1. Introduction
 A priori knowledge of motion properties: For the motion of environmental phenomena, a priori knowledge of the field motion properties can be assumed. For example, wind speed statistics exist or information on the advection of rainclouds. This domain knowledge can be used to specify the required parameters of the proposed algorithm.
 Temporal continuity and spatial uniformity of motion: In arbitrary images, moving objects, such as cars, can occur that change their direction, stop or accelerate rather quickly, e.g., within a couple of frames. Therefore, temporal continuity usually only holds for very short time periods [10]. For atmospheric or oceanographic fields, it is likely that the motion is rather consistent over long sampling periods, and therefore, integration of motion information over time gains importance. In addition, motion in images often exhibits sharp motion discontinuities, e.g., at boundaries of the moving objects such as cars with opposing motion that pass by each other. For atmospheric or oceanographic fields, sharp motion discontinuities can be considered unlikely: they might only arise at boundaries of the field, e.g., at the boundaries of rain clouds. However, then, they do not exist in reality (i.e., the atmosphere surrounding the rain cloud also moves), but affect the motion estimation algorithm that is based on field measurements. For example, while the atmosphere still moves when there is no rain, the motion can only be estimated when there is some rain, which is not uniform (the uniformity of the field values is commonly known as the “blank wall” problem in the optical flow literature).
 Controlled node deployment, sampling rate and irregular data: In imagebased optical flow, the pixels determine the sampling locations, and the motion speed and direction are influenced by numerous factors, such as spatial resolution and the orientation of the image, sampling rate, the spatial distance of the camera to the moving object, as well as the speed and direction of the moving object. When motion is to be estimated with a GSN deployed within (i.e., in situ) and sensing the field, the node deployment and sampling rate can be controlled to a certain extent, for example, the node spacing relative to the assumed field motion. However, while imagebased optical flow approaches usually rely on regular grids (i.e., images), a gridlike deployment of the nodes might not be possible.
 Decentralized estimation: Imagebased OF usually relies on a single computer that estimates the motion field from the images. With a GSN, the decentralized estimation of motion by each node is possible and potentially desirable for several reasons, which are described in the next section.
2. Contributions of This Work and Differences from Previous Work
 Previous work included a rather ad hoc error model for the individual observations. Here, a more sophisticated, probabilistic error model is introduced, and extensive evaluations illustrate the usefulness of the model.
 A more generic formalization of the optical flow algorithm is provided, which accounts for possible changes in motion over time.
 The algorithm is evaluated along the error measures of differences in motion speed and motion direction between true and estimated motion. The angular difference is a common error measure for optical flow approaches [15]. Motion speed is considered to be the other most important property of motion.
 More comprehensive simulations illustrate the performance of the algorithm in different conditions. Further, the algorithm is applied to simulated GSN sampling data from weather radar images collected during a precipitation event, and the motion estimation performance of the GSN is compared to that of a stateoftheart imagebased optical flow algorithm applied to the weather radar directly, which is a common approach from nowcasting [5]. In this way, the most complete information from the radar images can be compared to the sparse information from the nodes.
3. Background and Related Work
3.1. Related Work
3.2. Network and Field Model
3.3. Optical Flow: Basics
4. Methodology
 Gradient constraint estimation: Estimates of a gradient constraint at a node ${n}_{i}$ at time t, ${\widehat{z}}_{X}^{\prime}\left({u}_{i,t}\right)$, ${\widehat{z}}_{Y}^{\prime}\left({u}_{i,t}\right)$ and ${\widehat{z}}_{T}^{\prime}\left({u}_{i,t}\right)$, are calculated from sensor measurements of neighboring nodes. The error of a GC is derived from the spatial configuration of the node neighborhood. Details are given in Section 4.1.
 Motion estimation: A set of estimates of gradient constraints is integrated at each time step t by each node ${n}_{i}$ over its direct 1hop neighborhood to solve for the motion components. Further details are provided in Section 4.2.
4.1. Gradient Constraint Estimation
4.1.1. Stationarity of Nodes
4.1.2. Error in Derivative Calculation
4.1.3. Gradient Constraint Error Estimation
 Zerofield values: When the field values used for derivative calculation are zero and, hence, all derivatives are zero, the GC is not estimated at all. Usually, measurements of spatiotemporal fields are zeroinflated, meaning that the majority of samples are zero. In such a case, no motion can be estimated, and node energy can be saved.
 Node neighborhood extending field boundary: When only one of the sensor samples for estimating the GC is zero, the GC is not estimated at all, as the node neighborhood extends over the field boundaries. While the GC could still be estimated using the remaining sensor samples, this is not done, as the least squares matrices for estimating the partial derivatives are precomputed (see [14] for the details).
 Nonzero, but equal field values: When the field is completely flat, corresponding to the “blank wall” problem described previously, the GC is not estimated at all. This case can be recognized, when all of the neighboring sensor samples of a timestep t are larger than zero, but equal. Then, the derivatives are zero, and the GC does not contribute to the motion estimation.
4.2. Temporal Coherence: Kalman Filter for Motion Estimation
5. Empirical Evaluation
5.1. Study Area, Sensor Network and Deployment Strategy
5.2. Error Measures
5.3. Setting the Filter Parameters
5.4. Results: Simulated Field
5.4.1. Influence of Field Linearity, Field Speed and Node Density
5.4.2. Influence of Deployment Strategy
5.4.3. Influence of Kalman Measurement Noise
5.5. Results: Radar Field
6. Discussion and Conclusions
 Field properties and deployment density: The simulations have shown that the degree of field linearity in conjunction with the motion speed is an important factor and that the reachable accuracy decreases with increasing nonlinearity and motion speed (Figure 5). This is a fact that is known from work on imagebased optical flow, but has direct implications in a GSN setting where the deployment and sampling rate of the nodes can be controlled to a certain extent. Since small motion in space is advantageous due to the Taylor expansion of Equation (2), it is important that the field is sampled at a high sampling rate. In addition, it is also important that the nodes are deployed rather densely and close to each other (which is also beneficial concerning power consumption) since the accuracy of estimating partial derivatives decreases with increasing node distance.
 Deployment and stationarity of nodes: When the number of nodes and communication distance are held fixed, the deployment strategy of stationary nodes does not have a large influence on the motion estimation results (Figure 6). However, with decreasing node density, the performance of a random deployment will certainly decrease, since there will be disconnected nodes. Further, the algorithm has been developed for stationary nodes. Nonetheless, it can also be applied in a nonstationary setting, e.g., for cars. However, then the motion estimation accuracy decreases (Figure 6) due to the increased error in the temporal derivative estimate and the linear alignment of the cars along roads.
Appendix A. Algorithm Protocol
Protocol 1: Field motion estimation with a node ${n}_{i}$ 

INIT broadcast $({x}_{i},{y}_{i})$ if $\leftnbr\right({n}_{i}\left)\right<2$ become PROCESSING Receiving neighboring node position ($({x}_{j},{y}_{j})$) if $\leftnbr\right({n}_{i}\left)\right>1$ compute ${d}_{ij}$ and unit vector ${\widehat{u}}_{ij}$ add ${\widehat{u}}_{ij}$ as new row to ${A}_{i}$ add $r/{d}_{ij}$ as new diagonal entry to ${W}_{i}$ if rows(A_{i})=nbr_{>1}(n_{i}) compute normal equations ${M}_{i}={\left({A}_{i}^{T}{W}_{i}{A}_{i}\right)}^{1}{A}_{i}^{T}{W}_{i}$ compute ${\sigma}_{G{C}_{i}}^{2}$ (Equation (12)) add (i, ${\sigma}_{G{C}_{i}}^{2}$) to ${D}_{\sigma}$ broadcast ${\sigma}_{G{C}_{i}}^{2}$ to nbr_{>1}(n_{i}) Receiving ${\sigma}_{G{C}_{j}}^{2}$ of neighboring node ${n}_{j}$ add (j, ${\sigma}_{G{C}_{j}}^{2}$) to ${D}_{\sigma}$ if ${D}_{\sigma}={\mathrm{nbr}}_{>1}\left({n}_{i}\right)+1$ become PROCESSING PROCESSING at time step t

Acknowledgments
Author Contributions
Conflicts of Interest
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