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Uncertainty Wedge Analysis: Quantifying the Impact of Sparse Sound
Speed Profiling Regimes on Sounding Uncertainty
J. Beaudoin∗
J. Hiebert†
B. Calder‡
G. Imahori§
Abstract
Recent advances in real-time monitoring of uncertaintydue to refraction have demonstrated the power of esti-mating and visualizing uncertainty over the entire po-tential sounding space. This representation format,referred to as an uncertainty wedge, can be used tohelp solve difficult survey planning problems regard-ing the spatio-temporal variability of the watercolumn.Though initially developed to work in-line with un-derway watercolumn sampling hardware (e.g. movingvessel profilers), uncertainty wedge analysis techniquesare extensible to investigate problems associated withlow-density watercolumn sampling in which only a fewsound speed casts are gathered per day.
As uncertainty wedge analysis techniques require nosounding data, the overhead of post-processing sound-ings is circumvented in the situation when one needs toquickly ascertain the impact of a particular samplingregime. In keeping with the spirit of the underlyingreal-time monitoring tools, a just in time analysis ofsound speed casts can help the field operator assess theeffects of watercolumn variability during acquisitionand objectively seek a watercolumn sampling regimewhich would balance the opposing goals of maximizingsurvey efficiency and maintaining reasonable soundingaccuracy.
In this work, we investigate the particular problemof estimating the uncertainty that would be associatedwith a particular low-density sound speed samplingregime. A pre-analysis technique is proposed in whicha high-density set of sound speed profiles provides abaseline against which various low-density samplingregimes can be tested, the end goal being to ascertainthe penalty in sounding confidence that would be asso-ciated with a particular low-density sampling regime.
∗Ocean Mapping group, University of New Brunswick, Fred-ericton, NB, Canada
†noaa Hydrographic Systems and Technology Program, Sil-
ver Spring, MD, USA‡Center for Coastal and Ocean Mapping and noaa-unh Joint
Hydrographic Center, University of New Hampshire, Durham,NH, USA
§noaa Office of Coast Survey, Silver Spring, MD, USA
In other words, by knowing too much about the water-column, one can objectively quantify the impact of notknowing enough. In addition to the goal-seeking fieldapplication outlined earlier, this allows for more confi-dent attribution of uncertainty to soundings, a markedimprovement over current approaches to refraction un-certainty estimation.
1 Introduction
Uncertainty propagation allows for the estimation ofthe horizontal and vertical uncertainty of a depthsounding based on
• the uncertainties of all additional measurementsmade in support of the sounding, and
• a mathematical model that describes the relation-ship between the uncertainties of the supportingmeasurements and the uncertainty of the reducedsounding.
The total propagated uncertainty (tpu) of a soundingresults when one uses the mathematical model to es-timate how the uncertainties of observed parameterscombine to affect the uncertainty of sounding.
Although attribution of uncertainty to a soundingis a noble effort in and of itself, it has proved to beuseful for many applications in hydrography, partic-ularly in automated data validation and filtering [5].tpu estimates are also often used to reject data whosetpu exceeds a specified tolerance [15]. Manufacturersof mapping instrumentation strive to maintain sensoraccuracies within tolerable limits to minimize impacton tpu. There is clearly merit in having the tpu asaccurate as possible. This hinges on having (a) reason-able estimates of the uncertainties of the supportingmeasurements, and (b) an accurate uncertainty prop-agation model.
A weakness in many uncertainty models is the treat-ment of the spatio-temporal variability of the watercol-umn. For example, the Hare-Godin-Mayer [9] modelcombines the effects of sound speed instrumentation
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uncertainty and spatio-temporal variability of the wa-ter column into a single uncertainty estimate. Thesame model also limits the sound speed uncertaintyeffect to the deviation in refracted ray angle at theinterface of a two layer model of the ocean.
Sound speed instrumentation uncertainties are typ-ically available from instrument manufacturers or canbe estimated based on observed instrument drift orfrom repeated casts. The effect of systematically bi-ased sound speed instrumentation has been investi-gated by several researchers and is relatively well un-derstood [7, 8]. Uncertainty due to spatio-temporalwatermass variability, however, is markedly differentin nature from that of instrumentation uncertainty andcan potentially lead to large sounding biases if the wa-termass variability is inadequately sampled.
As with instrumentation uncertainty, the randomsmall-scale variations throughout the watercolumntend to have a negligible effect, but systematic largescale variations can lead to significant biases, with thenature of the bias varying dramatically with the verti-cal location and nature of variability in the water col-umn. Applying a single sound speed uncertainty valuefor the entire watercolumn likely remains a valid ap-proach for instrumentation uncertainty, however, thismay not suffice for spatio-temporal variability as thevertical location of the variability in the watercolumnhas a strong modulating effect on the magnitude ofrefraction artifacts. It is only those soundings thatare deeper than the water column variability that areimpacted by the variability. For example, consider theset of sound speed casts in Figure 1 where variability ispronounced between depths of 3 to 13 m. Though thevariability may be dramatic, it is of little consequencefor those soundings shoaler than 3 m. In this case, asingle all-encompassing uncertainty value for the speedof sound that applies to all depths could produce overlypessimistic or overly optimistic estimates of soundinguncertainty, depending on the sounding’s depth withrespect to the depth of the water column variability.
Recent work within noaa has attempted to sim-plify the task of estimating the component of tpu dueto variable watercolumn conditions. The EstimatingSound Speed uncertainty (ESS) method is based upona comparison of raypaths associated with a pair of syn-thetic sound speed casts statistically derived from aset of observed casts [10]. This work was an importantstep forward for two reasons. Firstly, it provided anobjective method to calculate the sound speed uncer-tainty from a set of observed casts, a process that waspreviously very subjective. The second, and perhapsmore important impact, was that it allowed field op-erators to objectively quantify the impact of varying
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Figure 1: Sound speed casts gathered over a 2.5 hourinterval near the mouth of the Rotterdam Waterwayin March 2009.
watercolumn conditions in the field, e.g. [17]. As theESS method was designed to generate uncertainty esti-mates to the Hare-Godin-Mayer model as implementedin the cube algorithm [5], its applicability was limitedin that it still reduced watercolumn variability to asingle uncertainty value. Additionally, the underlyingassumption that variability at the surface is perfectlycorrelated with variability at depth limits its use to avery limited set of oceanographic environments. Thesetypes of conditions are seen, for example, in very wellmixed environments where the temperature and salin-ity are more or less constant over the watercolumn andchange as a whole due to spatial or temporal variations.
Similar work done independently at the Universityof New Brunswick (unb) has demonstrated the valueof estimating and visualizing the uncertainty due todiffering watercolumn conditions over the entire po-tential sounding space, i.e. from sounder to seafloorand across the entire angular sector. Through a com-parative raytrace simulation process [3], uncertainty isestimated over the potential sounding space, generat-ing what we refer to as an “uncertainty wedge” [1].In this work, the unb method is used to (a) quantifythe impact of observed variability in terms of sounding
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uncertainty, and (b) analyze sounding uncertainties as-sociated with various sampling regimes, e.g. samplingonly once per day.
Fundamentals of uncertainty wedge calculation andanalysis are discussed first, followed by examinationof sample problems which demonstrate the applicationof uncertainty wedge analysis techniques to commontypes of problems in hydrographic surveying. Finally,the issue of integrating uncertainty wedges, and theanalysis techniques described herein, into existing al-gorithms and workflows is discussed.
2 Method
As mentioned earlier, the unb method is to simulatewhat sounding uncertainty would result from the useof alternates models of the watercolumn. The simu-lator requires only sound speed profiles as input andcan be tuned to match any mapping system. Variableparameters include draft, angular sector, range per-formance envelope, and the inclusion of surface soundspeed probe as an additional measurement (thoughsurface sound speed probe data are not required). Ap-pendix A discusses these parameters in further detail,Appendix B specifically treats the case of simulatingthe inclusion of a surface sound speed measurement.
The simulator is based upon monitoring the pro-gression of two or more acoustic raypaths, all sharinga common initial launch, or depression, angle and eachraypath being associated with a particular sound speedprofile. A constant velocity acoustic raytracing algo-rithm [14] is used to explore how differing models ofthe sound speed structure, e.g. the two sound speedprofiles shown in Figure 2, can alter the raypath, andultimately, the divergence of the set of raytraced so-lutions for a given two-way travel-time (TWTT) anddepression angle, as shown in Figure 3. By systemat-ically modifying the depression angle and TWTT, theentire potential sounding space is explored to populatea depth and distance indexed table of sounding depthand horizontal discrepancies, as shown in figures 4 and5. These tables are referred to as uncertainty wedgesthroughout the remainder of this work.
The raytrace simulator can be used to track thedispersion of raypaths associated with a set of sev-eral sound speed profiles representing a sample of thepopulation of possible water column conditions in agiven area. This type of analysis, referred to as a Vari-ability Analysis (va), allows for the construction of avariability wedge, or a v-wedge. The v-wedge cap-tures the “potential uncertainty” associated with watermass variability. Figure 6 demonstrates the principle
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Figure 2: Two sample sound speed profiles.
behind the estimation of the potential horizontal anddepth uncertainty for a single location in the potentialsounding space. Expanding the analysis for all nodesin the sounding space, one can construct a v-wedge.For example, Figure 7 shows a v-wedge constructedfor the set of sound speed casts shown in Figure 1.
An Uncertainty Wedge Analysis (uwa) consists ofcomparing two raypaths only, allowing for a quantita-tive answer to the following question: “What wouldthe bias be if sound speed profile B was used in theplace of sound speed profile A?”, where profile A rep-resents known conditions and profile B represents analternate model whose fitness is to be tested by a com-parison to A. As the comparison of two casts quanti-fies the sounding bias that would be introduced if onecast had been used in the place of the other, the re-sulting uncertainty wedge is more aptly named a biaswedge, or a b-wedge. By comparing many pairs ofcasts, a set of b-wedges can be generated; these canthen be averaged to provide a mean bias wedge alongwith a standard deviation wedge. These are respec-tively referred to as an m-wedge and s-wedge in thistext. By modifying the casts that are used as refer-ence and/or those that are used as test candidates, onecan perform sophisticated analyses. For example, onecan test the fitness of an oceanographic climatology as
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Figure 3: Raytrace solutions associated with sound speed casts in Figure 2; draft is 1.0 m, depression angle of 20◦
and TWTT is 0.051 s. The raytraces in Panel A demonstrate how dramatic variations in the watercolumn can causegreat divergence in the raypaths. Panel B demonstrates how using a surface sound speed probe has the potentialto mitigate the effects of surface variability in some cases. In this latter case, the solutions were computed using acommon surface sound speed value of 1455m s−1.
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Figure 4: Depth uncertainty wedges associated with casts in Figure 2; draft is 1.0 m and angular sector of 150◦. Asin Figure 3, Panel A and B show the cases of independent and common surface sound speeds, respectively. Notethe different colour scales for each panel.
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Figure 5: Horizontal uncertainty wedges associated with casts in Figure 2; draft is 1.0 m and angular sector of 150◦.As in Figure 3, Panel A and B show the cases of independent and common surface sound speeds, respectively. Notethe different colour scales for each panel.
a source of sound speed information, in this case thes-wedge represents the potential uncertainty associ-ated with using a climatology [3]. Another example isin real-time monitoring of uncertainty associated withhigh rate water column sampling, where b-wedges areused to ascertain whether or not appreciable changes inthe watercolumn are occurring during the interval be-tween casts. If so, the sampling rate is increased untilthe sounding discrepancy between casts is acceptableor at least within tolerable limits. If not, the operatormay choose to relax the profiling rate to reduce wearand limit exposure to risk of grounding or fouling ofinstrumentation [1].
In summary:
• v-wedge (variability wedge): measure of the po-tential uncertainty associated with the spatio-temporal variability of the water column.
• b-wedge (bias wedge): measure of the bias hadan alternative cast been used in place of an ob-served cast.
• m-wedge (mean bias wedge): arithmetic mean ofseveral b-wedges.
• s-wedge (sigma wedge): standard deviation asso-ciated with a set of b-wedges.
The following section demonstrates how these un-certainty representation formats and analysis tech-niques can be used to help the hydrographic surveyorachieve their accuracy requirements.
3 Sample Analysis Problems
Time spent on reconnaissance is seldomwasted.
British Army Field Service Regulations, 1912
Much can be learned about water column variabil-ity, and its impact on sounding accuracy, by heav-ily oversampling a water mass using underway or ex-pendable sound speed profiling instrumentation. Us-ing analysis techniques such as those presented in thiswork allows the hydrographic surveyor to perform anoceanographic pre-analysis of the survey area; insightsgained during such an effort can help direct immediateor future field operations.
For example, a va can quantify the potential uncer-tainty associated with watermass variability using theentire set of casts collected during such an sampling ef-fort. In the event that the va shows that the variabilitywill have little impact on sounding accuracy, the dataset can be artificially reduced, or thinned, to match aplanned operational sampling scheme, e.g., samplingevery six hours, to ascertain perhaps how many castsare actually required. uwa techniques can then beused to quantify the uncertainty cost associated witha given sampling regime in light of the potential uncer-tainty quantified by the v-wedge. On the other hand,if the va shows a potential for appreciable and intol-erable uncertainty, the surveyor may choose to sample
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Figure 6: Raypaths calculated for the 82 sound speed profiles shown in Figure 1 using a draft of 1.0 m, a depressionangle of 20◦, a TWTT of 0.051 s and a common surface sound speed of 1445m s−1. The inset panel (A) on theupper right corresponds to the rectangular box drawn near the termini of the raypaths shown in the main panel.The lower left panel (B) shows the ends of the raypaths only and demonstrates how the final raytraced solutionsdisperse depending on which sound speed profile is used for raytracing. The mean depth and position are indicatedby the yellow triangle, the error bars indicate the 95% confidence level. Note that the main panel and upper rightpanel share the same distorted aspect ratio whereas the aspect ratio of the lower left panel is correct.
the watercolumn more often or to accept the lose ofaccuracy in the outer portions of their mapping swathand reduce their line spacing accordingly. In eithercase, such a pre-analysis effort allows for a more in-telligent, efficient and systematic allocation of surveyresources. For example, a launch capable of under-way sampling might be sent to areas with dynamicoceanographic conditions whereas launches equippedwith standard sound speed sampling equipment can besent, with confidence, to areas where the oceanographyis perhaps more quiescent.
The following sample problems demonstrate thepower of such pre-analysis techniques. We begin byperforming a va for a set of locations in an area with
very dynamic oceanography and demonstrate how thepre-analysis can identify problematic areas. This isfollowed by an uwa in an area where oceanographicchanges have little impact on sounding accuracy. Inthis case the surveyor can ascertain just how few soundspeed casts would actually be required to maintain ac-curacy.
3.1 Variability Analysis
In this section, we demonstrate how v-wedges can beused to capture and quantify the spatial and tempo-ral variations of survey area’s watercolumn character-istics. In this case, we examine the Rotterdam Water-way in the Netherlands, where the Meuse river meets
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Figure 7: A Variability Wedge generated from the set of sound speed casts shown in Figure 1.
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the North Sea (see Figure 8). A field trial was con-ducted by the Dutch Public Works (Rijkswaterstaat,rws) in March of 2009 with an ODIM Brooke OceanMVP30 and a Kongsberg EM3002 (dual-head) multi-beam onboard the rws survey vessel Corvus. Thisarea was selected for the trial, because it tends to suf-fer from particularly strong refraction artifacts and be-cause high sedimentation rates and heavy vessel trafficnecessitate repeated surveys every four weeks.
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Figure 8: Overview map showing study area at theentrance to the Rotterdam Waterway.
Figure 9: Satellite image of sound speed cast locationsin Rotterdam Waterway, March 2009. Data courtesy ofRijkswaterstaat (Dutch Public Works). Cast locationsare indicated for several survey areas as black crosses.Text labels correspond to plots in Figures 13 through17.
Over the course of the seven day trial, 2,151 soundspeed casts were acquired in several test survey areasand over long sections running from Maassluis to anarea just offshore of the mouth of the Waterway (loca-
tions A and E in Figure 9, respectively). The temper-ature and salinity variations observed are consistentwith a salt wedge type estuary with a strongly strati-fied watermass. Fast flowing surface water is predomi-nantly fresh and bottom water is predominantly saltywith little mixing between the two watermasses. Saltwater intrudes upriver on the flood tide, acting like awedge and sliding underneath the surface fresh water.During a falling tide, strong river currents rapidly flushthe salty bottom water back to sea.
These types of environments are challenging to hy-drographers as the majority of the variability is in thedepth of the interface between the fresh and salt water,with the interfacial depth varying strongly spatiallyand temporally (note the turbulent interface betweenthe two layers in Figure 10). As the change in soundspeed can be quite dramatic at the interface betweenthe fresh and salt water, soundings can refract quitestrongly and can lead to significant sounding uncer-tainty with seemingly small variations in the interfa-cial depth. Figures 10 through 12 depict the intrusionof saltwater below the freshwater over various stagesof the tide for three days of the seven day trial.
Figure 10: Vertical salinity section (20 m deep) overa distance of 11.5 km from station A to C on March25th.
Figure 11: Vertical salinity section from station A toC on March 30th.
Plots of sound speed profiles and v-wedges derivedfrom them are shown for locations A through E inFigures 13 through 17, respectively (note that the lo-cations were sampled on different days). Examininglocation A first (Figure 13), the midwater interfacialdepth varies vertically by almost 5 m and introducessignificant outer beam uncertainty beyond depths of 5
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Figure 12: Vertical salinity section from station A toC on April 2nd.
m. Referring to the salinity section in Figure 11, it isclear that there are stages of the tide where the salt wa-ter is flushed away. Though the potential uncertaintyquantified by the v-wedge is non-negligible in for thestage of the tide over which the casts of location A weregathered, a patient surveyor might instead choose towait for an appropriate stage of the tide before survey-ing in this area. That is, armed with nothing but atide table and a conventional sound speed profiling in-strument, the surveyor could collect exploratory castsup the river on a falling tide until the terminus of thesalt wedge was found. Once found, a survey couldproceed slightly upstream of the salt wedge with littleconcern for the troublesome variability associated withthe wedge as it is safely downstream of the survey lo-cation and not likely to return until the tide begins torise after low water.
Turning to location B, the variability is more pro-nounced and dominates over a larger portion of thewatercolumn, perhaps half of the watercolumn as op-posed to one third as was observed at location A. Thevariability is predominantly spatial in nature and re-sults from survey lines running parallel to the riverflow, coinciding with the direction of maximum spatialvariability. In this case, none of the sections in figures10 through 12 show location B being free of the effectsof the migrating salt wedge. Location B can likely beclassified as a “trouble spot” in which underway sam-pling instrumentation is routinely used for all surveysin the area.
Compared to location A, the variability at locationC occurs deeper in the watercolumn. As a result, thepotential uncertainty is lower when compared to thev-wedge for location A, where the variability occursmuch shallower in the watercolumn. One can arguethat location B would benefit from this same logic asthere are stages of the tide where the variability ismuch lower in the watercolumn, for example, comparelocation B in Figure 10 and Figure 10.
The Caland Canal section of the Waterway (loca-tion D) is cut off from direct freshwater inflow fromthe river, it thus suffers less from the refraction prob-
lems that affect surveyors working in the main channelof the Waterway. In this area, it is likely possible tosample once every few hours and still maintain accu-racy. Given a fleet of survey vessels, one might assignthis site to the vessels equipped with static profilingsystems and focus vessels equipped with underway pro-filing systems in the main river channel.
At the mouth of the river (location E), the outflowof the river is limited to a thin surface layer. As thevariability is quite shallow in the watercolumn, the di-vergence of raypaths occurs early on during raytracing.Sounding uncertainty is thus introduced at a shallowdepth and persists over the remainder of the raytrace.In this area, it may be advisable to use a deeper draftvessel. A va was performed for a deeper draft vessel todemonstrate, the resulting v-wedge is shown in Figure17b. Though there is a loss in swath width associatedwith the deeper draft vessel, the survey system is muchless sensitive to the surface variability. A further ad-vantage to the deeper draft vessel is that surface soundspeed instrumentation uncertainties will have less of aneffect on electronically beam-formed systems as thereis less variability at the transducer depth.
3.2 Sampling Regime Analysis
In this section we examine an MVP data set consistingof 224 casts collected in Advocate Bay in the Bay ofFundy (Canada), by the CCGS Matthew in Octoberof 2008 (refer to figures 19 and 20). For research pur-poses, the profiling interval was set at two minutes.This rate is much higher than standard acquisitionrates onboard the Matthew: casts are typically onlycollected when significant changes in surface soundspeed are observed via the surface sound speed probe.The watercolumn was observed to be completely mixedwith temperature and salinity remaining more or lessconstant with depth, thus the only vertical variationin sound speed was due to the pressure increase withdepth. Small spatial variability in the temperature andsalinity led to small changes in the sound speed overthe survey area (approximately 10 km by 2.5 km), referto figures 21 and 22. A v-wedge has been constructedfor the set of casts (Figure 23). Clearly the variabil-ity here is not particularly troublesome. Contrastingstarkly with the previous pre-analysis problem donefor the Rotterdam Waterway, the problem here is in-stead to identify just how few casts would be requiredto maintain accuracy.
In this sample problem, a set of b-wedges is com-puted by comparing each cast in the full set of casts(the “truth”) to the profile that would have been usedin the artificially thinned set of casts (e.g., the first
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Figure 13: Sound speed casts and v-wedge for a 1.5 hour period near Maassluis in the Rotterdam Waterway (60casts, location A in Figure 9).
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Figure 14: Sound speed casts and v-wedge for a 2.5 hour period while surveying within the surge barrier structureof the Rotterdam Waterway (148 casts, location B in Figure 9).
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Figure 15: Sound speed casts and v-wedge for a 1.0 hour period at the entrance to the Rotterdam Waterway (52casts, location C in Figure9).
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Figure 16: Sound speed casts and v-wedge for a 2.5 hour period in the Caland Canal portion of the RotterdamWaterway (44 casts, location D in Figure9).
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Figure 17: Sound speed casts and v-wedge for a 2.5 hour period just off the entrance to the Rotterdam Waterway(106 casts, location E in Figure9).
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Figure 18: V-wedge for a deep draft vessel in LocationE, contrast with Figure 17b where a draft of 0.3 m wasused.
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Figure 19: Sound speed profiles acquired by the CCGSMatthew in Advocate Bay (224 casts).
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cast of the day, or six casts used with a nearest-in-distance selection algorithm). In this case, pairwisecomparisons are made that allow for the compilationof m-wedges and s-wedges that capture the com-bined effects of (a) the water column variability, and(b) the simulated sampling regime. It is proposed thatthe s-wedge derived from the set of b-wedges is a rea-sonable predictor of uncertainty as long as the rangeof oceanographic conditions remains similar from dayto day. As a counterexample, if the range of condi-tions (and their impact on accuracy) varied dramat-ically between spring tides and neap tides, then onemight achieve more accurate uncertainty estimates byperforming such an analysis twice: once during springtides and again during neap tides. By differing theparameters of the sub-sampling experiment, e.g., thenumber of casts per day or location, one can objectivelyseek a sampling regime which balances an increase inuncertainty against gains in survey efficiency througha hard cap on the number of casts to be collected. It isnot the intention of this work to find such an optimalsampling scheme, but instead to demonstrate how sucha scheme could be sought using the methods describedherein.
We begin by quantifying the reduction in potentialuncertainty achieved through the use of the MVP200.This is done by sequentially comparing each cast to itspredecessor in the set, yielding 223 b-wedges. Eachb-wedge represents the bias that affects the soundingsat the moment just prior to collection of each cast.By compiling the b-wedges into an m-wedge ands-wedge (see Figure 24), one can ascertain the reduc-tion in potential uncertainty gained through the use ofthe MVP system. In the well mixed, macro-tidal envi-ronment in Advocate Bay it is obvious that 224 castsprovided more than enough information: the depth un-certainty of the outermost beam is approximately 4 cmin 40 m of water, well within the most stringent of sur-vey specifications and a marked improvement over thepotential uncertainty of roughly 12 cm as depicted inthe v-wedge of Figure 23.
Would a single cast have done just as well? A similaranalysis procedure is followed in which 223 b-wedges
are computed by comparing each cast in the set to thevery first cast of the set, i.e. there exist 223 cases whereone can numerically quantify the bias that would occurif the first cast had been used for the entire survey.Compiling an m-wedge and s-wedge quantifies theaverage bias the data set would have suffered, as wellas the dispersion of the soundings about the averagebias (see Figure 25). In this case, using the first castof the day would have introduced a significant averagebias (see Figure 25a) and would have suffered the full
effects of the environmental variability (compare thes-wedge of Figure 25b to the v-wedge of Figure 19).Using the uncertainty wedges of Figure 25 as a lookuptable, the outermost beam in 40 m of water wouldhave been biased, on average, by approximately 0.10m +/- 0.13 m. This is still quite tolerable in the waterdepths encountered in the survey area and with the200% coverage typically achieved with CHS survey linespacing.
Although averaging b-wedges provides a useful“snapshot” of the potential uncertainty associatedwith a particular sampling regime, it masks potentiallyinteresting details one may see by examining all of theb-wedges that went into the average. A useful way tovisualize patterns and assess causes of bias is to geo-graphically map the outermost beam bias at full depthfor every b-wedge, as in Figure 26. In this case, thespatial variability in temperature and salinity has thepotential to cause up to 25 cm of bias in the outer-most beams had the first cast of the data set beenused throughout the entire area.
Though the single cast proved tolerable in termsof sounding bias, the situation can be improved withuse of only a few more casts. As the variability isprimarily spatial in nature and does not exhibit com-plex spatial patterns, it may be preferable to simply“box in” the survey area with a few casts along theperimeter. For this example, we have chosen six casts,three along the southern perimeter and three along thenorthern perimeter. Each cast in the entire set of castsis compared to its nearest neighbour from the subsetof six perimeter casts, generating 217 b-wedges. Anm-wedge and an s-wedge for this particular sam-pling regime is shown in Figure 28. Mapping the out-ermost beam bias for this scenario demonstrates thatthis method yields much less bias than the case of asingle cast as was examined previously (refer to Figure27).
The pre-analysis effort yields a few hints on how tosurvey in such an area, assuming of course that condi-tions remain similar at other times. Variability is notsignificant and is primarily spatial in nature. A fewwell chosen cast locations could have significantly im-proved upon the potential bias introduced through theuse of a single cast, even though a single cast providesreasonable accuracy over the range of water depths inthe survey area.
13
−65˚00' −64˚58' −64˚56' −64˚54' −64˚52'45˚15'
45˚16'
45˚17'
45˚18'
45˚19'
−60 −55 −50 −45 −40 −35
Depth
m
−66˚ −64˚
44˚
45˚
46˚
Figure 20: Survey area covered by CCGS Matthew during MVP200 data acquisition (224 casts, marked by yellowcircles).
4 Application and Integration
4.1 Integration into CUBE
cube [5] is a computer-assisted hydrography algorithmthat attempts to estimate the true depth at any givenposition within the survey area, and provide someguidance to the user as to how well that depth isknown; cube is an acronym for “Combined Uncer-tainty and Bathymetry Estimator”. The essence ofthe cube algorithm is the understanding, modelingand utilisation of the uncertainty of the measurementsthat go into the depth soundings that are collected bymbes equipment; for dense mbes data, cube can pro-vide very rapid, robust depth estimates from raw dataand assist the user in assessing which data needs at-tention, improving the data workflow.
While cube itself does not mandate how uncertain-ties are computed for the soundings that it ingests,it does require that the uncertainties are “reasonable”
in the sense that they need to reflect at least a first-order accurate depiction of the actual variability of thesoundings for the algorithm to operate as intended. In-tegration of the current work into cube therefore re-quires that we address the problem of how to integratethe uncertainty due to ssp spatio-temporal effects intothe computation of the tpu, and consider the implica-tions that this has for cube’s operation.
During development of the algorithm, the Hare-Godin-Mayer [9] model was used to construct tpu es-timates for the soundings, and although the model hasbeen refined since, it is still the most commonly usedmodel for tpu estimation in current use. In this model,the effects of sound speed uncertainty are containedin terms that affect the fundamental uncertainties ofrange and angle estimation. Due to the paucity ofinformation about the degree of spatio-temporal un-certainty at the time, however, the effects of measure-ment uncertainty in the ssp and the spatio-temporalvariability of the watermass as reflected in the profile
14
−65˚00' −64˚58' −64˚56' −64˚54' −64˚52'45˚15'
45˚16'
45˚17'
45˚18'
45˚19'
13.00 13.25 13.50
Temperature (deg. C) −66˚ −64˚
44˚
45˚
46˚
Figure 21: Map of temperature variation over survey area at 20 m water depth.
15
−65˚00' −64˚58' −64˚56' −64˚54' −64˚52'45˚15'
45˚16'
45˚17'
45˚18'
45˚19'
31.6 31.7 31.8 31.9
Salinity (psu) −66˚ −64˚
44˚
45˚
46˚
Figure 22: Map of salinity variation over survey area at 20 m water depth.
16
−50
−40
−30
−20
−10
0
Dep
th (
m)
0 25 50 75 100 125
Across−Track (m)
0.00 0.05 0.10 0.15 0.20
Depth Uncertainty (m)
Figure 23: A v-wedge showing the potential uncertainty in the survey area. Note that the jagged appearance inthe lower portion of the v-wedge is due to the decreasing number of casts at depths greater than 35 m. As fewerand fewer casts are available for the analysis at these depths, the standard deviation drops significantly as comparedto the shallower sections of the watercolumn.
−50
−40
−30
−20
−10
0
Dep
th (
m)
0 25 50 75 100 125
Across−Track (m)
−0.04 −0.02 0.00 0.02 0.04
Mean Depth Bias (m)
(a)
−50
−40
−30
−20
−10
0
Dep
th (
m)
0 25 50 75 100 125
Across−Track (m)
0.00 0.02 0.04 0.06 0.08 0.10
Standard Deviation of Depth Bias (m)
(b)
Figure 24: M-wedge and s-wedge showing uncertainty associated with using the entire set of casts. As withthe v-wedge of Figure 23, the deeper portions of the wedges suffer from sharp discontinuities as the number ofb-wedges contributing to the analysis for a given depth level declines with depth. In these cases, the shallowercasts (and the b-wedges that result from comparson to short casts) do not contribute and the sample mean andstandard deviation suddenly lock on to the potentially much tighter distribution of the deeper casts in the set.
17
−50
−40
−30
−20
−10
0
Dep
th (
m)
0 25 50 75 100 125
Across−Track (m)
−0.10 −0.05 0.00 0.05 0.10 0.15 0.20
Mean Depth Bias (m)
(a)
−50
−40
−30
−20
−10
0
Dep
th (
m)
0 25 50 75 100 125
Across−Track (m)
0.00 0.05 0.10 0.15 0.20
Standard Deviation of Depth Bias (m)
(b)
Figure 25: M-wedge and s-wedge showing uncertainty associated with using the first cast of the set instead ofthe 224 measured casts.
were combined into a single uncertainty term resultingin a simpler computational model, but a cruder repre-sentation of the true effects of the oceanographic envi-ronment. In particular, in environments where there issignificant spatial specificity to the degree of ssp vari-ation, the model is forced to adopt a pessimistic uncer-tainty analysis in order to cover the worst-case areas,and provides little assistance in assessing the effectsfor the user. This also results in reduction of process-ing power in cube since much of the algorithm’s abil-ity to determine likely depth reconstructions (which isessential to the efficiency of the algorithm) is predi-cated on the uncertainty reflecting the true nature ofthe soundings. By adopting a pessimistic analysis, therobustness of the estimation is weakened everywhere.
We now have the opportunity to refine this modelthrough use of the uwa to reflect the spatio-temporaluncertainty. The current model represents the depthof the sounding by
d = r cos(θ + ρ) cosφ
= r cos(θ′) cosφ (1)
with indicated range r and angle θ, roll ρ and pitch φ.(In the following, we work with the combined angle θ′
for simplicity of presentation.) Using the principle ofpropagation of uncertainty, the predicted uncertaintyin depth is
σ2d =
(
∂d
∂r
)2
σ2r +
(
∂d
∂θ′
)2
σ2θ′ +
(
∂d
∂φ
)2
σ2φ. (2)
Pitch and roll are unaffected by ssp effects, but if we
trace the effects due to range and angle uncertaintythrough the vertical model, we find that they are, re-spectively
σ2d(r) = cos2 φ cos2 θ′σ2
r (3)
σ2d(θ) = r2 sin2 θ′ cos2 φσ2
θ (4)
and that the effects of the various components of un-certainty in range and angle are linear once multipliedby their respective sensitivity factors [12]. Knowingthat Hare et al.[9] show that the uncertainty of rangeand angle measurement associated with ssp effects canbe approximated as
σ2r(ssp) =
( r
v
)2
σ2v(ssp) (5)
σ2θ(ssp) =
(
tan θ′
2v
)2
σ2v(ssp) (6)
respectively, we can refactor (3) and (4) to extract thessp specific terms, and evaluate the remainder of themodel as at present. Specifically, if we assume that
σ2v(ssp) = σ2
v(ssp−m) + σ2v(ssp−st) (7)
for measurement and spatio-temporal effects respec-tively, we can substitute in (5) and (6), extractthe spatio-temporal component and develop a modi-fied formulation that replaces the ssp spatio-temporalterms with the output of the uwa look-up tables, ne-glecting the terms
σ2d(ssp) =
1
4
(
4 cos4 θ′ + sin4 θ′) r2 cos2 φ
v2 cos2 θ′σ2
v(ssp-st).
(8)
18
−65˚00' −64˚58' −64˚56' −64˚54' −64˚52'45˚15'
45˚16'
45˚17'
45˚18'
45˚19'
0.00 0.05 0.10 0.15 0.20
Outer beam bias (m) −66˚ −64˚
44˚
45˚
46˚
Figure 26: Map of outer beam bias experienced when using the first cast of the survey for the entire survey area(location of first cast marked by yellow star).
19
−65˚00' −64˚58' −64˚56' −64˚54' −64˚52'45˚15'
45˚16'
45˚17'
45˚18'
45˚19'
−0.05 0.00 0.05 0.10 0.15 0.20
Outer beam bias (m) −66˚ −64˚
44˚
45˚
46˚
Figure 27: Map of outer beam bias experienced when using six casts along the perimeter of the survey area (locationsmarked by yellow stars). As expected the bias grows with distance from the cast locations.
20
−50
−40
−30
−20
−10
0
Dep
th (
m)
0 25 50 75 100 125
Across−Track (m)
−0.10 −0.05 0.00 0.05 0.10
Mean Depth Bias (m)
(a)
−50
−40
−30
−20
−10
0
Dep
th (
m)
0 25 50 75 100 125
Across−Track (m)
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Standard Deviation of Depth Bias (m)
(b)
Figure 28: M-wedge and s-wedge showing uncertainty associated with using six casts on the perimeter insteadof the 224 measured casts.
in the vertical and
σ2p(ssp) =
( r
v
)2
σ2v(ssp-st)×
[
(
1 − (cos θ′ cosφ)2)
+1
4
(
1 − (sin θ′ cosφ)2)
tan2 θ′]
(9)
in the horizontal from the original model. In practice,this can be done simply by setting the value of σ2
v(ssp)
used at present to represent only the measurement un-certainty of the ssp component, and then adding theuwa-derived look-up table values in vertical and hori-zontal to the output of the current model.
A less ideal, but simpler to implement, solutionwould be to compute an estimate of the combined un-certainty per the original model by equating (8) and (9)for σ2
v(ssp−st) first and then combining the values withthe measurement uncertainty for the ssp instrumentusing (7) before evaluating the model as currently im-plemented. This might be useful where the uwa showsthat there is little need to re-evaluate the look-up ta-bles per beam, since this approximation would providea significant time saving in the computation.
There are three main implications for this techniqueapplied to the cube algorithm. First, allowing the tpu
algorithm to properly reflect the uncertainty of the ssp
in the individual soundings should greatly assist thecube algorithm in maintaining robustness throughoutthe survey area. A great deal of the power of the cube
algorithm resides in its ability to automatically sepa-
rate groups of soundings that are mutually inconsis-tent, and his action relies on the uncertainties of thesoundings reflecting the variability expected in them.With empirical estimates of uncertainty applied, thenumber of stray soundings that are incorporated into ainternally consistent group should decrease, which willlead to better in-group depth estimates, and thereforeless interaction on the part of the user.
Secondly (and an immediate corollary of the firstimplication), in areas where there are significant ssp
spatio-temporal effects, the increase in uncertainty ap-plied in the uwa encourages the algorithm to con-sider soundings that are refracted as “sufficiently sim-ilar” that they can be accommodated as one consis-tent group, therefore reducing the number of spuri-ous secondary hypotheses on the actual depth that areformed. Reduction of the number of hypotheses thatare formed makes it simpler for the algorithm to as-sess which is most likely, and improves the algorithm’sability to choose the “right” hypothesis to report to theuser. The efficiency with which the user can processdata depends strongly on how often the algorithm cando this, so the improved uncertainty estimates shouldreduce operator time correspondingly.
Thirdly, the uncertainty that is assessed within thegroup of soundings that are combined to make thedepth estimate reported to the user as the primaryhypothesis will, under the proposed scheme, better re-flect the actual uncertainty in the data. At present, ifthe soundings have higher uncertainty than predicted(e.g., due to refraction), the algorithm tends to make
21
secondary hypotheses. The choice of which assess-ment of uncertainty to report to the user from thecube algorithm has been subject to some debate, butis currently most often an estimate of the standarddeviation of the soundings used to compute the hy-pothesis. Currently, therefore, the assessment withinthe secondary hypotheses will only reflect the uncer-tainty of one group of soundings in the area, typicallythose from one pass of the mbes, and therefore under-estimates the actual uncertainty of the data in thearea. In the proposed scheme, the increased uncer-tainty associated with the ssp spatio-temporal effectswould cause soundings with higher refraction effects tobe considered as one hypothesis (per the previous im-plication), and therefore the uncertainty reported tothe user would be correspondingly higher. This moreaccurately quantifies the actual uncertainty of the datain the area, and should result in better modeling andassessment of the value of the data in the area for thesurveyors and the end users. Note that this is not tosay that the data is necessarily useless under these con-ditions: depending on the survey standards in effect,the surveyor might assess the data as adequate, evengiven the increased uncertainty reflected in the finaloutput. The outputs of the algorithm will, however,better reflect reality.
Assessing the magnitude of the effects of the pro-posed scheme in cube processing is a non-trivial pro-cess, in great part because of the subjective natureof the operator-in-the-loop processing that is still re-quired by any practical processing system. It is possi-ble, however, to assess both the number of hypothesesthat the algorithm generates over a particular area,and measure how often the human operator agreeswith the assessment of which hypothesis the algorithmthinks is correct by measuring the number of rejectedhypotheses after operator intervention. Correlatingthese terms with a mapping of the uncertainty pre-dicted by the uwa might provide a semi-quantitativemethod to assess the utility of the proposed scheme,and we are actively pursuing this in current research.
4.2 Integration into NOAA Workflows
While the merits of the uncertainty wedge analysisare theoretically valuable in their own right, we un-derstand that a pragmatic approach be simultaneouslytaken in order for its significance to be realized. noaa’sOffice of Coast Survey has written a sound speed anal-ysis toolkit–VelociPy– that incorporates some of theuwa and va functionality of the unb SVP Toolkit [4].
At the beginning of 2008 an early version of the Ve-lociPy toolkit (named ESS) [10] was made available
to noaa field units, offering them a variation of thev-wedge described in section 1. This first pass atVelociPy was used by test-and-development personnelseeking to maximize efficiency of their survey opera-tions [17]. The current version of VelociPy implementsthe ability to do simple wedge analysis, calculating av-wedge for a set of profiles, calculating b-wedges
between profile pairs, temporal scatter plotting of pro-files and other tools that give the field hydrographerbetter oceanographic information. It is hoped that thecurrent version of the VelicPy toolkit will allow fieldunits to use uwa to better dynamically analyze thespatio-temporal variability of the sound speed in theirproject area and react accordingly to this new sourceof information.
noaa’s hydrographic survey workflow has yet to ad-just to the new tools and techniques available and willnow have several prospective options similar to thosedescribed above and by Beaudoin at Shallow Survey2008 [1].
• noaa may now have the ability to save on ex-cess ‘wear and tear’ of profiling sensors and re-lated equipment (e.g. moving vessel profiler fittedwith a sound speed sensor) by using a b-wedge
to determine whether the current sampling regimeis sufficient or if oversampling is occurring. If wa-ter column features correlate to tidal activity thenperiodic checks once or twice a day may suffice. Ifthe variability is occurring from multiplicative fac-tors, then more consistent sampling and b-wedge
analysis may be needed. The point is that thesampling rate may be adjusted on-the-fly.
• Likewise noaa may be able to determine wheremore sound speed samples are necessary and inwhat approximate geographic location.
• These new additions may also be used during theproject development stage using historical data.While it is possible that sound speed data is sim-ilar enough to weather in that there are no abso-lutes in determining exactly where a sample needsto be taken. However, by analyzing past vari-ability and driving oceanographic factors, projectmanagers may be able to instruct the field survey-ors as to where an area has typically needed moresampling (or less).
In the upcoming field season, noaa will be look-ing at simple test bed areas for the software testingand training purposes, then following with a few morecomplicated areas the next season to refine our prac-tices.
22
5 Conclusion
Having the tools to assess this type of sensitivity in thefield can help surveyors to make better decisions re-garding temporal and spatial sound speed profile sam-pling. Oceanographic pre-analysis campaigns can beundertaken in areas where repetitive, high accuracy,surveys are the norm for safety of navigation or secu-rity.
The most important benefit of this approach is thatuncertainty is estimated from watercolumn conditionsdirectly, circumventing the problem of oversimplify-ing the water column in a model. As the uncertaintywedge representation provides estimates of uncertaintyfor each sounding based on its unique depth and across-track position in the potential sounding space, it hasthe potential to augment the fidelity of the water col-umn variability component of uncertainty models usedin the hydrographic community.
6 Future Work
6.1 Integration with Ocean Modeling
These new analytical techniques have given hydrogra-phers the ability to better understand ocean variabil-ity and have greatly improved how noaa can adjustsound speed sampling frequency. However, certain ar-eas have such significant variability which are eitherimpossible to capture or resource restrictions may pre-vent the use of a representative sampling regime thatthere may never be enough data to fully understandocean dynamics.
For this reason, noaa is supplementing the investi-gation of this technique with a model-based approachfor predicting the level of variability in a survey area[11]. Rutgers University’s ROMS numerical ocean cir-culation, forecast model has been used successfully tomodel shallow waters where a significant portion ofnoaa hydrography occurs. For areas where a halo-cline or thermocline creates problems for multibeamsystems, the model’s horizontal and vertical grid reso-lution can be adjusted to capture the necessary physicsassociated with the area. Also, any additional post-processing metric requirements can be computed viaFortran routines and added to the existing ROMS soft-ware package.
noaa is currently developing operational circula-tion/forecast models for Chesapeake Bay, Tampa Bay,and Delaware Bay and has future hopes of completinga model for Cook Inlet, AK.
For surveys which fall within areas for which thereexist validated ROMS models, we envision the models
to be used during the survey planning process. Dur-ing survey planning, output of the model may predictthe driving factors of the local oceanography, allowingthe determination of an appropriate sampling regime.ROMS includes the ability for data assimilation, so atypical first day of surveying may include oversamplingthe water column profiles for the purposes of modelvalidation.
The caveat of this approach is that producing an op-erational ROMS model for each particular geograph-ical area can take an extended duration (especiallyif data is not readily available) and requires a greatamount of resources. However, once the model is val-idated, it may run in perpetuity. For this reason theregion-by-region predictive modeling approach for cre-ating a sound speed sampling regime may have limitedapplication, until a larger basin-scale set of oceano-graphic models can be developed and validated.
6.2 UWA and VA with sparse data sets
Though the value of this new method of estimating un-certainty due to refraction is attractive, it relies heavilyon having an oversampled watermass, something whichis not always practical to collect. The question thenbecomes: can this be done, or at least approximated,with an undersampled watercolumn only? Future workwill investigate whether uwa and va gives reasonableresults when given much smaller data sets of casts.Of course both methods can provide results with smalldata sets, but how uncertain are we willing to be aboutour uncertainty?
7 Acknowledgments
This work was supported by the sponsors of the omg
and by noaa under grant NA05NOS4001153. The au-thors would like to thank Rijkswaterstaat for the op-portunity to participate in the Rotterdam Waterwaytrial as it provided the perfect data set for which toshowcase the analysis methods presented in this work.The Canadian Hydrographic Service is thanked for ac-quiring the Advocate Bay data set on our request.Thank you to Dr. Lyon Lanerolle for his supportingwork with the ROMS ocean circulation model and forreviewing the related sections of this article.
References
[1] J. D. Beaudoin. Real-time monitoring of uncer-tainty due to refraction in multibeam echosound-ing. In Proc. Shallow Survey 2008, Oct 2008.
23
[2] J. D. Beaudoin, J. E. Hughes Clarke, andJ. Bartlett. Application of surface sound speedmeasurements in post-processing for multi-sectormultibeam echosounders. International Hydro-graphic Review, 5(3), 2004.
[3] J. D. Beaudoin, J. E. Hughes Clarke, andJ. Bartlett. Usage of oceanographic databasesin support of multibeam mapping operations on-board the CCGS Amundsen. Lighthouse, Journalof the Canadian Hydrographic Association, (68),2006.
[4] J. D. Beaudoin, J. E. Hughes Clarke, J. Bartlett,S. Blasco, and R. Bennett. Mapping canada’s arc-tic seabed: Collaborative survey processing anddistribution strategies. In Proc. Canadian Hydro-graphic Conference 2008, May 2008.
[5] B. R. Calder and L. A. Mayer. Automaticprocessing of high-rate, high-density multibeamechosounder data. Geochem., Geophys. andGeosystems (G3) DID 10.1029/2002GC000486,4(6), 2003.
[6] D. S. Cartwright and J. E. Hughes Clarke. Multi-beam surveys of the frazer river delta, copingwith an extreme refraction environment. In Proc.Canadian Hydrographic Conference 2002, May2002.
[7] D. F. Dinn, B. D. Loncarevic, and G. Costello.The effect of sound velocity errors on multi-beamsonar depth accuracy. In Proc. IEEE Oceans 1995,volume 2, pages 1001–1010. IEEE, Oct 1995.
[8] E. Hammerstad. Multibeam echo sounder accu-racy. Technical report, Kongsberg Maritime AS,2001.
[9] R. Hare, A. Godin, and L. A. Mayer. Accu-racy estimation of canadian swath (multibeam)and sweep (multitransducer) sounding systems.Technical report, Canadian Hydrographic Service,1995.
[10] G. Imahori and J. Hiebert. An algorithm forestimating the sound speed component of totaldepth uncertainty. In Proc. Canadian Hydro-graphic Conference 2008, May 2008.
[11] G. Imahori, L. Lanerolle, S. Brodet, R. Downs,J. Xu, R. Becker, and L. Robidoux. An inte-grated approach to acquiring sound speed pro-files using the Regional Ocean Modeling System(ROMS) and an Autonomous Underwater Vehicle(AUV). In Caris Conference, 2008.
[12] International Organisation for Standardization.Guide to the expression of uncertainty in mea-surements. Technical report, ISO, Geneva, 1995.ISBN 92-67-10188-9.
[13] Kongsberg Maritime AS. Operators Manual - EMSeries Datagram Format (20-01-06), 2006.
[14] H. Medwin and C. S. Clay. Fundamentals ofAcoustical Oceanography. Academic Press, 1998.
[15] National Oceanic and Atmospheric Administra-tion. NOS Hydrographic Surveys Specificationsand Deliverables, 2008.
[16] Reson Inc. SeaBat 8101 Multibeam EchosounderSystem Operators Manual Version 2.20, 2000.
[17] D. Wright. Analysis of NOAA methods of ap-plication of sound speed uncertainty and alterna-tives derived from use of the Estimating SoundSpeed uncertainty software. In Proc. Shallow Sur-vey 2008, Oct 2008.
A Raytracing Simulator Func-
tionality
In order to serve as a reasonable predictor of sound-ing uncertainty, the simulator must honour the real-time sounding geometry as much as possible. The ray-tracing procedure thus requires reasonable estimates ofseveral parameters, some of which simply modify therange of depths and angles to be investigated whereasothers fundamentally change the behaviour of the ray-tracing algorithm. These are listed below along withexplanation of how they affect the fidelity of the sim-ulation.
Surface speed probe. A surface sound speedprobe is often required to ensure correct electronicbeam steering angles with linear transducer arrays. Itis also often used to augment the sound speed profileduring raytracing by (1) using the measured surfacevalue as “the initial entry in the sound speed profileused in the raytracing calculations” [13] or (2) cal-culating Snell’s constant, or the ray parameter, withthe observed surface value prior to raytracing [2]. Aspointed out by Cartwright and Hughes Clarke [6], theincorporation of the surface sound speed measurementhas a significant effect on the behaviour of a raytrac-ing algorithm, in some cases it allows for a gracefulrecovery from surface layer variability as long as thedeeper portion of the watermass is relatively invariant.Regardless of this potential gain, the inclusion of the
24
surface sound speed as an additional measurement fun-damentally changes the behaviour of a raytracing al-gorithm thus its effect must be included in uncertaintymodels.
The unb method mimics the use of a surface soundspeed probe by retrieving the sound speed at trans-ducer depth from the reference profile and using thisto compute the ray parameter for the test cast ray-trace without modifying the test cast (i.e. replacingthe measurement at the draft depth with the surfacesounds speed). One must take care, however, to onlyperform this additional step if the acquisition and/orpost-processing software can accommodate the surfacesound speed as an additional aiding measurement dur-ing sounding reduction, specifically the raytracing por-tion of the procedure. For example, a surface soundspeed value may be input into a Reson 8101 MBESfor use in pitch stabilization [16]. Though this value islogged in the data stream, it is not used in subsequentraytracing calculations performed in post-processing inCaris HIPS (Wong, personal comm.). In this case, thesimulator should not be configured to mimic a surfacesound speed probe as this would give inaccurate re-sults, especially in the case where surface variability issignificant.
Angular sector. The nominal angular sector thatcan be achieved by the sounder controls the shallowestdepression angle that must be investigated and heav-ily influences a mapping systems overall sensitivity tovariable watercolumn conditions. As the outermostedges of the swath are typically the most sensitive torefraction, the predictive ability of the simulator de-pends heavily on having an accurate estimate of theoutermost beams depression angle. The outermost de-pression angle can be easily underestimated and over-estimated in various conditions. These two cases areexamined in turn below.
In dynamic roll conditions, a system that is not roll-stabilized can experience larger refraction artifacts inthe outer portions of the swath due to smaller thannormal depression angles associated with extremes invessel roll. By limiting the investigation to the nomi-nal angular sector, the simulator would underestimatethe refraction in the outermost beams during largeroll events and the output would be overly optimistic(though this would only apply to one side of the swath).If the outermost soundings must be retained to main-tain overlap between survey lines, then the simulatorshould allow for an artificial increase to the angularsector to allow for large roll events. It should be notedthat in particularly large roll events (10◦ − 15◦) andwith large angular sector systems (e.g. ±75◦), theoutermost rays will tend to horizontal and will not
likely have a bottom return. With an unstabilized sys-tem, the operator must make an effort to estimate thelargest achieved angular sector instead of simply in-creasing the angular sector by adding the largest ex-pected roll value. Vessel pitch can also reduce the out-ermost depression angles though the influence is notnearly as pronounced as that of vessel roll.
In the case that the outermost edges of the swathfall beyond the maximum achievable range of a map-ping system, the achieved angular sector can be signif-icantly smaller than the nominal case. In this case, thesimulator must allow for a reduction of angular sectorwith increasing depth, otherwise the uncertainty esti-mates would be overly pessimistic. This can be donemanually by adjusting the angular sector to match thesector achieved under actual working conditions. Thiswould also apply in the case where filtering applied inpost-processing would artificially reduce the angularsector, e.g. filtering all soundings outside of ±60◦.
Draft. A particular mapping system’s susceptibil-ity to surface variability can vary dramatically depend-ing on the depth of the transducer in the watercolumn.The transducer draft should therefore be used as thestart point of the raytrace. The simulator currentlydoes not allow for vertical motion of the transducerthrough the watercolumn and all analyses are basedon a static draft assumption.
B Simulation of Surface Sound
Speed Probe
Variability Analysis (va) is based upon examining thedivergence of a bundle of raypaths, with each raypathtied to a different depiction of the watercolumn. For agiven travel time, depression angle and surface soundspeed, the bundle of rays will land at some locationin the potential sounding space. The scatter of theirsolutions about their mean position in the potentialsounding space serves as an indicator of the sensitiv-ity to watercolumn variability. The problem is that weneed to simulate the use of a common surface soundspeed measurement as the initial entry into the water-column during the raytrace for each ray in the bundle,but which sound speed should be used? It turns outthat it doesn’t matter.
Consider a raytrace with a depression angle of 20◦
(incidence angle of 70◦) and sound speed of 1445m s−1.The ray parameter used in the raytrace is calculatedas:
k1 = sin(70◦)/1445
As the ray parameter is a function of depression an-
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gle and sound speed, there exists other angle/soundspeed pairs that would yield the same rayparame-ter. For example, consider a surface sound speed of1440m s−1. Snell’s law is applied to determine whichangle would give the same ray parameter:
k2 = sin(θ)/1440 = k1
⇒ sin(θ)/1440 = sin(70◦)/1445
so: θ = arcsin(1440 sin(70◦)/1445)
≈ 69.462◦
∴ ψ = 90◦ − θ ≈ 20.538◦
where ψ is the depression angle.If one were to perform an acoustic raytrace with
a common sound speed profile and differing surfacesound speed/depression angle pairs, the rays wouldshare the exact same raypath, despite having differentdepression angles and different surface sound speeds.In essence, it is possible to get to the same locationon the seafloor through a different launch angle andsurface sound speed combination.
How does this apply to the simulation of the use ofa surface sound speed probe in va? If the above ex-ercise is true for one ray, then it is true for all rays ina bundle of rays being investigated in a va. One canarrive at the mean location by investigating a givendepression angle and surface sound speed from one ofthe casts, or by using a different depression angle anda different surface sound speed, chosen from a differentcast in the set. As all of the rays in the bundle will allarrive at their same respective positions in either case,then their relative positions with respect to their meanposition will remain the same. It follows that the dis-persion of the solutions about the mean location wouldalso remain the same regardless of how the bundle ofrays arrived at the mean location. In other words,any one of the casts can be chosen as a measurementof truth and we would eventually, through some com-bination of surface sound speed and depression angle,arrive at the same mean location and witness the samedispersion of the raytraced solutions. So, an arbitrarysurface sound speed value could be chosen, and witha systematic sweep across the angular sector we willhave visited every spot in the potential sounding spaceand calculated the dispersion in the same manner hadanother surface sound speed been chosen.
Note that the exact matching of raytraced solutionsdepends heavily on how the raytrace algorithm uses theadditional surface sound speed measurement to aug-ment the sound speed profile. The following procedureis used with the UNB method: (1) the surface soundspeed and depression angle are used to define the ray
parameter, and (2) the ray is immediately refracted atthe beginning of the raytrace as if an infinitesimallythin layer of water exists at the transducer face inwhich the speed of sound is the measured surface soundspeed.
Deviation from this methodology will result in smalldiscrepancies in the equality of a ray solutions whenone modifies the surface speed or depression angle aswe have in this exercise. For example, simply replac-ing the sound speed at the transducer depth in thewatercolumn can yield slight inconsistencies in the re-sults and is sensitive to layer thickness in the raytracealgorithm.
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