Filtering Depth Measurements in Underwater Vehicles for Improved Seabed Imaging

Transcription

1 #64-6 Filtering Deth Measurements in Underwater Vehicles for Imroved Seabed Imaging Are B. Willumsen, Ove Kent Hagen and Per Norvald Boge Abstract Underwater vehicles are commonly used for detailed seabed maing. To derive a bathymetric ma, the deth of the underwater vehicle must be known. The offshore survey industry has often strong requirements on both absolute and relative accuracy. Deth is usually deduced from a measurement of absolute ressure. Regardless of the quality of the ressure sensor, the ressure measurement is influenced by environmental factors, such as waves, tide, atmosheric ressure and sea water density rofile. In the aer we analyze these factors with resect to their behavior and effect on the measured deth. An underwater vehicle equied with an aided inertial navigation (AINS) system has through its accelerometers and Doler velocity log redundant information on the movement of the underwater vehicle. Using this information, an AINS is well suited to filter out dynamic ressure sensor noise. Linear wave theory is used to model the effects of the waves on the ressure sensor, and it is suggested how the arameters of the ressure sensor noise model can be deduced from the mean wave amlitude and eriod. AINS erformance is often substantially imroved by ost-rocessing. The aer will therefore examine the deths oututted from the ressure sensor, from the AINS real-time filter, and from ost rocessing the AINS measurements. Index Terms Kalman Filtering, Imaging, Inertial Navigation, Pressure, Deth, Waves I I. INTRODUCTION n underwater navigation the main focus has been to establish the vehicle s horizontal osition using a variety of different aiding techniques []. In many alications the vertical osition can be deduced directly from ressure measurements with sufficient accuracy. However, with the new requirements on seabed maing accuracy, the accuracy of straightforward conversion from ressure to deth is challenged. A stable and accurate deth estimate is required for any high accuracy bathymetric maing sensor mounted Manuscrit received March 9, 7. Are B. Willumsen is with Kongsberg Maritime AS, PO Box, 39 Horten, Norway ( ; fax: ; are.b.willumsen@kongsberg.com). Ove Kent Hagen is with FFI, PO Box 5, 7 Kjeller, Norway ( Ove-Kent.Hagen@ffi.no). Per Norvald Boge is with Geoconsult AS, Nedre Åstveit, 56 Øvre Ervik Bergen Norway ( Per.Norvald@geoconsult.no). on an underwater vehicle. Otherwise, the resulting bathymetry estimate gets distorted with unwanted riles. The introduction of terrain navigation also sets high demands on real-time vertical channel accuracy in certain tyes of bathymetry []. Vehicle deth errors can reduce the ability of the terrain navigation system to correlate bathymetric measurements with re-stored digital terrain models. The conversion of ressure measurements to deth estimates has traditionally been solved based on an assumtion of hydrostatic ressure in the water column above the vehicle [3]. Since 984 the UNESCO formula [4], has rovided a standard for erforming this conversion. However, in shallow water oerations the influence of a dynamic ressure comonent induced by surface waves has been observed. In this aer we suggest that the wave induced ressure oscillations can be modelled as noise on the deth measurements fed into an aided inertial navigation system (AINS). This way one combines the suerior INS short term estimates with the stable long term estimates from the hydrostatic conversions, and the dynamic ressure variations can be filtered out. The rincile of combining inertial and ressure measurements was used for submarines in [5]. The objective and aroach there was quite different from this aer. A. Vertical datum Deth (and height) is always referenced to a secific vertical datum. Ocean vehicles and airlanes have traditionally been using the mean sea level (MSL). This is the level of the sea at a articular horizontal osition, when the effects of waves and tides have been averaged out. The WGS-84 is another imortant vertical datum and is the one used by GPS. It is defined by the surface of the WGS-84 ellisoid aroximation of the earth geoid. In this article we will rely uon the MSL vertical datum. However the choice of datum is not imortant to the task, as conversion from one vertical datum to another is usually a straight forward rocedure. II. TOTAL PRESSURE FIELD Assuming the underwater vehicle is at rest, and there is no sea current, then according to linear theory of fluid dynamics the total ressure measured by a sensor on the vehicle, B s

2 #64-6 TABLE I A CLASSIFICATION OF OCEAN WAVE TYPES BASED ON THEIR PERIODS, Period Wavelength Tye -. s Centimeters Riles, caillary waves.-9 s To about 3 m Wind waves 9-5 s Hundreds of meters Swell 5-3 s Several hundreds of Long swell, forerunners meters.5 min To thousands of Long eriod wave, tsunami hours.5, 5 h, Adoted from [6] kilometers Thousands of kilometers Tides can be described as a suerosition of the stationary hydrostatic ressure equilibrium h and the dynamic ressure field w caused by surface waves B = s h + w. (II.) The incoming waves can be categorized by their eriod, see Table I. Riles and caillary waves both have small amlitudes. They decay fast with deth due to viscous forces, and are therefore ignored in this analysis. The incoming wind waves and swell is treated searately from tides in section D. The wave-body interaction is commented on in section E. A. Tidal wave elevation Tides act in two natural eriods: semi-diurnal (about.5 hours, common) and diurnal (about 5 hours, uncommon). They can in extreme areas be in the order of 7 m amlitude (Bay of Fundy, Canada [6]). In addition, local toograhy might contribute through to the effect of traing energy from tides in resonating bays [6]. The local rise and fall of sea level can also be contributed to horizontal variations in atmosheric ressure, and storm surges in costal regions. On the timescale of wind waves and swells, tides will seem like a stationary rocess, and the effect of its dynamics is therefore ignored. Hydrostatic ressure to deth conversions will always give an answer relative to current tidal elevation level. To be able to fully utilize deth data, they need to be transformed to the vertical datum. Tide elevation with resect to MSL at the osition of the underwater vehicle should therefore be measured or estimated. It can be measured from a surface shi using a high quality GPS system, or by bottom mounted ressure sensors, or wave gauges at shore. Local tide estimates can also be based on of the official services for tidal redictions, ossibly corrected by measured data. The hydrostatic ressure to deth calculation has its zero level at the tidal level and the estimate of the tide elevation above MSL must be subtracted to yield the deth below MSL. B. Atmosheric ressure At the sea surface the boundary condition on the subsurface ressure field imlies that it by continuity must equal the sea surface atmosheric ressure. The atmosheric ressure must be subtracted from the absolute ressure measured by the ressure sensor on the underwater vehicle. If a surface vessel is resent, the atmosheric ressure can easily be measured and udates can be sent to the underwater vehicle, if needed. If the vehicle is oerating autonomously, the vehicle itself can measure the atmosheric ressure when surfacing, or it can use the best available weather forecast for the oeration area. If none of these data are available, the best one can do is to subtract the standard atmosheric ressure of 35 Pa (.35 bar). Assuming that this is an unbiased estimate, the maximum error in doing so is in the order of.35 m. For the case of ost-rocessing, better data can be obtained by including measurements from the nearest weather stations as well. If the weather station is situated above the MSL, the ressure also needs to be comensated for this altitude. As an aroximation we will consider the atmosheric ressure to be measured at the tidal elevation level instead of the actual sea surface level. The error in doing so is less than. m in most cases []. C. Hydrostatic ressure field For the hydrostatic ressure field, wave elevation and motion is neglected excet for wavelengths in the order of tidal waves. Since the wavelength of the tidal wave is much larger than the horizontal extent of the water column above the vehicle, the ressure field in the water column above the vehicle is assumed to only deend on deth. The main contributions to change in density with deth are from comressibility of seawater at high ressure, change in salinity S with deth, and finally change in temerature T with deth. The secific volume V( S, T, ) is defined as a function of salinity, temerature and ressure based on the equation of state EOS8. In [4] it is shown that the relationshi between ressure and deth z under these assumtions is aroximated by V(35,, ) d z = + δ ( d ) 9.8. g ( µ ) γ ( ) + (II. ) Here g ( µ ) is sea surface gravity at latitude µ, and γ is the mean vertical gradient of gravity with resect to ressure in the water column. The first integral of (II.) reresent the assumtion of standard ocean (S= 35 su and T = C). In [4] a 4 th order olynomial fit to this integral has been calculated, and this serves as the standard of today. The integral of the secific volume anomalyδ in (II.) is the geootential anomaly, and reresents corrections according to the actual density rofile of the water column. The integral can be calculated numerically from a CTD rofile of the water column The error in ignoring this term can in some areas be substantial (e.g..3% of deth in the Baltic Sea []). The integral can also be aroximated by geograhically deendent equations of the ressure [8]. D. Waves The dynamic ressure field generated by surface waves can

3 # be analyzed using linear wave theory. For simlicity we restrict the analysis to D. Assuming the wave velocity vector field is curl-free, there exists a velocity otential φ ( x w, zt, ). For an incomressible fluid the velocity otential solves Lalace s equation in the fluid volume φ w =. (II.3) The following boundary conditions are given: No normal velocity at bottom deth z = h. The surface z = η( xt, ) and the fluid articles have the same vertical velocity at the tidal elevation level. Continuity in surface ressure For a sinusoidal surface wave η( x, t) = asin( kx ωt) with amlitude a, wave number k and angular frequency ω, the solution to the PDE in (II.3) with these boundary conditions is, ag φw ( x, zt, ) = cosh( kh ( z))cos( kx ωt). (II.4) ω cosh( kh) Here g denotes the gravitational acceleration, assumed to be constant. The corresonding disersion relation can be found from the boundary conditions at the surface, and is given by ω = gk tanh( kh). (II.5) Now, using the linear version of Euler s equations we get the following relationshi between the velocity otential field and the ressure field w w = ρ φ t. (II.6) Here ρ denotes the seawater density. By inserting (II.4) into (II.6), we obtain the subsurface dynamic ressure field caused by a sinusoidal surface wave, cosh( kh ( z)) w( xzt,, ) = ρg η( xt, ). (II.7) cosh( kh) These equations are all valid for the linear case only. The requirement for linear wave theory to be an adequate model is given by a restriction on wave steeness (ratio of wave amlitude to wavelength) given by ak, for both the deewater and shallow water cases. Additionally it is required that the wave amlitudes are much smaller than the water deth, so a h for the shallow water case. Also the ideal fluid aroximation leads to an unhysical condition at the seafloor, where a no sli condition is a more realistic condition to aly. The assumtion of uniform bottom deth is in general not good for littoral waters, as the interactions with bathymetry severely influence wave amlitude, eriod and direction of roagation. The linear model will however rovide a first order aroximation and insight into the effects there. ) Ocean wave sectrum The free surface is now considered as a suerosition of sinusoidal waves with random amlitude and hase. In [7] a formal reresentation of the irregular sea surface is given. It is common to describe the surface statistically by the ocean wave sectrum ( ω ), which is a distribution describing how the S η wave energy is distributed with resect to wave frequency. In 3D the energy sectrum is tyically directional, so the above sectrum is considered as the result by integration over all wave directions. The characterizing moments of the sectrum are defined by i mi Sη ω ( ω) dω, i =,,,.... (II.8) From the wave sectrum moments, assuming a Rayleigh distribution of wave amlitudes [7], several characterizing variables may be derived using (II.8), such as Average wave height, H = m. Significant wave height, H s = 4 m. m Average angular frequency, ω =. m One of the most used sectrums to describe a not fully develoed state at sea is given by the JONSWAP sectrum [9]. By using the subsurface ressure field reresentation (II.7), it is straightforward to derive the sectrum for the surface wave induced ressure oscillations as a function of deth []. cosh( k( ω)( h z)) Sh, ( ω, z) = ρ g Sη ( ω). (II.9) cosh( k( ω) h) Here the wave number is a function of the angular frequency imlicitly defined through the disersion relation (II.5). In numerical calculations, the wave number can be found using Newton s method. Using hydrostatic conversions for these ressure oscillations, the sectrum of the corresonding deth oscillations S ( ω, z) is found from the aroximation h cosh( k( ω)( h z)) Sh ( ω, z) = Sη ( ω). (II.) cosh( k( ω) h) A demonstration of how a JONSWAP sectrum of surface waves attenuates with deth is shown in Fig.. It shows that the high frequency waves attenuate faster with deth than the S(ω) [m s] Significant wave height: 5 m, Peak time eriod: 8 s, Water deth: 8 m 5 JONSWAP at surface Frequency [Hz] JONSWAP at 5 m deth JONSWAP at m deth JONSWAP at 5 m deth Fig.. JONSWAP sectrum at different deths

4 # low frequency waves, so the energy is shifted relatively towards lower frequency with deth. E. Other asects If the ressure transducer is ositioned at areas on the hull exosed to stagnation ressure effects, this will inflict an error in the measurement. Also the non-zero velocity of the vehicle with reference to the wave field causes a Doler shift of the measured ressure waves. The wave-body interaction is also a source of deviation in the measured ressure. All these asects are closely discussed in []. The effects are not imortant to this analysis and are as such omitted. III. FILTERING WITH AN AINS The aided inertial navigation system (AINS) integrates accelerations and angular rates from an inertial measurement unit (IMU) into seed, orientation and osition. It has low short term errors, but will drift in the long term. It is therefore deendent uon aids to reduce the integrated error. For underwater vehicles, such aids are usually ressure sensor, DVL, LBL and USBL. A. Static and semi-static errors For underwater vehicles, there is almost no other measurement that can rovide an accurate deth reference. It is therefore usually not ossible to calibrate the ressure to deth conversion, nor is it ossible to on-line estimate a (semi- ) static error in the ressure sensor deth measurement. The correction for atmosheric ressure, tide, and geootential anomaly must be done to get an unbiased global deth. This could be done both on- and off-line, since this causes the AINS to have a close to constant deviation from the true deth. B. Dynamic errors The dynamic errors caused by waves of different sort (see Table I), can though be estimated in the AINS. In essence it means that long term accuracy of the ressure sensor and the short term accuracy of AINS are combined to give an accurate short and long term deth estimate. In order to get an otimal integration of the ressure sensor into the AINS, one must rovide a model of the develoment of the dynamic deth error of the ressure sensor. One of the simlest and easiest aroximations of the error roagation is the first order Gauss-Markov rocess, which is comletely secified by the autocorrelation function: τ T ( z) r(, τ z) = σ () z e. (III.) The standard deviation σ and the inverse of the correlation time T are as a first suggestion the mean wave amlitude and half average wave eriod resectively at the vehicle s deth. However closer investigations suggest a lower T, in the range -3 seconds. In [] an otimal choice for these arameters is derived for the case of regular sinusoidal waves. At sea one usually does not have as detailed information as a wave sectrum, but as we will see in section IV, the AINS is robust with resect to these arameters. Observations of significant wave height and eak time eriods for the waves will robably suffice in a realistic scenario. IV. SIMULATIONS NavLab a generic navigation tool [], has been used as the environment for the simulations. In the simulation scenario the INS is aided by ressure sensor, DVL, and USBL. The vehicle is ket stationary at 5 m deth, and the ocean deth is 8 m. The surface waves are assumed sinusoidal, and the corresonding effect on the ressure sensor is calculated using (II.7). The resulting deth estimates of a m surface wave with a eriod of 9 s are shown in Fig. and the standard deviations from true deth is shown in Table II. The simulations are erformed with a navigation grade IMU, standard tye of DVL and high class deth sensor. Both the effect on the ressure at the deth of the vehicle Period [s] TABLE II SIMULATED WAVES AND RESULTING STANDARD DEVIATIONS Surface amlitude [m] Measured [m] Real- Time [m] Post- Processed [m] and the arameters of the surface waves themselves can be hard to measure or estimate. It is therefore imortant to examine the filtering methods robustness against inaccurate or wrongful arameter estimates. For that reason the filter was tested with arameters that did not match the waves in the simulation. The result is seen in Fig. 3, where a surface wave of m amlitude and 9 s eriod (corresonding to.67 m Deth [m] Surface wave: a= m and λ=6 m, induced oscillations at 5 m deth Measurement SD =.67 m Real-time SD =.6 m Smooth SD =.4 m True Fig.. Simulated deth estimates of waves with eriod 9s amlitude and 9 s eriod at 5 m deth) is simulated. The arameters are set according to a tyical configuration of.5 m standard deviation and s correlation time. The results show that the real-time solution is sensitive to arameter errors whereas the smoothed solution is not.

5 # Surface wave a= m λ=6 m, induced oscilations at 5 m deth Measurement SD =.67 Deth of ROV mission Realtime SD =.47 Smooth SD =.3 True 5 Deth [m] Deth [m] Fig. 3 Simulated deth estimates, with waves (amlitude m, eriod 9s), but AINS arameters corresonding to a tyical configuration (.5m, s) Fig. 4 Deth of ROV trajectory V. RESULTS FROM COLLECTED DATA Instead of dedicated sea-trials we will use navigation data from a survey missions erformed with ROV. This is a real mission and we have no true reference. Therefore the smoothed solution is what is closest to the truth and will be used as a reference. The ROV survey was erformed from 9 th of January 7 at about meters deth. Fig. 4 shows the mission deth. The deviation from ost-rocessed deth is shown in Fig. 5. It shows that the wave effect is only artially removed from the data with the real-time filtering of the AINS. Fig. 7 is showing DTM generated with the measured deth and with the ostrocessed deth of Fig. 6 (same time frame). The figure shows a clear rile effect in the former, whereas in the latter this effect is almost entirely filtered out. Besides showing the imrovement in using AINS filtered deth, this suorts our claim of using the ost-rocessed deth as a reference. We also tried using different arameter sets for these data. Although not shown here, the results verified the findings in the simulations. The real-time solution is sensitive to arameter settings whereas the ost-rocessed solution is robust and will generate the same result with different arameters. VI. CONCLUSIONS This aer has shown that it is advantageous to use AINS in order to obtain a smooth and more correct deth estimate than the one given from a ressure sensor. With the AINS one is able to filter out the effects of ressure fluctuations caused by surface waves. The fluctuations decrease with deth, and so also the benefit of the AINS. Real-time estimates are usually a clear imrovement over the measured ones. However simulations and exeriments show that these results can often be well imroved by the use of ost-rocessing. Both simulations and exerimental data show that ost rocessed estimates are smooth, recise, and as oosed to real-time Deth deviation [m] Deth [m] Deviation from ost-rocessed deth Fig. 5 The deviation of measured and real-time estimated deths from ost rocessed estimate Deths of ROV mission 7--9 Measured Real-time Post-roc Measured Real-time Fig. 6 Measured, real-time and ost-rocess estimated deths of mission 7--9

6 # Fig. 7 DTM generated by the use of measured deths (to) and ost-rocessed deths (bottom) shown in Fig. 6 estimates, robust with reference to arameter settings. Mas generated with AINS and ressure sensor are clearly better than the ones generated with only a ressure sensor. Rile caused by surface waves are almost eliminated when using ost-rocessed navigation data to generate the ma. REFERENCES [] B. Jalving, K. Gade, O. K. Hagen and K. Vestgård, A Toolbox of Aiding Techniques for the HUGIN AUV Integrated Inertial Navigation System, Proceedings from OCEANS 3, [] O. K. Hagen and P. E. Hagen, Terrain Referenced Integrated Navigation (TRIN), BP MREA 3 Conference Proceedings, NATO Undersea Research Centre, La Sezia, Italy, May [3] B. Jalving, Deth Accuracy in Seabed Maing with Underwater Vehicles, Proceedings from Oceans 99, Seattle, WA, USA [4] N. P. Fofonoff and R. C. Millard Jr., Algorithms for comutation of fundamental roerties of seawater, Unesco technical aers in marine science 44, 983 [5] M. Glass and J. A. M. ter Horst, Combining ressure and Inertial Measurements For Submarine Deth Determination, Naval Engineers Journal, vol 98, no. 6, 6-68, 986 [6] S. Pond, G. L. Pickard, Introductory Dynamical Oceanograhy, Oxford, UK: Pergamon Press, 983. [7] J. N. Newman, Marine Hydrodynamics, The MIT Press, Cambridge, Massachusetts 977. [8] C. C. Leroy and F. Parthiot, Deth-ressure relationshis in the oceans and seas, Journal of the Acoustical Society of America, vol 3, no. 3, , March 998 [9] K. Hasselmann, T. P. Barnett, E. Bouws, H. Carlson, D. E. Cartwright, K. Enke, J. A. Ewing, H. Giena, D. E. Hasselmann, P. Kruseman, A. Meerburg, P. Møller, D. J. Olbers, K. Richter, W. Sell, and H. Walden, Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP), Ergnzungsheft zur Deutschen Hydrograhischen Zeitschrift Reihe A(8) (Nr. ): 95. [] K. Gade, NavLab, a Generic Simulation and Post-rocessing Tool for Navigation, Euroean Journal of Navigation, vol, no. 4,. 5-59, 994. [] O. K. Hagen and B. Jalving, Converting Pressure to Deth for Underwater Vehicles, Norwegian Defence Research Establishment (FFI), Tech. Reort., unublished