MARCH 2014 MATTE ET AL. 729 A Robust Estimation Method for Correcting Dynamic Draft Error in PPK GPS Elevation Using ADCP Tilt Data PASCAL MATTE AND YVES SECRETAN Eau Terre Environnement, Institut National de la Recherche Scientifique, Quebec, Quebec, Canada JEAN MORIN Hydrology and Ecohydraulic Section, Meteorological Service of Canada, Environment Canada, Quebec, Quebec, Canada (Manuscript received 11 June 2013, in final form 2 December 2013) ABSTRACT Measuring temporal and spatial variations in water level with high resolution and accuracy can provide fundamental insights into the hydrodynamics of marine and riverine systems. Real-time kinematic global positioning systems (RTK GPS), and by extension postprocessed kinematic (PPK) positioning, have provided the opportunity to achieve this goal, by allowing fast and straightforward measurements with subdecimeter accuracy. However, boat-mounted GPS are subject to movements of the water surface (e.g., waves, longperiod heaves) as well as to the effects of dynamic draft. The latter contaminate the records and need to be separated and removed from the data. A method is proposed to postcorrect the elevation data using tilt information measured by an attitude sensor—in this case, an acoustic Doppler current profiler (ADCP) equipped with internal pitch and roll sensors. The technique uses iteratively reweighted least squares (IRLS) regressions to determine the position of the center of rotation (COR) of the boat that leads to optimal corrections. The COR is also allowed to change in time by performing the IRLS analyses on data subsamples, thus accounting for changes in weight distribution, for example, due to personnel movements. An example of application is presented using data collected in the Saint Lawrence fluvial estuary. The corrections exhibit significant reductions associated with the boat motion while keeping subtle variations in water levels likely related to local hydrodynamics. 1. Introduction In recent years, real-time kinematic global positioning systems (RTK GPS), and by extension postprocessed kinematic (PPK) positioning, have played an increasingly important role in the study of marine and riverine environments. Aboard survey vessels or mounted on buoys, kinematic GPS record any motion of the water surface and measurement platform across the whole frequency spectrum. Relying on differential carrier-phase measurements, they are capable of subdecimeter accuracy in both horizontal and vertical positioning (Awange 2012; Ghilani and Wolf 2012). Such vertical accuracy is required in order to measure variations in water level associated with waves, long-period heave, tides, and nonperiodic motions Corresponding author address: Pascal Matte, Eau Terre Environnement, Institut National de la Recherche Scientifique, 490 rue de la Couronne, Qu ebec QC G1K 9A9, Canada. E-mail: [email protected] DOI: 10.1175/JTECH-D-13-00133.1 Ó 2014 American Meteorological Society such as those due to morphological features, currents, and dynamic draft effects (Bisnath et al. 2004b). When acquired with precision, temporal and spatial variations in water level can provide new insights into dynamical processes (e.g., Sime et al. 2007) and can be very helpful in the calibration of numerical models (e.g., Church et al. 2008; Capra et al. 2010) or satellite altimetric measurements (e.g., Watson et al. 2003). More specifically, RTK and PPK GPS have proven very beneficial in hydrographic surveys for the determination of chart and tidal datum (e.g., Riley et al. 2003; Bisnath et al. 2004a; Moegling et al. 2009) and for monitoring changes in seabed morphology when combined with multibeam echosounders (e.g., Church et al. 2009). In bathymetric soundings, they represent an advantage over traditional tidal and heave compensation methods typically restricted only to long- and shortperiod motions, respectively (Work et al. 1998; Blake 2007). Kinematic GPS have also been used for the measurement of tides as a complement to shore-based 730 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY gauges or in offshore regions where stations are remote or nonexistent (e.g., Hess 2003; Zhao et al. 2004; Hughes Clarke et al. 2005). Notably, centimeter-level differences were obtained by Rocken et al. (1990) and Bisnath et al. (2003) between GPS and tide gauge water levels. Furthermore, reliable estimates of the vertical motion of the sea surface have been achieved with buoy-mounted GPS for mean sea level and ocean wave measurements (e.g., Rocken et al. 1990; Kelecy et al. 1994). Similarly, estimates of wave heights made by Bender et al. (2010a, b) in the context of Hurricane Katrina were shown to be as accurate as data obtained from a strapped-down, oneaxis accelerometer, if properly corrected for tilting. In rivers and estuaries, local variations in water-level profiles are important to characterize. For example, Bauer et al. (2007) were able to identify pools with potential sand deposition in the Sandy River, by comparing water surface profiles taken at different years with a RTK GPS. Similarly, Sime et al. (2007) surveyed the main thalweg and branch channels of the lower Fraser River using RTK GPS and observed strong streamwise fluctuations in water surface slope, reflecting the pronounced riffle-pool variations as well as the backwater and acceleration effects induced by diagonal bar complexes. Likewise, Rennie and Church (2010) identified local variations in water surface slope over a succession of riffles and pools in the Fraser River, using RTK GPS. During vessel surveys, pitch and roll motions modify the reported water levels, as they both rotate and translate the GPS antenna. The measured vertical positions can fluctuate by a few decimeters as the boat shifts position due to changes in boat loading, weight distribution, heading, speed, wave, wind, and current action. These effects are referred to as dynamic draft, which is the sum of static draft, settlement, and squat. Static draft is the draft of the vessel at rest, when fully loaded (with equipment, fuel, and personnel). Settlement is the vertical lowering of the moving vessel, relative to what its level would be if it were motionless; it is measured at the vessel’s center of rotation (COR). Squat refers to the sinking of the vessel’s stern into the water as speed increases; it acts as a lever arm from the COR to the mounted instrument, thus changing its angle and draft (CHS 2008; NOAA 2010). Random oscillations, for example, due to waves, can easily be removed from the records by smoothing. However, nonperiodic and low-frequency vertical displacements can either be attributed to variations of the water surface elevation or to dynamic draft effects, and need to be accounted for. While it is crucial to keep the actual displacements of the water surface, shifts arising from the boat movements have to be identified and removed from the records. VOLUME 31 In many of the above-mentioned studies (Kelecy et al. 1994; Work et al. 1998; Bisnath et al. 2003; Hess 2003; Riley et al. 2003; Zhao et al. 2004; Hughes Clarke et al. 2005; Church et al. 2009; Moegling et al. 2009; Bender et al. 2010a,b), the effects of pitch and roll and/or dynamic draft on the measured water levels have been examined, by use of either squat models or sensors for attitude determination. Because squat depends on several factors, such as channel depth and cross section, shape of the ship’s hull, and ship speed (Barrass 2004), estimating the vessel squat characteristics as a function of speed through the water is not a trivial task (e.g., Beaulieu et al. 2012). On the other hand, using attitude sensors implicitly means that the position of GPS antennas relative to COR of the measurement platform is known and constant over time, two conditions not easily met in many surveys. In fact, the position of the COR, if not a priori known, must first be determined to calculate the vertical displacements of the antennas induced by pitch and roll motions. For this purpose, Alkan and Baykal (2001), for example, lifted their survey boat from the sea to the shore and mapped it in three dimensions with all the equipment in place. However, this technique is unpractical and most often impossible to achieve. Furthermore, the weight distribution, and thereby the position of the COR, may change in time due to fuel consumption and personnel movements on board, which limits the applicability of the method. As an alternative to attitude sensors, Beaulieu et al. (2009) applied the on-thefly (OTF) GPS technology, which is a class of RTK surveys, using the Canadian Coast Guard’s GPS network to measure ship squat in the Saint Lawrence waterway. The ship and escort boat were each equipped with two OTF GPS antennas on the longitudinal axis (bow and stern) and two others on the starboard and port sides to ensure simultaneous measurements of all vessel movements (rolling, sinkage, trim, etc.). Although the vertical accuracy can be high (65 cm in 95% of all cases, as confirmed by validation pretests), the required number of antennas (four per boat) makes this technique less attractive. A method is presented here for postcorrecting systematic errors in GPS elevations associated with dynamic draft effects, using tilt information measured by an attitude sensor—in this case, an acoustic Doppler current profiler (ADCP) equipped with internal pitch and roll sensors. It is assumed that the low-frequency motions of the water surface do not induce changes in the pitch and roll angles of the boat and that these rotations are exclusively related to dynamic draft effects. Hence, high-frequency oscillations are first removed from the records by smoothing, and the resulting lowpassed PPK GPS and ADCP time series are resampled to a common time vector and lagged to eliminate any MARCH 2014 MATTE ET AL. 731 synchronization issues. The implemented procedure applies iteratively reweighted least squares (IRLS) regressions to determine the position of the COR of the boat that leads to optimal tilt corrections. The COR is also allowed to change in time, by performing the IRLS analyses on smaller segments of the time series, thus accounting for changes in weight distribution over time. The method is tested using data collected in the Saint Lawrence fluvial estuary along repeated transects, aimed at documenting the lateral and intratidal variations in water levels and currents, at cross sections characterized by complex geometries (e.g., river bends, tidal flats) and in regions of contrasting tidal ranges and/or degrees of ebb–flood asymmetry. The paper is divided as follows: section 2 details the implemented method, section 3 applies the procedure to the Saint Lawrence River data, section 4 discusses the results, and section 5 follows with concluding remarks. 2. Methods To calculate the vertical corrections, a coordinate system is defined with the origin located at the COR of the boat (Fig. 1). The x axis is relative to the centerline of the boat and is positive to the bow. The y axis is perpendicular to the x and is positive to port. The z axis points to the nadir direction and is positive upward. Pitch is described as the forward and backward rotation of the boat about the transverse y axis and is positive when the bow of the boat goes up. Roll is described as the side-to-side rotation about the longitudinal x axis and is positive when the port of the boat goes up. The corrections applied to the data are aimed at reducing systematic errors in the recorded GPS elevations by minimizing the variations in water levels that correlate the most with the long-period movements induced by pitch and roll. The input time series of observations (i.e., water surface elevations, and pitch and roll angles) are thus filtered versions of the original time series. The GPS elevations are detrended to remove the effects of tides, by subtracting a cubic smoothing spline function from the original time series. The cubic smoothing spline s is constructed for the specified smoothing parameter p and weights wi so that it minimizes (Reinsch 1967; de Boor 1978) ð p å wi [yi 2 s(ti )] 1 (1 2 p) (d2 s/dt2 )2 dt , 2 (1) i where yi represents the observed water levels measured at times ti. Here, p 5 0 would produce a straight-line fit to the data, while p 5 1 corresponds to the cubic spline (exact) interpolant. The csaps Matlab function FIG. 1. Configuration of the GPS and ADCP on the boat and its associated coordinate system, with the origin located at the COR. (MathWorks 2012) is used with a very low smoothing parameter (p 5 1027), well suited to the slowly varying character of tides. The wi are set to 1 for all data points. Similarly, mean values in the pitch and roll data are subtracted to correct for sensor misalignment, using the detrend Matlab function (MathWorks 2012). The input time series are then smoothed to remove random oscillations due to waves, using cubic smoothing spline functions [cf. Eq. (1)] with a smoothing parameter of p 5 0.05, thus keeping enough variations in the records to allow corrections for systematic errors. As a result of filtering, the computed corrections only become effective when the low-passed pitch and roll signals are departing from zero. The ADCP time series are also reinterpolated to a common time vector (that of the GPS), and lagged in such a way that the correlation between observed water levels and vertical corrections for pitch and roll is maximal, thus eliminating any synchronization issues. Moreover, to account for temporal variations in the position of the COR—for example, due to changes in the weight distribution of the boat—the time series are divided into subsets, or transects, each of which being analyzed separately. Mathematically, the observed water levels can be represented by a linear model of the form 732 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY h 5 Ax 1 e , (2) where h is an n 3 1 vector of the filtered GPS elevations, A is an n 3 m tilt correction matrix (here m 5 3), x is an m 3 1 vector of unknown COR parameters, and e is an n 3 1 vector of observational error. Corrections for the motion of a measurement platform can be performed by use of rotation matrices (e.g., Edson et al. 1998; Miller 2 sinP(t1 ) 6 .. Dz(t) [ Ax 5 4 . sinP(tn ) VOLUME 31 et al. 2008). To transform the GPS elevations from ship coordinates to earth coordinates, a first rotation about the x axis (roll), followed by a rotation about the intermediate y axis (pitch), are calculated (cf. Teledyne RD Instruments 2010). A translation along the z axis is then applied to transform GPS-rotated heights into true GPS orthometric heights. The resulting tilt corrections Dz applied to the data are defined as 3 2 3 sinR(t1 ) cosP(t1 ) 1 2 cosR(t1 ) cosP(t1 ) a 7 4 5 .. .. 3 b , 5 . . c sinR(tn ) cosP(tn ) 1 2 cosR(tn ) cosP(tn ) (3) where P and R are the low-passed pitch and roll time series of length n, respectively; t is an n 3 1 time vector; and a, b, and c are the x–y–z coordinates of the GPS antenna relative to the COR, respectively (Fig. 1). Hence, the farther the antenna is from the COR, the larger the displacements Dz are for given angles of rotation. The first two columns of A correspond to vertical translations of the antenna due to the pitch and roll movements, while the third column is the vertical shift associated with its rotation. The latter correction is either strictly positive or strictly negative, depending on the sign of c, and is up to two orders of magnitude smaller than the translations, for small P and R angles. Robust parameter estimation models can be used to solve Eq. (2) in a way to reduce the influence of variations in water levels other than those associated with dynamic draft effects. A number of techniques with various levels of efficiency and effectiveness have been proposed, some of which were described by Huber (1996), Awange and Aduol (1999), and Goncalves et al. (2012). Among them, IRLS regression analysis (Holland and Welsch 1977; Huber 1996) has successfully been applied to geophysical problems (see, e.g., Bube and Langan 1997; Leffler and Jay 2009). The IRLS algorithm reduces the influence of high-leverage data points that increase residual variance by downweighting the outliers. The level of confidence in the computed parameters is therefore increased compared to ordinary least squares (OLS) analyses. The IRLS solution to Eq. (2) is given by convergence of the residual. At each iteration j, the following steps are repeated: x 5 (AT WA)21 AT Wh , where ri 5 R0i /st. 5) A new solution is obtained by application of Eq. (4), with wi 5 diag(W). (4) where W is an n 3 n diagonal weight matrix. The initial solution is obtained from OLS regression by setting the weight matrix to the identity matrix, that is, W 5 I. Iterations are then performed on x and W until there is 1) The residual R is computed from previous fit [i.e., Eq. (4)]: Rj 5 Wj21 (h 2 Ax)j21 . (5) 2) The residuals are adjusted using leverage li, as advised by DuMouchel and O’Brien (1990), which is a measure of the influence of each point i on the least squares fit: R0i 5 Ri / qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2 li , (6) where Ri are the ith elements of vector R, R0i is the adjusted residuals, and li 5 diag[A(AT A)21 AT ]. (Weisberg 2005). 3) A standard deviation estimate s is computed using the median absolute deviation (MAD) of adjusted residuals from zero: s 5 MAD(R0i )/0:6745, (7) where the constant 0.6745 makes the estimate unbiased for the normal distribution. 4) New weights are calculated using the specified weight function f and tuning parameter t: wi 5 f (ri ) , (8) IRLS regression analyses are performed using the robustfit Matlab function (MathWorks 2012). A bisquare weighting function is used, defined as MARCH 2014 MATTE ET AL. ( wi 5 (1 2 ri2 )2 , 0, jri j , 1 , jri j $ 1 (9) with a (default) tuning constant t of 4.685. The latter can be adjusted to penalize the outliers more or less heavily, depending on the level of filtering needed. In the present application, the default Matlab value is used, yielding coefficient estimates that are approximately 95% as statistically efficient as the OLS estimates, assuming that the response has a normal distribution with no outliers (MathWorks 2012). In general, for small pitch and roll angles, departure from the default function and parameter suggested by Matlab has little effect on the corrections (i.e., of the order of a few millimeters). A list of available weight functions can be found in Matlab documentation (MathWorks 2012); also summarized in Leffler and Jay (2009). To assess the significance of the corrections Dz, the p values are computed for the relative standard errors of coefficient estimates a, b, and c. They represent the probability of obtaining a test statistic at least as extreme as the observed error. For p values less than 0.001, the null hypothesis is rejected—that is, the correlation between the calculated corrections and the observations is highly unlikely to be the result of random chance alone. Otherwise, for p values superior to 0.001, the correlation is not significant and the coefficients are rejected (i.e., set to zero), in which case the solution is recalculated using only the columns of A corresponding to nonzero coefficients x. Threshold values on the coefficients can also be used to ensure that realistic distances between the antenna and the COR are obtained. 3. Data analysis Data were collected at 13 cross sections of 1–4-km width on the Saint Lawrence fluvial estuary, Quebec, Canada, during the summer of 2009. Each cross section was surveyed repeatedly over a period of approximately 12 h, corresponding to the semidiurnal tidal period. Boat speed was maintained at 1–2 m s21 on average to ensure data of good quality. Two Trimble R6 RTK GPS receivers—the base, located on the shore, and the rover, mounted at the rear of the boat (Fig. 1)—were used for positioning and water-level measurements along the transects (Trimble 2003). They were operated simultaneously, collecting data at a frequency of 1 Hz. Tilt information was obtained from the internal pitch and roll sensors of a 600-kHz RD Instruments Rio Grande ADCP, mounted on the side of the boat (Fig. 1). The frequency of acquisition was set to 2.5 Hz. 733 For the purpose of testing, the elevation data measured during one crossing at Portneuf were used, which present systematic shifts typically encountered during the campaign due to dynamic draft effects. As shown in Fig. 2a, measured water levels are indeed highly correlated with the variations in pitch and roll angles. Before using the data, the input time series were detrended and/or demeaned to remove variations associated with tides or sensor misalignment and then smoothed to remove random oscillations due to waves. Note that GPS elevations are detrended solely for the sake of regression analysis; once the correction Dz is made, the trend is reapplied to the elevation time series. The resulting filtered time series are shown in Fig. 2b. It can be seen that even the sharpest variations appearing in the original records (Fig. 2a) are preserved in the filtered time series. The first entry of Table 1 shows the regression coefficients [a, b, c] obtained for the chosen transect, representing the position of the antenna that leads to the optimal corrections. Because the pitch and roll angles are relatively small (cf. Fig. 2b), vertical corrections from the third column of A in Eq. (3) were very small too. Consequently, the value of c was not significant according to its p value, which was much higher than 0.001; it was thus set to zero. The zero time lag indicates that the elevation and pitch and roll time series were synchronous. Furthermore, the correlation coefficient (0.790) highlights the strong relation that exists between the computed correction and the elevation time series. The impact of dividing the time series into smaller subsets and performing successive IRLS analyses was assessed. In Table 1, results are presented from consecutive 5-min intervals of the same transect as in the first entry. The optimized coefficients slightly differ from each other, highlighting the respective influence of pitch and roll on each subsample. The corresponding standard deviations are thus smaller than for the total transect (24.6-min interval), with the exception of the fourth 5-min subset, because coefficients are adjusted to the local conditions prevailing during each time interval. Correlation coefficients are also higher in subsets where water-level variations induced by dynamic draft movements are the strongest. Overall, the displacement of the COR is within 61 m in all directions, reflecting the combined action of external factors (such as currents, winds, weight distribution, etc.) on the position of the COR. Despite these dissimilarities, maximum absolute differences in water levels between the analyses performed using the whole transect (first entry) and using 5-min intervals were less than 1 cm (given maximum absolute corrections of ;3 cm; cf. Fig. 3). 734 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 31 FIG. 2. (a) Elevation data for a transect at Portneuf measured with a kinematic GPS, compared to pitch and roll data from an ADCP. (b) Filtered time series of elevations, and pitch and roll angles. The corrections obtained by IRLS analyses for the transect at Portneuf are shown in Fig. 3. The computed correction is shown in the top panel, while the signals before and after correction are presented in middle panel. Variations associated with pitch and roll motions are significantly reduced, although not completely eliminated in some instances. The fluctuations, however, remain within an interval of 61 cm once corrected, which is comparable to the data vertical accuracy at Portneuf, as generally expected for kinematic GPS surveys (Ghilani and Wolf 2012). The weight time series appearing in the bottom panel of Fig. 3 follows the variations in the residual and shows how the IRLS regression analysis reduces the influence of nonzero residuals on calculated coefficients. Lower weights are attributed to portions of the signal where variations caused by pitch and roll motions still remain as well as where variations with no apparent correlation with the boat motion appear, possibly related to local hydrodynamics. 4. Discussion GPS elevations are subject to errors other than those related to dynamic draft effects. These include data latency associated with the motion of the rover during data transmission from the base receiver, reception, and processing at the rover. In the present application, positioning errors caused by this time difference tend to be small, since boat speed was maintained around 1–2 m s21 on average. Other factors that limit the positioning accuracy of kinematic surveys are errors associated with spikes in positional dilution of precision (PDOP), tropospheric and ionospheric refraction, weak MARCH 2014 MATTE ET AL. TABLE 1. Regression coefficients for a transect at Portneuf, along with their associated statistics. First entry shows results for the entire crossing (24.6-min interval), while the other entries correspond to 5-min subsets of the transect. Time interval (min) 24.6 5 5 5 5 4.6 [a, b, c] (m) Entire transect [20.854, 0.468, 0.000] 5-min subsets [0.000, 0.673, 0.000] [20.865, 0.630,0.000] [0.394, 0.409, 0.000] [20.629, 0.000, 0.000] [0.782, 0.782, 0.000] Std dev (m) Time lag (s) Corr coef 0.0047 0 0.790 0.0041 0.0028 0.0025 0.0050 0.0039 0 0 0 0 0 0.880 0.809 0.695 0.277 0.948 satellite geometry, ephemeris error, multipathing, obstructions to satellite signals due to topography or infrastructure, base station coordinate errors, firmware algorithms, and weather (Blake 2007; Ghilani and Wolf 2012). These factors should be taken into account when possible in the planning of a field campaign. During the Saint Lawrence campaign, the number and configuration of available satellites was not considered due to time and resources constraints—only the weather was, 735 mainly for security reasons. However, the number of available satellites was always high (.10 in the example shown, with PDOP , 2) due to a very open survey environment, thus limiting the associated errors. Moreover, errors associated with tropospheric delay were limited by short-baseline surveys (,4 km). Some of the remaining errors were directly filtered out in the receivers (e.g., multipath effects), the rest being partly cancelled by data smoothing. In the robust model, only the variations in measured heights that are significantly correlated with pitch and roll (through the p value criterion) were corrected. Also, the weights attributed to uncorrelated variations were reduced, leaving them almost unchanged. This allows a separation between dynamic draft and GPS-related errors. The latter perturb the robust model only if they are synchronized with the boat movements, which is improbable. Because of the strategy of repeated transects used in the Saint Lawrence fluvial estuary, data from each river crossing was analyzed and corrected separately. Although smaller regression intervals than the transect length might be desired (e.g., Table 1), care must be taken at the junctions of neighboring subsets because of discontinuities appearing between successive corrections. FIG. 3. (top) Computed correction for a transect at Portneuf, (middle) original (filtered) and corrected elevations, and (bottom) bisquare weights from the IRLS analysis. 736 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY More tests are needed to determine the optimal time interval to apply to each subset at a given location. Although the GPS antenna is fixed on the boat, the position of the COR is generally unknown and can move due to changes in weight distribution, for example, caused by personnel movements on board. The use of IRLS weighting functions allows for obtaining robust coefficients, corresponding to the optimal distances between the antenna and the COR in terms of residual variance reduction. More work, however, is necessary to assess the performance of other weighting functions and to define practical limits for the tuning parameters, which may vary from place to place and as a function of the conditions that prevail. In the example shown, pitch and roll angles were relatively small, so that the impact of the choice of the weighting function and tuning constant on the corrected water levels was minor (of the order of a few millimeters). This may not be the case under more extreme dynamic draft conditions. The regression approach put forward here is intended to be used in a context where the position of the COR is unknown and/or changes in time. Its position can only be determined when time variations in tilt angles occur and when these variations are significantly correlated with changes in the measured GPS elevations. Therefore, the presented method does not apply in contexts where the boat is tilted by a constant angle during the whole survey, unless the ADCP is perfectly aligned in the vertical. In this case, the average pitch and roll can be used to shift the GPS data, using the known or iteratively determined COR. 5. Conclusions ADCPs are used in a wide variety of applications, from discharge monitoring to the investigation of sediment transport, turbulence, or habitat quality (e.g., Lu and Lueck 1999; Yorke and Oberg 2002; Shields and Rigby 2005; Rennie and Church 2010). In the present application, the tilt information provided by an ADCP was used in place of traditional attitude sensors, thus broadening its range of applicability. In contrast, the use of kinematic GPS technology in the study of riverine and marine systems is relatively new. As experience is gained in the field, measurement techniques and data analysis procedures are refined. As argued by Work et al. (1998), extreme care and detail must be maintained during data collection and processing to yield useful data. A simple method was presented to postcorrect elevation data obtained from a boat-mounted GPS. Using data collected in the St. Lawrence fluvial estuary, contamination of the measured elevations was shown, arising VOLUME 31 from dynamic draft effects undergone by the boat as it followed its survey path. The observed systematic errors were correlated with pitch and roll data obtained from an ADCP, which were used to correct the PPK GPS data, thereby reducing the error to within instrumentation accuracy. This improved accuracy demonstrates the potential of using boat-mounted PPK GPS in a variety of environments and conditions, as long as there is no signal obstruction and sufficient information is available to compensate for boat movements. Even when the measured tilt angles are relatively small, there is no reason not to perform the corrections if tilt information is available. Obviously, under more extreme dynamic draft conditions, as the tilt angles increase, the need for such a correction becomes increasingly important. The combination of kinematic GPS and ADCP technologies allows simultaneous acquisition of both water level and velocity data, which is crucial for the calculation of accurate discharges in rivers. This becomes especially useful when there is no water-level gauge close to the study site, or when cross-sectional variations cannot be adequately captured by shore-based gauges. Likewise, local variations in water levels either due to local geographic features or transient processes can only be precisely measured using instruments capable of high resolution and accuracy, such as PPK GPS. Acknowledgments. This work was supported by scholarships from the Natural Sciences and Engineering Research Council of Canada and the Fonds de recherche du Qu ebec—Nature et technologies. The field campaign was funded by Environment Canada (Meteorological Service of Canada). Special thanks go to Guy Morin, Jean-Franc¸ois Cantin, Patrice Fortin, Olivier Champoux, and Catherine Leblanc for their contribution to the field work as well as to two anonymous reviewers for their constructive comments on the manuscript. REFERENCES Alkan, R. M., and O. Baykal, 2001: Survey boat attitude determination with GPS/IMU systems. J. Navig., 54, 135–144. Awange, J. L., 2012: Environmental Monitoring Using GNSS. Springer-Verlag, 386 pp. ——, and F. W. O. Aduol, 1999: An evaluation of some robust estimation techniques in the estimation of geodetic parameters. Survey Rev., 35, 146–162. Barrass, C. B., 2004: Ship Design and Performance for Masters and Mates. Elsevier, 264 pp. Bauer, T. R., J. Bountry, and R. C. Hilldale, 2007: 2007 profile survey of the Sandy River for Marmot Dam removal. U.S. Department of the Interior, Bureau of Reclamation, Technical Service Center, 25 pp. MARCH 2014 MATTE ET AL. Beaulieu, C., S. Gharbi, T. B. M. J. Ouarda, and O. Seidou, 2009: Statistical approach to model the deep draft ships’ squat in the St. Lawrence waterway. J. Waterw. Port Coastal Ocean Eng., 135, 80–90. ——, ——, T. Ouarda, C. Charron, and M. Aissia, 2012: Improved model of deep-draft ship squat in shallow waterways using stepwise regression trees. J. Waterw. Port Coastal Ocean Eng., 138, 115–121. Bender, L. C., N. L. Guinasso Jr., J. N. Walpert, and S. D. Howden, 2010a: A comparison of methods for determining significant wave heights—Applied to a 3-m discus buoy during Hurricane Katrina. J. Atmos. Oceanic Technol., 27, 1012–1028. ——, S. D. Howden, D. Dodd, and N. L. Guinasso, 2010b: Wave heights during Hurricane Katrina: An evaluation of PPP and PPK measurements of the vertical displacement of the GPS antenna. J. Atmos. Oceanic Technol., 27, 1760– 1768. Bisnath, S., D. Wells, S. Howden, and G. Stone, 2003: The use of a GPS-equipped buoy for water level determination. Proceedings of OCEANS 2003: Celebrating the Past, Teaming toward the Future, Vol. 3, MTS, 1241–1246. ——, ——, ——, D. Dodd, D. Wiesenburg, and G. Stone, 2004a: Development of an operational RTK GPS-equipped buoy for tidal datum determination. Int. Hydrogr. Rev., 5, 54–63. ——, ——, M. Santos, and K. Cove, 2004b: Initial results from a long baseline, kinematic, differential GPS carrier phase experiment in a marine environment. PLANS 2004: Position Location and Navigation Symposium, IEEE, 625–631. Blake, S. J., 2007: Heave compensation using time-differenced carrier observations from low cost GPS receivers. Ph.D. thesis, Institute of Engineering, Surveying and Space Geodesy, University of Nottingham, 209 pp. Bube, K. P., and R. T. Langan, 1997: Hybrid l1/l2 minimization with applications to tomography. Geophysics, 62, 1183–1195. Capra, H., and Coauthors, 2010: Consequences de l’artificialisation de l’hydrologie du Rh^ one sur la structuration des communautes d’invert ebr es et de poissons—Rapport final. Cemagref Framework Agreement, Agence de l’Eau RM&C Action 15, 104 pp. CHS, 2008: Hydrographic survey management guidelines. Canadian Hydrographic Service, Fisheries and Oceans Canada, 63 pp. Church, I., J. E. Hughes Clarke, S. Haigh, M. Santos, M. Lamplugh, J. Griffin, and R. Parrott, 2008: Using globally-corrected GPS solutions to assess the viability of hydrodynamic modeling in the Bay of Fundy. Proc. Canadian Hydrographic Conf. and National Surveyors Conf., Victoria, BC, Canada, IIC Technologies, 4-2. [Available online at http://www.omg.unb.ca/ omg/papers/Ian_Church_CHC08_Paper.pdf.] ——, S. Brucker, J. E. Hughes Clarke, S. Haigh, J. Bartlett, and T. Janzen, 2009: Developing strategies to facilitate long term seabed monitoring in the Canadian Arctic using post processed GPS and tidal models. Proc. U.S. Hydro 2009 Conf., Norfolk, VA, Hydrographic Society of America, 0513A_06. [Available online at http://www.thsoa.org/hy09/0513A_06. pdf.] de Boor, C., 1978: A Practical Guide to Splines. Springer-Verlag, 372 pp. DuMouchel, W. H., and F. L. O’Brien, 1990: Integrating a robust option into a multiple regression computing environment. Computing Science and Statistics: Proceedings of the 21st Symposium on the Interface, L. Malone and K. Berk, Eds., American Statistical Association, 297–301. 737 Edson, J. B., A. A. Hinton, K. E. Prada, J. E. Hare, and C. W. Fairall, 1998: Direct covariance flux estimates from mobile platforms at sea. J. Atmos. Oceanic Technol., 15, 547– 562. Ghilani, C. D., and P. R. Wolf, 2012: Elementary Surveying: An Introduction to Geomatics. 13th ed. Prentice-Hall, 983 pp. Goncalves, R. M., J. L. Awange, C. P. Krueger, B. Heck, and L. S. Coelho, 2012: A comparison between three short-term shoreline prediction models. Ocean Coastal Manage., 69, 102–110. Hess, K., 2003: Water level simulation in bays by spatial interpolation of tidal constituents, residual water levels, and datums. Cont. Shelf Res., 23, 395–414. Holland, P. W., and R. E. Welsch, 1977: Robust regression using iteratively reweighted least-squares. Commun. Stat. Theory Methods, 6, 813–827. Huber, P. J., 1996: Robust Statistical Procedures. 2nd ed. CBMSNSF Regional Conference Series in Applied Mathematics, Vol. 68, SIAM, 67 pp. Hughes Clarke, J. E., P. Dare, J. Beaudoin, and J. Bartlett, 2005: A stable vertical reference for bathymetric surveying and tidal analysis in the high Arctic. Proc. U.S. Hydro Conf., San Diego, CA, Hydrographic Society of America, 09_05. [Available online at http://www.thsoa.org/hy05/09_5.pdf.] Kelecy, T. M., G. H. Born, M. E. Parke, and C. Rocken, 1994: Precise mean sea level measurements using the global positioning system. J. Geophys. Res., 99 (C4), 7951–7959. Leffler, K. E., and D. A. Jay, 2009: Enhancing tidal harmonic analysis: Robust (hybrid L1/L2) solutions. Cont. Shelf Res., 29, 78–88. Lu, Y., and R. G. Lueck, 1999: Using a broadband ADCP in a tidal channel. Part II: Turbulence. J. Atmos. Oceanic Technol., 16, 1568–1579. MathWorks, 2012: R2010a MathWorks documentation. [Available online at http://www.mathworks.com/help/releases/R2010a/ helpdesk.html.] Miller, S. D., T. S. Hristov, J. B. Edson, and C. A. Friehe, 2008: Platform motion effects on measurements of turbulence and air–sea exchange over the open ocean. J. Atmos. Oceanic Technol., 25, 1683–1694. Moegling, C. H., J. L. Dasler, J. C. Creech, and P. Canter, 2009: GPS derived water levels for large scale hydrographic surveys: Implementation of a separation model of the Columbia River datum, a case study. Proc. U.S. Hydro 2009 Conf., Norfolk, VA, Hydrographic Society of America, 0513A_04. [Available online at http://www.thsoa.org/hy09/ 0513A_04.pdf.] NOAA, 2010: Field procedures manual. National Oceanic and Atmospheric Administration, Office of Coast Survey, 326 pp. Reinsch, C., 1967: Smoothing by spline functions. Numer. Math., 10, 177–183. Rennie, C. D., and M. Church, 2010: Mapping spatial distributions and uncertainty of water and sediment flux in a large gravel bed river reach using an acoustic Doppler current profiler. J. Geophys. Res., 115, F03035, doi:10.1029/2009JF001556. Riley, J. L., D. G. Milbert, and G. L. Mader, 2003: Hydrographic surveying on a tidal datum with kinematic GPS: NOS case study in Delaware Bay. Proc. U.S. Hydro 2003 Conf., Biloxi, MS, Hydrographic Society of America, 16 pp. Rocken, C., T. M. Kelecy, G. H. Born, L. E. Young, G. H. Purcell, and S. K. Wolf, 1990: Measuring precise sea level from a buoy using the global positioning system. Geophys. Res. Lett., 17, 2145–2148. 738 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY Shields, F., and J. Rigby, 2005: River habitat quality from river velocities measured using acoustic Doppler current profiler. Environ. Manage., 36, 565–575. Sime, L. C., R. I. Ferguson, and M. Church, 2007: Estimating shear stress from moving boat acoustic Doppler velocity measurements in a large gravel bed river. Water Resour. Res., 43, W03418, doi:10.1029/2006WR005069. Teledyne RD Instruments, 2010: ADCP coordinate transformation: Formulas and calculations. 36 pp. Trimble, 2003: Real-Time Kinematic Surveying: Training Guide. Part 33142-40, Revision D, Trimble Navigation Limited, 368 pp. Watson, C., R. Coleman, N. White, J. Church, and R. Govind, 2003: Absolute calibration of TOPEX/Poseidon and Jason-1 VOLUME 31 using GPS buoys in Bass Strait, Australia. Mar. Geod., 26, 285–304. Weisberg, S., 2005: Applied Linear Regression. 3rd ed. Wiley, 336 pp. Work, P. A., M. Hansen, and W. E. Rogers, 1998: Bathymetric surveying with GPS and heave, pitch, and roll compensation. J. Surv. Eng., 124, 73. Yorke, T. H., and K. A. Oberg, 2002: Measuring river velocity and discharge with acoustic Doppler profilers. Flow Meas. Instrum., 13, 191–195. Zhao, J., J. E. Hughes Clarke, S. Brucker, and G. Duffy, 2004: On the fly GPS tide measurement along the Saint John River. Int. Hydrogr. Rev., 5, 48–58.

© Copyright 2017 ExploreDoc