nonlinear optimization of autonomous undersea vehicle...

Nonlinear Optimization ofAutonomous Undersea Vehicle

Sampling Strategies forOceanographic

Data-Assimilation

Kevin D. HeaneyOcean Acoustical Services and InstrumentationSystems, Inc.11006 Clara Barton Dr.Fairfax Station, Virginia 22039

Glen Gawarkiewicz and Timothy F. DudaWoods Hole Oceanographic InstitutionWoods Hole, Massachusetts 02543

Pierre F. J. LermusiauxDivision of Engineering and Applied SciencesHarvard UniversityPierce Hall, 29 Oxford St.Cambridge, Massachusetts 02138

Received 2 May 2006; accepted 22 December 2006

The problem of how to optimally deploy a suite of sensors to estimate the oceanographicenvironment is addressed. An optimal way to estimate �nowcast� and predict �forecast�the ocean environment is to assimilate measurements from dynamic and uncertain re-gions into a dynamical ocean model. In order to determine the sensor deployment strat-egy that optimally samples the regions of uncertainty, a Genetic Algorithm �GA� approachis presented. The scalar cost function is defined as a weighted combination of a sensorsuite’s sampling of the ocean variability, ocean dynamics, transmission loss sensitivity,modeled temperature uncertainty �and others�. The benefit of the GA approach is that theuser can determine “optimal” via a weighting of constituent cost functions, which caninclude ocean dynamics, acoustics, cost, time, etc. A numerical example with three glid-ers, two powered AUVs, and three moorings is presented to illustrate the optimizationapproach in the complex shelfbreak region south of New England. © 2007 Wiley Periodicals,Inc.

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Journal of Field Robotics 24(6), 437–448 (2007) © 2007 Wiley Periodicals, Inc.Published online in Wiley InterScience (www.interscience.wiley.com). • DOI: 10.1002/rob.20183

1. INTRODUCTION

The problem of accurately predicting the ocean tem-perature and salinity fields for use by sonar perfor-mance prediction algorithms is addressed �Abbot &Dyer, 2002; Heaney & Cox, 2006; Robinson & Lermu-siaux, 2004; Robinson, Lermusiaux & Sloan, 1998�.Current meso-scale oceanographic models have in-corporated much of the relevant physics so that if ini-tial and boundary conditions are known well, thesemodels can provide useful predictions of the oceantemperature, salinity, and velocity fields. Recent de-velopments in data assimilation �optimal interpola-tion, extended Kalman filtering, and error subspacemethods� have shown that models can predict theocean state better with the inclusion of field data, andthe use of models can improve mapped fields gener-ated from measured data.

The challenge to this situation is limited re-sources. Working in the ocean is expensive and tech-nically challenging. Methodologies that determineideal ways to measure the ocean with the fewestwisely placed sensors are needed. An integratedoptimization-assimilation-modeling system is beingdeveloped to perform optimal ocean sampling andocean prediction. The system involves the combina-tion of three subsystems or algorithms: �1� an oceanmodel with data-assimilation capabilities �in our casethe Harvard Ocean Prediction System �HOPS� �Haley,Lermusiaux, Leslie & Robinson, 2006; Lermusiaux etal., 2006; Patrikalakis et al., 2004; Robinson et al.,1998��, with the Error Subspace Statistical Estimationalgorithm �Lermusiaux, 2004� �ESSE��; �2� an en-semble of measurement platforms �i.e., moorings,gliders, powered autonomous undersea vehicles�;and �3� a nonlinear constrained global optimizationalgorithm, which determines the optimal placementof these sensors. In this paper, we present the mea-surement subsystems and the nonlinear constrainedglobal optimization subsystem.

The optimization approach is an application ofthe Genetic Algorithm �GA� �Goldberg, 1989�. TheGA solves complex global optimization �minimiza-tion in our case� problems by generating a populationof individuals �possible solutions� and within eachsuccessive population of individuals �iteration�choosing the survivors based upon the value of thecomplex cost-function surface. Mutation and cross-over, as well as other analogies to natural selection,are implemented to prevent the solution leading to alocal minimum. The implementation of the GA solu-

tion requires a model for the search parameter spaceand the definition of a cost function to be minimized.The parameter space will involve a vector for eachplatform defining its search geometry. Examples ofvector values for a sensor deployment are initial po-sition, initial direction, range of sample path, andnumber and direction of turns. Once a sensor deploy-ment scheme is defined �an individual in the GA no-menclature� the cost function is evaluated using thevalues from the oceanographic model that this par-ticular deployment of sensors would sample. Thecost function is evaluated for this individual, andthen the GA generates another set of search laydownsbased upon these results. In order to define the globalcost function for the GA to optimize, a set of scalarcost function constituents has been developed. Thesolutions of the GA optimization are only as optimalas the cost function that is defining them. It is a cur-rent research program to build and define the best setof cost functions for this problem.

Significant work has been done in the past 15years on developing and deploying AUV samplingnetworks �Curtin, Bellingham, Catipovic & Webb,1993�. The issues associated with optimal deploy-ment and oceanographic sampling have also beencovered �Bellingham & Wilcox, 1996; Wilcox, Belling-ham, Zhang & Baggeroer, 2001�. Real-time, adaptiveexperiments using AUVs and ocean models havebeen conducted. Other powerful techniques for adap-tive control of underwater assets based on coveragemetrics are described in Leonard �Leonard et al., 2006;Ogren, Fiorelli & Leonard, 2004; Paley, Zhang & Le-onard, 2006�.

The outline of this paper is as follows. In Section2 we use measurements and a fully data-assimilatedmodel from the ONR Shelfbreak PRIMER Experi-ment �Linder, Gawarkiewicz & Pickart, 2004� as thetest-bench for the new methods we have developed.This includes a description of the ocean model runsthat were used for optimization. In Section 3 the Ge-netic Algorithm optimization for multiple sensor de-ployment is presented and applied to these data-driven model runs. Section 4 is the conclusion, whereplans for the future testing of this algorithm are dis-cussed.

2. MODEL PROBLEM DEFINITION

In this paper, a data-driven model of the ocean in theMiddle Atlantic Bight region �shelfbreak south of

438 • Journal of Field Robotics—2007

Journal of Field Robotics DOI 10.1002/rob

New England� will be used to illustrate the new op-timal adaptive sampling approach. This is a region ofhigh oceanographic variability that has been studiedby US oceanographers and ocean acousticians for thepast 15 years �Apel et al., 1997; Gawarkiewicz et al.,2004�. The shelfbreak is located 100 km offshore andinteractions of the shelfbreak front, Gulf Stream me-anders, and wind-driven response to storms create acomplicated environment. The bathymetry for the re-gion is shown in Figure 1�a�. The model grid has beenrotated to align with the local isobaths.

Using measured environmental data �tempera-ture, salinity, and density� from the ShelfbreakPRIMER experiment from July 2006, a 30 day simu-lation of the shelfbreak south of New England wasgenerated using the Harvard Ocean Prediction Sys-tem �HOPS� along with data assimilation via the Er-ror Subspace Statistical Estimation �ESSE� algorithm.

A significant amount of measured data went into thismodel forecast, including a week of SeaSoar high-resolution sampling, which resolved the fields interms of both the spatial and temporal correlationscales Gawarkiewicz et al., �2004�. The resulting sur-face temperature and salinity fields �for day 30� areshown in Figure 1 �upper right and lower left panels�.The remnants of a slope eddy are visible in the sur-face temperature and salinity maps �at y=−50 km N/S, x=−100 km E/W�. The surface expres-sion of the shelfbreak front is visible as the hightemperature/salinity gradient line at 0 km N/S inthe western half of the model domain. In the easternhalf, the shelfbreak front extends much further off-shore, up to 100 km seaward relative to the westernportion of the model domain. To illustrate the HOPS-ESSE estimate of ocean field uncertainties, the stan-dard deviation of the surface temperatures of the en-

Figure 1. Illustration of the HOPS-ESSE simulations with assimilation of data from the ONR Shelfbreak PRIMER Ex-periment. Shown are the bathymetry �m� and horizontal snapshots of the surface temperature �°C� and salinity �PSU� onday 30 and of the modeled temperature uncertainty of the surface temperature �°C� on day 15.

Heaney et al.: Sensor Planning Optimization For Data Assimilation • 439


semble of model simulations on day 15 is shown inthe lower right panel of Figure 1. On the shelf, the sur-face temperature uncertainty is around 0.2 to 0.4°C.Uncertainties are largest �above 1°C� around the sur-face expression of the shelfbreak front and its eddies,from about y=0 to −100 km, with a band of localizeduncertainty near y=−100 km N/S, where the shelf-break front extends far offshore. Such complex time-and depth-dependent ocean fields �temperature, sa-linity, velocity� and their uncertainties will be used inthe estimation of an optimal sampling strategy in thenext section.

3. GENETIC ALGORITHM OPTIMIZATION

A numerical experiment was undertaken to demon-strate the use of the nonlinear optimization algo-rithm. The task is to optimally deploy eight sensorsfor a period of 1 week in this simulation. For thisproblem, we optimize the deployment of the follow-ing assets: three fixed moorings, three slow moving�30 cm/s� gliders, and two powered REMUS �Re-mote Environment Monitoring UnitS AutonomousUndersea Vehicles developed by Woods HoleOceanographic Institution� AUVs. The REMUS ve-hicles are only deployed for the initial 24 h of opera-tions. The optimization algorithm is tasked with de-termining the best way to deploy all of these sensorsto optimally sample the simulated ocean variability.

The specifications of each platform for this par-ticular numerical example are as follows:

• Moorings—fixed position �x ,y� temperaturesensor for 7 days

• Gliders—drifting temperature sensor �with a0.3 m/s velocity in addition to the local cur-rents� deployed for 7 days. Input parametersare x, y, and direction vector.

• REMUS powered AUV vehicles—temperature sensor with a velocity of 4 m/sdeployed for 1 day �two 12 h legs with aturn�. Search parameters: deployment x, y,and direction vector and turn at 12 h.

This ensemble of platforms leads to a parametersearch of 23 independent variables. The samplingconstraints for each platform were coded in MATLABand the cost function was defined, for use within theMATLAB Genetic Algorithm Toolbox.

3.1. Cost Function Constituents

There are currently five constituent cost functions,evaluated for a specific sensor path �x�. The first isthe oceanographic variability. The specific functionis the negative average standard deviation �f=−std�T�x�� of the measured temperature field foreach sensor. Intuitively, the GA optimization �mini-mization� leads us to sensor sample patterns whichsample the most dynamic ocean regions includingfronts and eddies. Note that this function, as well asmany of the other cost function constituents used, isnot positive definite. The second cost function is theoceanographic temperature range �related to thestrongest dynamical features�. The negative averageof the maximum minus the minimum measuredtemperature for each platform is used �f=−�max�T�x��−min�T�x�� . This function seeks re-gions where there is a very strong change in the tem-perature field—an indication of strong fronts orother ocean features. The third constituent cost func-tion is the negative of the integrated uncertaintyalong each platform path �f=−sum�U�x��. The ESSEscheme outputs an estimate of the uncertainty foreach region and for each field, in space and time.This uncertainty-based function seeks to place sen-sors at positions where the uncertainty estimate isthe largest, helping reduce the largest uncertaintieswith subsequent measurements. The fourth costfunction is the transmission loss �TL� sensitivity. Foreach platform path, the acoustic field is computedfor the first day of the measurement and for the finalday of the measurement �using the model-onlyrange-dependent sound speed field�. The negativeaverage TL difference is the value of the cost func-tion. This function seeks to measure places wherethe sound speed field changes in such a way as toimpact the ocean acoustic propagation. Regions ofhigh ocean variability, where the acoustic field is notaffected, are not important for our use of the oceanmodel as an input to sonar performance prediction.The final cost function option is a distance potential,where each platform is punished for close proximity�approximately 1.5 to 2 times the baroclinic Rossbyradius of deformation in the region, or 20 km in thissituation� to other platforms, by the inverse of therange. This function maintains sensor distance tophysically intuitive lengths and minimizes possiblecollisions between sensor platforms.

The global scalar cost function is what the Ge-netic Algorithm seeks to minimize. It is a function of



the search parameters, in this case the positions anddirections of the sensor platforms. In defining the“optimal” deployment configuration of this diversesuite of sensors, it is up to the user to linearly com-bine the defined cost-function components to build aglobal scalar cost function for the GA to minimize.This permits the user to weigh ocean dynamics,ocean variability, model uncertainty, TL sensitivity,etc. and produce a platform deployment plan that isconsistent with this view of optimal. The cost func-tion E we will use is a weighted combination of thefive cost functions described above. The weighted,normalized cost function E�x� is computed as fol-lows:

E�x� = �i

Wi fi�x��fi�

�1�

where Wi are weights and ��fi� are normalizationfunctions, each explained in the next paragraph. Theunderlying principle is that for a specific surveyplan, or sensor geometry �x�, a variety of constituentfunctions �f� can be used at the discretion of the user,each evaluated from the modeled values of the field.In real-time, with data coming in, a residual vectorcan be computed between the measured and themodeled ocean fields can be included. This will re-duce the value of measurements where the model iscorrectly predicting the true ocean, even in dynamicplaces.

The primary input by the user are the weightsWi. Prior to a genetic algorithm run, a large sam-pling of the space is conducted and the variance �en-ergy� of each of the cost functions is computed. Thecumulative cost function in E is then obtained bysumming the weighting value of each cost function�Wi fi� divided by the prior computation of the stan-dard deviation of ��fi�. This normalization takes careof the difference in magnitudes of the various costfunctions as well as the units. Each cost function is adimensionless, normalized scalar, permitting theuser defined weighted sum to be the global costfunction used for optimization. Many options existfor the constituent functions. The five that we haveused, which were introduced above, are next de-scribed in more detail.

3.1.1. Oceanographic Variability ��T�The first constituent cost function is the oceano-graphic variability. It is computed by taking the

negative of the standard deviation of the measuredtemperatures. The underlying principle here is thatwe want to measure where the ocean is changing asa function of space and time. Regions with oscillat-ing temperature fields or strong fronts will yieldhigh standard deviations and the GA will prefer tosample these locations. To illustrate the GA searchfor this function, a simplified optimization with asingle REMUS vehicle is computed. The results areshown in Figure 2 �upper-left panel� for day 30 ofthe ocean model simulation. The REMUS vehiclewas placed by the GA such that it transected theremnant of a warm-core eddying feature and thenmoved into colder water. The standard deviation ofthe measured temperature for this path was 2.851°,which is large relative to other possible paths in themodel domain. This solution agrees well with ourintuition for where the most oceanographic variabil-ity can be expected.

3.1.2. Oceanographic Dynamic Range �FeatureStrength�

The oceanographic dynamics cost function measuresthe temperature drop across the sensor path. Specifi-cally it is the negative of the Max�T�−Min�T�. Thiscost function is based upon the notion that we wantto measure where the ocean has the strongestchanges in temperature. This cost function works ef-fectively as a front detector. The results for the singleREMUS case are shown in Figure 2 with the tem-perature differential across the path of −8.7°. Thispath is very similar to the path found using onlyoceanographic variability, in Section 3.1.1, exceptthat the path is moved slightly north to capture thecold water just north of the eddy.

3.1.3. Oceanographic Uncertainty

One of the outputs of statistical ocean circulationmodels �from a model ensemble�, such as the HOPS-ESSE system, is the forecasted uncertainty for eachspecific location and variable as a function of time.In order to improve accuracy, measurements shouldbe taken at positions �and times� that reduce theseuncertainties the most. This problem has been ad-dressed for example by Yilmaz �Yilmaz, 2005; Yil-maz, Evangelinos, Lermusiaux, & Patrikalakis, 2006�and Lermusiaux �Lermusiaux, 2006�. The definitionof the cost function is the negative of the integrateduncertainty across the path. This makes the mini-



mum of the cost function the path in which the high-est uncertainty is measured. The results for a singleREMUS deployment are shown in Figure 2 �lowerleft panel�. The REMUS sample position is overlayedon the uncertainty map for day 15 of the HOPS-ESSEsimulation. Clearly the highest uncertainty at thesurface is where the shelfbreak front extends a largedistance offshore. The uncertainty cost functioncomplements the previous two cost functions. Thesolution is to run the REMUS through the peak ofthe uncertainty map both ways. The REMUS vehiclepath doubles back on itself to resample the region ofhighest temperature uncertainty. This could beeliminated by applying the distance potential con-straints to regions where a single sensor has previ-ously gone. Current implementation of the distancepotential is between separate sensors.

3.1.4. Temperature Minimum

Following optimal adaptive sensor planning anddata assimilation, one of the final products of theoceanographic model is the prediction of soundspeed fields for acoustic computations. Acoustic en-ergy refracts away from warm water and thereforepropagation is best in regions of a sound channelaxis or a sound minimum. An efficient proxy foracoustic propagation, and therefore a candidate for acost function, is the minimum temperature mea-sured by a particular sensor.

3.1.5. Transmission Loss Sensitivity

The final product of the anti-submarine-warfare�ASW� system that uses oceanographic predictions isa sonar performance prediction. It is the stated goal

Figure 2. Single REMUS results for ocean variability, ocean dynamic range, ocean uncertainty and transmission losssensitivity. The color maps are temperature �top two and bottom bottom right, °C� with color axis: �15 26� and tempera-ture uncertainty �bottom left, °C� color axis: �0 2�.



of this work that the oceanography should be accu-rate enough to permit accurate acoustic propagationmodeling �Abbot & Dyer, 2002; Heaney & Cox, 2006;Robinson, Abbot, Lermusiaux & Dillman, 2002; Rob-inson and Lermusiaux, 2004�. Ocean acoustic trans-missions are commonly characterized by their trans-mission loss �TL� as a function of range �Jensen,Kuperman, Porter & Schmidt, 1997�. TL is the at-tenuation of an acoustic signal due to geometricalspreading, absorption, and scattering. TL is sensitiveto the oceanographic environment �sound speed� aswell as the sediment, which along with the surfaceforms the boundaries of the ocean sound channel.Experimentally, TL is measured by taking the re-ceived acoustic intensity �RL� and subtracting �indB� the source level �SL� within any prescribed fre-quency band. Thus TL=RL-SL. Often in the litera-ture, TL is positive �SL-RL�. With accurate acousticpredictions in mind, we seek to include acousticpropagation sensitivity in the cost function for theplatform deployment plan. Clearly regions wherethe acoustics are insensitive to the exact nature of theoceanographic variability need not be measured.

To begin to investigate this problem, the trans-mission loss �TL� was computed using the parabolicequation �Navy Standard RAM �Collins, 1993�� foreach platform path �a 40 km path was used for thefixed moorings�. The three-dimensional sound speedfield and bathymetry �Figure 2� were used to gener-ate two 2-D environments for the acoustic propaga-tion computation. The TLs computed using thesound speed profile �SSP� from day 1 and day 5were then compared �difference averaged over rangeand depth�, to compute a single measure of theacoustic sensitivity to oceanographic variability. Thismethod should be normalized so that bathymetricdifferences and geo-acoustic differences between re-gions are removed from the sensitivity computation.In order to compare transmission loss and not bedominated by the interference of multipath �whichDyer �Dyer, 1970� showed should lead to a standarddeviation around 5.6 dB�, we performed rangesmoothing �10% of the range�. This is equivalent to a20 Hz bandwidth average at 200 Hz �Harrison &Harrison, 1995�. Range smoothing is significantlymore efficient than computing the broadband acous-tic field. The cost function metric is the average ofthe point-by-point difference for regions with TL lessthan 110 dB. This last threshold is put in to limit theregions of variability to be those where we expect

signal. Note that in the nulls of a TL field, the differ-ences could easily be 10–15 dB, and this result is in-significant to the problem.

The GA result for a single sensor using only theTL cost function is shown in Figure 2 �lower rightpanel�. This is not a location of high surface dynamicoceanography nor of significant surface uncertainty.This is because the TL depends on the vertical varia-tions �not shown� of the temperature, salinity, andpressure, and not so much on the surface values ofthese properties �fields at depths were not used inthe previous GA examples�. The value of the costfunction for this path is −11.041 dB, which is a sig-nificant average TL difference �remember we are do-ing the equivalent of a 20 Hz band averaging�. Toillustrate the magnitude of this TL difference, inmany environments, the TL drops as a function ofrange �cylindrical spreading� according to 10 log �r�.In this environment, a 10 dB difference in TL is anorder of magnitude difference in range of detectionof a quiet target �all other variables being fixed�.Thus a target would be detectable at 20 km ratherthan 2 km for a 10 dB error in prediction.

To diagnose what happened to produce an aver-age TL difference of 11 dB, the range- and depth-dependent oceanography along the path of the ve-hicle was examined. The mixed layer is deeper�30–50 m� at t=0, and there is the presence of theshelf-break front for the last 30 km of the path. Theshelf-break front is not affecting the net TL differ-ence, however. It is clear that the deepened mixedlayer at t=0 leads to the situation where energy isrefracted towards the bottom and stripped out of thewater column. With a shallower mixed layer at t=5days, and the presence of some warm water near thesea-floor at 20–25 km, the sound is able to survivethe bottom interactions and propagation over theshelf into deeper water.

This example illustrates the need to incorporateacoustic sensitivity, or at least three-dimensionalocean fields, in any oceanographic measurement andmodel system, usually referred to as Ocean Observ-ing and Prediction System �Lermusiaux, 2006�. If thisparticular portion of the environment is not wellmodeled, acoustic predictions can be off by as muchas 15 dB.

3.1.6. Distance Potential

In the situation where multiple platforms are de-ployed it is important to apply a constraint to the



deployment scheme that does not permit the plat-forms to overlap each other. Without this sort of con-straint, the GA will attempt to put all of the sensorsin the same location, which minimizes the cost func-tion. This constraint can be applied in two differentways. The first is in the sensor deployment param-etrization. A scheme could be devised to move a sen-sor if another one was already there. This leads tounphysical dependencies between variables in thesearch space, in addition to being a challenge to pro-gram. A simpler approach, the one taken here, is toapply a distance potential and incorporate this intothe cost function �Leonard et al. 2006�. Simply stated,the distance potential cost function penalizes anyparticular individual �a specific deployment schemeof multiple platforms� if two or more of the plat-forms come within a certain range of each other. Ifthe platforms stay beyond the specified potentialdistance, there is no penalty. The potential distancewas chosen to be 1.5 times the baroclinic Rossby ra-dius. The ocean is expected to be coherent withinthis range and, if another sensor has measured thisarea, there is no value in adding a second one. Theequation for the distance potential cost function is:

f�rij� =ro

rij+ 1 �2�

where ro is the distance potential width and wastaken to be 20 km. The distance between each sensorpair for all times is rij. The distance potential there-fore asymptotically approaches 1 for large rij, andgrows exponentially when rij�ro. By adding the dis-tance potential to the cost function, the GA solutionnaturally chooses solutions that have platforms re-maining beyond the distance potential width ro.

3.2. Multiple-Platform Combined Cost FunctionRun

Prior to running the multi-platform, multi-parameter cost function, we perform a simple ex-amination of the topography of the cost function. Ingeneral the cost function is a function of multipleparameters �launch position, launch direction, turns�and multiple vehicles. To examine the form of thecost function, we will search over the initial positionof a single platform, which samples a 20 km pathoriented due North. The cost functions associatedwith the temperature dynamics, the minimum tem-

perature, and the integrated uncertainty, as well asthe weighted sum of the normalized cost functions,are shown in Figure 3. With uniform weighting, thetopology of each cost unction is visible in the com-bined cost function. In particular, the shelf breakfront minima in the STD T panel around �y=0 km, x=−150 km� and the strong T-Uncertaintyminima �y=−120 km, x=110 km� are visible.

With the cost function constituents now defined,a run of the GA for a full suite of eight sensor plat-forms and a linear combination of all five cost func-tions is conducted. The sensor suite includes threemoorings, simultaneously deployed with three glid-ers for 5 days and two REMUS vehicles �for 24 honly�. Each of the first five cost functions ocean vari-ability, ocean dynamics, ocean uncertainty, TL sensi-tivity, and sensor range potential, was given anequal weighting �Wi=1 for all i�. The sensitivity ofthe solution to this weighting can be evaluatedthrough multiple GA runs, or can be estimated fromFigure 3. The ocean variability �and ocean dynamicrange, not shown� cost function is deepest near theremnant of the gulf stream eddy; the ocean T uncer-tainty is focused on the shelf-break front and thetemperature minima is generaly spread north-east ofthe front.

The full GA solution is shown in Figure 4. Theproof that this is the optimal sensor deployment willrequire a significant amount of Monte Carlo model-ing and ocean data assimilation and is left for futurework. The results are qualitatively appealing, how-ever. Note that all three moorings have been placeddirectly on the shelf-break front �which passes overthem during the 5 days of deployment�. The REMUSvehicles both pass over manifestations of the breakand the gliders track large gradients with time. Thedistance potential also appears to be performing itstask of maintaining sensor placement beyond aRossby radius �15 km in this example�. The temporaldependence of the ocean field strongly influencesthe location of the moorings �which are fixed inspace� and the gliders �which are slowly drifting andare driven by currents�. The REMUS vehicles, on theother hand, are quickly moving and therefore areinsensitive to time-dependent oceanographic effects.

Note that every sensor begins its survey in coldwater and ends the survey in warm water, highlight-ing the effect of the min/max value in the cost func-tion as well as the standard deviation. The exceptionto this is platform 4 �a 24 h REMUS deployment�,which does the reverse.



4. CONCLUSION

By combining model forecasts, nonlinear optimalsensor deployment and data assimilation, a coupledapproach to optimally estimating the ocean environ-ment is presented. The technique involves a nonlin-ear constrained optimization utilizing the Genetic Al-gorithm. The technique was applied to a simulatedexample involving the placement of eight measure-ment platforms in a dynamic region. Five differentconstituent cost functions were developed and ap-plied separately to this problem. The functions areocean variability, ocean dynamic range, ocean uncer-tainty, sensor distance potential, and transmissionloss sensitivity. A major advantage of this techniquerests in the ability to add multiple normalized scalar

cost functions, providing the user with the ability toweight the final cost function based upon his ownspecific needs.

One of the significant questions to be answered inmodel-based optimization is how “optimal” is the so-lution. Without an extensive set of measurements thisis difficult to verify. Note that this is still true if amodel is not formally utilized in the planning: onecannot quantitatively determine if chosen measure-ment sites are ideal without additional �large� datasets. Another question relates to the optimality of thesolution computed by the GA algorithm. Eventhough the algorithm is global and designed to avoidlocal minima, its final solution is not guaranteed to bethe optimum of the chosen cost function. In addition,the final generation of the GA depends on the rules

Figure 3. Cost function topology for the standard deviation of the measured temperature on day 30, the integrateduncertainty on day 15, and the minimum measured temperature on day 30. The normalized, combined cost-function isplotted in the lower right panel. The only independent variables in the computation are the single platform startingposition.



chosen to determine how new generations areformed during the optimization process. Nonethe-less, there are counter-arguments to these two ques-tions. First, the use of computational models to pre-dict oceanic and acoustic states is quantitative and isnow beginning to provide more skill than guessing atwhat the future ocean and acoustic conditions will be.Similarly, the sampling strategies provided by ourGA approach so far are intuitively pleasing, combineall of the features we wish to optimize, are consistentwith the deployment constraints and are obtained au-tomatically and efficiently by a computer.

The examples provided here to to illustrate theapproach all used computationally simulated ratherthan experimentally derived fields. To further evalu-ate the capabilities and validity of the approach, realat-sea exercises should be carried out. Such exerciseswould sample the ocean using GA-based path plan-ning and the GA would define its cost function basedon the real ocean measurements and the data-drivenreal-time model predictions of ocean and acousticfields and uncertainties. Pretest surveys would be re-quired to provide initial conditions for the model.With the ocean model up and running, a flexible sam-pling architecture would then be necessary. Thiswould hopefully include various surface ships for de-

ploying fixed moorings and deploying and recover-ing AUVs. Glider deployment can be done at the startof the test and the gliders can be guided via satellitecommunications. Redeploying the gliders with sur-face ships is a significant plus, but for large opera-tional areas this can be very difficult. Acoustic propa-gation measurements, both as data for assimilationand/or post-test verification, would also be part ofthis demonstration test. A comprehensive set ofoceanographic measurement �moorings/air XBTs�could be used to validate the results, after the test.The suite of measurements will provide the opportu-nity, after the test, to perform a numerical study todetermine the value added to the ocean prediction foreach measurement.

The example field measurement optimizationthat has been described here was designed to samplevariability and to sample regions where the model ispoorly constrained. This is only one of many possibleapproaches to survey design. Because the ocean sup-ports many physical and biogeochemical processes,optimization of field measurements is not uniquelydefined because various scientific objectives wouldeach be consistent with different sets of optimizationcriteria. Thus, a suitable oceanographic questionmust be posed before an optimal survey can be com-puted.

Note that only features having scales resolved bythe model can enter into the optimization. Consis-tency between data and model is important, and candetermine the role of unresolved features �such asnonlinear internal waves�. This information can thenbe used in continuing data collection, filtering, andassimilation.

In the absence of a reliable model, the situationmay arise where the importance of a specific dynami-cal effect or feature is to be determined, sometimescalled the classification process. An example is deter-mining the scale length of dominant features. In thesecases, the statistical functions used in the example op-timization do not provide the necessary information,and other cost functions would need to be employed.Estimating spatial scales using data from fixed andmoving platforms required careful analysis and is aproblem in its own right. Methods have been devel-oped for such estimation �Zhang, Bellingham & Bag-geroer, 2001�, and for determining the measurementcapabilities of moving platforms for certain types ofprocesses �scales� �Wilcox et al., 2001�. The capablitiesof platforms as defined in prior work can be coupledwith knowledge of the resolved energy-containing

Figure 4. Genetic Algorithm solution for three moorings,three gliders, and two REMUS vehicles. Note how themoorings �o� are located near the shelf-break front, theREMUS vehicles and the gliders cross over major fronts.The blue tracks are REMUS vehicles and the magentatracks are gliders.



scales �perhaps through an iterative or feedbackmethod� to constrain the optimization process. Forexample, the distance potential constituent functionmay be adjusted based on this, or an AUV/glider sur-vey generation algorithm �i.e., lawnmower patterngenerator� could be tuned appropriately. In this paperwe have presented a methodology for automatedoptimal sensor planning for data-assimilation indynamical ocean modeling. The solution involvesperforming a Genetic Algorithm optimization simul-taneously on a deployment of multiple sensors, withthe ability of using multiple platform types. The ver-satity of the approach is the ability to use multipleuser defined scalar cost-functions with user definedweights. Thus the user has the control to define whatis meant by ‘‘optimal’’ and the algorithm search for adeploment scheme which optimizes the user definedglobal cost function.

ACKNOWLEDGMENTS

We appreciate the useful comments from the anony-mous reviewers. P.F.J.L. thanks P. J. Haley and W. G.Leslie for discussions. This work was funded undera Small Business Innovative research �SBIR� contractthrough the Office of Naval Research, Space and Na-val Warfare Center �SPAWAR-San Diego�, contractgrant numbers N00039-06-c-0063 �Heaney andDuda�, N00014-05-1-0413 �Gawarkiewicz�, andN00014-05-1-0335 and N00014-05-1-0370 �Lermu-siaux�.

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