4.1 Speed inaccuracies and VP corrected dead-reckoning
It is notable that our examples of animals travelling on land (lions and walking penguins), had far less net error per given VP correction rate than the animals travelling in fluid media (swimming penguins, cormorants and tropicbirds). The paradox however, is that dead-reckoning seems to be most valuable for animals in fluid media (particularly for 3-D movement), even though the inaccuracy is greatest at such times. There are three reasons for this:
The DBA approach of deriving speed estimates is temporally highly resolved and more accurate than GPS-derived estimates (used for tropicbirds) and the constant values used for part of the paths calculated for penguins and cormorants.
Typically, terrestrial species move slower than aerial/marine equivalents and thus incorporate less spatial error per unit time
External current flow vectors can cause the relationship between an animal’s (longitudinal axis) powered direction of travel and their true vector of travel to deviate [83, 84]. Indeed, in one of the earliest considerations of dead-reckoning for animals, Wilson et al.  noted that ocean currents were likely to be the greatest source of inaccuracy for positional fixes because of this.
Although convenient and powerful, DBA-derived speed has its own inaccuracies. The proposed linear relationship between DBA and mechanical power [cf. 63, 85] presumably changes when the animal is load-bearing [86, 87], moving over a deformable substrate, over varying incline [88–90] or changing gait , or the attached logger undergoes motion independent of the body frame (e.g., collar roll), whilst even stationary behaviours can impart appreciable DBA, all of which may affect the relationship between DBA and speed [42, 90].
DBA-derived speed estimates can sometimes break down for species that hold appreciable quantities of air underwater (such as birds ) due to the compression of the air that takes place with increasing depth, with consequent changes in upthrust and power allocation according to swim angle  (cf. the difference of VeDBA magnitude between ascents and descents – Fig. 6). In addition, DBA does not scale reliably with speed for animals that glide, use thermals (cf. Figures 5 & 7), or bank and turn sharply [69, 92] because the more a gliding bird pitches down, the faster it will travel, even though there is no change in DBA. The same is true of animals with a higher density than water, such as elasmobranchs [cf. 93], although in both cases the speed can be determined using the rate of change of altitude/depth if the pitch angle is known and it is high enough ). Recent research into fluid media speed sensors [e.g., 94] may eventually provide systems that could markedly enhance the dead-reckoning process for animals travelling in water or air. Beyond this, the principal low-resolution methods for determining a speed proxy involve GPS-derived speed estimates or constant/simulated values according to behaviour-type or topological whereabouts (the primary approach used here for aquatic/aerial movement). Clearly, using constant speed estimates (even if they are a mode of the true value) quickly give erroneous integrated travel vectors quickly, which emphasises the importance of appropriately spaced VP-correction, particularly when speed is highly variable.
Generally, environmental covariate maps are typically given with lower resolution in aerial and aquatic domains, so location errors seem less important because space-use is, anyway, typically considered at larger scales [cf. 95]. For example, foraging ‘hotspots’ can be obtained based from 3-D dive profiles and even if dead-reckoning accuracy had an approximate 500 m error radius (e.g., 1 fix/hour VP correction rate for penguins (Fig. 1)), such errors seem more acceptable in an apparently predominantly featureless ocean. The same reasoning applies to most flying species although, because so many fly over land, higher absolute resolution is often required in order to map out the specifics of land-based features, such as wind turbines [96, 97] or thermals , that are relevant for bird (or bat) movement. Airflows themselves represent dynamic environments and assessing fine-scale dead-reckoned tracks in 3-D may reveal important interactions between animal and airscape and the energetic consequences involved [cf. 49, 72, 99, 100]. Most importantly, although dead-reckoning for fliers can incur substantial wind-based drift, GPS-based VP is usually accurate, because of the open sky which enhances signal transmission , and this can help correct tracks accordingly.
Whilst the accuracy of current flow vectors may be imprecise, their integration (see Gunner et al.  for method) can improve dead-reckoning estimates substantially (both pre- and post-VP correction; Fig. 8), which is especially important when VPs are scarce. It is worth noting however, that using GPS-derived speed and/or output from speed sensors estimating parameters of flow incorporates the speed of any current flow. Against this, assessing dead-reckoned travel vectors alongside VPs and external current flow vector estimates can provide insights into movement strategies of animals compensating for current drift [cf. 83, 84].
The greater accuracy of VP-corrected dead-reckoning in terrestrial movement compared to fluid media is important because covariates of interest on land are typically highly resolved with, for example, habitat use [101, 102], conspecific interactions  and the effect of man-made structures [104, 105] (e.g., roads, fences etc.) being of interest. Unlike most aquatic and aerial species, DBA can be continuously applied as the speed proxy for land animals, and the DBA ~ speed regressions (m and c values) can be modelled according to behaviour/terrain type, for higher resolution estimates . In this, a primary factor in maximising dead-reckoning accuracy in a speed context, is to ensure that only periods of genuine traveling movement are dead-reckoned, since even stationary behaviours (e.g., grooming, feeding, rolling over etc.) can impart appreciable DBA, which can inaccurately advance the vector of travel (cf. Figure 3). Though notably, this is harder to achieve for animals in/on fluid media.
The VP correction procedure outlined in Walker et al.  (and used here), divides the distance between consecutive VPs with the corresponding distance between temporally aligned dead-reckoned positions to obtain a distance correction factor (ratio) that is multiplied to all intermediate dead-reckoned distance moved estimates. This method has the advantage that the periods when the dead-reckoned vectors are not advanced (e.g., by allocating zero speed values for stationary behaviours, which can be determined from inertial data [e.g., 38]), are not subsequently expanded out in the linear drift correction procedure (since multiplying by zero achieves a zero-correction factor). Notably though, this method of correction can inflate error, beyond the normal linear vector expansion or contraction [cf. 106]. This is particularly problematic in small looping movements because if there is a disparity in the distance estimates between successive VPs and the corresponding dead-reckoned positions, path segments may be disproportionately expanded (and even inappropriately rotated) in order for the endpoints of both to align (e.g., Fig. 7), even though such path segments may simply be an artefact of wrongly assigned speed values, VP inaccuracy or heading error. This has consequences for space-use estimates and thus drives home the importance of initial behavioural identification, speed allocation, and VP screening prior to the VP-correcting dead-reckoning procedure.
4.2 Heading inaccuracies and VP-corrected dead-reckoning accuracy
Heading is calculated using the arctangent of the ratio between two orthogonal components of the magnetic vector when the magnetic field sensor is lying flat and parallel to the Earth’s frame of reference . The tilt-compensated compass method rotates the attached tag’s magnetic vector coordinates and subsequently converts values of each magnetic vector channel to the corresponding Earth’s reference coordinate system, using the angles between the tag’s magnetic- and the gravity vector. These angles are typically expressed as pitch and roll (Euler angles), which are resolved from the static component of acceleration (the gravity vector). The difficulty can be separating the static- (due to gravity) and dynamic (due to the animal’s movement) components of acceleration [cf. 108]. Although various methods have been proposed to do this (e.g., using a running mean [59, 109] or high-pass filter ), estimates are problematic during periods of; high centripetal acceleration (‘pulling g‘; e.g., rapid cornering ), free-falling (no discernible or low gravity-component)  and highly dynamic movements . Consequently, azimuth measurement error can be inflated at times when derived static acceleration estimates break down as a proxy of tag attitude relative to the Earth’s fixed reference frame.
Incorporating gyroscopes can improve the accuracy of computed heading, since they accurately reconstruct gravity-based attitude, irrespective of acceleration . However, gyroscopes suffer from drift, high-power requirements and rapid memory consumption [113, 114]. Complex data processing makes them unappealing in most free-ranging bio-logging studies, particularly when information gains may be limited [cf. 115]. Further work should assess the extent to which gyroscopes do improve (species-specific) VP-corrected dead-reckoning accuracy, particularly at fine-scales (e.g., during fast, transient manoeuvres such as prey pursuit).
The usual method to derive Euler angles is to determine a set vectoral orientation with each orthogonal channel representing a particular body plane (anterior-posterior, medio-lateral and dorsal-ventral) with respect to the earth’s frame of reference [41, 60, 116], and the order of these channels is pivotal for deriving correct estimates of body rotation about the three axes  (for equations see; Gunner et al. ). However, this assumption breaks down for animals (or attached tags) that change orientations frequently at angles greater than perpendicular from their longitudinal and lateral axes of ‘normal’ posture due to the singularity issues (Gimbal lock) that arise when using the Euler sequence of 3-D vector rotation . This problem can be mitigated by using a quaternion-based orientation filter [118, 119], however such an approach requires complex mathematical processing which may, in part, explain why Euler rotations are favoured (at least in biologging studies). We suggest that quaternion estimated heading should be compared with Euler angle-derived heading within the dead-reckoning framework, to assess the extent of error that occurs during times when the Euler sequence for determining attitude/orientation is likely to break down (e.g., during high centripetal acceleration). At the very least (when using Euler angles), inertial measurement coordinate frame adjustments of the tag frame (reflecting the body frame) relative to the Earth should be carried out [cf. 58] for animals that carry out ≥ 90o body inversions (e.g., a penguin walking vs swimming). Small discrepancies between the tag and animal body coordinate frames are not as vital to correct for deriving heading since the tilt-compensated compass only concerns the attitude of the tag relative to the Earth so any required heading offset between the tag and animal’s body frame can be subsequently applied. In fact, consistent biases in tag heading are easily corrected for within the VP-corrected dead-reckoning framework, with the difference of heading from true North between consecutive VPs and corresponding dead-reckoned positions being applied as the correction factor. However, there is no straight-forward solution to correcting heading from tags that move independently of the body (e.g., through partial dislodgment).
Animals that undertake long migrations can be subject to variations in the strength and declination of magnetic fields and this can be difficult to account for, because the magnetometry calibration procedure , required for correcting soft and hard iron distortions , is typically performed prior to deployment and is therefore only relevant according to the specific magnetic conditions of that area. Even after sufficiently calibrating magnetometry data, local changes in the magnetic field (e.g., due to the presence of ferrous material) and temperature-induced offsets [58, 122] can introduce channel bias in measured magnetism, confounding heading output. Moreover, the horizontal components of the magnetic field become small when the magnetic-field inclination angle increase towards the poles, which can also result in heading measurement error . Lastly, heading estimates assume the animal moves in the direction of its longitudinal axis, which is not always the case .
4.3 VP inaccuracies and VP-corrected dead-reckoning accuracy
Collar data generally shows more variation in acceleration and magnetic field intensity values than data obtained from loggers deployed near an animal’s Centre of Mass (CoM), because collars can roll independent of the body frame. That our net error estimates plateaued for (collared) lions at ca. 10 m, with a 1 fix/30 mins VP correction rate demonstrates though, the value that VP-corrected dead-reckoning can have for constructing long-term, fine-scale terrestrial movement. Indeed, across all VP correction rates, distance moved estimates alone were more consistent (and higher) when estimated between dead-reckoned positions than VPs (Fig. 2). The sharp increase that occurs in distance moved estimates (at the highest VP correction rate) stems principally from incorporating all the VP locational error. Notably, the temporal sub-sampling intervals of VP correction were not always exact because fix success can fail for periods longer than the set VP correction rate (e.g., during submersion in water) . As such, we advocate that the VP correction rate should not be treated literally between species with the number and regularity of VP correction generally lower for aquatic animals per set VP sampling rate. Indeed, dead-reckoned distance moved estimates were generally much higher than the equivalent VP distance in aquatic and flying species. This is because VPs can fail for extended periods while dead-reckoning is continuous.
It is worth reemphasizing that across all travel media, dead-reckoning accuracy as assessed via net error must not be taken literally (particularly at high VP correction rates), since VP error can also be appreciable (cf. Figure 4), whilst net error does not account for inaccuracies between VPs (cf. Figure 7) and extremely high values at single points in time (likely due to VP error) may increase overall net error estimates (cf. Figure 3). Only including fixes where genuine travelling movement occurred (e.g., as assessed from motion sensor data) can help remove GPS error that occurs when animals are stationary or extremely slow-moving (e.g., tortoises) where the disparity between VP error and genuine travelling movement become disentangled (even at low VP correction rates).
4.4 Deciding drift correction rates
The specific number of VPs that are required to drift correct are obviously species-specific and there are many confounds to this process that we outline above, including user-defined track-scaling and initial VP screening, that will change on a case-by-case basis. The scenarios outlined above should provide a general idea of the required correction rates for the resolution that is required in aerial, aquatic and terrestrial domains. In essence, we suggest that VP correction should be undertaken as little as possible, but as much that is required. For investigating highly defined scales of movement (for example here, lion-fence boundary interactions or penguin navigation strategies on land) then 1 fix/15 mins or more may be required. For longer-term studies (e.g., weeks to months) general movement networks and distance moved estimates, where net errors of ca. two hundred metres, may be deemed reasonable definition for the questions being asked, much lower VP correction rates could be used to preserve battery life, allowing animals to carry smaller tags. Importantly, even when high VP correction rate is possible (e.g., ≥ 0.1 Hz), corrections should only be carried out at times of genuine traveling movement, whereby distance moved between VPs exceeds the positional error radius stemming from the precision of their measurement.
4.5 The utility of dead-reckoning
The vast majority of animal tag studies investigating space-use have done so subject to the resolution of the VP system utilised (typically GPS), something that has generally resulted in low-aspect ratio location-based point density [cf. 124] or diffusive straight-line movements [cf. 3]. VP-corrected dead-reckoning provides a means to incorporate all the various scales and directions of movement between VPs (rather than just linear interpolation ) and thus has the capacity to map out movement patterns to a hitherto-unrealized degree . Such expansion of the resolution of animal space-use into fine-scale, uninterrupted movement path networks can enhance insight into a number of fundamental concepts considered important in structuring movement paths and space-use by animals, including energy landscapes , landscapes of fear  and accident landscapes . VP-corrected dead-reckoning has particular relevance for marine underwater studies because 3-D movement can be reconstructed [54, 67] at times when VPs cannot be obtained  (e.g., Fig. 6).
The immediate benefits of using VP-corrected dead-reckoning are:
That it can reconstruct continuous, fine-scale 2-/3-D movement paths, irrespective of the environment and at higher resolution than any VP system [42, 50]
That it provides a means to reduce the recording frequency of GPS locations, thus extending battery life and/or reducing deployment bulk/weight 
That it prevents/limits positional noise (‘jitter’) of ’high-res’ (e.g., ≥ 1 Hz) GPS datasets, which is most apparent during non-moving behaviours such as rest and in highly heterogenous environments where radio signal can be easily obstructed [cf. 22, 23]
In particular, the scales of tortuosity exhibited between VPs, as defined with VP-corrected dead-reckoning, irrespective of the drift from true location (net error), can highlight behaviours that VPs alone cannot. For example, we demonstrate here that circling behaviour  can easily be distinguished in dead-reckoned tracks from tropicbirds (Fig. 5), even when the circling duration is as low as 10 s. VP-corrected dead-reckoning can also greatly improve the accuracy of space-use estimates by limiting the inclusion of positional noise via advancing travel vectors and carrying out VP correction only at times when the animal is determined to be travelling. In fact, we believe that a particular value of VP-corrected dead-reckoning, is that it will provide important detail about the effects of humans and anthropogenic landscape features on animal movements, a topic that is increasingly germane [128–130]. For example, understanding the extent of the permeability of anthropogenic barriers (e.g., fences, roads) and the hazards that they pose to specific animals [131–133] is key to proper livestock and wildlife management [134–137]. Our work demonstrates that this approach details the intricacies of animal-barrier interactions, including the locations of barrier transgression as well as movement paths pre-, during and post- barrier transgression. Moreover, VP-corrected dead-reckoning should also elucidate animal foraging- and predator avoidance strategies as well as provide vital information that will help us understand how animals respond to, and navigate through (air/tidal) current flows . Beyond this, dead-reckoning has been demonstrated to have high welfare value in zoos, by enabling continuous assessments of enclosure space-use relative to enrichment regimes and the possible occurrence of stereotypical behaviours such as pacing . Importantly, this approach has implications for informing conservation management. For instance, the impacts of free-ranging forest elephants depend largely on what they are doing at very specific localities [139, 140]. At present, GPS is mostly used to reflect on where elephants move as a general response to the availability of resources such as food, water and safety [e.g., 139, 141, 142]. Drift-corrected dead-reckoning can highlight the specifics of behaviours and localities, and therefore, for example, allow researchers to retrace elephant movements to determine what elephants feed on and where they do it, which has obvious management value. Lastly, alongside capturing under-water movements, dead-reckoning may prove effective for elucidating movement-specific behaviours in other habitats that have poor signal reception, such as within caves and burrows.
4.5 Key considerations governing the relationship between VP correction rate and dead-reckoning accuracy
To improve VP-corrected dead-reckoning estimates (assuming the accelerometer-magnetometer Euler angle approach) the minimum pre-routine should consider the following:
Screening for, and removal of, erroneous VP estimates
A suitable magnetometer calibration [120, 143] with correction of acceleration and magnetometry data for any discrepancies between the tag coordinate frame and body coordinate frame, relative to the Earth’s fixed frame of reference [e.g., by visually taking note of the deployment angle offset and derotating using rotation matrices as outlined in 58]
Application of any required magnetic declination offsets (and approximate yaw offset if step 2 was not carried out)
Computation of suitable estimates of speed (possibly modulated according to identified behaviour and/or terrain type)
Integration of external current flow vectors where appropriate (and when reasonably modelled/measured)
Post-examination of dead-reckoned tracks (both pre- and post-VP correction), relative to VPs, visually to examine and readjust aspects of the initial track scaling.
Further advances could include additional limb-borne logger deployments that may decipher limb stride frequency via clearer stylised patterns of inertial measurement [39, 144, 145]. Such counts per unit time, may themselves be used as a speed proxy . Whilst not covered here, investigation of extremely high or biased distance (speed) and heading correction factors may be used to aid in identifying inaccuracies originating from tag performance (heading, speed and/or VP inaccuracy) . Very low distance correction factors (< 1) either indicate inaccurately identified bouts of travelling movement or supplying inaccurately high-speed estimates. On the other hand, very high distance correction factors (> 1) again, could indicate inaccurately identified bouts of travelling movement, or supplying inaccurately low-speed estimates or, the most likely cause is due to VP error. Consistency in the direction of heading correction factors either indicate a yaw offset of the tag relative to the animal’s coordinate frame, a hard iron offset in magnetic data (or a required summation of the magnetic declination), or due to external current flow drift.
Generally, the factors that affect dead-reckoning and VP accuracy are illustrated in Fig. 9, with the level of obtainable dead-reckoning accuracy depending on the user-defined initial track scaling, VP screening and the study species.