Original: Physics-informed inference of  animal movements from weather radar data
Revised: Physics-informed inference of aerial 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 and conflict mitigation.

Original: For aerial movements, operational weather radars have becomean indispensable data source in this respect.
Revised: For aerial movements, operational weather radars have become an 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 it difficult 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 from high-dimensional radar measurements to low-dimensional latent representations using 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, we perform 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: A convolutional decoder maps the inferred latent system states back to the physical space in which the known radar observation model can be applied, enabling fully unsupervised training.

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

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 in terms 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 movement of 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 not only 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 invaluable opportunities for ecologists to study mass movements of birds, bats and insects that otherwise remain hidden 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 measurement range of radars is limited, the spatial coverage of operational weather radar networks is typically incomplete [15].

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 can be translated directly into animal density estimates, movements can be captured only partially by measuring the radial velocity, i.e. the component of movement along the direction of the radar beam, based on the Doppler shift [7].

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 weather radar networks remains a challenging inverse problem that requires additional knowledge to constrain the 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 vary substantially 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 lacking well-established mathematical equations that can be used as dynamical prior in data assimilation frameworks such as [23, 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, 17], leaving more fine-scale local movements unresolved.

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 can learn physically consistent spatio-temporal dynamics from high-dimensional data sets by integrating (partial) domain knowledge into the training process [14, 8, 25].

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 to ecological systems, and animal movements in particular, is hindered by the size of available data sets (particularly in Europe [21]), 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 state space models [4, 13, 20, 19], we propose a data-efficient approach to reconstruct high-dimensional time-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 model densities and velocities in a learned feature space in which the dynamics are well approximated by 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 the classical Kalman filter or smoother [10, 18].

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 is learned together with the latent dynamics in an unsupervised way, minimizing both the reconstruction loss and a physics-informed loss that exploits known mass conservation constraints based on the continuity 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-temporal dynamics.

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 and discuss how weather radars can be used to quantify such processes.

Original: Modelling 
Revised: Modelling mass movements

Original: mass 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. during migration, can be described in terms of individuals moving along a shared velocity field v that changes over time depending on weather conditions, landscape features, food availability, social interactions, etc [17].

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 the underlying 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 have become 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 angles to sample the three dimensional airspace around the antenna.

Original: With every sweep, several data productsare obtained.
Revised: With every sweep, several data products are 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 density ρ can be estimated, and radial velocity , describing associated movements towards or away from the antenna derived from the Doppler 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 velocity r ( 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 of radar measurements in combination with scattering effects restricts data collection to some maximum distance 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 and ρ 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 from the radar, and the rows of matrix A n contain the unit vectors a n ( x ) associated with radial velocity measurements 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, the reconstruction 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 v then reduces to a single vector which can easily be estimated with velocity volume profiling techniques [24, 9].

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 et al.

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 simultaneously modelling radial velocity measurements from many spatially distributed radars with a Gaussian process 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 allows ecologists 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 radial velocity measurements ˜ r t ∈ R K × L × N and log-transformed density measurements ˜ q t

Original:  R K × L × N on a two-dimensional K × L grid, taken by N different radars with known projections
Revised: = log(˜ ρ t ) ∈ 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 at reconstructing 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 for high-resolution fluid dynamics [5, 6], we assume that animal movements can be compressed to a much 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 convolutional encoder 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 the associated 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 features of 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 contain artefacts 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 dynamics and measurement errors, we model latent measurements ( w 1 , ... w T ) with a linear Gaussian state space model (LGSSM)

Original: [Equation]
Revised: [Equation]

Original: which links latent measurements to latent representations z t ∈ Z ⊆ R D of the true physical state( 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 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.
Revised: Here, H and P are predefined, µ 1 , Σ 1 , and R are learned parameters, and F t is a linear combination of C constant transition matrices F ( k ) with coefficients defined by a small neural network 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, we can perform efficient inference in latent space using the standard Kalman smoother [18].

Original: This resultsin latent state posterior estimates z +1 , ..., z
Revised: This results in 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, to recover the physical system states from some latent state z t , we use a convolutional decoder network to 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 physical space 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 known forward 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 physical consistency, i.e. compliance with conservation laws, we regularize our model based on the continuity equation (see section 4.1).

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 to densities ρ t , and discretize equation 1 in time (using forward Euler) and in space (using finite differences): [Equation]

Original: [Equation]
Revised: [Equation]

Original:  with j ( k,l ) t = (cid:104)
Revised: 6) with j ( k,l ) t = (cid:104)

Original: [Equation] [Equation] · ρ ( k
Revised: j ( k,l ) t,x , j

Original: [Equation]
Revised: [Equation]

Original: l ) t the flux at time t and grid cell ( x k , y l ) .
Revised: k,l ) t,y (cid:105) T = v ( k,l ) t · ρ ( k,l ) t the flux at time t and grid cell ( x k , y l ) .

Original: The resultingphysical loss term
Revised: The resulting physical 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 4.1 Synthetic data generation

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: Besides noisy and partial measurements, working with real weather radar data comes with several additional challenges, such as aliasing, clutter effects, irregular sampling, 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 of model 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 proposed methodology (see appendix for details).

Original: 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.
Revised: In particular, we generate 1000 training sequences andtest sequences of time-varying 2D velocity and animal density fields with a resolution of 32 ×and 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 noisy measurements { ˜ ρ n , ˜ r n } 3 n =1 of 3 randomly positioned radars are obtained by applying the respective forward model and adding independent Gaussian noise with σ = 0 . 4.2 Architecture and hyperparameters

Revised: Convolutional encoder and decoder

Revised: In all our experiments, we use a convolutional encoder network consisting of 3 layers with 3 × 3 filters, 1 × 1 stride, ReLU non-linearities, and 2 × max-pooling.

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

Original: Evaluation and baselines We compare our method against the following baselines:
Revised: A fully-connected layer with linear output maps the encoder output to a M = 128 dimensional latent measurement. Following the structure of the encoder, we decode latent states with dimension D = 128 by first applying a fully-connected layer with linear output, followed by 3 convolutional layers where in each layer we first upsample the input by factor 2, and then apply a 3 × 3 filter with 1 × 1 stride, followed by a ReLU non-linearity (for all but the last layer). LGSSM We assume both P and R to be diagonal covariance matrices. We define P = 0 .

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

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

Revised: Z to have the same dimensionality

Revised: M = D = 128 .

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

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

Revised: [Equation]

Revised: with C = 8 and α 1 , ..., α C the outputs of a 2-layer MLP with softmax activation.

Revised: Training For all experiments, we trained 3 models with different random seeds for a maximum of 100 epochs using the Adam optimizer [11] with default settings.

Revised: Based on a grid search, the learning rate is set to 0.001.

Revised: 4.3 Evaluation and baselines

Revised: 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 [12] is used to jointly model velocities and log-densities, without capturing temporal dependencies.

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 space matches 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) [24] is used to reduce the set of radial velocity measurements of each radar to a single velocity vector describing a uniform velocity field 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 the resulting 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 d n = 2 ( 98 ± 4% coverage) to d n = ∞ ( 100% coverage).
Revised: Comparison to baselines We evaluate all methods for radar ranges varying from d n = 1 ( 64 ± 13% coverage) to 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 and log-densities.

Original: Fig.
Revised: Fig.

Original: 1 visually compares the different methods based on an example velocity field fromthe test set.
Revised: 2 visually compares the different methods based on an example velocity field from the 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. [Table]

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: Data-efficiency In a second experiment, we evaluate the data-efficiency of our method by gradually changing the number 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 , ∞} are considered.

Original: While in general the RMSE increases as the number of training sequences decreases,[Table] 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: While in general the RMSE increases as the number of training sequences decreases, we are able to obtain more accurate velocity reconstructions than the VVP baseline (which does not require any training) using as little as 200 training sequences for ranges d n ∈ { 2 , ∞} andsequences for d n = 1 (see fig. 3).

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 when trained on 1000 sequences.

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

Revised: Computing the average per-grid-cell standard deviation for each time point reveals higher uncertainties at the beginning and end of each sequence, for velocities and log-densities respectively (see fig. 4).

Revised: This matches the generally higher RMSE at the beginning and end of each sequence and indicates that our method utilizes information about both past and future to generate accurate reconstructions.

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 state space models to reconstruct high-dimensional time-varying animal movement patterns from partial and noisy Doppler weather radar data, without requiring in-depth knowledge about the underlying processes 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, while requiring 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 are limited to synthetic radar data, and several additional challenges need to be tackled before applying the proposed method to real weather radar data.

Revised: Firstly, reliable separation of biological echoes from irrelevant signals (weather, clutter, etc.) is crucial for our model to learn spatial features that are indeed typical for animal movements, and to use mass conservation constraints effectively.

Revised: Although there has been significant progress in in this regard, available classification methods [22, 16] remain to be adjusted and thoroughly evaluated for European weather radar data.

Revised: Secondly, in contrast to our synthetic data generated on a 2D regular grid, real weather radar data consists of 3D polar volumes for which the resolution decreases with distance to the antenna.

Revised: To apply our method to this data, we either need to project it onto a regular grid (and thus lose information) or adjust the encoder network to work with irregularly structured data (think of point clouds or graphs) directly.

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: Finally, radial velocity measurements may be subject to aliasing, meaning that the true radial velocity is only known up to an additive multiple of the Nyquist velocity, which depends on the radar and operating mode. Training our model based on these aliased measurements will inevitably result in erroneous reconstructions. To account for aliasing effects, the deterministic forward model mapping full velocity and density fields to radar measurements needs to be replaced by a probabilistic one (e.g. using a wrapped normal likelihood similar to [2]). We are hopeful that future work in these directions will eventually help uncover aerial movement patterns of birds, bats and insects at various spatial scales.

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 insights into the underlying behaviours and movement strategies, and to develop more effective strategies for species conservation and mitigation of humanwildlife conflicts.

Original: AppendixA Data generation
Revised: Appendix A 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 between attraction to areas with food, mild climate, etc. and repulsion from geographical barriers, predators and 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 scalar value 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 a mixture 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 some fixed 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 as v ( 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 with dt = 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-varying velocity 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 distributed inputs, 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 are masked 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 toas well.

Original: B Architectures and hyperparameters

Original: 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.

Original: The number of output channels are 32, 64, and 128 respectively. A fully-connected layer with linear output maps the encoder output to a M = 128 dimensionallatent measurement. Following the structure of the encoder, we decode latent states with dimension 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: B Additional figures

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

Original: We define P = 0 .

Original: 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 Z to have the same dimensionality M = D = 128 . 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: Figure 7 shows an example time series of ground-truth velocity and density fields together with reconstructions generated by our proposed model, a convolutional VAE, and a linear interpolation based on velocity volume profiling (VVP).

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

Original: F t =

Original: C (cid:88) k = α k ( z t ) F ( k ) , with C = 8 and α 1 , ..., α C the outputs of a 2-layer MLP with softmax activation. Training We trained all models for a maximum of 100 epochs using the Adam optimizer [10] withdefault settings. For all experiments, the learning rate is set to 0.001 (based on grid search).
Revised: Our model reconstructs both velocities and densities more accurately than the considered baseline models.

Original: C Additional results

Original: 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 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 atthe beginning and end of each sequence and indicates that our method relies on information aboutboth past and future to generate accurate reconstructions.