This rate varies between different parts of the genome, and its inference is sensitive to the timescale and diversity spanned by the sequence data used. Higher rates are estimated on short timescales and particularly for within-host evolution, while rate estimates spanning decades or the entire HIV-1 pandemic tend to be lower. The underlying causes of this difference are not well understood. We investigate here the role of rapid reversions toward a preferred evolutionary sequence state on multiple timescales. We show that within-host reversion mutations are under positive selection and contribute substantially to sequence turnover, especially at conserved sites. We then use the rates of reversions and non-reversions estimated from longitudinal within-host data to parameterize a phylogenetic sequence evolution model. Sequence simulation of this model on HIV-1 phylogenies reproduces diversity and apparent evolutionary rates of HIV-1 in gag and pol, suggesting that a tendency to rapidly revert to a consensus-like state can explain much of the time dependence of evolutionary rate estimates in HIV-1.Human immunodeficiency virus 1 (HIV-1) is a rapidly evolving virus able to evade host immunity through rapid adaptation during chronic infection. The HIV-1 group M has diversified since its zoonosis into several subtypes at a rate of the order of 10 The human immunodeficiency virus 1 (HIV-1) is a prime example of such a rapidly evolving virus. The life-long infections it causes are characterized by a large viral population that accumulates diversity at a high rate to constantly evade host immunity . This coQuantifying the rate of viral evolution, however, is surprisingly difficult and different approaches yield different answers. Most importantly, the timescale across which sequences are compared strongly affects the estimates, sometimes by orders of magnitude: the longer the timescale, the lower the estimate . These dHIV-1 is an ideal system to study these effects in detail as the rate discrepancies among the WH, pandemic, and broader scales are well documented , the panWe use WH longitudinal deep-sequencing data to explore how the rapid evolutionary processes within hosts can give rise to apparently lower rates on longer timescales. These results suggest that the ‘adapt and revert’ mechanism can explain most of the rate mismatch observed at different timescales of the HIV-1 pandemic. We, firstly, show that HIV-1 sequence evolution shows strong signs of site saturation while distance relative to the root of the tree increases much more slowly than expected based on the rate of evolution. Similar signatures are observed in longitudinal WH data, suggesting that this saturation is independent of whether evolution is quantified along transmission chains or within hosts. Secondly, we investigate the cause of this saturation and find that WH reversion toward the HIV-1 consensus is more common than expected and that such reversions are positively selected. Lastly, we use simulations of evolution to quantify the impact of rapid reversions on rate estimates for timescales of decades or more. These simulations show that the degree of reversion observed within hosts can explain the phylogenetic patterns observed in the pandemic. More generally, our results highlight the evolutionary bias of viruses toward a state of high intrinsic fitness in a changing environment.2.We use (1) a set of sequences representative of the HIV-1 pandemic spanning multiple decades and (2) a longitudinal data set following the evolution of the virus within individual hosts to investigate patterns of evolution on multiple timescales. The former between-host (BH) data set contains 1,000 HIV-1 group M sequences from the Los Alamos National Laboratory (LANL) HIV database . This suOur WH analysis is based on the HIVEVO data set , a wholeenv, pol, and gag genes of HIV-1 pol region in the main text and present analogous results for the env and gag regions in the Supplementary Materials.We analyze the evolution of the 2.1The ‘adapt-and-revert’ mechanism to explain the rate mismatch within and between hosts assumes that reversions during WH evolution ‘shadow’ previous changes, resulting in very low rate estimates. The ‘store-and-retrieve’ mechanism postulates that many WH changes are not transmitted and thus irrelevant for the evolution on longer timescales . To lookThe rate at which divergence between sequences increases decreases with distance as more and more sites are hit multiple times by mutations . For verSequences in the HIV-1 pandemic differ from each other at about 10–20 per cent of sites and we would naively expect that saturation effects are small unless rate variation is very strong or reversion is a substantial contribution to evolution. d(t) relative to a reference sequence, such as the root of the tree, according to: N is the length of the region and i and time t in the viral population. This founder nucleotide at each position i is approximated as the majority nucleotide at the first time point t0, and its frequency at each time point t is used to compute d(t). Details about the computation of the founder sequence can be found in i is the same as in the reference sequence and We then performed a similar analysis on WH data on a 5-year timescale to determine whether similar rates and saturation effects exist within hosts. We compute WH evolutionary rates by measuring the divergence over time in d(t) measures the Hamming distance to the reference sequence expected for a randomly chosen sequence from the viral population of a sample. We then averaged the divergence trajectories of different patients and estimated uncertainty by bootstrapping groups of samples from the same patient with replacement. In analogy to the BH analysis, we use the root of the HIV-1 group M tree and subtype consensuses as reference sequences, supplemented by the founder sequence of each patient. Results are shown in In all cases, the quantity e higher . Hammingenv and gag regions shown in A ‘store-and-retrieve’ mechanism to explain the discrepancy between rate estimates should not only result in differences between BH and WH rate estimates (the rates at which the RTT distance and the distance to the founder sequence increase), but also for the rates at which Hamming distances to HIV-1 root or subtype consensuses increase. Since divergence to reference sequences decades in the past is increasing at compatible rates within and between hosts, these analyses suggest similar modes of divergence accumulation and do not support ‘store-and-retrieve’ as a primary mechanism to explain the discrepancy in rate estimates. In contrast, rate variation or rapid reversion is not expected to affect Hamming distance dynamics to fixed reference sequences like the HIV-1 (M) root. RTT distance estimates, however, are expected to be biased downward since rapid back-and-forth mutations are unaccounted for by the substitution models and do not contribute to the RTT distance. The observations in 2.2pol and gag, while this figure is 76 per cent for env. Mutations to a third state are thus a minor contribution. The sevenfold increased rate at 6 per cent of the sites that are initially non-consensus (in pol) implies that about one in three mutations bring the sequence closer to the HIV-1 root sequence . This strong tendency to revert can explain the difference in evolutionary rates observed on WH and BH scales and is consistent with the threefold difference in slope between the divergence relative to the founder or HIV-1 group M root shown in Next, we explored the evolution toward and away from consensus within hosts in pol.This accelerated evolution could be due to (1) reversion to an ancestral state to increase fitness or (2) reduced purifying selection at sites with high levels of diversity in global HIV-1 population. In order to differentiate between these possibilities, pol, env, and gag. Details on the computation of these rates can be found in 2.3t = 0 corresponds to their first data point seen in the frequency interval and compute the mean frequency of the trajectory group over time. The small minority of trajectories where both the initial nucleotide and the target nucleotide differ from the consensus sequences are classified as ‘non-reversions’ in this analysis. More details about the definition of trajectories and the methodology are in If a lot of reversions are driven by selection, as the codon-position-specific analysis above suggests, effects of selection should be detectable in the dynamics of intra-host single nucleotide variants (iSNVs). Specifically, we expect to see a tendency of reversion mutations to increase in frequency and fix. We analyzed the frequency trajectories of iSNVs to look for such features. Similar to the previous analysis, we separate all trajectories into reversion and non-reversion groups and compare their evolutionary dynamics in t = 0 along with their average. The mean frequencies for different initial conditions and groups of trajectories are shown in t = 0, we expect that trajectories tend to rise for t < 0, as is indeed observed. The dynamics at t > 0, i.e. after the time of conditioning, are informative about the selection of the iSNV. We do not expect any consistent trend to rise or fall in frequency for neutral mutations, hence their average frequency should be constant for t > 0. Contrary to that, we show in t > 0, suggesting that such mutations tend to be slightly selected against or are neutral—at least those that reach high frequency in the first place.gag region, see t > 0, while on average non-synonymous mutations increase, see We note that the selection for reversion mutations is strongest for the 2.4Over longer timescales, the rapid reversions we observe within hosts will lead to undetected substitutions along branches of the phylogeny whenever a mutation and its corresponding reversion happen on the same branch. When such reversion dynamics are not captured by the substitution models, the evolutionary rate inferred by phylogenetic methods will be too low . Here wej to i at position α as: We quantify the impact of reversions on the BH evolution rate using an evolutionary model that accounts for the reversion bias we observed within hosts. We use the TreeTime library  to definµ is the mean mutation rate per site per year, i at site α, and Wij accounts for the overall variation in rate between different nucleotide pairs i and j independent of position (i.e. the differences between transitions and transversions). We use α, enabling us to skew the frequencies toward the consensus nucleotide at each position. Contrary to common evolutionary models that include rate variation between sites, we keep the evolutionary rate constant across positions and vary α that is a first codon position where We then used this model to simulate evolution along an HIV-1 phylogeny and generate a multiple sequence alignment (MSA) using TreeTime and the inferred HIV-1 root sequence . For each region, Subtype B represents approximately 40 per cent of all sequences, Subtype C approximately 15 per cent, and the rest encompass the other subtypes or unlabeled subtypes. Subtype B sequences are more common in early years while Subtype C sequences represent a larger proportion in recent years. We then performed an MSA, including the reference HIV-1 HXB2 sequence, using Multiple Alignment using Fast Fourier Transform  and the 4.1.2p7 and p10 from our analysis as their samples were very uneven in time or because there was evidence of multiple founder sequences.Our WH analysis leverages the time resolution of the HIVEVO data set . This daenv gene, which is the region with the most noise.The estimates of nucleotide frequencies are unbiased in the fficient . We appl4.2The first result section gives an overview of the method used to compute the distance and divergence over time in Hamming distances were computed by counting the number of sites that do not match the reference sequence for each sequence in the data set. We then divide this number by the length of the sequence to obtain the relative distance to the reference. Hamming distances were computed using three reference sequences. The first is the root sequence of the tree. The tree was inferred using the IQ-TREE GTR+F+R10 model , while tThe RTT distances shown in 4.3t = 0, which could bias our evolution rate estimates. Consequently, we decided to only use data starting from 200 days into the infection for the fit, which is more than enough to get an accurate estimate of the slope. For the WH rate estimates, we estimate the error by bootstrapping patients.The evolutionary rates in Estimating confidence intervals for evolutionary rates at the level of the pandemic is challenging because of the phylogenetic relationship and shared ancestry of the sequences. Instead of using probabilistic phylogenetic models, which suffer from residual recombination and model inadequacies, we opted for phylogenetic boot-strapping procedure for the BH rate estimates. Specifically, we cut all branches of the time-scaled phylogenetic trees at the year 1980 and thereby obtain a collection of subtrees. Sequences in the same subtree are correlated, but they are not correlated with sequences on another subtree . We performed bootstrapping to estimate distances and rates by sampling with replacement from sequences in these subclades. The errors provided for the rate estimates in 4.4Consensus sequences were computed from our BH data sets. We computed three consensus sequences for each region studied. The first is the HIV-1 group M global consensus, which is the majority nucleotide of the alignment at each position. The second and third are the Subtype B and C consensus sequences. These were computed in the same way, using a subset of the alignment that contains only the sequences of the subtype in question.The founder sequence is an approximation of the sequence of the virus at the time of infection in a patient. They are computed from our WH data set for each patient separately. The founder sequence is the majority nucleotide in each position from the first sample of each patient. In this sense, it is the consensus sequence obtained from the first sample of each patient. For most patients in our data set, the first sample is taken at approximately 90 days after infection and no data are available on the early phase of infection. Consequently, the founder sequence computed is an approximation of the original virus.4.5A trajectory is a sequence of nucleotide frequencies and associated time. Each trajectory corresponds to one genome position and one nucleotide only. We extracted trajectories from our WH data set according to several criteria. Firstly, every trajectory must be extinct before the first point, i.e. we consider only new mutations. This is to avoid biases that could be due to immune interaction existing already. Secondly, frequencies must be between 0.01 and 0.99 at all time points. The trajectory is considered extinct if it is below 0.01 and fixed if above 0.99. Lastly, we apply a mask to data points according to what is shown in Every trajectory extracted according to the criteria above is coupled with its metadata. This contains all the relevant information, such as whether the mutation is a reversion or not and whether it fixed or was lost. This information is used to create subgroups of trajectories. From these subgroups, one can study the impact of a trait associated with a mutation for WH evolution, as shown in 4.6While looking at divergence values informs us about the global evolution of the WH population, it cannot tell us whether the mutations we see on non-consensus sites are actually reversions to the consensus state or simply mutations to another nucleotide. This motivated us to look directly at the evolution of new mutations independently by observing their frequency trajectories in time. Trajectories were extracted and filtered according to Due to the limited number of trajectories available and the often lack of information about trajectory fixation, for example, because it is still active at the last sample, the probability of fixation plots were not adequate for our analysis. We, thus, decided to pay attention to the evolution of the mean frequency in time for groups of trajectories. Trajectories were grouped in frequency bins, as described in the main text, to avoid bias toward positively or negatively selected trajectories. f = 1 to time bins subsequent to their fixation. Similarly, lost trajectories contribute f = 0 to time bins subsequent to their disappearance in the viral population. Trajectories that are still active after their last data point (because the study stopped before it could fix or be lost) contribute the frequency of their last data point to the following time bins.We then created time bins of 400 days from 600 days before up to 3,000 days after a trajectory is seen in a frequency window. We compute the average frequency of all trajectories belonging to the same group in each time bin. A trajectory contributes its current frequency if a data point is available at this time and does not contribute if no data are available in that time bin. Trajectories that fixed in the population contribute with a frequency of veac118_SuppClick here for additional data file.


In this paper, the dynamic response of a damaged double‐beam system traversed by a moving load is studied, including passive control using multiple tuned mass dampers. The double‐beam system is composed of two homogeneous isotropic Euler–Bernoulli beams connected by a viscoelastic layer. The damaged upper beam is simulated using a double‐sided open crack replaced by an equivalent rotational spring between two beam segments, and the lower primary beam is subjected to a moving load. The load is represented by a moving Dirac delta function and by a quarter car model, respectively. Road surface roughness (RSR) is classified as per ISO 8606:1995(E). The effect of vehicle speed of the moving oscillator and variable RSR profiles on the dynamics of this damaged double Euler–Bernoulli beam system for different crack‐depth ratios (CDRs) at various crack locations is studied. It is observed that coupling of two beams leads to a vehicular effect on the damaged beam, even when no vehicle on it is present. The effects of single and multiple tuned mass dampers to control the vibrational responses of the primary beam due to damage on the secondary beam is studied next. The performance of tuned mass dampers to reduce the transverse vibrations of the damaged double‐beam system and of the quarter car is investigated. The paper links the coupling between the two levels of double beam with the inertial coupling of the vehicle to the double‐beam system. For these types of bridges, the connected beams can accommodate traversing moving loads and their bending stiffness are connected by the continuous spring–dashpot system, which has a vertical stiffness. In reality, the vertical stiffness comes with some horizontal stiffness, which resists the vertical–horizontal shear pair effects . Consequently, there is an interpretation of the canonical double‐beam model, whose interpretation goes beyond typical viscoelastic railway track–subgrade interaction. For example, Takemiya and BianTo better understand how such dynamics can be relevant for structural health monitoring and control for double‐decker bridges, it is important to model damage in such double‐beam systems, investigate effects of moving load for the coupled model in the presence of damage and also assess the impact of control considering both moving load and damage. Despite this clear need, such an investigation has not been considered. This paper attempts to bridge this gap.Existing literature indicates that there are works that have considered certain aspects of this problem, but they are in a disjointed manner and without the impact of moving loads. TakahashiIn terms of structural control, unsurprisingly, passive damping has been considered for double‐beam systems. OniszczukFor bridge‐like systems, the work by Abu‐HilalEfforts in modelling damage in double‐beam system has been observed typically through Nguyen,It is observed that the literature has not considered the possibility that the moving load can have on a damaged structure for a double‐beam model as a health monitoring tool. Moving loads are typically modelled as Dirac delta functions, and the sampling property of a Dirac delta function ensures that spatial changes like damage will not only be sampled in time history from such moving loads but also be filtered to both beams due to coupling effect. Under such circumstances, the damage information from one beam to another will be transferred, through a bridge–vehicle interaction, even though no vehicle is physically present on it. On the other hand, the undamaged beam can be loaded with a vehicle to probe the damage in the other beam through coupling, without physically requiring a vehicle on it. This ‘shadow’ effect or ‘phantom’ effect of moving vehicles is particularly interesting and useful in the context of monitoring, and there exists no work investigating and modelling it. This paper, as a first, models such damaged coupled beam interacting with moving vehicles, including inertial coupling and surface roughness, and investigates damage effects and detection possibilities. An elastically connected two identical Euler–Bernoulli beams has been used to represent a bridge with two decks (a bilevel bridge). The viscoelastic layer coupling the two beams is replaced by a spring–damper system whose coupling parameters like stiffness, damping and material properties can be changed. The upper beam is a cracked structural member with a double‐sided open crack, and its damage is simulated by an equivalent rotational spring Narkis63 whoseExisting work on passive control of such systems is then extended by considering single and multiple tuned mass dampers (TMDs) to assess their performance for varied vehicular speed and damages. A tuning criterion has been shown here to perform well for different damage locations and extent. This investigation not only considers the control of both beams but also provides comparative control performance of the displacement, velocity and acceleration responses as a function of the level of coupling between the beams via damage parameter, along with the vehicular coupling aspects. Such a set of result is not investigated to date, and this paper provides a first and practical set of estimates, including the impact of varying mass ratios of the dampers.This systematic numerical study provides the first model combining damage, vehicular effects and control for a wide range of practical scenarios, including road surface roughness (RSR), damage location and extent, TMD mass ratio and the effect of multiple TMDs. It then provides ideas of how to use such a model for bridge–vehicle interaction‐based structural health monitoring and provides performance estimates of TMDs under a wide variation of parameters. The first section models a damaged double‐beam system where the primary beam is subjected to a moving load. The second section follows the exact procedure incorporating a quarter car toy model. Surface roughness of the primary beam has been incorporated in this case, and the extent of the damage is studied here for various crack locations and different crack‐depth ratios (CDRs). A range of velocity parameter up to the critical velocity of the vehicle model is simulated. In the last section, a numerical study has been performed where single versus multiple TMDs with different mass ratios has been implemented as a passive control and a tuning criterion has been shown to reduce the overall vibrational responses of the double‐beam system for a varied velocity range and damage extent.22.1A, geometric moment of inertia I, Young's modulus E and mass density ρ. The lower beam is designated as the primary beam and is subjected to a moving load represented by a moving Dirac delta function. The upper beam is considered to be a damaged structural member thereby weakening or ‘softening’ the entire double‐beam structure. It is assumed that the effect of the crack damage is local to its immediate neighbourhood and can be modelled as a rotational spring.The double‐beam structure is modelled as two parallel Euler–Bernoulli beam placed on top of one another and coupled by a viscoelastic layer. Both the beams in this distributed spring–damper system has identical material properties , is divided into two subbeams and is separated at the crack location x=xa. The transverse vibration of the subbeam to the left of the crack location is designated as wl, and the subbeam to the right is wr. The crack location is at a distance of a from the origin (left pinned end of a simply supported Euler–Bernoulli beam) as shown in Figure w given by Equation  can be controlled and changed accordingly. The double Euler–Bernoulli beam model are represented by Equation (The lower beam is subjected to a moving load Equation . (2)EThe dynamics of the transverse vibrations of the beams are functions of space and time which need appropriate methods for solving. As per standard literature,Xln(x),  Xrn(x) and Xn(x) in Equation , that is, The mode shapes Equation  is the hx=0 and at x=L have been chosen. The damage in the structural member is due to the presence of a double‐sided open crack. It is assumed that the vertical deflection across the open crack is continuous. The influence of the structural damage in the secondary beam is local and affects only its immediate neighbourhood. Thus, at the crack location for the upper beam, the displacement, moment and shear forces at the crack junction are continuous. However, the slope of the displacement at the crack location is discontinuous. The crack is modelled by an equivalent rotational spring with stiffness being dependent on beam parameters, crack geometry and nondimensional crack flexibility, θ  and is defined ash is the height of the beam, L is the span of the beam and δ is the CDR.The nondimensional crack flexibility Ai,  Bi and Ci, we have δ and α are functions of the beam length L, natural frequency λn and crack location a as Applying the boundary conditions Equation  on EquatThe natural frequencies for the mode shapes in Equation  are solvTun(t) and Tln(t)) are acquired, by substituting Equation (Xn(x)) using Equation m(x)Xn(x)dx=(L/2)δnm, as Once the mode shapes have been obtained from the boundary conditions, the dynamics of the modal amplitude   9) has b3.1a=0.05L,0.25L and 0.50L for varying CDR δ ranging from 0.05 to 0.35.Natural frequencies are numerically obtained by solving for the determinant of the coefficient matrix  Equation  for the 3.3The transverse vibration of both the beams , the absolute displacement for the upper and lower beam depend on the damping coefficient c. Smaller values of stiffness make the double‐beam system weakly coupled to each other. This leads to a weak elastic coupling between the primary and secondary beams. By increasing the damping coefficient, the secondary beam can be made to vibrate more. This is due to the fact that under weak elastic coupling, there is more transfer of mechanical energy from the primary beam to the secondary beam for higher values of damping coefficient. Thus, in case of weak elastic coupling, the damping coefficient provides coupling between the beams.For smaller values of stiffness Figure Figure 3.4DAF is a dimensionless quantity defined as the ratio of dynamic displacement of the beam to the static displacement of the beam when it is subjected to a static load at the midspan of the beam, that is, DAF describes how much factor should be multiplied to the transverse vibration caused by static load to obtain the transverse vibration due to dynamic forces in the system. Static deflection for an Euler–Bernoulli beam for a pinned–pinned support at both ends for a static load at midspan location is by Fstatic Equation v0, of the moving load ranging from (0–180) km/h . Figure DAF is computed by normalising the transverse vibrations of the beams  about its own weight. The vehicle moves with a constant velocity with no acceleration and its location on the beam is described by the moving Dirac delta function. The upper beam designated as the secondary beam has an open crack which has been modelled similarly as a rotational spring. RSR is classified as per ISO 8606:1995(E).4.2Fv and Fdi correspond to the external excitation force due to the vehicle and the ith TMD. The location of the vehicle on the primary beam and the TMDs are given by xv=v0t and xdi=xi (placed equidistant below the primary beam). The dynamical equations of the vehicle and the ith TMD are mv,  kv,  cv and md,  kd,  cd are the mass, stiffness and damping of the vehicle and the TMD. The RSR is given by r(xv=v0t). The external excitation forces Fv and Fdi are The dynamical equations for the double‐beam system excited by a quarter car and controlled by a TMD or vibration absorber is given by Equations (14)–. (14a)Xn) and integrating from 0 to L. Using the orthogonality conditions (Equation Tun(t) and Tln(t)) of the solution for the double‐beam system. Since only the first mode shape is significant in contributing to the transverse vibrations of the beam  in Equations (ith TMD are The solutions for the double‐beam system (Equation 14) is given by Equation . ApplyinEquation  and the Equation . Expandiquations  and 16)Xn) anThe above system  cycles/m. α is an exponent for the PSD and for a vehicle with constant velocity, α=2. Sd(f0) is the roughness coefficient in m3/cycle which describes the quality of the road. Five qualities of RSR profiles, Appendix A in the ϕn uniformly distributed between 0 and 2π. The RSR profile in its discrete form is given byN is the number of discrete points on the surface profile, Lc is twice the length of the bridge and ϕn are random phase angles uniformly distributed between 0 and 2π.RSR used to generate a road surface profile for the primary beam has been classified as per ISO 8606:1995(E).4.4TMD or vibration dampers are used to reduce the vibrational responses of a system comprising a vibrating oscillator driven by an external force. For effective control of the vibrations of the primary system, due to excitations by the force, the parameters of the TMD have to be cleverly chosen or tuned. Parameters like natural frequency of both the primary system and the vibration absorber play an important role. Incorporation of one or many TMDs to control vibrational responses of systems exist in literature ,sin(λ1L2)≈1, Equation (23) takes the form of the single degree of freedom system considered by Ghosh and Basux(t) to the forcing term f(t) in the frequency domain  and F(ω)) are H1(ω),  H2(ω) are Since Hx(ω) is not dependent on damping of TMD), the peaks of Hx(ω) are equal, thus minimising the responses of the primary structure. The values of ω1 and ωd that satisfy this condition were solved analytically by Ghosh, and the optimum tuning frequency νopt=ωdω1 has been implemented here where Ghosh and Basuνopt in terms of partial fractions, followed by a binomial expansion, Pakrashi et alνopt as Expressing νopt is linear in μ0 and quadratic in ξ1 neglecting higher order terms O(μ0ξ12).Further performing a binomial expansion reveals that Hx(ω). The chosen value of damping  agrees with expressions for tuning the optimal damping ratio (ξd) of a TMD available in literature. A simple expression has been shown to work effectively by LuftWhile choosing the damping parameter of the TMD, several values of damping has been simulated here to minimise the peaks of 55.1Simulations for the transverse vibrations of the beams and vehicle motion were performed using the beam and vehicle parameters mentioned in Appendix A in the sin(λnx)). Figure The mode shape difference function has been evaluated by calculating the difference between the mode shapes for the double‐beam system  at the crack location in the secondary beam. The change in mode shape difference increases as the crack location moves towards the centre of the beam. For different CDRs, there is an increase in the mode shape difference function of the damaged secondary beam at the crack location. For the primary beam, there is an increase in the difference of the mode shapes as the crack location moves towards the centre of the secondary beam. The difference also increases as the CDR increases in the secondary beam. The damage is located in the secondary beam; however, due to both the beams being coupled to one another by means of the viscoelastic layer, the effects of the damage are reflected in the mode shape function of the primary undamaged beam.Figure Xn(x)Tn(t) for the primary beam and the secondary beam for varying CDRs and at different crack locations on the secondary beam. The velocity of the quarter car is chosen to be a moderate value of 50 km/h. The RSR profile used corresponds to an average quality of road surface. It is observed that for damage locations near the midspan of the secondary beam, the mean and RMS values increase along with increasing CDRs. The effect of the crack location on the vibration response is significant for the primary beam in comparison to the secondary beam although the damage is on the secondary beam. This is seen in the gradual increase on the mean and RMS values of both the beams as the crack location nears the centre of the beam. Hence, from now onwards, all the figures shown correspond to a damaged secondary beam with crack location at the midspan of the upper beam.Figure Figure v0=50 km/h. The quality of the surface roughness is controlled by roughness coefficient Sd(f0). Five different values of roughness coefficient has been chosen as mentioned in Appendix A in the Figure Figure 5.2μ to 0.03 or 3% for every TMD while several values of the damping ratio of the TMD are investigated. For the optimal tuning frequency, Equation  km/h, the control of RMS of displacement for the primary beam, secondary beam and the vehicle is not significant using either single or multiple TMDs. For velocity range of around (30–45) km/h, the TMDs detune in this region and adversely effect the displacement response by around 2%. The TMDs start performing better for higher range of velocity around (60–180) km/h where the displacement response control is around 4% using multiple TMDs and around 1% for a single TMD. The control of velocity responses of the system using TMDs is more desirable. For lower ranges of velocities around (1–30) km/h, there is better control of the velocity response in comparison to the displacement response using TMDs. Using multiple TMDs, the velocity response of the primary beam and the vehicle reduces by 6% while a single TMD located at the midspan reduces the response by 2% for higher velocity range. The performance of the TMDs is most effective in controlling the acceleration responses of the system and the vehicle. For lower velocities ranges around (1–50) km/h, the acceleration response reduces by 15% using multiple TMDs placed at regular intervals in comparison to around 10% when using a single TMD. For higher range of velocities (50–100) km/h, the control of acceleration response is around 10% using multiple TMDs. The figure also shows the performance of multiple TMDs versus a single TMD.Figure % and 3%. Figure %. Figure Figure %.Figure 6The concept of a ‘phantom’ vehicle on damage detection and passive control is considered. The effect of damage at the crack location on the entire double beam indicates that as the extent of damage of the secondary beam increases, the stiffness of the entire system increases thereby resulting in the decrease in the natural circular frequency of the double‐beam set‐up. This results in an increase in the peak vibration responses of both the beams.The effect of coupling and stiffness reveals that for lower values of stiffness between the beams, that is, when the beams are under weak elastic coupling, there is lesser transfer of mechanical energy from the primary beam to the secondary beam. During such a set‐up, the damping coefficient of the viscoelastic layer between both the beams provides extra coupling between the double‐beam system. For higher values of stiffness, that is, when the primary beam is strongly coupled to the secondary damaged beam, there is a transfer of mechanical energy from the primary beam to the secondary; hence, as a result, the vibrations of both the beams increase. This leads to the idea of passage of a ghost or ‘phantom’ vehicle on the secondary beam even though no vehicle traverses it. The effect of damping coefficient on the dynamics of the double beam is negligible when the system is strongly coupled to one another. The effect of damage on the secondary beam  leads to an increase in the vibration responses of the double‐beam system. Although the primary beam is healthy, the transverse vibration increases due to deterioration of health of the secondary beam.The dynamics of the system is studied for various velocity ranges of the vehicle. A surface roughness profile on the primary beam is incorporated. The extent of damage on the vibration responses for different damage conditions is studied. It is found that as the CDR in the secondary beam increases, the peak vibration of displacement response of both the beams increases. By choosing different surface profiles, the effect is studied in a similar fashion. Poor quality of surface roughness leads to an increase in the overall vibration response of the double‐beam set‐up.A comparative study between single and multiple TMDs has been carried out to control the vibrational responses of the damaged double‐beam set‐up. Control of RMS values of vibration responses of the system is studied for a wide range of vehicle velocity up to its critical speed. The tuning parameters for the TMDs to control the vibrational responses of the system have been chosen with respect to the parameters of the undamaged double‐beam set‐up. However, the same tuning criteria for single and multiple TMDs reduce the vibrational responses of the double‐beam set‐up for various damage conditions as well. For lower range of vehicle velocity, the TMDs control the velocity and acceleration of the vibrational responses of both the beams and the quarter car. For higher range of vehicle velocity, the TMDs attenuate the displacement responses of both the beams for every damage condition. It is found that multiple TMDs perform better to reduce vibrations in comparison to an implementation of a single TMD. However, due to physical restrictions of lower mass ratios of TMDs in most bridges, it is preferable to use a single TMD with a slightly higher mass ratio located at the midspan of the primary system rather than using a multiple TMD set‐up with lower mass ratio. Finally, it is found that the acceleration of the quarter car is also effectively attenuated after the implementation of single or multiple TMDs for a wide range of vehicle velocity. Thus, passenger comfort is improved using TMDs on the damaged double‐beam model.A systematic vibration analysis of a damaged double Euler–Bernoulli beam traversed by a moving quarter car is performed followed by a control analysis of the system using a TMD. The following conclusion are derived.All authors contributed equally to the work.STC_2933‐sup‐0001‐Appendix.pdfClick here for additional data file.


Electrochemical discharge machining (ECDM) is a promising non−traditional processing technology used to machine non−conductive materials, such as glass and ceramic, based on the evoked electrochemical discharge phenomena around the tool electrode. The discharge in ECDM is a key factor that affects the removal of material. Moreover, the discharge current is an important indicator reflecting the discharge state. However, the discharge characteristics remain an open topic for debate and require further investigation. There is still confusion regarding the distinction of the discharge current from the electrochemical reaction current in ECDM. In this study, high−speed imaging technology was applied to the investigation of the discharge characteristics. By comparing the captured discharge images with the corresponding discharge current, the discharge can be classified into three types. The observations of the discharge effect on the gas film indicate that a force was exerted on the gas film during the discharge process and the shape of the gas film was changed by the force. In addition, the energies released by different types of discharge were calculated according to the voltage and current waveforms. The discharge frequency was found to increase with the increase in applied voltage and the frequency of the second type of discharge was approximately equal to that of the third type when the applied voltage was higher than 40 V. Recently, electrochemical discharge machining (ECDM) has been proposed as a potential technology for processing non−conductive materials, such as glass, quartz and ceramics . These mECDM is a non−traditional processing method that is based on the electrochemical discharge phenomenon around the tool electrode . The tooECDM is a complex process that includes gas film formation and discharge generation ,10. The As the discharge plays a critical role in ECDM, many efforts have been devoted to understanding the nature and mechanisms of discharge generation. Crichton and McGough  found thIn this study, a high−speed camera was applied to capture the discharge phenomenon around the tool electrode tip. The captured discharge images were compared with the corresponding current waveforms, and different types of discharges were then classified based on the discharge currents. The energies released by the different discharges in ECDM were calculated and their frequencies were estimated.For the general cylindrical tool electrode, the discharge location was random. The discharge may occur at the bottom of the tool electrode or on the side wall of the tool electrode. In order to clearly capture the image of the discharge, it is necessary to fix the discharge to a point in the visible region. The conical electrode can concentrate the discharge at the tip of the tool electrode, ensuring that each discharge can be captured. Therefore, the conical electrode was used in this paper.The electrolyte used in the experiments was sodium hydroxide with a concentration of 6 mol/L. The power supply was a smooth DC power with an applied voltage in the range of 30 V and 40 V. The conical electrode used in this study is small. When the applied voltage is higher than 30 V, the time for the gas film formation is very short, which makes it difficult to capture the evolution phenomena of the gas film using the camera. In order to obtain the complete evolution of the gas film formation process around the tool electrode, a voltage of 30 V was applied. A relatively high voltage (40 V) was used in the experiment to capture the discharge around the electrode tip since bright discharges can be produced under this condition benefiting the capture of obvious discharge images. The camera parameters were set before the experiments by adjusting the time interval between any two adjacent images and the exposure time. Once the power is applied, the camera, triggered by the real−time current signal, begins to capture the gas film and discharge images.Since each image was generated within a time period of 50 μs, the corresponding current for Three pulses appear in the discharge current for The corresponding current in The current also increased in After these several leap currents, the current dropped to 40 mA. As shown in According to the above analysis, the discharge currents in ECDM can be classified into three categories, as shown in The second type of discharge current is shown in As described in previous study, a spark is a sudden transient and noisy discharge between two electrodes, and an arc shows a stable thermionic phenomenon . TherefoE is the discharge energy, U is the discharge voltage, I is the discharge current and T is the discharge time. According to the literature [U between electrodes can be regarded as the voltage applied between the tool electrode and the auxiliary electrode. It can be observed from i(t) is the time varying discharge current and t0 is the discharge time which can be obtained from the discharge current curve. Since the discharge energy of the first type of discharge is very small and has a minor contribution to the material removal, this article focuses on the calculation of the discharge energies of the second and the third types. According to Equation (2), the discharge energy of the second type can be calculated to be in the range of 1.76 × 10−5 J–1.2 × 10−4 J and the discharge energy of the third type is within the range of 2.0 × 10−4 J–7.2 × 10−4 J.The discharges in ECDM were categorized into three types in the previous section. Based on the voltage and current curves, the energy released by each type of discharge can be calculated in accordance with the following formula:terature , the disQ is the total electric charge, i is the time varying discharge current and t0 is the discharge time. According to Equation (3), the total amount of electric charge released by the second type of discharge can be calculated. This is found to be in the range of 4.4 × 10−7 C–3.0 × 10−6 C, while the total amount of electric charge released by the third type of discharge is in the range of 5.0 × 10−6 C–1.8 × 10−5 C.The total amount of electric charge released per discharge can be obtained from the following formula:n is the number of moles of electrons involved in the discharge, Q is the total electric charge released by one discharge, NA is Avogadro’s number and e is the electric charge carried by one electron. According to Equation (4), the number of electrons participating in the second type of discharge can be calculated to be in the range of 4.6 × 10−12 mol–3.1 × 10−11 mol and for the third type of discharge, the number of electrons is calculated to be in the range of 5.2 × 10−11 mol–1.87 × 10−10 mol.According to the total amount of electric charge, the number of electrons participating in the discharge can be obtained using the following formula:According to the above analysis, the energy released by the third type of discharge is the largest among the three studied forms. When the discharge is generated by the breakdown of the gas film, a force is produced which will change the shape of the gas film and further affect the current signal.It can be concluded from the above phenomenon that a force will is on the electrolyte during the discharge process that thus changes the shape of the gas film. For the third type of discharge, the effect of the discharge on the gas film is obvious. The shapes of the gas film in The big oval gas film moves upwards owing to the buoyancy, as shown in In order to count various discharge types in ECDM, the ECDM current was recorded over a prolonged time period. The pulse current with a peak value around 1 A in (1)The discharge can be classified into three types according to the brightness of the discharge images and the corresponding discharge current. The first type of discharge presented the characteristics of a low−amplitude current with a narrow pulse duration and with current intensity values in the range of 100 mA to 200 mA. The pulse duration was 1 μs and the discharge was very weak. The second type of discharge showed the characteristics of a high−intensity current and short pulse duration. The peak current of this type was in the range of 200–600 mA and the pulse width was approximately 8 μs. The discharge brightness was stronger than the first type. The third type of discharge was characterized by a high−intensity current and long pulse duration. The peak current of this type of discharge was in the range of 200–300 mA and the pulse duration ranged from 20 μs to 90 μs. The third type of discharge resembled an arc and it was the brightest.(2)The observations of the effect of discharge on the gas film indicate that a force was exerted on the electrolyte during the discharge process and that the gas film was dragged into a large oval bubble owing to the bombardment of anions on the electrolyte. Meanwhile, the current dropped to almost 0 mA during the deformation process of the large oval bubble.(3)−5 J–1.2 × 10−4 J, and the discharge energy of the third type was in the range of 2.0 × 10−4 J–7.2 × 10−4 J. Experimental results showed that the discharge frequency increased with increasing applied voltages and the third type of discharge occurred only when the applied voltage was high enough. In addition, it was found that the frequency of the second type of discharge was approximately equal to that of the third type when the applied voltage was higher than 40 V.The energies of different types of discharges were calculated according to the voltage and current waveforms. The discharge energy of the second type was calculated to be in the range of 1.76 × 10A high−speed imaging technology was employed to investigate the characteristics of the discharge in ECDM. The captured discharge images were compared with the real−time discharge current. The following conclusions can be drawn:


Regulatory agencies are required to evaluate the impacts of thousands of chemicals. Toxicological tests currently used in such evaluations are time‐consuming and resource intensive; however, advances in toxicology and related fields are providing new testing methodologies that reduce the cost and time required for testing. The selection of a preferred methodology is challenging because the new methodologies vary in duration and cost, and the data they generate vary in the level of uncertainty. This article presents a framework for performing cost‐effectiveness analyses (CEAs) of toxicity tests that account for cost, duration, and uncertainty. This is achieved by using an output metric—the cost per correct regulatory decision—that reflects the three elements. The framework is demonstrated in two example CEAs, one for a simple decision of risk acceptability and a second, more complex decision, involving the selection of regulatory actions. Each example CEA evaluates five hypothetical toxicity‐testing methodologies which differ with respect to cost, time, and uncertainty. The results of the examples indicate that either a fivefold reduction in cost or duration can be a larger driver of the selection of an optimal toxicity‐testing methodology than a fivefold reduction in uncertainty. Uncertainty becomes of similar importance to cost and duration when decisionmakers are required to make more complex decisions that require the determination of small differences in risk predictions. The framework presented in this article may provide a useful basis for the identification of cost‐effective methods for toxicity testing of large numbers of chemicals. A 2019 survey of 22 chemical inventories from 19 countries found that more than 137,000 unique organic chemicals and more than 9,300 unique inorganic chemicals were registered in one or more of these inventories  is a decision‐support tool that systematically evaluates different courses of action to determine the option that achieves a specified desirable outcome at the lowest cost. Unlike benefit‐cost analysis, it is not necessary to assign a monetary value to the benefits achieved in a CEA; however, outcomes are quantified and consistently defined across the various courses of action. CEAs place costs and outcomes into a common temporal framework that enables the determination of the net present value of the costs and outcome measures  analyses  reduction of adverse effects from current exposures; (2) prevention of adverse effects from future exposures; (3) reduction of societal concerns by the conformation that current and future commercial uses of chemicals are safe; and (4) securing economic benefits from the use of chemicals incorrectly identified as being of concern. All these benefits require a determination of the acceptability of risks from different doses of the chemical. Therefore, within the context of this framework, the outcome is defined as the ability of risk managers to make “correct” risk‐based decisions for a tested chemical. As discussed below, risk‐based decisions are made using objective rules that link decisions to risk findings. A “correct” decision is defined as reaching the same decision under a given decision rule that would have been reached if the decisionmaker had perfect knowledge of the toxicity of the relevant chemical.The outcome is further defined as the number of years in the time horizon where toxicity data are available to support correct decisions. The duration of this period is used as the outcome (rather than simply the date when the data first become available) because while some toxicity findings will be of immediate value , others will find value in assessing exposures that occur from future uses of the chemicals. In addition, findings of safety for existing exposures provide an ongoing assurance to the exposed population that the chemicals in the products that they use have been evaluated and found to be safe. Thus, the benefits of toxicity data are best viewed as an ongoing resource that enables society to assess risks from current and future commercial use of chemicals.The duration of toxicity testing affects the characterization of the net present costs and outcomes of testing. Since some toxicity tests may require multiple years to complete, a CEA will need to depreciate costs incurred in the future. Similarly, when the outcomes are delayed by toxicity‐testing methodologies with longer durations the value of the outcome will need to be discounted. To account for the effects of delay, a time horizon is defined for the CEA that is sufficiently long that it allows the measurement of both the costs and benefits of the toxicity‐testing methodology. In this article, future costs and outcomes are discounted using a fixed annual rate of reduction in value in the second and subsequent years of the time horizon.2.2CER). Unlike many CEAs, the framework does not attempt to determine an incremental cost‐effectiveness ratio (ICER) for testing. An ICER measures changes in the CER that result from testing by comparing a revised CER to a CER based on the prior understanding of the toxicity of a chemical. In this framework, we assume that in the absence of toxicity testing no decision on the risk posed by an exposure to a chemical is possible and as a result no prior value of the CER can be determined. Once a chemical has been tested, however, the framework could be extended to include an ICER that evaluates the incremental benefits of additional testing. For example, if an in silico technique such as Wignall et al. is then given by:Cyjis the cost of performing the jth testing methodology in the yth year (millions of dollars); DMVyj|l is the probability of correctly making the lth type of regulatory decision in the yth year given the findings of the jth testing methodology (unitless); yT,j is the time it takes to perform the jth testing methodology (years); yTH is the time horizon of the analysis, where yTH must be greater than the largest yT,j of the methodologies evaluated (years); y is the time since the beginning of the toxicity testing (years); and  r is the annual discount rate (fraction reduced per year)In this framework, theCER because DMV appears in the denominator ofCER and the values of DMV are reduced by uncertainty. In order to define DMV, the framework needs to link the toxicity data from a testing methodology to a decision‐making process. This is done by first linking the toxicity test data to a risk finding and then linking the risk finding to a decision‐making process. Once this relationship is in place, the uncertainty in the toxicity findings can be propagated through to the decision‐making process and the impact on decision making can be determined.Uncertain toxicity data that increases the likelihood of incorrect decisions will generate higher values ofCER for testing a chemical using a methodology taking ten years and costing five million dollars. The DMV for each year is 0.9 indicating that the estimate of toxicity has some uncertainty and could result in an incorrect decision 10% of the time. Assuming a discount rate of 3% and a time horizon of 20 years, the resulting value of CER is $0.85 million.Table 2.3R. The model assumes that the doses of a chemical necessary to cause a toxicological response and doses of the chemical received by the members of the exposed population both follow lognormal distributions  be the probability density function of doses received by individuals in an exposed population over the duration. The cumulative distribution function, G(x), constitutes a population‐level dose–response function describing the incidence of adverse effects in the population if the entire population received a dose of x for the specific duration of time. With the assumption of a lognormal distribution for G(x), the dose–response function has a monotonically increasing and symmetric sigmoidal shape when viewed on the logarithmic scale, with the population incidence of 50% at the median (and geometric mean) of G(x). The function g(x), represents the probability density function associated with the interindividual variability in the bodyweight‐adjusted dose at, or above, which adverse effect(s) occur to the individual . The function G(x) is hereafter referred to as a chemical's toxicity distribution. While the model assumes that individuals have thresholds, it does not assume that the population distributions have thresholds is unbounded).In this article, we illustrate the framework using a quantitative model of risk, θ denote the set of parameters that determine the variability in the toxicity and exposure factors that determine the variability in the distribution of risk levels across individuals in an exposed population, where θ can be partitioned into separate parameters for the toxicity and exposure distribution, θ=. The function f(x) is defined by a parametric distribution (f(x|θexp)=LN) and the function g(x) is defined by a parametric distribution (g(x|θtox)=LN), where the parameters μtox and μexp are the log10 values of the geometric means of g(x) and f(x) and σtox and σexp are the log10 values of the geometric standard deviations of g(x) and f(x). Assuming that individuals’ exposures and toxicity thresholds are statistically independent, the average risk for the population can be expressed as:Let g(x) and f(x) for three populations with different levels of risk. Larger risks occur when large fractions of the population receive doses that have a large probability of causing an adverse effect (the distributions overlap). The assumption of lognormality allows the determination of R for a population using the analytical solution to Equation (2).Φ(·) denotes the cumulative distribution function of the standard normal distribution. The derivation of this solution is given in Appendix 1 of this article.Fig. R is the fraction of individuals in a population who experience one or more adverse effects resulting from exposures to a chemical that occur over a specific period of time. Examples of this approach have been developed using risk models based on quantitative aggregate exposure and adverse outcome pathways  of the parameter values θtox. With J testing methodologies, there are J sets of estimates of the toxicity parameters μ^toxj and σ^toxj. If we assume that μ^toxjand σ^toxj have uncertainty distributions that are lognormal, the uncertainty from thejth methodology for toxicity testing can be characterized by the parameters μ(μ^toxj),σ(μ^toxj), μ(σ^toxj), and σ(σ^toxj).The lack of perfect information leads to uncertainty in estimates of θ^toxj and θtox for an individual chemical cannot be empirically determined by any toxicity test. Information on the uncertainty in the predictions of specific methodologies, however, can be characterized by the application of the methodology to chemicals where human toxicity data are available. Alternatively, information on the testing methodology itself can allow a characterization of the uncertainty in the values of θ^toxj. For example, error propagation should be possible in estimates of θ^toxj based on the quantitative adverse outcome and aggregate exposure pathway models. As a result, the framework assumes that the analyst is able to develop an estimate of the relevant uncertainty parameters (μ(μ^toxj),σ(μ^toxj), μ(σ^toxj), and σ(σ^toxj)).The relationships between μ0,exp and σ0,exp. The parameters that describe the doses that occur under the K regulatory actions are defined as μk,exp and σk,exp.Regulatory actions reduce a population's exposure to a chemical by changing events in the exposure pathways for one or more sources of the chemical. These actions result in lower expected values of dose for the exposed population and values for exposure parameters that differ from the parameter values of the uncontrolled exposures. The parameters that describe the doses received from uncontrolled exposures in our framework are defined as μ^0,exp, σ^0,exp,μ^k,exp, and σ^k,exp that are uncertain. These uncertainties would be characterized by the parameters μ, σ, μ, σ, μ, σ, μ, and σ. While the proposed framework can account for uncertainty in estimates of exposure and the CER can be determined for different exposure measurement methodologies, the focus of this article is on the application of CEA to the selection of a preferred toxicity‐testing methodology. The issue of selection of a methodology to characterize exposure  is, therefore, beyond the scope of the article. In this article, we make the simplifying assumption that there is a single method of determining the exposure parameters and the uncertainties in the estimates of the exposure parameters do not materially affect the uncertainty in the estimates of risk (R^0j and R^kj).The exposure parameters under different regulatory actions as measured using a specific exposure assessment methodology will result in estimates of jth toxicity‐testing methodology, θ^toxj, and the estimate of exposure under no action and the kth regulatory action, θ^0,exp andθ^k,exp, are used in Equation (R^0j and R^kj) that reflect the choices of toxicity‐testing methodology, exposure measurement methodology, and regulatory action.The estimated parameters derived from the Equation  to gener2.5θ^toxj will result in the same decision as those based on θtox. Decisions reflect the characteristics of the sources of a chemical, the populations that are exposed, available controls, and the physicochemical, toxicokinetic, and toxicodynamic properties of the chemical. Regulatory decisions are also determined by the legal and regulatory frameworks for specific sources of exposure and specific populations  and is expressed as:k is the selected regulatory action and ACk is the cost of the kth regulatory action.Decisions become more complicated when the population risk exceeds the TRL and requires regulatory control. The following “complex decision rule” (CDR) is an example of these more complex decisions. Under this rule, the decisionmaker selects a regulatory action that reduces a population's risk to a level equal to, or below, the TRL. When there are multiple regulatory actions available that reduce the risk to levels equal to, or below, the TRL, the decisionmaker selects the action with the lowest cost. The rule also assumes that there will be at least one methodology that can reduce the risk to levels equal to, or below, the TRL. It should be also noted that only the rank of the costs of the regulatory actions is used in this process and not the specific costs. As with the SDR, the CDR is based on the estimated risks . Under the CDR, regulatory action #1 would be selected for the population. Now consider a second population with a value of R0 that is eight times the TRL. Regulatory action #1 would not be included in the pool of possible actions because Rk=Action1 would exceed the TRL by a factor of 1.6. Let regulatory action #2 have the lowest cost of the smaller pool of actions with values of Rk that are less than the TRL when R0 is eight times the TRL. Because regulatory action #1 was the lowest cost action in the original pool and has now been excluded, the cost of the lowest cost action in the new pool (regulatory action #2) will be higher than regulatory action #1. Let regulatory action #2 reduce risk by a factor of 50. Now consider a third population where R0 is 100 times the TRL. Regulatory action #2 would not be included in the pool of possible actions and a third action (regulatory action #3) would be selected. regulatory action #3 would be more be more effective, reducing risks by a factor of at least 100, and more expensive than regulatory action #2.This process of applying the CDR results in a selection of either no action or one of a series of increasingly stringent and increasingly costly regulatory actions that is a function of the size of n et al.  as the “R0 is more than the TRL but less than five times the TRL; regulatory action #2 is selected where R0 is more than five times the TRL but less than ten times the TRL; and regulatory action #3 is selected where R0 is greater than ten times the TRL. Regulatory actions #1, #2, and #3 make up the choice set of K=3 actions for this chemical and its exposures. Risks of one, five, and ten times the TRL are values of risk where the selection of the regulatory action changes. These values are decision points (DPs) in the risk findings for the population.Regulatory action #1 is selected by the CDR for populations where 2.6DMVj|l is defined as the conditional probability of making a correct decision for the lth decision rule using information from the jth toxicity‐testing methodology, where the correct decision is the decision that would be made using θtox. DMVj|lfor a simple decision rule (DMVj|SDR) can be expressed as:DR(R^j|TRL) is the decision from a decision rule (DR) that is based on R^j and TRL. If the jth toxicity‐testing methodology perfectly predicts toxicity ( θ^toxj=θtox) the value of DMVj|SDRis 1. The uncertainty in θ^toxj translates into uncertainty in the estimates of R^0j and R^kj used in decision making and ultimately uncertainty in the choice of regulatory action. This uncertainty may reduce the probability of making a correct decision and if this occurs, the resulting values of DMVj|l are less than 1.DMVj|l and a given level of uncertainty in θ^toxj varies with the nature of the decision. If the decision rule is the SDR , then there are three possible relationships between TRL and the DPs. First, TRL may be greater than R0. Second, the TRL may be greater than RK but less than R0. Finally, RK may be greater than TRL. As a result, DMVj|CDR is given by:If the decision rule considers two risk findings, such as the CDR where there is a single regulatory action (DMVj|CDR is given by:When there is more than one possible regulatory action in a choice set (K>1), 2.7CERj|lvaries with the toxicity of the chemical being tested. As a result, a toxicity‐testing methodology that has the lowest CERj|l for one chemical may not have the lowest CERj|l for a second chemical. The relationship between CERj|l and toxicity is determined by the relationship between DMVj|l and R0 and Rk and by extension with θ.The value of DMVj|l varies with θ is given in Fig. θ. In the figure, the population risk is presented on the x axis and the decision points and TRL are identified by vertical lines. The range of uncertainty in R^0j resulting from the uncertainty in θ^toxj is presented for each of the four chemicals. In this figure, chemicals A and B have similar degrees of uncertainty in the estimates of R^0j, but have different values of DMVj|SDR. The range of R^0jvalues for chemical A are all below the TRL. Since the corresponding uncertainty in R^0j does not affect the finding that the risk is below the TRL, the uncertainty in θ^toxj for chemical A has no impact on the decision and the value of DMVj|SDR is 1. In contrast, chemical B's range of R^0j values include levels that are above and below the TRL. As a result, some of the estimates of R^0j for chemical B led to an incorrect decision.  This results in a value of DMVj|SDRthat is less than 1. As decisions become more complex, the impact of uncertainty in θ^toxj on DMVj|l increases. In the complex decision shown by Fig. θ^toxj are uncertain and when small differences in toxicity parameter values affect a decision, the probability of a correct assignment can be low. For chemical C, the uncertainty in θ^toxj is sufficiently large to make it unclear which of the four possible levels of control are required. As a result, there will be a low value of DMVj|CDRfor chemical C. Finally, it is also possible for uncertainty in θ^toxjto become less important for highly toxic chemicals. The estimate of toxicity for chemical D in Fig. DMVj|CDR of 1.An example of how The finding that the cost‐effectiveness ratio of a toxicity‐testing methodology applied to a given risk‐based decision will vary across chemicals with different toxicities and/or different exposures suggests that the selection of a toxicity‐testing methodology will not result in the identification of a single preferred methodology for all untested chemicals. As a result, a testing program may find it beneficial to investigate which methodology is preferred for a chemical. This point is examined in more detail in the discussion section below.3j=1,…,5) are used, resulting in five values of CER for each of the two decision rules CERj=1,…,5|SDR and CERj=1,…,5|CDR.To demonstrate how the proposed framework operates, we present two example CEAs that address the use of toxicity data in the SDR and CDR described above. Data from five hypothetical toxicity‐testing methodologies , nor are any of the five methodologies intended to represent an existing or new testing methodology. Instead, the characteristics of the five methodologies are designed to explore the impacts of differences in the cost of the tests, the duration of the tests, and the level of uncertainty in the values of CER generated in these examples are the cost of the program divided by the sum of the outcomes for the tested chemicals over the time horizon.In these examples, we assume that a regulatory agency is tasked with assessing the toxicity of a large number of chemicals that have varying levels of toxicity. While Equation  provides3.1jth toxicity‐testing methodology, yT,j is completed. TheDMV and the cost of testing are reduced by the same annual discount rate for year 2 and later years. The program starts testing a new group of chemicals each year. The cost of the toxicity testing is fully funded using the budget that is available the year the test is initiated. The size of the group is a function of the cost of the testing of a single chemical and the annual budget.The toxicity testing program in the illustrative examples is assumed to have the following characteristics. The program is funded on an annual basis at a constant amount over the period spanned by the time horizon and the cost of testing a chemical remains constant for each of the methodologies for the same period. The data become available when the testing of the chemical using the R0 and Rk. To maintain values of Rk that are no more than the TRL requires more stringent controls for chemicals with higher toxicity. Therefore, the values of CERj|CDR vary across chemicals with different toxicities and a range of toxicity values are used to explore how this variation occurs. In the CDR we assume that there are three regulatory actions in the choice set (K=3). The values of θ^0,expand θ^k,exp are defined for the uncontrolled exposures and for the exposures under each of the three regulatory actions, respectively. For simplicity, the uncertainties in θ^0,exp and θ^k,exp are not considered in these analyses. The values of θ^toxj, θ^0,exp,and θ^k,exp are used to determine values of R^0j and R^kj. These values along with information on the costs and durations of the testing are used to determine the values of CERj=1,…,5|SDR and CERj=1,…5|CDR.As described above, for a fixed initial exposure, chemicals with higher toxicity will pose larger values of –6. Different values for TRL could be used depending on the severity of a chemical's endpoint. Table The values of the parameters that are common to the two CEAs are given in Table σ(μtox) by a factor of five. Toxicity‐testing methodology #5 differs from the base case by a combination of a fivefold reduction in cost, duration, and σ(μtox).The parameter values for the toxicity‐testing methodologies are designed to explore the impacts of changes in cost, testing duration, and uncertainty Table . ToxicitTH) influences the relationship between duration of testing and the value of the CER. The shortest TH possible in the two examples is 11 years. This is the minimum length of time needed for generating toxicity data for toxicity‐testing methodologies #1, #2, and #4 and for promulgating regulations that use the data. The maximum value of TH used in the examples is 20 years. This upper limit for the value of TH is selected because it is expected that the methodologies used by testing programs will evolve over time, making the assumption that current toxicity‐testing methodologies will be used without modification for more than 20 years unlikely.As discussed in Appendix B, the selection of the time horizon  is then determined based on the uncertainty in the measurement and the nature of the decision. This process is repeated using a wide range of values for μtox. All modeling is performed using Excel software.Fig. μtox for the tested chemicals are assumed to follow a log uniform distribution bounded by −5 and 2 (0.00001 to 100 mg/kg/day). We note that this range is likely to be wider than the expected range of chemical toxicities, but the wide range is helpful in demonstrating the relationship between CER and μtox. A value of μtox is randomly selected from this range and values of DMVj|SDR and DMVj|CDRare determined using the processes described below. The process is repeated 5,000 times to produce a data set of matched values of μtox, DMVj|SDR,DMVj|CDR, CERj|SDR, and CERj|CDR.In this article, values of μtox that corresponds to the TRL (TRLμtox). This value can be derived using Equation  denote the cumulative distribution function of a normal distribution with a mean of μ and standard deviation of σ for a value x. Let μ equal μ^toxj, σ equal σ(μ^toxj), and x equal TRLμtox. DMVj|SDR is then given by:The probability that μtox≤TRLμtox, we have:Similarly, when DMVj|SDR is then:I[·] denotes an indicator function for the condition [·]. Using Equation  is then:The corresponding equation for DMVj|SDR and DMVj|CDR are determined for a chemical using Equations (σ(μ^toxj), and values in Tables The values of quations  and (17)DMVj|SDR and DMVj|CDR are determined, the corresponding values of CERj|SDR and CERj|CDRare calculated using Equation . The changes in the value of CER with μtox occur as a result of the decreases of DMVj|SDR, and DMVj|CDR for chemicals with values of μtox that are close to TRLμtox (because the DMV occurs in the denominator of the CER [Equation (CER). The decreases in DMV occur at such values of μtox because at such values even small differences between μtox and μ^toxj can result in an incorrect decision.Figs. CER for chemicals with values of μtox that are greater than TRLμtox. For values that are less than the TRLμtox, the values of CERj|CDR diverge from CERj|SDR with the values of CERj|CDR increasing steeply. This occurs because a given level of uncertainty in values of the toxicity parameters leads to incorrect decisions more often with the CDR than with the SDR for such chemicals.Fig. CER with toxicity for the SDR and CDR. Toxicity‐testing methodology #1 (base case) has the largest number of undesirable characteristics  of the five methodologies considered. As a result, the values of CER1 are the highest of the five methodologies for both the SDR and CDR. Toxicity‐testing methodology #5 has the largest number of desirable characteristics and the values of CER5 are the lowest of the methodologies for both SDR and CDR.Figs. CERj|CDR with the toxicity of the chemical appears as a series of separate peaks when the toxicity‐testing methodology has low uncertainty (toxicity‐testing methodologies #4 and #5) and as a step function when the methodology has higher uncertainty . Both the peaks and the edges of the steps occur at the values of μtox that correspond to the TRL and the decision points . At higher uncertainties, the shape is driven by the relative difficulty of identifying the correct regulatory action. The ability of a testing methodology to generate a toxicity estimate that identifies the correct action is smaller when the range of μtox values that require a certain regulatory action is small. This can be seen in Fig. CER occur for values of μtox where regulatory action #2 is the correct action. Regulatory action #2 is only correct for a small range of values of μtox (0.0002–0.0004 mg/kg/day). Regulatory action #1 is correct for a larger range (0.0004–0.002 mg/kg/day) and the values of CER for that action are lower.In Fig. CER by a factor of five. This decrease is independent of the values of the toxicity parameters of the chemicals and the decision‐making rule.In these examples, we assume a fixed budget for testing. Reducing the cost of testing for a single chemical (toxicity‐testing methodology #2), therefore, increases the number of chemicals tested and increases the output  by a factor of five. As a result, the fivefold decrease in cost reduces the value of CER. Like the impact of a reduction in cost, this decrease is independent of the values of the toxicity parameters of the chemicals and the decision‐making rule. Unlike cost, the impact of the reduced duration of testing is not direct but is mediated through its impact on the times when the test findings are available.The reduction in testing duration by a factor of five (toxicity‐testing methodology #3) also resulted in a reduction in the value of yT,1, r, and TH, with the value of TH having the largest effect. There is no “correct” value for TH. In this assessment we use a range of values to determine the sensitivity of the results to the assigned value of TH. As shown in Appendix B, at the lowest value of TH (11 years), the value of CER decreases 49‐fold from a fivefold reduction in duration. At the largest value of TH (20 years), the reduction is 3.4‐fold. Since it is not clear which value of TH is most appropriate, the plots of CER in Figs. CER for values of TH of 11 through 20 years (a 14‐fold reduction).As discussed in Appendix B, the size of the impact is affected by σ(μ^toxj)) fivefold (toxicity‐testing methodology #4) lowers the value of the CER by raising the values of DMV that had been affected by the uncertainty in μ^toxj. As discussed in Section DMV varies with the chemical‐specific value of μtox and changes in DMV are limited to chemicals with values of μtox that are close to the TRLμtox. This can be seen in Figs. CER for chemicals with low toxicity (μtox>TRLμtox) are similar for both levels of uncertainty.Reducing the uncertainty in the toxicity data  results in a 1.1‐fold reduction in the average value of CERj,l for the 5,000 chemicals under the SDR and 1.6‐fold reduction under the CDR. The largest reduction in any of the 5,000 chemicals is 1.5 and 4.6‐fold for the SDR and CDR, respectively.The average and maximum values of 5CER, that is quantitative, objective, and transparent. The framework can be adapted to incorporate the specific characteristics of different toxicity‐testing methodologies, the nature of the chemicals under consideration, the chemical and endpoint‐specific dose‐response models, and the various decision rules used by different regulatory programs.The focus of this article is the development of a framework for the selection of cost‐effective toxicity‐testing methodologies for the systematic testing of the large numbers of chemicals in commerce that have little or no toxicity data. The framework addresses the issue of how to perform tradeoffs between varying levels of uncertainty, cost, and duration by placing the three elements into a common metric, CER suggest that a fivefold change in cost consistently provides a fivefold benefit to decisionmakers by increasing the throughput of a testing program. The benefits of a fivefold reduction in duration varied with the duration of the time horizon. For shorter time horizons, the benefits are many times larger than those from reductions in cost. For longer time horizons, the benefits are smaller. Shorter time horizons are more appropriate for testing programs for groups of priority chemicals where there are immediate concerns about health risks. Longer time horizons are more appropriate for testing programs that systematically test chemicals of various levels of concern. It should be noted that cost and duration are likely to be positively correlated, since longer duration tests will require more resources and tie up laboratory space for longer periods of time than shorter duration tests. As a result, studies of a shorter duration are likely to have lower costs, resulting in larger reductions in CER. Using the assumptions in Table CER for both the simple and complex decisions.The illustrative applications presented in the article are designed to explore the relative impacts of fivefold differences in costs, durations, and uncertainties of different toxicity‐testing methodologies. Results from these examples show that the reductions in all three elements provide benefits to decisionmakers. The relative changes in σ(μ^toxj) used in the examples is more complex than the reductions in cost and duration. Like duration of testing, uncertainty in test results is not a direct input to the CER equation. Instead, the effects of uncertainty are mediated through changes in the DMV. The value of the reduction in the examples varies across chemicals since the likelihood of a given level of uncertainty in a toxicity finding resulting in an incorrect decision is a function of how close the risks posed by a chemical are to the TRL or a decision point. For the simple decision, the reduction of uncertainty reduced the CER by at most 1.5‐fold for chemicals with certain toxicities. The impact of the reduction in uncertainty is more important in the complex decision than the simple decision. The largest reduction in CER in the complex decision from the uncertainty reduction is 4.6‐fold. For many chemicals, including those that posed no risk, there is little or no reduction in CER with the reduction in uncertainty. These findings suggest that regulatory programs making simple screening decisions (are risks acceptable or unacceptable?) on large numbers of chemicals, and where the chemicals have a wide range of toxicities, may be better served by testing methodologies with shorter durations and lower costs, rather than longer and more expensive testing methodologies that result in greater reductions in uncertainty.The benefit of the fivefold reduction of The findings of this article provide support for the concept of tiered approaches to testing of chemicals . This suggests that the DMV (a measure of the probability of a correct decision) might be used to determine if a methodology would generate large numbers of incorrect decisions for a specific decision‐making rule and, if so, it may not be “fit for purpose.” The framework also does not discriminate between errors that lead to over‐regulation and errors that lead to under‐regulation. Future versions of the framework can include the use of uncertainty factors , two specific risk‐based decision‐making approaches (SDR and CDR), and five hypothetical toxicity‐testing methodologies. These examples demonstrate how the framework addresses a simple and a more complex decision; however, the examples do not fully explore the range of risk‐based decision‐making approaches that might be encountered in practice, and additional work in this area is needed. In addition, only two values for cost, duration, and uncertainty are modeled. Future work could examine the impact of these elements in greater detail and over a wider range of values.Finally, the framework can be extended to consider uncertainty in the variation of toxicity across individuals and the uncertainty in estimates of the mean and variation in exposure across individuals. For example, the use of probabilistic techniques to characterize risk allows the consideration of multiples sources of uncertainty in toxicity and exposure data and does not require the assumption of lognormality in inter‐individual variation in dose and toxicity.In summary, the framework presented here represents an attempt to use CEA to address the issue of selecting a toxicity‐testing methodology that can be applied to the large numbers of untested chemicals in commerce. Two example CEAs created using the framework found that the three elements  are all important considerations in selection of a testing methodology. Under the specific assumptions underlying the analyses conducted, reductions in testing duration and testing costs are more important than reductions in uncertainty in toxicity findings.


Letter to the Editor concerning “Describing center of pressure movement in stabilometry by ellipse area approximation” from Agnieszka Gołąb. Dear Editors,N is the length of the signal, σML and σAP are the standard deviations of the mediolateral and anteroposterior time series, respectively, and σML−AP is the covariance of the ML and AP time series.We thank Agnieszka Golab for the opportunity to bring precisions and additional definitions and formulas concerning the features used in stabilometry that are based on ellipse area calculations. Our choice was actually to present the formula of the prediction ellipse area in the article, as it indeed does not strongly depend on the sample size as the confidence ellipse area does. However, we first created a undesirable confusion by choosing the name “confidence ellipse area”—which is the most common name used in posturography regardless of the method actually used. Secondly, the definition at the beginning of the paragraph does not correspond to the formula that is provided, the later is the one used in the articles results and available in the open‐access code. Therefore, we agree that a clarification is required and take the opportunity of this letter to present rigorously the two different definitions of the confidence ellipse area and the prediction ellipse area. The proofs of these formulas are given in (Schubert & Kirchner, The prediction ellipse area, which is the feature computed in the open‐access code proposed in (Quijoux et al., We agree with Agnieszka Golab that the prediction ellipse area formula, which shows only marginal differences with variations in sample size (Schubert & Kirchner,


We study 10 years (1995–2004 inclusive) of auroral kilometric radiation (AKR) radio emission data from the Wind spacecraft to examine the link between AKR and terrestrial substorms. We use substorm lists based on parameters including ground magnetometer signatures and geosynchronous particle injections as a basis for superposed epoch analyses of the AKR data. The results for each list show a similar, clear response of the AKR power around substorm onset. For nearly all event lists, the average response shows that the AKR power begins to increase around 20 min prior to expansion phase onset, as defined by the respective lists. The analysis of the spectral parameters of AKR bursts show that this increase in power is due to an extension of the source region to higher altitudes, which also precedes expansion phase onset by 20 min. Our observations show that the minimum frequency channel that observes AKR at this time, on average, is 60 kHz. AKR visibility is highly sensitive to observing spacecraft location, and the biggest radio response to substorm onset is seen in the 21:00–03:00 hr local time sector. Auroral kilometric radiation (AKR) observations made over 10 years are compared with four event lists of substorm onsets using superposed epoch analysesOn average, AKR power increases and source region extends to higher altitudes in the 20 min prior to onsetThe occurrence of AKR power at higher altitudes is sensitive to the substorm size ElectroBZ in geocentric‐solar‐magnetic (GSM) coordinates) component is present in the IMF, magnetic flux is loaded into the magnetotail as it lengthens and the plasma sheet compresses. This is the growth phase of the substorm . When the dayside reconnection rate is high, often when a southward  index ) on the dawn and dusk flanks. From mid‐2003 to mid‐2004, Wind explored the nightside magnetosphere, being placed in a trajectory that sent it downtail to the Lagrangian point L2. The nightside location of the source regions and the highly anisotropic beaming of the emission has consequences for the viewing of AKR for a remote sensing spacecraft such as Wind, and so the spacecraft position at the time of substorm onset must be considered. The RAD1 receiver of the Wind/WAVES instrument takes a variable number of samples (between 1 and 4) of 32 frequency channels, between 20 and 1,040 kHz, over a ∼3 min sweep cycle , and shows the general bias toward measurements made in the Northern magnetic hemisphere. The majority of observations are made between −40° and 40°, while Wind infrequently reached higher latitudes to a maximum magnitude of 80°. The appearance of the average data at higher latitudes is due to slow, sparse passes of the spacecraft through these regions. The general trend in the AKR power shows an average increase when viewed from higher latitudes in either hemisphere, with decreased power and dependency on latitude in dayside LTs; agreeing with previous observations  to determine substorm onset, using a typical signature in both the Northward and Eastward components of the magnetic field to derive the mid‐latitude positive bay (MPB) index. McPherron and Chu (McPherron and Chu  publishe and Chu  use a stBZ < 0), as magnetic flux is loaded into the nightside magnetosphere via dayside reconnection and convection across the polar cap. This is seen prior to onset for each of the included lists, which see southward IMF for an hour prior to onset; SOPHIE lists are displayed with their EPT values in percentages.Each of the lists used in this study are represented in Figure The average SML profile from the SOPHIE 75% and MPB lists are comparable in magnitude, with minimum deviations in SML of between −250 and −150 nT. The same is true for the SOPHIE 90% and LANL lists with minimum deviations between −400 and −300 nT. As such, the selection criteria for these latter lists tend to favor larger substorm events. The SOPHIE lists show the effect of using the rate of change in SML as a threshold for event selection, as SML begins to decrease sharply before the epoch. While the SML response from the LANL event list begins to decrease more than 40 min before onset, falling gradually compared to the other lists, this is due to the coarse resolution of the event list as previously discussed.BZ shows the comparative magnitude of the events that are retained by the respective event selection; those with more negative BZ prior to onset are assumed to produce a greater disturbance within the magnetosphere as this allows for longer periods of ideal IMF conditions to provide magnetic flux to the magnetotail via dayside reconnection. Comparing the median BZ from the SOPHIE 90% list with that from the LANL list, for example, which have minimum BZ of between −2.5 and −2.0 nT, suggests that these lists contain larger events. Given that the LANL event list is based on particle injections at geosynchronous orbit, for an event to be retained it requires a substorm of a magnitude that will allow the Earthward‐traveling electron population to reach a distance of at least six RE. It can be assumed that not all substorms will be of the energy to meet this criteria, and so the comparison between the LANL and the SOPHIE 90% event lists is warranted given the 90% quantile threshold applied to SML deflections in the SOPHIE algorithm. Other lists show a more pronounced minimum, with the 75% and 90% SOPHIE lists having similar profiles.The median reponse of the IMF Figure 3Intensifications of AKR are known to coincide with auroral brightenings; it is expected that the average apparent power of the AKR will increase around substorm onset as the auroral oval expands and becomes brighter, signifying the presence of a substorm‐injected electron population which subsequently leads to the generation of AKR. Integrating the AKR power over a particular spectral range gives a proxy of the extent of the source regions along a field line; an increase in power integrated over a given observation frequency range implies the ignition of AKR source regions within an altitude range given by the corresponding electron gyrofrequencies.With an appropriate list of substorm onsets, such that Wind is an appropriate viewing position, a superposed epoch analysis can be performed on the AKR power. In this way, the average variations in the AKR power with respect to the substorm timeline can be deduced, removing any variations that could be present for single events and not representative of the typical AKR response. For each of the onset lists described in Section Table Figure The comparative magnitude of events selected by each list is seen in the AKR response, with the median power for the LANL and SOPHIE 90% lists greatly exceeding that for the SOPHIE 75% and MPB lists, which each have a similar response in magnitude. In the HF, each list sees a gradual increase in the AKR power from 20 to 0 min before the epoch, with an increasingly steep rise to a clear peak in the 20 min after the epoch. For the LF, each list also sees an increase in the AKR power from 20 to 0 min before the epoch, but the peak is seen up to an hour after the epoch. The profiles are noisier in the LF, however, which could be due to the inclusion of less AKR observations at low frequencies and subsequent influence of powerful bursts for a given observation. Particularly prominent peaks appear in the LT ranges 18:00–21:00 hrs for the SOPHIE 90% list, and 21:00–00:00 hrs for the SOPHIE 90% and LANL lists, at ∼20–40 min after onset. This could be indicative of further substorm intensifications occurring. Although changes in the average AKR power profile gained using events from the LANL list tend to precede those from other lists, this is assumed to be due to the aforementioned coarse resolution of the selection algorithm used. While there is a clear gradual rise in AKR power in the 20 min preceding onset, the beginning of the steep increase to the peak is clearly seen between ∼ −20 and −5 min, which could indicate the ignition of more powerful AKR source regions prior to other observable signatures of substorm onset. The idiosyncracies of the lists may have a greater influence than a true AKR response, however. For SOPHIE lists, the AKR response here could reflect the median SML response in Figure Figure Comparing the AKR response for the SOPHIE 75% and 90% event lists, for both frequency ranges across the epoch, we see that the power decreases more gradually after the peak at onset for the SOPHIE 75% (weaker substorms) compared to the SOPHIE 90% (stronger substorms) lists. For the average LF AKR power from the SOPHIE 75% list, considering the 21:00–00:00 LT range, this continues to increase past the epoch. This could be due to the fact that intensifications (expansion phase onsets following recovery phases) are removed from the analyses, but we have not discriminated substorms that are followed by an intensification. In these cases, the intensification that follows later in the epoch will have associated auroral dynamics, and so also AKR dynamics. Considering the SOPHIE 75% and 90% event lists are derived from a quantile threshold of the rate of change of SML , it follows that the former list will include more events in total than the latter as events with smaller magnetic fluctuations are retained in the event selection. If those events are also the initial expansion phase onset before multiple intensifications, which may be more likely for a smaller EPT value, then their combined, average effect could produce this.Table 4The determination of AKR bursts allows us to quantify spectral features such as the bounding frequencies of the bursts and their spectral extent  where observation channels are logarithmically spaced. At higher altitudes however, corresponding to lower frequency channels between 60 and 148 kHz, the spacing between channels is between 8 and 20 kHz, with an average spacing of approximately 12 kHz between the eight frequency channels in this range. This corresponds to an altitude range of ∼1,000 km, assuming the source location is given by the electron gyrofrequency equivalent to the emission frequency and lies on a magnetic field line with an invariant latitude of 70° (as included in and estimated from Figure 3 of (Morioka et al., The bottom panel of Figure These results show conclusively that the response of the range of emission frequencies of AKR begins to extend around 20 min prior to substorm expansion phase onset as determined by SOPHIE. As well as the results from Section 5BZ component of the IMF, shown in Figure AKR sees enhancements in intensity and changes in frequency, and has been postulated to be associated with other dynamics in the terrestrial magnetosphere such as auroral brightenings and discrete arcs, earthward bulk flows of electrons and strengthenings of high latitude current systems. Previous studies have explored the AKR variability alongside these phenomena, which are also closely associated with substorm dyanmics, but have used AKR observations that cover a few months or studies that include only a few tens to a hundred events. Here we use observations of AKR from Wind, made between 1995 and 2004, that coincide with published lists of substorm events to expand upon previous studies and further examine the average AKR response during the substorm timeline. We integrate the AKR power over two important frequency ranges that best characterize the spectrum, covering higher and lower frequency portions. To infer the evolution of the acceleration region, we also examine the observed spectral extent of the AKR bursts as well as the minimum and maximum bounding frequency. We use a variety of substorm lists, including those output from the SOPHIE algorithm at EPT values of both 75% and 90%, that derived from the MPB index, and the list derived from the geosynchronous LANL satellites and their measurements of electron populations. The SOPHIE and MPB index lists are themselves derived from ground magnetometer observations. As an initial comparison of the substorm lists themselves, we perform a superposed epoch analysis of the Figure The results of comparing the AKR burst parameters with the SOPHIE 90% event list show the average evolution of the nightside AKR source region, viewed remotely, as it extends vertically. Figure


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