Proc. of the IEEE/OES Seventh Working Conference on Current Measureent Technology UNCERTAINTIES IN SEASONDE CURRENT VELOCITIES Belinda Lipa Codar Ocean Sensors 15 La Sandra Way, Portola Valley, CA 98 blipa@pogo.co Abstract This paper describes the ethods used to derive uncertainties in SeaSonde radial and total velocity vectors. Studies of baseline deviations are used to illustrate/validate the results. Recoendations are ade for further work leading to reduction of the uncertainties. INTRODUCTION A easureent is incoplete without an estiate of its uncertainties. SeaSonde current vector output now includes uncertainties in both radial and total velocity vectors. This paper describes how current vector uncertainties are derived. A single SeaSonde unit easures the coponent of the velocity radial to the radar. Most uncertainty in these radial vectors is due to spatial and teporal variations of the current field and use of non-optial analysis paraeters. Spatial uncertainty results because there can be any different current velocities present in the radar scatter patch due to horizontal shear; these velocity values are calculated during analysis and are averaged to produce the value output for that location. The uncertainty is estiated by calculating the standard deviation of all the velocities that fall on a particular location. This spatial uncertainty usually increases with distance fro the radar as the size of the radar scatter patch increases proportionally with range. Uncertainty can also arise fro variations in the current velocity field over the duration of the radar easureent, and fro assuptions and siplifications that are ade during the analysis process. The uncertainty due to these effects is estiated by calculating the standard deviation of radial velocities resulting fro analysis of successive short spectral averages. The total current vector is obtained by leastsquares fitting to the radial vectors fro two or ore radar sites on a grid that extends over the joint coverage area. The uncertainties in the total velocities follow fro those in the radials using standard linear error propagation, which includes the effects of radial uncertainties as well as the geoetry. We illustrate and validate our results using data obtained along the baseline joining two radar sites. Ideally, radial velocity coponents fro the two sites are equal and opposite at each point on the baseline; in practice they are not, due to data iperfections. Baseline deviations give a quantitative easure of the degree of iperfection and can be used to test various hypotheses. The rest of the paper is structured as follows: Section A. describes sources of radial velocity uncertainties and their calculation. Section B describes the derivation of the total velocities and their uncertainties. Section C describes tests along a baseline between two Long Island Sound SeaSondes. A: UNCERTAINTIES IN RADIAL VECTORS ii Sources of uncertainty Assuing that the radar is operating correctly, we can identify the following sources of uncertainty in the radial velocities: (a Variations of the radial current coponent within the radar scattering patch. (b Variations of the current velocity field over the duration of the radar easureent. (c Errors/siplifications in the analysis. For exaple these ay include the use of incorrect antenna patterns and analysis paraeters, and errors in epirical firstorder line deterination. (d Statistical noise in the radar spectral data. ( Coputation of radial velocity uncertainties The radar analysis proceeds as follows: In the first step, fro analysis of a 1-inute voltage spectral average, we collect all the radial vectors falling on a given radar cell, as defined by range 1
Proc. of the IEEE/OES Seventh Working Conference on Current Measureent Technology y Fig. 1: Geoetry of a backscatter radar ξ O Receiver fro the radar and a finite angular increent (usually 5 around a central aziuth angle. We then calculate the ean and standard deviation of these vectors. We believe that this uncertainty is due to ainly horizontal velocity shear within the radar cell (Source (a above and will refer to the standard deviation as the spatial uncertainty in what follows. Support for this hypothesis will be given in Section C. In step, successive 1-inute radial files are erged over an hour to produce the final output result, which consists of the ean radial velocity and the standard deviation of the ean, obtained fro the 1-inute radial velocities at each location. This uncertainty results fro the reaining sources (b (d above. The spatial uncertainty is not at present included in the standard Seasonde output but is available in the unerged radial files. It is not included because the output radial current vectors represent the area average of the radial velocities over the radar scatter patch; the corresponding uncertainty is the standard deviation of this ean, which is calculated fro successive tie saples. Thus even if the spatial unceratinties are high, the current aps ay look uch the sae fro tie to tie, as they represent spatial averages. SECTION B. UNCERTAINTIES IN TOTAL VELOCITIES Firstly, we define a rectangular grid covering the joint coverage area with cell size typically 1.5 x 1.5 k. In practice, all radial coponents that fall within a cobining circle of defined radius (typically k are cobined to for a single total current vector at the grid point. We define these coponents as having agnitudes w i and inclinations ξ i for i=1,,..., if the coponent is produced by backscatter, it is noral to the circular range cell (see Fig. 1. We define a total velocity vector with coponents U and V along the x, y axes, which are w Scatter Point x deterined by iniizing the su of weighted deviations given by: where the weights are the uncertainties in the radial velocities. The solutions to this iniization are given by: w i cos(ξ i Σ i=1 U = cos (ξ i Σ i=1 V = S= Σi = 1 Σ i=1 w i sin(ξ i w i Ucos (ξ i V sin (ξ i cos (ξ i i sin (ξ i Σ i=1 cos (ξ i w i sin(ξ i Σ i=1 sin(ξ i cos(ξ i Σ i=1 sin (ξ i w i cos(ξ i i sin (ξ i with the corresponding variances obtained fro linear error propagation (see for exaple Brandt: Statistical and Coputational Methods in Data Analysis. When a data atrix W is defined in ters of a paraeter atrix P through a linear transforation atrix T i.e. W = TP (4 then the covariance atrix C P is defined in C P = T' C W 1 T 1 ters of the covariance atrix of W, C W by the relation In our case, P is a ( x1 atrix with coponents u, v (the coponents of the total velocity vector and W is the atrix containing the radial velocities fro the different radar sites at that geographical location. Ignoring correlations between radial velocities easured by the different radars, the variances in U and V and their covariance are given by: (1 sin(ξ i cos(ξ i i (5 ( (3
Proc. of the IEEE/OES Seventh Working Conference on Current Measureent Technology U V = sin (ξ i Σ i=1 i Σ i=1 cos (ξ i For the case of two backscatter units along the x axis, it follows fro (7 that the variance in the coponent V along the baseline between the sites becoes infinite. This ust be true because all radial velocity coponents are parallel to the baseline and therefore the input data provides no inforation on the perpendicular coponent of the total velocity vector. This calculation is siilar to the that tered geoetrical dilution of precision. The latter however assues that errors in the radial velocities are constant, which of course they are not; see for exaple: K.-W. Gurgel, Shipborne easureents of surface current fields by HF radar, L'Onde Electrique, Septeber-October 1994, Vol. 74, No. 5. SECTION C: BASELINE ANALYSES This section describes analysis of radial velocities easured at points along the baseline between radar sites at Montauk and Misquaicut. The baseline lies over about k of open ocean. The current velocity and horizontal velocity shear increase close to Montauk, as the current swirls around the Montauk point. Ideally, radial vectors fro the two sites would be equal and opposite at a point on the baseline, so the agnitude of their su represents a easure of iperfection in the data. In Subsection 1 we describe the data fro the individual radars, and in Subsection we give results of tests based on baseline deviations. 1 Radial data and their uncertainties In Fig., we plot for the two sites at points along the baseline the rs radial velocity, together Σ i=1 (8 with the ean spatial uncertainty and the standard deviation of the ean velocity. Values are obtained by averaging hourly data over the period of a week. Ideally, the rs speeds fro the two sites would be identical. Clearly however, close to Montauk, the Montauk speed is uch larger than for Misquaicut and the Misquaicut spatial uncertainty becoes alost as large as the speed itself. We can explain this as follows: (a there is a large horizontal velocity shear close to Montauk (b the radar cell size is large as it is distant fro the radar. Therefore when the velocities fro the radar cell are averaged in the first step described in Section A, there is a lot of averaging-down, yielding a value uch lower than the extree value. In contrast, at this location, the Montauk radar cell size is sall, as it is close to the radar, and there is uch less averaging-down, leading to a higher velocity value and a lower spatial uncertainty. Baseline deviations We define the ean-square baseline deviations at points on the baseline as B = <(v 1 + v > (9 Where v 1, v are the Montauk, Misquaicut radial velocities and the average is taken over tie (in our case, the period Noveber 11 to Noveber 17,. Ideally B is zero; its agnitude provides a useful paraeter to aid in aking analysis decisions. We now give two exaples of this. i The effect of spectral averaging tie on accuracy. In Fig. 3 (upper, the rs baseline deviations fro equation (9 ( B are plotted versus distance along the baseline for 1-hour and 1- inute spectral averages. If the baseline deviations were due to statistical noise, one would expect the 1-hour results to be a factor of 6 lower than the 1-inute deviations. As these curves are alost identical, it can be concluded that statistical noise is not a significant contributor. We note that the baseline deviations increase as Montauk is approached, corresponding to the divergence between the velocities fro the individual sites shown in Fig.. 3
Proc. of the IEEE/OES Seventh Working Conference on Current Measureent Technology Misquaicut Velocity, c/s 3 1 5 1 15 5 3 Montauk Velocity, c/s 3 1 5 1 15 5 3 Distance fro Montauk, k Fig. : RMS radial velocities and their uncertainties for Misquaicut (upper and Montauk (lower obtained by averaging over SeaSonde hourly results obtained alone the baseline between the sites fro Noveber 11 to Noveber 17,. Solid line: (dark rs radial velocity; dashed line: (ediu averaged spatial uncertainty; dotted line (light: averaged standard deviation in the ean velocity. ii Coparison of baseline deviations and spatial uncertainties We now show how baseline deviations can be used to validate our uncertainty calculations. Expanding about the ideal velocities gives: v 1 = v 1 ideal + v 1 ; v = v ideal + v (1 Substituting (1 into (9, and using the fact that the su of v 1 ideal and v ideal is zero, gives the following expression for the ean-square baseline deviation: B = < (v 1 ideal + v ideal + v 1 + v > or B = σ ( v 1 + σ (v +ρ( v 1,v σ (v 1 σ (v (11 where σ is the rs value and ρ( v 1,v is the correlation coefficient. Ignoring correlations between v 1 and v, it follows the rs deviations obey the approxiate relation: B (σ ( v 1 + σ (v 1/ (1 Fig. 3 (lower shows the two curves representing the right and left side of (1, v 1, v are the Montauk, Misquaicut radial velocities and the average is taken over tie and where spatial uncertainties only are incuded on the righthand. The good agreeent between the curves indicates that the large baseline deviations are explained by the disparity in the sizes of the radar cels for the two sites, together with the large horizontal velocity shear. 4
Proc. of the IEEE/OES Seventh Working Conference on Current Measureent Technology RMS Velocity Deviations, c/s 3 1 EFFECT OF SPECTRAL AVERAGING TIME ON BASELINE DEVIATIONS 5 1 15 5 3 RMS Velocity Deviations, c/s 3 1 COMPARISON OF BASELINE DEVIATIONS AND SPATIAL UNCERTAINTIES 5 1 15 5 3 Distance fro Montauk, k Fig. 3: BASELINE TESTS Upper: : Coparison of the rs baseline deviations for one-hour spectral averages (solid line and 1-inute spectral averages (dotted line Lower: Coparison of the rs baseline deviations (solid line and the rs su of the spatial uncertainties (crosses CONCLUSIONS In conclusion: we have described the uncertainties inherent in SeaSonde current easureents: any of these are failiar: statistical uncertainties, analysis errors and so on. There is an additional source of uncertainty coon to all systes that ake polar easureents: as the area covered by the easureent increases with range, so does the span of values contained within the area. This should be taken into consideration whenever easureents with a different footprint are being cobined or copared. We have shown in this paper that large baseline deviations can occur when velocities obtained fro large and sall radar cells are copared, even though the central points of the cells coincide on the ocean surface. Siilarly errors ay occur when radial vectors corresponding to very different cell sizes are cobined to for a total velocity vector, as the velocity fro the larger cell size is ore area-averaged. This will be significant in regions of high current shear. When coparing Seasonde and ADCP easureents, it ust be born in ind that the SeaSonde gives an area easureent and the ADCP gives a point easureent; it is only in regions of low current shear and/or a sall 5
Proc. of the IEEE/OES Seventh Working Conference on Current Measureent Technology radar cell size that results fro the two instruents can be expected to be in good agreeent. When aking such a coparison, the SeaSonde spatial uncertainties should be exained if they are high at the ADCP location, one can not/should not expect good agreeent. This paper presents evidence fro baseline tests that indicate that the spatial standard deviation in SeaSonde radial velocity easureents is a easure of the horizontal velocity shear in the radar cell; However, this hypothesis awaits experiental verification, for exaple by coparison with siultaneous output of two or ore ADCPs in the sae radar scatter patch. In future studies, the effects on SeaSonde results of reducing the angular and radial size of the radar cell will be exained. In addition, we will be studying the effect of reducing the easureent duration on the radial velocity uncertainties. The result is not iediately obvious, as there is a tradeoff: reducing the easureent tie will increase the uncertainty due to statistical noise, and reduce the uncertainty due to variation of the ocean current pattern. At present, uncertainties in the total vector coponents do not include contributions fro spatial uncertainties in the radial vectors. These need to be included, because, as for the baseline deviations, significant errors result when radial vectors corresponding to radar cells of different sizes are cobined to for a total velocity vector. Finally, it is clear that a coplete calculation of total vector uncertainties ust include the uncertainties in the radial velocities as well as the effects of geoetry. Calculations that include only the latter (coonly tered geoetrical dilution of precision are necessarily incoplete. Recoendations for future work: 1. Verification that SeaSonde spatial uncertainties represent horizontal current velocity shear using ADCPs located in a single radar cell.. Reduction of SeaSonde spatial uncertainties by reducing the size of the radar scatter patch. 3. Studies to deterine whether SeaSonde uncertainties can be reduced by decreasing the tie duration of a easureent. 4. Inclusion of spatial uncertainties in the total vector output. Acknowledgeents: Many thanks to David Kaplan, for his critical reading of this anuscript and any valuable suggestions, to Don Barrick and David Ullan for helpful discussions and to David Ullan for providing the data used in the baseline analyses. 6