Original: Physics-informed inference of animal movements from weather radar data
Revised: Physics-informed inference of animal movements from weather radar data

Original: Abstract
Revised: Abstract

Original: Studying animal movements is essential for effective wildlife conservation andconflict mitigation.
Revised: Studying animal movements is essential for effective wildlife conservation andconflict mitigation.

Original: For aerial movements, operational weather radars have becomean indispensable data source in this respect.
Revised: For aerial movements, operational weather radars have becomean indispensable data source in this respect.

Original: However, partial measurements, incomplete spatial coverage, and poor understanding of animal behaviours make itdifficult to reconstruct complete spatio-temporal movement patterns from available radar data.
Revised: However, partial measurements, incomplete spatial coverage, and poor understanding of animal behaviours make itdifficult to reconstruct complete spatio-temporal movement patterns from available radar data.

Original: We tackle this inverse problem by learning a mapping fromhigh-dimensional radar measurements to low-dimensional latent representationsusing a convolutional encoder.
Revised: We tackle this inverse problem by learning a mapping fromhigh-dimensional radar measurements to low-dimensional latent representationsusing a convolutional encoder.

Original: Under the assumption that the latent system dynamics are well approximated by a locally linear Gaussian transition model, weperform efficient posterior estimation using the classical Kalman smoother.
Revised: Under the assumption that the latent system dynamics are well approximated by a locally linear Gaussian transition model, weperform efficient posterior estimation using the classical Kalman smoother.

Original: Aconvolutional decoder maps the inferred latent system states back to the physical space in which the known radar observation model can be applied, enablingfully unsupervised training.
Revised: Aconvolutional decoder maps the inferred latent system states back to the physical space in which the known radar observation model can be applied, enablingfully unsupervised training.

Original: To encourage scientific consistency, we additionallyintroduce a physics-informed loss term that leverages known mass conservationconstraints.
Revised: To encourage scientific consistency, we additionallyintroduce a physics-informed loss term that leverages known mass conservationconstraints.

Original: Our experiments on synthetic radar data show promising results interms of reconstruction quality and data-efficiency.
Revised: Our experiments on synthetic radar data show promising results interms of reconstruction quality and data-efficiency.

Original:  Introduction
Revised:  Introduction

Original: Doppler weather radars provide high-resolution information about the distribution and movementof objects in the atmosphere.
Revised: Doppler weather radars provide high-resolution information about the distribution and movementof objects in the atmosphere.

Original: Although designed to monitor weather, radar beams are reflected notonly by precipitation but also by animals passing the airspace around the radar.
Revised: Although designed to monitor weather, radar beams are reflected notonly by precipitation but also by animals passing the airspace around the radar.

Original: This offers invaluableopportunities for ecologists to study mass movements of birds, bats and insects that otherwise remainhidden due to low light conditions and high flight altitudes [3].
Revised: This offers invaluableopportunities for ecologists to study mass movements of birds, bats and insects that otherwise remainhidden due to low light conditions and high flight altitudes [3].

Original: However, since the measurementrange of radars is limited, the spatial coverage of operational weather radar networks is typicallyincomplete [14].
Revised: However, since the measurementrange of radars is limited, the spatial coverage of operational weather radar networks is typicallyincomplete [14].

Original: Moreover, while the amount of energy reflected back to the radar antenna canbe translated directly into animal density estimates, movements can be captured only partially bymeasuring the radial velocity, i e. the component of movement along the direction of the radar beam.
Revised: Moreover, while the amount of energy reflected back to the radar antenna canbe translated directly into animal density estimates, movements can be captured only partially bymeasuring the radial velocity, i e. the component of movement along the direction of the radar beam.

Original: Inferring the complete underlying density and velocity fields from partial observations of weatherradar networks remains a challenging inverse problem that requires additional knowledge to constrainthe solution space.
Revised: Inferring the complete underlying density and velocity fields from partial observations of weatherradar networks remains a challenging inverse problem that requires additional knowledge to constrainthe solution space.

Original: Unfortunately, the movement of animals is much harder to understand than that of physical particles:individual behaviours depend on a wide range of environmental and social factors, and can varysubstantially within a population.
Revised: Unfortunately, the movement of animals is much harder to understand than that of physical particles:individual behaviours depend on a wide range of environmental and social factors, and can varysubstantially within a population.

Original: That means that, unlike meteorologists, ecologists are lackingwell-established mathematical equations that can be used as dynamical prior in data assimilation frameworks such as [20, 1].
Revised: That means that, unlike meteorologists, ecologists are lackingwell-established mathematical equations that can be used as dynamical prior in data assimilation frameworks such as [20, 1].

Original: Instead, strong spatio-temporal smoothness assumptions have been used to reconstruct continental-scale movement patterns [2, 15], leaving more fine-scale local movementsunresolved.
Revised: Instead, strong spatio-temporal smoothness assumptions have been used to reconstruct continental-scale movement patterns [2, 15], leaving more fine-scale local movementsunresolved.

Original: Machine learning opens up new opportunities in this respect: deep neural networks canlearn physically consistent spatio-temporal dynamics from high-dimensional data sets by integrating(partial) domain knowledge into the training process [13, 7, 22].
Revised: Machine learning opens up new opportunities in this respect: deep neural networks canlearn physically consistent spatio-temporal dynamics from high-dimensional data sets by integrating(partial) domain knowledge into the training process [13, 7, 22].

Original: However, the application toecological systems, and animal movements in particular, is hindered by the size of available data sets(particularly in Europe [19]), as well as the lack of ground-truth data.
Revised: However, the application toecological systems, and animal movements in particular, is hindered by the size of available data sets(particularly in Europe [19]), as well as the lack of ground-truth data.

Original: Inspired by recent work on combining deep learning with exact inference in linear Gaussian statespace models [4, 12, 18, 17], we propose a data-efficient approach to reconstruct high-dimensionaltime-varying animal density and velocity fields from partial and noisy radar data.
Revised: Inspired by recent work on combining deep learning with exact inference in linear Gaussian statespace models [4, 12, 18, 17], we propose a data-efficient approach to reconstruct high-dimensionaltime-varying animal density and velocity fields from partial and noisy radar data.

Original: We jointly modeldensities and velocities in a learned feature space in which the dynamics are well approximatedby a locally linear Gaussian transition model.
Revised: We jointly modeldensities and velocities in a learned feature space in which the dynamics are well approximatedby a locally linear Gaussian transition model.

Original: In this space, inference can be performed with theclassical Kalman filter or smoother [9, 16].
Revised: In this space, inference can be performed with theclassical Kalman filter or smoother [9, 16].

Original: The mapping between physical space and feature space islearned together with the latent dynamics in an unsupervised way, minimizing both the reconstructionloss and a physics-informed loss that exploits known mass conservation constraints based on thecontinuity equation.
Revised: The mapping between physical space and feature space islearned together with the latent dynamics in an unsupervised way, minimizing both the reconstructionloss and a physics-informed loss that exploits known mass conservation constraints based on thecontinuity equation.

Original: This allows us to reconstruct high-resolution movement patterns while having(i) incomplete knowledge about the underlying process, (ii) no access to ground-truth fields, and(iii) only a limited number of training samples to learn a surrogate for the complex spatio-temporaldynamics.
Revised: This allows us to reconstruct high-resolution movement patterns while having(i) incomplete knowledge about the underlying process, (ii) no access to ground-truth fields, and(iii) only a limited number of training samples to learn a surrogate for the complex spatio-temporaldynamics.

Original:  Background
Revised:  Background

Original: In this section, we briefly describe relevant prior knowledge about mass movements of animals anddiscuss how weather radars can be used to quantify such processes.
Revised: In this section, we briefly describe relevant prior knowledge about mass movements of animals anddiscuss how weather radars can be used to quantify such processes.

Original: Modelling
Revised: Modelling

Original: mass
Revised: mass

Original: movements
Revised: movements

Original: The coordinated movement of large number of animals, e.g. duringmigration, can be described in terms of individuals moving along a shared velocity field v that changesover time depending on weather conditions, landscape features, food availability, social interactions,etc [15].
Revised: The coordinated movement of large number of animals, e.g. duringmigration, can be described in terms of individuals moving along a shared velocity field v that changesover time depending on weather conditions, landscape features, food availability, social interactions,etc [15].

Original: The resulting spatio-temporal distribution of animals is restricted by the continuity equation
Revised: The resulting spatio-temporal distribution of animals is restricted by the continuity equation

Original: [Equation]
Revised: [Equation]

Original: where ρ denotes animal density.
Revised: where ρ denotes animal density.

Original: The spatio-temporal dynamics of the velocity field v and theunderlying drivers, however, remain largely unknown and need to be estimated from data.
Revised: The spatio-temporal dynamics of the velocity field v and theunderlying drivers, however, remain largely unknown and need to be estimated from data.

Original: Radar basics For birds, bats and insects, continent-wide networks of Doppler weather radars havebecome an invaluable tool to quantify mass movements over large spatial and temporal extents [3].
Revised: Radar basics For birds, bats and insects, continent-wide networks of Doppler weather radars havebecome an invaluable tool to quantify mass movements over large spatial and temporal extents [3].

Original: Every 5-15 min, Doppler weather radars perform multiple 360 ◦ sweeps at different elevation anglesto sample the three dimensional airspace around the antenna.
Revised: Every 5-15 min, Doppler weather radars perform multiple 360 ◦ sweeps at different elevation anglesto sample the three dimensional airspace around the antenna.

Original: With every sweep, several data productsare obtained.
Revised: With every sweep, several data productsare obtained.

Original: In this work, we are specifically interested in reflectivity , from which the animal densityρ can be estimated, and radial velocity , describing associated movements towards or away from theantenna derived from the phase shift of reflected radio waves.
Revised: In this work, we are specifically interested in reflectivity , from which the animal densityρ can be estimated, and radial velocity , describing associated movements towards or away from theantenna derived from the phase shift of reflected radio waves.

Original: More precisely, the radial velocityr ( x ) = a ( x ) T v ( x ) of a moving object at location x is the projection of the full velocity vector v ( x )onto the unit vector a ( x ) pointing in the direction of the radar beam.
Revised: More precisely, the radial velocityr ( x ) = a ( x ) T v ( x ) of a moving object at location x is the projection of the full velocity vector v ( x )onto the unit vector a ( x ) pointing in the direction of the radar beam.

Original: In general, the geometry ofradar measurements in combination with scattering effects restricts data collection to some maximumdistance from the antenna.
Revised: In general, the geometry ofradar measurements in combination with scattering effects restricts data collection to some maximumdistance from the antenna.

Original: Hence, we can define the forward model H mapping the true fields v andρ to measurements taken by radar n :
Revised: Hence, we can define the forward model H mapping the true fields v andρ to measurements taken by radar n :

Original: [Equation]
Revised: [Equation]

Original: where f <d n ( · ) represents the partial (i.e. masked) observation of a quantity up to distance d n fromthe radar, and the rows of matrix A n contain the unit vectors a n ( x ) associated with radial velocitymeasurements taken by radar n .
Revised: where f <d n ( · ) represents the partial (i.e. masked) observation of a quantity up to distance d n fromthe radar, and the rows of matrix A n contain the unit vectors a n ( x ) associated with radial velocitymeasurements taken by radar n .

Original: Velocity field reconstruction Unless measurements of multiple radars are spatially overlapping, thereconstruction of v from { ˜ r n } Nn =1 presents an under-determined inverse problem.
Revised: Velocity field reconstruction Unless measurements of multiple radars are spatially overlapping, thereconstruction of v from { ˜ r n } Nn =1 presents an under-determined inverse problem.

Original: To constrain the solution space, ecological studies traditionally assume movements to be uniform.
Revised: To constrain the solution space, ecological studies traditionally assume movements to be uniform.

Original: The vector field vthen reduces to a single vector which can easily be estimated with velocity volume profiling techniques[21, 8].
Revised: The vector field vthen reduces to a single vector which can easily be estimated with velocity volume profiling techniques[21, 8].

Original: In practice, however, animal movements are heterogeneous: local weather conditions, barriers,food availability, etc. lead to high variability in both movement directions and speeds.
Revised: In practice, however, animal movements are heterogeneous: local weather conditions, barriers,food availability, etc. lead to high variability in both movement directions and speeds.

Original: Angell etal.
Revised: Angell etal.

Original: [2] made a first step towards reconstructing spatially detailed velocity fields by simultaneouslymodelling radial velocity measurements from many spatially distributed radars with a Gaussianprocess model that encodes smoothness assumptions about the latent velocity field.
Revised: [2] made a first step towards reconstructing spatially detailed velocity fields by simultaneouslymodelling radial velocity measurements from many spatially distributed radars with a Gaussianprocess model that encodes smoothness assumptions about the latent velocity field.

Original: While this allowsecologists to recover large-scale movements across continents, more fine-scale patterns are neglected.
Revised: While this allowsecologists to recover large-scale movements across continents, more fine-scale patterns are neglected.

Original:  Method
Revised:  Method

Original: Consider a time series of radar observations ( o 1 , ..., o T ) with o t =
Revised: Consider a time series of radar observations ( o 1 , ..., o T ) with o t =

Original: [˜ r t , ˜ q t ] consisting of radialvelocity measurements ˜ r t ∈ R K × L × N and log-transformed density measurements ˜ q t = log(˜ ρ t ) ∈
Revised: [˜ r t , ˜ q t ] consisting of radialvelocity measurements ˜ r t ∈ R K × L × N and log-transformed density measurements ˜ q t = log(˜ ρ t ) ∈

Original: R K × L × N on a two-dimensional K × L grid, taken by N different radars with known projections
Revised: R K × L × N on a two-dimensional K × L grid, taken by N different radars with known projections

Original: A =
Revised: A =

Original: [ A 1 , ..., A N ] and ranges d = [ d 1 , ..., d N ] .
Revised: [ A 1 , ..., A N ] and ranges d = [ d 1 , ..., d N ] .

Original: Given this partial and noisy data, we aim atreconstructing the true underlying velocity and log-density fields { v t , q t } Tt =1 .
Revised: Given this partial and noisy data, we aim atreconstructing the true underlying velocity and log-density fields { v t , q t } Tt =1 .

Original: Encoding radar measurements Inspired by traditional reduced order modelling approaches forhigh-resolution fluid dynamics [5, 6], we assume that animal movements can be compressed to amuch lower dimensional space.
Revised: Encoding radar measurements Inspired by traditional reduced order modelling approaches forhigh-resolution fluid dynamics [5, 6], we assume that animal movements can be compressed to amuch lower dimensional space.

Original: To learn this unknown space from data, we use a convolutionalencoder network f enc ( o t , A )
Revised: To learn this unknown space from data, we use a convolutionalencoder network f enc ( o t , A )

Original: = w t that maps high-dimensional measurements
Revised: = w t that maps high-dimensional measurements

Original: o t together with theassociated radar projections
Revised: o t together with theassociated radar projections

Original: A to a latent measurement w t ∈ W ⊆
Revised: A to a latent measurement w t ∈ W ⊆

Original: R M capturing key spatial featuresof the measured fields.
Revised: R M capturing key spatial featuresof the measured fields.

Original: Linear Gaussian state space model
Revised: Linear Gaussian state space model

Original: Apart from spatial features, animal movements are characterised by strong temporal dependencies.
Revised: Apart from spatial features, animal movements are characterised by strong temporal dependencies.

Original: Moreover, radar measurements can be noisy or containartefacts due to reflections by other objects in the atmosphere.
Revised: Moreover, radar measurements can be noisy or containartefacts due to reflections by other objects in the atmosphere.

Original: To account for both temporal dynamicsand measurement errors, we model latent measurements ( w 1 , ... w T ) with a linear Gaussian statespace model (LGSSM)
Revised: To account for both temporal dynamicsand measurement errors, we model latent measurements ( w 1 , ... w T ) with a linear Gaussian statespace model (LGSSM)

Original: [Equation]
Revised: [Equation]

Original: which links latent measurements toa latent representation z t ∈ Z ⊆ R D of the true physicalstate ( v t , q t ) .
Revised: which links latent measurements to latent representations z t ∈ Z ⊆ R D of the true physical state( v t , q t ) .

Original: Here, H and P are predefined, µ 1 , Σ 1 , and R are learned parameters, and F t is alinear combination of C constant transition matrices F ( k ) with coefficients defined by a small neuralnetwork that takes the current latent state z t as input.
Revised: Here, H and P are predefined, µ 1 , Σ 1 , and R are learned parameters, and F t is a linearcombination of C constant transition matrices F ( k ) with coefficients defined by a small neuralnetwork that takes the current latent state z t as input.

Original: Inference Given a sequence of encoded measurements ( w 1 , ..., w T ) and the LGSSM from eq. 3, wecan perform efficient inference in latent space using the standard Kalman smoother [16].
Revised: Inference Given a sequence of encoded measurements ( w 1 , ..., w T ) and the LGSSM from eq. 3, wecan perform efficient inference in latent space using the standard Kalman smoother [16].

Original: This resultsin latent state posterior estimates z +1 , ..., z
Revised: This resultsin latent state posterior estimates z +1 , ..., z

Original: + T and corresponding covariances
Revised: + T and corresponding covariances

Original: Σ +1 , ..., Σ + T .
Revised: Σ +1 , ..., Σ + T .

Original: Finally, torecover the physical system states from some latent state z t , we use a convolutional decoder networkto learn the inverse mapping f dec ( z t ) =
Revised: Finally, torecover the physical system states from some latent state z t , we use a convolutional decoder networkto learn the inverse mapping f dec ( z t ) =

Original: [ v t , q t ] from the latent state space Z back to the physicalspace P ⊆
Revised: [ v t , q t ] from the latent state space Z back to the physicalspace P ⊆

Original: R K × L × 3 .
Revised: R K × L × 3 .

Original: Training Since in practice it is impossible to obtain data on the true physical states { v t , q t } Tt =1 ,we rely on unsupervised learning.
Revised: Training Since in practice it is impossible to obtain data on the true physical states { v t , q t } Tt =1 ,we rely on unsupervised learning.

Original: That means, we minimize the reconstruction loss
Revised: That means, we minimize the reconstruction loss

Original: [Equation]
Revised: [Equation]

Original: where we first map each posterior estimate
Revised: where we first map each posterior estimate

Original: z + t back to physical space, and then apply the knownforward model H (see eq. 2) to obtain the corresponding reconstruction of radar measurements.
Revised: z + t back to physical space, and then apply the knownforward model H (see eq. 2) to obtain the corresponding reconstruction of radar measurements.

Original: Incorporating physical constraints To encourage scientific consistency, we regularize our modelbased on the continuity equation (see section 4).
Revised: Incorporating physical constraints To encourage scientific consistency, we regularize our modelbased on the continuity equation (see section 4).

Original: In particular, we map log-densities q t back todensities ρ t , and discretize equation 1 in time (using forward Euler) and in space (using finitedifferences):
Revised: In particular, we map log-densities q t back todensities ρ t , and discretize equation 1 in time (using forward Euler) and in space (using finitedifferences):

Original: [Equation]
Revised: [Equation]

Original: with j ( k,l ) t = (cid:104)
Revised: with j ( k,l ) t = (cid:104)

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: · ρ ( k
Revised: · ρ ( k

Original: [Equation]
Revised: [Equation]

Original: l ) t the flux at time t and grid cell ( x k , y l ) .
Revised: l ) t the flux at time t and grid cell ( x k , y l ) .

Original: The resultingphysical loss term
Revised: The resultingphysical loss term

Original: [Equation]
Revised: [Equation]

Original: is combined with the reconstruction loss to form the final training objective
Revised: is combined with the reconstruction loss to form the final training objective

Original: [Equation]
Revised: [Equation]

Original: where θ includes encoder, decoder, and LGSSM parameters.
Revised: where θ includes encoder, decoder, and LGSSM parameters.

Original:  Experiments and results
Revised:  Experiments and results

Original: Synthetic data generation Besides noisy and partial measurements, working with real weatherradar data comes with several additional challenges, such as aliasing, clutter effects, and discriminating biology from weather.
Revised: Synthetic data generation Besides noisy and partial measurements, working with real weatherradar data comes with several additional challenges, such as aliasing, clutter effects, and discriminating biology from weather.

Original: Moreover, the lack of ground truth fields prevents a thorough evaluation ofmodel reconstructions.
Revised: Moreover, the lack of ground truth fields prevents a thorough evaluation ofmodel reconstructions.

Original: We thus generate a synthetic radar data set to train and evaluate our proposedmethodology (see appendix for details).
Revised: We thus generate a synthetic radar data set to train and evaluate our proposedmethodology (see appendix for details).

Original: In particular, we generate 1000 training sequences and 100test sequences of time-varying 2D velocity and animal density fields with a resolution of 32 × 32and length T = 20 (each corresponding to ca.
Revised: In particular, we generate 1000 training sequences and 50test sequences of time-varying 2D velocity and animal density fields with a resolution of 32 × 32and length T = 20 (each corresponding to ca.

Original: 2-6h of radar data).
Revised: 2-6h of radar data).

Original: The associated partial and noisymeasurements { ˜ ρ n , ˜ r n } 3 n =1 of 3 randomly positioned radars are obtained by applying the respectiveforward model and adding independent Gaussian noise with σ = 0 .
Revised: The associated partial and noisymeasurements { ˜ ρ n , ˜ r n } 3 n =1 of 3 randomly positioned radars are obtained by applying the respectiveforward model and adding independent Gaussian noise with σ = 0 .

Original: Evaluation and baselines We compare our method against the following baselines:
Revised: Evaluation and baselines We compare our method against the following baselines:

Original: A convolutional VAE [11] is used to jointly model velocities and log-densities, without capturing temporaldependencies.
Revised: A convolutional VAE [11] is used to jointly model velocities and log-densities, without capturing temporaldependencies.

Original: The encoder and decoder architecture as well as the dimensionality of the latent spacematches our proposed model.
Revised: The encoder and decoder architecture as well as the dimensionality of the latent spacematches our proposed model.

Original: Further, velocity volume profiling (VVP) [21] is used to reduce the setof radial velocity measurements of each radar to a single velocity vector describing a uniform velocityfield around the radar.
Revised: Further, velocity volume profiling (VVP) [21] is used to reduce the setof radial velocity measurements of each radar to a single velocity vector describing a uniform velocityfield around the radar.

Original: An estimate of the full velocity field is obtained by linearly interpolating theresulting N velocity vectors in space.
Revised: An estimate of the full velocity field is obtained by linearly interpolating theresulting N velocity vectors in space.

Original: All methods are evaluated based on their reconstruction quality,measured by the root mean squared error (RMSE) between model reconstruction and ground-truth,for velocities and log-transformed densities respectively.
Revised: All methods are evaluated based on their reconstruction quality,measured by the root mean squared error (RMSE) between model reconstruction and ground-truth,for velocities and log-transformed densities respectively.

Original: Experiments We evaluate all methods for radar ranges varying from d n = 1 ( 64 ± 13% coverage)to
Revised: Experiments We evaluate all methods for radar ranges varying from d n = 1 ( 64 ± 13% coverage)to

Original: d n = 2 ( 98 ± 4% coverage) to d n = ∞ ( 100% coverage).
Revised: d n = 2 ( 98 ± 4% coverage) to d n = ∞ ( 100% coverage).

Original: The results, summarized in table 1,show that for all considered ranges our method yields the lowest RMSE for both velocities andlog-densities.
Revised: The results, summarized in table 1,show that for all considered ranges our method yields the lowest RMSE for both velocities andlog-densities.

Original: Fig.
Revised: Fig.

Original: 1 visually compares the different methods based on an example velocity field fromthe test set.
Revised: 1 visually compares the different methods based on an example velocity field fromthe test set.

Original: Our method is well able to capture the patterns of both direction and speed of movement,whereas the two baseline methods fail to reconstruct some of the local structures.
Revised: Our method is well able to capture the patterns of both direction and speed of movement,whereas the two baseline methods fail to reconstruct some of the local structures.

Original: In a second experiment, we evaluate the data-efficiency of our method by gradually changing thenumber of training sequences from 100 to 1000.
Revised: In a second experiment, we evaluate the data-efficiency of our method by gradually changing thenumber of training sequences from 100 to 1000.

Original: As before, radar ranges d n ∈ { 1 , 2 , ∞} areconsidered.
Revised: As before, radar ranges d n ∈ { 1 , 2 , ∞} areconsidered.

Original: While in general the RMSE increases as the number of training sequences decreases,
Revised: While in general the RMSE increases as the number of training sequences decreases,

Original: [Table]
Revised: [Table]

Original: we are able to obtain more accurate velocity reconstructions than the VVP baseline (which doesnot require any training) using as little as 200 training sequences for ranges d n ∈ { 2 , ∞} and 500sequences for d n = 1 (see fig. 2).
Revised: we are able to obtain more accurate velocity reconstructions than the VVP baseline (which doesnot require any training) using as little as 200 training sequences for ranges d n ∈ { 2 , ∞} and 500sequences for d n = 1 (see fig. 2).

Original: In comparison, the VAE cannot beat the VVP baseline, even whentrained on 1000 sequences.
Revised: In comparison, the VAE cannot beat the VVP baseline, even whentrained on 1000 sequences.

Original:  Conclusion
Revised:  Conclusion

Original: In this work, we combined deep spatial feature learning with exact inference in linear Gaussian statespace models to reconstruct high-dimensional time-varying animal movement patterns from partialand noisy Doppler weather radar data, without requiring in-depth knowledge about the underlyingprocesses or access to ground-truth data.
Revised: In this work, we combined deep spatial feature learning with exact inference in linear Gaussian statespace models to reconstruct high-dimensional time-varying animal movement patterns from partialand noisy Doppler weather radar data, without requiring in-depth knowledge about the underlyingprocesses or access to ground-truth data.

Original: Our results indicate that this approach is able to reconstruct movement patterns in more detail than traditional approaches based on uniformity assumptions, whilerequiring less data than models neglecting spatio-temporal movement dynamics.
Revised: Our results indicate that this approach is able to reconstruct movement patterns in more detail than traditional approaches based on uniformity assumptions, whilerequiring less data than models neglecting spatio-temporal movement dynamics.

Original: Our experiments arelimited to synthetic radar data, and several additional challenges need to be tackled before applyingthe proposed method to real weather radar data.
Revised: Our experiments arelimited to synthetic radar data, and several additional challenges need to be tackled before applyingthe proposed method to real weather radar data.

Original: However, we are hopeful that future work in thisdirection will eventually uncover aerial movement patterns of birds, bats and insects at various spatialscales.
Revised: However, we are hopeful that future work in thisdirection will eventually uncover aerial movement patterns of birds, bats and insects at various spatialscales.

Original: In the long term, this opens up opportunities for ecologists to obtain novel scientific insightsinto the underlying behaviours and movement strategies, and to develop more effective strategies forspecies conservation and mitigation of human-wildlife conflicts.
Revised: In the long term, this opens up opportunities for ecologists to obtain novel scientific insightsinto the underlying behaviours and movement strategies, and to develop more effective strategies forspecies conservation and mitigation of human-wildlife conflicts.

Original: AppendixA Data generation
Revised: AppendixA Data generation

Original: We generate synthetic velocity fields by describing animal movements as the interplay betweenattraction to areas with food, mild climate, etc. and repulsion from geographical barriers, predatorsand hostile environmental conditions.
Revised: We generate synthetic velocity fields by describing animal movements as the interplay betweenattraction to areas with food, mild climate, etc. and repulsion from geographical barriers, predatorsand hostile environmental conditions.

Original: That means, we assign each point in time and space a scalarvalue ϕ ( t, x ) , representing the associated attractiveness (or potential).
Revised: That means, we assign each point in time and space a scalarvalue ϕ ( t, x ) , representing the associated attractiveness (or potential).

Original: We initialize ϕ (0 , x ) as amixture of 10 two-dimensional Gaussians with randomly sampled mean and diagonal covariance.
Revised: We initialize ϕ (0 , x ) as amixture of 10 two-dimensional Gaussians with randomly sampled mean and diagonal covariance.

Original: Given ϕ ( t, x ) , we generate auto-correlated potentials at time t + 1 , by shifting each mode by somefixed displacement vector.
Revised: Given ϕ ( t, x ) , we generate auto-correlated potentials at time t + 1 , by shifting each mode by somefixed displacement vector.

Original: Finally, the velocity fields associated with ϕ ( t, x ) are computed asv ( t, x ) =
Revised: Finally, the velocity fields associated with ϕ ( t, x ) are computed asv ( t, x ) =

Original: −∇ ϕ
Revised: −∇ ϕ

Original: ( t, x ) .
Revised: ( t, x ) .

Original: Similar to ϕ (0 , x ) , we initialize animal densities as a mixture of 10 two-dimensional Gaussians.
Revised: Similar to ϕ (0 , x ) , we initialize animal densities as a mixture of 10 two-dimensional Gaussians.

Original: Then,we simulate eq.
Revised: Then,we simulate eq.

Original: 1 forward in time
Revised: 1 forward in time

Original: using a forward time centered space finite difference scheme withdt = 0 .
Revised: using a forward time centered space finite difference scheme withdt = 0 .

Original: 001 to generate a time-varying density fields ρ ( t, x ) that are consistent with the time-varyingvelocity field v .
Revised: 001 to generate a time-varying density fields ρ ( t, x ) that are consistent with the time-varyingvelocity field v .

Original: The resulting fields are sampled at ∆ t = 0 .
Revised: The resulting fields are sampled at ∆ t = 0 .

Original: 025 to obtain the final time series.
Revised: 025 to obtain the final time series.

Original: In practice, density distributions are highly skewed.
Revised: In practice, density distributions are highly skewed.

Original: To obtain approximately normally distributedinputs, we work with log-transformed densities instead.
Revised: To obtain approximately normally distributedinputs, we work with log-transformed densities instead.

Original: Before feeding v and log( ρ ) to our model,we rescale them to values between -1 and 1.
Revised: Before feeding v and log( ρ ) to our model,we rescale them to values between -1 and 1.

Original: Measurements outside the range d n of radar n aremasked by setting them to zero.
Revised: Measurements outside the range d n of radar n aremasked by setting them to zero.

Original: The corresponding vectors in the projection matrix A n are set to 0as well.
Revised: The corresponding vectors in the projection matrix A n are set to 0as well.

Original: B Architectures and hyperparameters
Revised: B Architectures and hyperparameters

Original: Convolutional encoder and decoder
Revised: Convolutional encoder and decoder

Original: In all our experiments (including VAE baseline), we usea convolutional encoder network consisting of 3 layers with 3 × 3 filters, 1 × 1 stride, ReLU nonlinearities, and 2 × 2 max-pooling.
Revised: In all our experiments (including VAE baseline), we usea convolutional encoder network consisting of 3 layers with 3 × 3 filters, 1 × 1 stride, ReLU nonlinearities, and 2 × 2 max-pooling.

Original: The number of output channels are 32, 64, and 128 respectively.
Revised: The number of output channels are 32, 64, and 128 respectively.

Original: A fully-connected layer with linear output maps the encoder output to a M = 128 dimensionallatent measurement.
Revised: A fully-connected layer with linear output maps the encoder output to a M = 128 dimensionallatent measurement.

Original: Following the structure of the encoder, we decode latent states with dimension
Revised: Following the structure of the encoder, we decode latent states with dimension

Original: D = 128 by first applying a fully-connected layer with linear output, followed by 3 convolutionallayers where in each layer we first upsample the input by factor 2, and then apply a 3 × 3 filter with1 × 1 stride, followed by a ReLU non-linearity (for all but the last layer).
Revised: D = 128 by first applying a fully-connected layer with linear output, followed by 3 convolutionallayers where in each layer we first upsample the input by factor 2, and then apply a 3 × 3 filter with1 × 1 stride, followed by a ReLU non-linearity (for all but the last layer).

Original: LGSSM We assume both P and R to be diagonal covariance matrices.
Revised: LGSSM We assume both P and R to be diagonal covariance matrices.

Original: We define P = 0 .
Revised: We define P = 0 .

Original: 1 · I D ,while the diagonal σ R is learned.
Revised: 1 · I D ,while the diagonal σ R is learned.

Original: The latent measurement operator H is set to the identity matrix I D ,assuming the latent measurement space W and the latent state space
Revised: The latent measurement operator H is set to the identity matrix I D ,assuming the latent measurement space W and the latent state space

Original: Z to have the same dimensionality
Revised: Z to have the same dimensionality

Original: M = D = 128 .
Revised: M = D = 128 .

Original: Further, we define the prior for time point t = 1 as N ( z 1 | 0 , 10 · I D ) , expressinghigh uncertainty about the initial state.
Revised: Further, we define the prior for time point t = 1 as N ( z 1 | 0 , 10 · I D ) , expressinghigh uncertainty about the initial state.

Original: Finally, we define the transition matrix F t as
Revised: Finally, we define the transition matrix F t as

Original: F t 
Revised: F t =

Original: [Equation]

Original:  (cid:88) k
Revised: C (cid:88) k

Original: [Equation]

Original: [Equation] F [Equation] with C = 8 and α 1 , ..., α C the outputs of a 2-layer MLP with softmax activation.
Revised: = α k ( z t ) F ( k ) , with C = 8 and α 1 , ..., α C the outputs of a 2-layer MLP with softmax activation.

Original: Training We trained all models for a maximum of 100 epochs using the Adam optimizer [10] withdefault settings.
Revised: Training We trained all models for a maximum of 100 epochs using the Adam optimizer [10] withdefault settings.

Original: The learning rate is set to 0.001 (based on grid search).
Revised: For all experiments, the learning rate is set to 0.001 (based on grid search).

Original: C Additional results
Revised: C Additional results

Original: Based on  latent posteriorestimates N ( z + t , Σ + t ) , we can quantify the uncertainty associated with thereconstructed velocity and log-density fields by repeatedly sampling ˆ z t ∼ N ( z + t , Σ + t ) and decodingˆ z t to obtain a set of physical reconstructions.
Revised: Based on the estimated latent posterior N ( z + t , Σ + t ) , we can quantify the uncertainty associatedwith the reconstructed velocity and log-density fields by repeatedly sampling ˆ z t ∼ N ( z + t , Σ + t ) anddecoding ˆ z t to obtain a set of physical reconstructions.

Original: Computing the average per-grid-cell standard deviationfor each time point reveals higher uncertainties at the beginning and  of each sequence, for velocitiesand log-densities respectively (see fig. 5).
Revised: Computing the average per-grid-cell standarddeviation for each time point reveals higher uncertainties at the beginning and end of each sequence,for velocities and log-densities respectively (see fig. 5).

Original: This matches the generally higher RMSE at the beginningand end of each sequence and indicates that our method relies on information about both past andfuture to generate accurate reconstructions.
Revised: This matches the generally higher RMSE atthe beginning and end of each sequence and indicates that our method relies on information aboutboth past and future to generate accurate reconstructions.