long distance from the source.

A morpho-climatic instantaneous unit hydrograph model for urban catchments based on the kinematic wave approximation

Summary Stormwater models are valuable tools in urban planning as well as stormwater system design and management. Although the hydraulic simulation of pipes and channels in these models is often quite sophisticated, the hydrologic simulation of the flow contributing to the hydraulic elements is frequently greatly simplified. Hydrologic simulation of urban catchments is made particularly complex due to the presence of features such as streets, small pipes, and channels. In this study, we develop a new model called the U-McIUH (Urban Morpho-climatic Instantaneous Unit Hydrograph), which defines the IUH as the probability density function of the travel time from a random location in the urban terrain to the outlet. Flow paths are extracted from a specially processed digital elevation model that incorporates hillslopes, streets, pipes, and channels, and travel times are computed in each cell using the average wave celerity from kinematic wave theory. These expressions depend on the upstream contribution of flow and the excess rainfall intensity, so they incorporate the so-called climatic dependence of the IUH, which is the dependence of the unit hydrograph on the rainfall intensity. Rainfall pulses of different intensities are convoluted with their respective IUH and superimposed to generate the response to a given storm event. The application of the model to a real catchment provides good reproduction of observed hydrographs, suggesting that the U-McIUH is able to capture some significant hydrologic properties of the catchment. The model is studied by analyzing the effects of (1) the upstream contribution of flow on the travel time formulation, (2) the variation of flow velocities within the urban catchment, (3) the non-linear dependence of the IUH on the rainfall intensity (i.e. the climatic dependence), and (4) the spatial distribution of imperviousness. Overall, these results suggest that the presence of artificial elements in urban watersheds has a significant role in determining the travel times and thus the hydrologic response of the watershed.

A generalized two point ellipsoidal anisotropic ray tracing for converted waves

Rocks can be anisotropic due to a variety of reasons. When estimating rock velocities from seismic data, failure to introduce anisotropy into earth models could generate distortions in the final images that can have enormous economic impact. To estimate anisotropic earth velocities by tomographic methods, it is necessary to trace rays or to solve the wave equation in models where anisotropy has been properly considered. Thus, in this work we present a 3-D generalized ellipsoidal travel time formulation that allow us to trace rays in an anisotropic medium. We propose to trace rays in anisotropic media by solving a set of nonlinear optimization problems, where the group velocities for P and S wave propagation modes are 3-D ellipsoidal approximations that have been recently obtained. Moreover, we prove that this 3-D ellipsoidal anisotropic ray tracing formulation is a convex nonlinear optimization problem, and therefore any solution of the problem is a global minimum. Each optimization problem is solved by the global spectral gradient method, which requires first order information and has low computation and low storage requirements. Our approach for tracing rays in anisotropic media is a generalization in the sense that handles titled axis of symmetry and, close to the axis of symmetry, it is an accurate formulation for 2-D transversely isotropic media and 3-D orthorhombic media, depending on the input parameters. Moreover, this formulation gives the exact ray trajectories in 2-D and 3-D homogeneous isotropic media. The simplicity of the formulation and the low computational cost of the optimization method allow us to present a variety of numerical results that illustrate the behavior and computational advantages of the approach, and the difficulties when working in anisotropic media.

1-D layered media: Part 2, Layer‐based waveform inversion

We are interested here in the inversion of multioffset data to determine 1-D medium properties. The forward modeling technique accounts for 3-D wave propagation effects. The inversion is performed by minimizing the difference between modeled and reference data using a least‐squares technique. The gradient is computed by the adjoint state technique adapted to the first‐order hyperbolic system and to the Hankel transform. To reduce the number of optimization parameters and to optimize along reflected events from some selected interfaces, we propose a layer‐oriented reparameterization in the full‐waveform inversion. Then we discuss the effect on the inversion results of an incorrect number of layers. The local minima of the least‐squares functional are experimentally investigated and physically interpreted. This sensitivity study has shown that it is necessary to have an accurate initial model for the position of the layer interfaces. In principle, seismic data give access directly to this information. This is why we have formulated the inverse problem in one‐way vertical traveltime as opposed to depth. Using an example, we show that with this parameterization the variation of the least‐ squares misfit function when we change the interval velocity for a given layer tends to a quasi quadratic form. In practice, a layer‐based waveform inversion (LBWI) procedure is proposed to overcome the inherent problem of local minima. The procedure attempts to determine interval velocities, impedance contrasts, and interface positions for a limited number of layers to explain the major seismic events. The LBWI procedure is based on four key elements: traveltime formulation, layer‐based reparameterization, progressive enrichment of the model, and interpretative approach. This procedure is evaluated for a large‐scale acoustic synthetic example.

A stochastic‐convective transport representation of dispersion in one‐dimensional porous media systems

A stochastic‐convective transport formulation based upon solute travel time probability is presented and is shown to include Fickian transport as a special subcase. It is further demonstrated that a travel time probability associated with a lognormal distribution of hydraulic conductivity will yield concentration breakthrough curves nearly equivalent to those of Fickian transport when the coefficient of variation in travel time is sufficiently small, i.e., less than unity. When applied to column tracer experiments, the results suggest that typical laboratory‐measured hydrodynamic dispersion may be ascribed to local variations in hydraulic conductivity. Moreover, stochastic‐convective transport is shown to conserve solute mass under a flux boundary condition but to fail to do so under a held concentration condition. This result indicates the importance of boundary conditions for properly formulated stochastic transport. The travel time formulation is shown to provide a direct link between measured dispersivity and the autocovariance of local flow velocity variations that are a consequence of media inhomogeneity and the system boundary conditions. Dispersivity is shown to manifest a scale effect by increasing in proportion to the length of a system when the velocity correlation range is greater than that length. An expression for the effective dispersivity is derived for the case of long‐range velocity correlations and is shown to represent non‐Fickian behavior.