Precise Determination of Sediment Dynamics Using ...

Precise Determination of Sediment Dynamics Using Low-cost GPS-floaters Markus Vennebusch, Lena Albert Steffen Sch¨on, Franziska Kube

Nils Goseberg, Anna Zorndt, Torsten Schlurmann

Institut f¨ur Erdmessung (Institute of Geodesy) Leibniz University Hanover Schneiderberg 50, D-30167 Hannover, Germany Email: {vennebusch,schoen,kube}@ife.uni-hannover.de

Franzius-Institute for Hydraulic, Waterways, and Coastal Engineering Nienburger Strasse 4, D-30167 Hannover, Germany Email: {goseberg,zorndt,schlurmann}@fi.uni-hannover.de

Andreas Wurpts Coastal Research Station Lower Saxony Water Management, Coastal Defence and Nature Conservation Agency (NLWKN) An der M¨uhle 5, D-26548 Norderney, Germany Email: [email protected] Abstract—For many water engineering tasks, precise information about the water flow characteristics is required to model discharge and sediment transport. In many cases, the Global Positioning System (GPS) is used to measure floater trajectories and to derive flow velocities indirectly from floater positions. In this paper, we describe a new GPS-based measurement system that avoids the use of expensive equipment and which uses a direct velocity determination approach. Despite its low costs, the new floater system achieves position accuracies of several meters and accuracies of about 5 cm/s for velocities. We describe both hardware and software details of the new system. Results from a first measurement campaign show the system’s capabilities for hydrometric applications.

I. I NTRODUCTION Alluvial water courses, such as rivers or coastal waters show a permanent cycle of sediment erosion, sediment transport, and sediment deposit. Varying flow velocities, either caused by nature (wet and dry seasons) or by human interference (e.g. river constructions) influence the magnitude of suspended particles and the sedimentation process ([4]). Depending on the hydrodynamic conditions and the type of suspended particles, sedimentation might influence safe navigation on rivers or in harbours. For the assessment of those conditions, measurements of flow velocities and sediment transports are an important prerequisite for coastal or river engineering tasks as well as for irrigation agriculture. To guarantee minimum navigation depths time-consuming and expensive dredging operations are needed on a regular basis (and could often be avoided by appropriate constructional approaches such as bulkheads or wider port entrances). Sedimentation is often reduced by means of constructional measures on the basis of physically-sound understanding of flow pattern and erosion potential. Therefore, sediment transport mechanisms must first be determined by measuring flow velocities and sediment concentration and finally by numerical modelling of the morphodynamics. Up to now, most flow velocity measurements have been based on e.g. tacheometric tracking of floaters, propeller

gauges, electromagnetic current meters, or acoustic methods (e.g., acoustic Doppler current profiler, ADCP), while sediment concentration is being measured by manual samples or optical backscatter sensors ([17]). In many cases, however, velocity measurements as a basis for sediment transport processes have been quite inaccurate, mainly due to simple measurement principles, like e.g., manual stop-watch measurements of floater passages, etc. Since the advent of the Global Positioning System (GPS), flow velocity measurements have been measured with floaters equipped with GPS receivers of different quality. For example, expensive realtime kinematic systems (RTK) ([16]) or low-cost systems ([18], [13]) can be used to measure floater trajectories with lengths of hundreds of kilometers. In all cases, velocities are based on an ’indirect’ approach by deriving them from more or less precise floater positions. Obviously, the accuracy of velocities derived in this way is always worse than the GPSderived position accuracy. This explains the common use of expensive RTK systems (with costs of 20.000 $ or more). For various hydrometric applications the Institut f¨ur Erdmessung and the Franzius-Institute for Hydraulic, Waterways, and Coastal Engineering (both university of Hannover, Germany) developed a system of several floaters with a positioning accuracy of several meters and a velocity accuracy of about 5 cm/s. In this paper, both hardware and software of a low-cost system with a ’direct’ velocity determination approach will be described. In contrast to other GPSbased systems this system is based on both pseudoranges (for position determination) and pseudorange-rates/Doppler observations (for velocity determination) of low-cost GPS receivers. Besides the technical aspects, this paper focuses on the theoretical background and practical applications (incl. validation) of the system. The first chapter provides a general description of velocity determination with GPS, the second chapter describes the position-velocity-time (PVT) filter as used in the floater system that is being described in the third chapter. The fourth chapter shows results of a measurement

campaign in a river and the capabilities of the new flow velocity measurement system. We close with a small summary and a look into the future. II. V ELOCITY DETERMINATION BASED ON GPS PSEUDORANGE - RATES /D OPPLER MEASUREMENTS A. GPS observables GPS has been mainly developed for positioning and timing applications using pseudoranges derived from the signal propagation time τ , i.e. the difference between the signal transmit time tT and the signal reception time tR ([19], [15]). The pseudorange observation ρiu between a user u and a satellite i is basically the signal propagation time τ scaled by the speed of light c and can be related to the user position ru and the receiver-clock offset bu via: ρiu = kri − ru k + bu + ρi .

(1)

The satellite position ri at transmit time tT can be computed e.g. from the ephemeris parameters of the broadcast message ([24]) while the other parameters (usually expressed in terms of cartesian coordinates x, y, z and the metric clock offset bu ) are being estimated on an epoch-by-epoch basis or within a filter adjustment procedure. After correcting the observations for tropospheric and ionospheric refraction, the observation errors ρi contain all remaining effects such as receiver noise, satellite ephemeris errors, multipath, etc. Depending on the receiver quality and the modelling efforts a positioning accuracy of several meters can be obtained. In addition to pseudoranges, most receivers also output the phase of the carrier signal Φ. In this paper only the carrier phase of the L1 signal is considered. Scaled to meters these measurements can be expressed as: Φiu = kri − ru k + bu + Nui + ρi .

(2)

The main difference compared to Eq. (1) is the ambiguity term Nui . Further details can be found in general textbooks on GPS (such as [19], [12], or [15]). Beside positioning and timing, another designated GPS application is the precise determination of object velocities with an accuracy of several cm/s ([14]). Velocity determination is based on the pseudorange-rate, or equivalently, the Doppler shift due to the relative motion of a satellite and a user which results in changes in the observed frequency of the satellite signal ([15]). The range-rate observation is basically the projection of the relative velocity vector on the satellite line-of-sight vector, biased by the receiver clock error rate (i.e., a frequency offset). The observation equation for rangerates can be obtained by differentiating Eqs. (1) or (2) with respect to time and reads (in unit of m/s): ri − ru + f + ρ˙i , (3) ρ˙ iu = (ν i − ν u ) · kri − ru k where f is the receiver clock drift and ρ˙i is the observation error, both in meters per second. While the satellite position ri and satellite velocity ν i can be computed with sufficient accuracy from e.g. the broadcast ephemeris (see e.g. [24]) the user

velocity ν u (usually expressed in cartesian velocities x, ˙ y, ˙ z) ˙ and the receiver clock drift f are the unknown parameters to be estimated. Due to the explicit estimation of x, ˙ y˙ and z, ˙ instead of deriving it from x, y and z of sucessive epochs, this approach is called ’direct’ velocity estimation. The differencing process eliminates most of the error effects that are commonly present in GPS positioning. In the current context, differential tropospheric and ionospheric refraction can be neglected and constant contributions such as ambiguities or other slowly changing errors cancel if the sampling rate is sufficiently high ([3], [22]). B. Range rate measurements from central differences The signal Doppler shift is measured within the receiverinternal signal processing components and is usually output together with the pseudorange and carrier-phase observations. The internal (or raw) Doppler shift can be considered as an ’instantaneous’ measurement that is derived from typically 120 ms of signal data and also depends on the specific tracking loop design i.e., integration time, loop filter type, and noise bandwidth. Due to the typical noise level of dm/s to m/s these observations can hardly be used to achieve the required velocity accuracy level of a few cm/s ([22]). Several authors (e.g., [5]) propose alternative methods for the derivation of range-rates based on central differences of carrier-phase observations Φ. Range rate measurements from this approach are proportional to the average range rates over the interval used and are thus much smoother than the internal Doppler measurements. A first-order central difference approximation to the derivative of a signal x is given by: x(t + δt) − x(t − δt) , (4) 2δt with δt being the time difference between two successive observations (i.e., the sampling interval). Second-order and third-order central difference approximations are: x0 (t) =

x0 (t) =

−x(t + 2δt) + 8x(t + δt) − 8x(t − δt) + x(t − 2δt) , 12δt (5)

and x(t + δt) − x(t − δt) x(t + 2δt) − x(t − 2δt) −9 (6) 60δt 60δt x(t + 3δt) − x(t − 3δt) + . 60δt Further details about central differences can be found in e.g. [21]. Applications to carrier-phase observations and more sophisticated filter designs can be found in e.g. [5].

x0 (t) = 45

III. PARAMETER ESTIMATION IN A PVT FILTER A. Point solution vs. filter solution In principle, four pseudorange observations and four rangerate observations of one epoch are sufficient to estimate the eight parameters x, y, z, x, ˙ y, ˙ z, ˙ bu and f in a so-called point solution. However, a disadvantage of this approach is that it discards useful information from previous measurement epochs and that it does not take any known user dynamics

into account. It also heavily depends on the instantaneous satellite geometry so that small changes (e.g., a setting or rising satellite) might lead to unrealistic parameter changes ([2]). A filtered solution overcomes these problems and includes knowledge of the deterministic and stochastic properties of the system parameters and the measurements to obtain optimal estimates. Further advantages of the filter estimation algorithm are that the system can handle small data lags with less than four observed satellites and that carrier-derived rangerate observations smooth out the pseudorange tracking noise on the position solution ([10]). For a GPS-based velocity measurement system the eight parameters of position, velocity and clock behaviour form the filter state vector x. Since pseudoranges and rangerates, modelled by Eqs. (1) and (2), are non-linear functions of the state estimates, the total state vector is estimated by a discrete, extended Kalman filter (EKF, [10], [8]) that needs ˆ− to be initialised by an initial point solution x 0 and the − corresponding state covariance matrix P0 . The filter proceeds by propagating the state estimate and its covariance matrix from epoch k to epoch k + 1 (time update step) and then by including all measurements (measurement update step). In the following, only a short summary of the implemented EKF is given. Further general details can be found in textbooks on filtering, such as [20] or [9]. B. General EKF filter steps The four key matrices that are needed for a discrete EKF implementation are ([2]): H - the measurement or design matrix containing the partial derivatives of the observation equation(s) h(x) with respect to each of the states, R - the measurement noise covariance matrix, Φ - the state transition matrix that describes the user process model(s), Q - the process or system noise covariance matrix. The time update step from epoch k to k + 1 for the state vector and its covariance matrix are: ˆ− ˆ+ x k+1 = Φk x k

(7)

+ T P− k+1 = Φk Pk Φk + Qk .

(8)

and Quantities indicated by a superscript ’-’ denote time propagated estimates and covariances; a superscript ’+’ denotes quantities after the measurement update step. The measurement update consists of the following steps: • Computation of the measurement matrix or design matrix H (based on the respective functional model h(x)): ∂h(ˆ x− k) (9) ∂x Computation of the expected (’computed’) pseudorange and rangerate observables based on the current satellite position and the propagated vehicle state: Hk (ˆ x− k)=

•

ˆ k = h(ˆ y x− k)

(10)

•

Computation of the Kalman gain matrix Kk : − T T −1 Kk = P− k Hk (Hk Pk Hk + Rk )

•

Measurement update of the state vector: ˆ+ ˆ− ˆk ) x k =x k + Kk (yk − y

•

(11)

(12)

Measurement update of the covariance matrix: − T T P+ k = (I − Kk Hk )Pk (I − Kk Hk ) + Kk Rk Kk . (13)

Position and velocity determination in an EKF relies on the appropriate choice of process models for each parameter (group). For the applications considered in this context and due to the moderate vehicle dynamics (low velocity, no sudden shifts), the positions are modelled as the integral of the respective velocities and the clock offset as the integral of the clock drift. The velocities and clock drift are not functions of any of the Kalman filter states ([10]). The state transition matrix Φ thus reads:   1 0 0 δt 0 0 0 0  0 1 0 0 δt 0 0 0     0 0 1 0 0 δt 0 0     0 0 0 1 0 0 0 0  .  (14) Φk =    0 0 0 0 1 0 0 0   0 0 0 0 0 1 0 0     0 0 0 0 0 0 1 δt  0 0 0 0 0 0 0 1 The system noise covariance matrix Qk is a function of the power spectral densities of the system state parameters, i.e. Sp for the positions, Sv for the velocities, Sb for the clock offset, and Sf for the clock rate ([6], [3], I is an 3×3-identity matrix):  2 3 0 0 (Sp δt + Sv δt3 ) · I (Sv δt2 ) · I 2  (Sv δt2 ) · I (Sv δt) · I 0 0  Qk =  2 δt3  0 0 Sb δt + Sf 3 Sf δt2 2 0 0 Sf δt2 Sf δt (15) Appropriate coefficients of the clock terms can be derived from Allan variance parameters ([7]). For the GPS receiver clock used in the velocity measurement system (see below) standard Allan variance parameters for an uncontrolled quartz oscillator have been used (see e.g., [6]). C. Observation weighting In order to reduce the impact of noisy or weak observations, usually from satellites observed at low elevations, observation variance models usually use the satellite elevation Ei or the signal strength (i.e., C/N0 value, [22]) to compute the elements of the observation covariance matrix R. For the current application, an elevation-dependent weighting scheme has been used and observations have been considered as uncorrelated, i.e.: σ02 and σij = 0 (16) σi2 = sin2 Ei

   . 

Different σ02 -values for both pseudoranges and rangerates define the overall variance level, in this case: σ02ρ = 1 m2

and

σ02ρ˙ = 0.04 m2 /s2

GPS antenna position light

(17)

water level

D. Outlier checks

data logger

Outlier detection algorithms usually use observation residuals (in the case of point solutions) or the differences between observed and predicted observations (=innovations) of a filter approach. In both cases, first a ’global’ or chi-squared test is performed to check the overall model consistency based on the square sum of the residuals or innovations. If the global test fails, a second step consists of a ’local’ test of the individual observations. Details about these tests can be found in [10].

electronic components

Fig. 1. floater.

E. Horizontal constraints The geometry matrix or design matrix H contains the partial derivatives of the pseudorange and rangerate observation equations with respect to the unknown parameters. In order to apply different constraints on the horizontal and vertical coordinate component a topocentric (north-east-down, NED) representation should be used instead of an earth-centeredearth-fixed (ECEF) implementation. In this case the partial derivatives consist of the topocentric components of the usersatellite line-of-sight unit vector ([10], [8]). Thus,

• • •

          

battery

GPS floater and details of electronic components in bottom part of

a data logger (for GPS NMEA and binary raw data logging) a position (flash) light a 12V-battery.

With a total prize of about 1000$ the entire system can still be considered as ’low-cost’. V. R IVER MEASUREMENTS / R ESULTS

H=



uBlox GPS receiver

−ui,N −uj,N .. . −uk,N 0 0 .. . 0

−ui,E −uj,E .. . −uk,E 0 0 .. . 0

−ui,D −uj,D .. . −uk,D 0 0 .. . 0

0 0 .. . 0 −ui,N −uj,N .. . −uk,N

0 0 .. . 0 −ui,E −uj,E .. . −uk,E

0 0 .. . 0 −ui,D −uj,D .. . −uk,D

1 1 .. . 1 0 0 .. . 0

0 0 .. . 0 1 1 .. . 1

with uk,y being the y-component of the line-of-sight unit vector to the k-th satellite. IV. GPS FLOATER HARDWARE Current velocity measurement systems usually rely on the Lagrangian principle, i.e. on freely moving platforms equipped with positioning sensors, data loggers and/or telemetry devices. For best current representations, floaters should possess optimal hydrodynamical properties such as stable and upright floating positions, optimal waterflow surfaces, low lying center of mass, and smallest interactions with wind and high-frequent capillary waves. Based on these requirements, the FranziusInstitute developed six floaters ([11], Fig. 1) each consisting of: • a PVC-U pipe housing (weight: 8.3 kg, height: 99 cm, 84 cm below water level and 15 cm above water level, bottom part diameter: 12.5 cm, upper part diameter: 7.5 cm) • a L1-GPS receiver (uBlox LEA-4T) and patch antenna on a 5 mm aluminium plate (for multipath reduction)

After first equipment tests in the wave flume Schneiderberg (WFS) a measurement campaign was conducted on the Weser   river (52.2 N, 9.2 E, [1]). Four floater trajectories both along   or close to the river thalweg and within the recirculation zone   near the harbor entrance (which are known for their complex   flow patterns, [23]) were measured with a specially prepared   GPS floater (see Fig. 2 and Tab. 1). This floater was equipped  with a 360o prism for automatic tracking by a Leica TS30 total station that also time-tagged each floater position with a GPS time stamp (see Fig. 3). Due to the high angular accuracy of 1 arcsec and an distance accuracy of 0.6 mm + 1 ppm these tacheometric floater trajectories served as references for position and velocity comparisons. The following results are based on the algorithms described above and were computed with a postprocessing C++ software developed at the Institut f¨ur Erdmessung. Original C/A-codepseudoranges and L1-carrierphase-derived rangerates using third-order central differences were used as input observables. An elevation cut-off angle of 15o was applied to avoid noisy and multipath-contaminated (reflected) low observations. Refraction effects were modelled by the ’Goad & Goodman’ tropospheric delay model and the standard ’Klobuchar’ ionospheric delay model (see [19], [15]). For each epoch an independent PVT point solution and a (forward) filtered PVT solution was computed, both with outlier checks. The main filter settings and Allan variance parameters for the derivation of system noise coefficients are: Sp = 0.02 [m2 /Hz], Sv = 0.005 [m2 /Hz], h0 = 2e-10 [-], and h2 = 2e-20 [-]. 

Fig. 2. Four floater trajectories (A-D) on the river Weser: Trajectories A and B on the laminar main stream and trajectories C and D near port entrance vortices.

Fig. 3. Automatic floater tracking with a Leica TS30 total station (incl. GPS antenna for GPS-time tagging of tacheometric floater positions). Inset: GPS floater equipped with 360o prism and GPS patch antenna.

A. Position and velocity comparisons Although the tacheometric positions were time-tagged with GPS time stamps, the temporal differences between individual positions were not identical. Thus, for position comparisons, tacheometric positions had to be interpolated to full seconds. After these interpolations, position comparisons (see Tab. 1) show an agreement in the range of ±3 m. Since both the tacheometric reference trajectories and the GPS results were determined in the WGS84/UTM system the position differences are a direct ’accuracy’ measure. The position difference standard deviations (column 6 in Tab. 1) describe the average scatter of the GPS trajectories about the reference trajectory

and thus describe the ’precision’ of the GPS trajectory ([12]). Considering the quality of the used GPS receiver and due to the fact that the broadcast ionosphere model only accounts for approximately 50% of the ionospheric refraction these results are plausible. Mean differences between the indirect velocities derived from tacheometric positions and direct GPS-based floater velocities are below the 1 cm/s level with standard deviations (1σ) below 6 cm/s (Fig. 4). In addition, the histograms and the cumulative distribution functions in Fig. 5 also show that approximately 80% of the velocity differences are below the ±5 cm/s level and that in general there is a better velocity agreement for slow trajectories.

Trajectory A B C D

Length [m] 356 454 563 319

Duration [s] 157 214 573 392

Velocity range [m/s] 1.75-2.35 1.41-2.31 0.01-0.96 0.02-0.83

Mean position deviation [m] 1.71 1.52 -2.35 -3.62

Position deviation stddev [m] 0.31 0.71 3.06 0.44

Mean velocity deviation [m/s] 0.004 0.007 0.005 0.009

Velocity deviation stddev [m/s] 0.056 0.059 0.041 0.033

Comment: river thalweg river thalweg recirculation zone recirculation zone

TABLE I T RAJECTORY DETAILS

Obviously, these results depend on the PVT filter settings, the tacheometer tracking capabilities, the floater velocity, and the tacheometer-floater geometry. Best results are obtained for tracking with small direction changes due to the known issues of synchronisation between distance and angle measurements. This explains, for example, the low scatter of the velocity differences of trajectory D. This floater only had low velocities (usually less than 0.6 m/s) and was close to the total station (distance less than 10 m). B. Epoch solution vs. filtered solution Differences between epoch solutions and filtered solutions can be seen in coordinates, velocities and clock parameters. For example, Fig. 6 shows details of both solution types for approximately 13 epochs. It can be clearly seen that the epoch solution deviates from the filter solution by up to 1 m and that the filter solution is much smoother. The same effect can be observed in receiver clock offsets and clock rates (see Fig. 7). Differences between the two solution types for the clock offset (black crosses and black dots with errorbars on the straight line) can hardly be recognized due to the large clock rate. However, for the clock rate the epoch solutions (red diamonds) are significantly more noisy than the filtered clock rate values. For the current filter settings, differences of up to 0.2 m/s can be seen. Again, these differences heavily depend on the chosen filter settings. VI. C ONCLUSIONS & O UTLOOK We described hardware and software developments of a low-cost flow velocity measurement system. Nowadays, flow velocity determination is based on rather inaccurate measurement approaches or makes use of expensive GPS-RTK systems. In contrast to other GPS-based systems, we use a ’direct’ approach that processes pseudoranges and rangerates from low-cost GPS receivers with software optimised for hydrometric applications. Some of the benefits are: • due to the ’direct’ estimation of floater velocities, there is no need for high-precision positions and thus for the use of expensive GPS equipment, • the use of central differences of carrier phase measurements for the derivation of satellite-user rangerates allows the use of ’future’ measurements and thus leads to a smoother floater trajectory, • compared to commercial solutions, our postprocessing software allows for more flexible data processing such as user-selectable central difference order, filter settings, or horizontal constraints.

We validated the system with classical geodetic terrestrial measurements. With a position accuracy of approximately 3 m and a velocity accuracy of about 5 cm/s, the system fullfills the requirements of most hydrometric applications. The performance may change if obstructions occur. The low costs enable measurement campaigns with several floaters. In the future, we intend to equip floaters with high rate GPS receivers, acceleration sensors (i.e., MEMS inertial measurement units) and very precise, miniaturised frequency standards. This will provide even more wave/flow details and will especially enhance the precision of the height component. Further research is focused on analysis methods of (sensor) networks of GPS floaters. Together with appropriate sediment concentration sensors the newly developed floater system might serve as a powerful and economic hydrometric measurement system. ACKNOWLEDGMENT The authors would like to thank Leica Geosystems for providing a Leica TS30 total station for the Weser measurement campaign. We also thank the surveying authorities of Lower Saxony (LGLN Niedersachsen) for providing digital maps. We acknowledge the contribution of Christine Hegemann as well as the technicians Kurt Grube and Thomas Mathyl (FranziusInstitute). R EFERENCES [1] L. Albert, Analyse und Validierung eines GPS-basierten Fliessgeschwindigkeits-Messsystems, unpublished master thesis, Institut f¨ur Erdmessung, Leibniz University Hanover, 2010. [2] P. Axelrad, R.G. Brown, GPS Navigation Algorithms, in: Global Positioning System: Theory and Applications, Ed. by B.W. Parkinson, J.J. Spilker, Progress in Astronautics and Aeronautics, Vol. 163, Chapt 9, American Institute of Aeronautics and Astronautics, 409-433, 1996. [3] T. Beran, D. Kim, R.B. Langley, High-Precision Single-Frequency GPS Point Positioning, ION GNSS, 2003. [4] W. Boiten, Hydrometry, 3rd ed., Boca Raton: CRC Press Inc., 2008. [5] A.M. Bruton, C.L. Glennie, K.P. Schwarz, Differentiation for HighPrecision GPS Velocity and Acceleration Determination, GPS Solutions, Vol. 2, No. 4, pp. 7-21, 1999. [6] R.G. Brown, P. Hwang, Introduction to Random Signals and Applied Kalman Filtering, 3rd ed., Wiley, 1997. [7] A.J. van Dierendonck, J.B. McGraw, R.B. Brown, Relationship Between Allan Variances and Kalman Filter Parameters, Proceedings of the 16th Annual Precise Time and Time Interval (PTTI) Applications and Planning Meeting, NASA Goddard Space Flight Center, Nov. 27-29, 273-293, 1984. [8] J.A. Farell, M. Barth, The Global Positioning System & Inertial Navigation, McGraw, 1998. [9] P.B. Gibbs, Advances Kalman Filtering, Least-Squares and Modeling, Wiley, 2011. [10] P.D. Groves, Principles of GNSS, Inertial, and Multisensor Integrated Navigation Systems, Boston, USA: Artech House, 2008.

Velocity [m/s]

GPS velocities 2.5 2.0 1.5 1.0 0.5 0.0

Trajectory A Trajectory B Trajectory C Trajectory D 0

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Velocity difference [m/s]

Trajectory duration [s] GPS velocity - tacheometer velocity 0.20 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15 -0.20

0

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Trajectory duration [s] Fig. 4.

Upper plot: GPS velocities and errorbars, lower plot: differences between GPS and tacheometer velocities.

Trajectory B

Trajectory D

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Cumulative frequency

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-0.1

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Velocity difference [m/s] Fig. 5.

0 -0.2

-0.1

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Histograms and cumulative distribution functions of GPS-tacheometer velocity differences, left: trajectory B (fast), right: trajectory D (slow).

[11] C. Hegemann, Konzeption, Implementierung und Anwendung eines GPS gest¨utzten Schwimmersystems, unpublished diploma thesis, FranziusInstitute for Hydraulic, Waterways, and Coastal Engineering, 2010. [12] B. Hofmann-Wellenhof, H. Lichtenegger, E. Wasle, GNSS - Global Navigation Satellite Systems, Wien: Springer, 2008. [13] D. Johnson, R. Stocker, R. Head, J. Imberger, C.B. Pattiaratchi, A Compact, Low-cost GPS Drifter for Use in Oceanic Nearshore Zone, Lakes and Estuaries, Journal of Atmospheric and Oceanic Technology, Vol. 20, 1880-1884, 2003.

[14] M.B. May, Measuring Velocity Using GPS, in: GPS World, Innovation, September issue, 58-65, 1992. [15] P. Misra, P. Enge, Global Positioning System: Signals, Measurements, and Performance, 2nd ed., Lincoln, MA: Ganga-Jamuna Press, 2006. [16] H. Nasner, Hydromechanische und Morphologische Vorgnge in Brackwasserbeeinflussten Vorh¨afen - in Situ Messungen, Die K¨uste, No. 68, 1-65, 2004. [17] L.J. Olive, W.A. Rieger, An Examination of the Role of Sampling Strategies in the Study of Suspended Sediment Transport, in: Sediment Budgets,

epoch solution PVT filter solution

Fig. 6.

Trajectory detail: Noisy epoch solution velocity vectors vs. smooth filter velocity vectors.

Receiver clock offset and clock rate (Trajectory C) Clock offset (epochwise) Clock offset (filter) Clock rate (epochwise) Clock rate (filter)

Clock offset [m]

-550000

-40.4 -40.2 -40

-555000

-39.8 -560000 -39.6 -565000

-570000

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-545000

-39.4

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Duration [s] Fig. 7. Receiver clock offsets and receiver clock rates of floater trajectory C. Due to the large receiver clock rate the plots of epoch-wise and filtered clock offsets can not be distinguished (straight line).

IAHS Publications No. 174, International Assosciation of Hydrological Sciences, Washington DC, 259-267, 1988. [18] C. Perez, J. Bonner, F.J. Kelly, C. Fuller, Development of a Cheap, GPSbased, Radio-tracked, Surface Drifter for Closed Shallow-water Bays, Proceedings of the IEEE/OES Seventh Working Conference on Current Measurement Technology, 66-69, 2003. [19] G. Seeber, Satellite Geodesy, 2nd ed., Berlin: de Gruyter, 2003. [20] D. Simon, Optimal State Estimation - Kalman, H Infinity, and Nonlinear Approaches, Wiley, 2006. [21] G. Strang, Introduction to Applied Mathematics, Wellesley-Cambridge

Press, 1986. [22] A. Wieser, GPS-based velocity estimation and its application to an odometer, Aachen: Shaker Verlag, 2007. [23] A. Wurpts, Numerical Simulation of Combined Flow-, Density- and Tide-effects, Third Chinese-German Joint Symposium on Coastal and Ocean Engineering, National Cheng Kung University, Tainan, 2006. [24] J. Zhang, K. Zhang, R. Grenfell, R. Deakin, GPS satellite velocity and acceleration determination using the broadcast ephemeris, Journal of Navigation, Vol. 59, 293-305, 2006.