Geophysical Prospecting, 2011, 59, 176187 doi: 10.1111/j.1365-2478.2010.00900.xComparing gravity and gravity gradient surveysGary J. Barnes, John M. Lumley, Phill I. Houghton and Richard J. GleaveARKeX Ltd, Newton House, Cambridge Business Park, Cowley Road, Cambridge CB4 0WZ, UKReceived November 2009, revision accepted April 2010ABSTRACTNoise levels in marine and airborne full tensor gravity gradiometer surveys togetherwith conventional land, marine and airborne gravity surveys are estimated and anal-ysedingriddedform, resultinginrelationsthatdetail howthesedifferentsurveysystemscanbecomparedanalytically. Afterdefiningsurveyparametersincludingline spacing, speed and instrument bandwidth, the relations estimate the noise levelsthat result on either grids of gravity (gz) or gravity gradient (Gzz) as a function of thespatial filtering often applied during geological interpretation. Such comparisons arebelieved to be a useful preliminary guide for survey selection and planning.Key words: Gravity gradiometry, Instrumentation noise, Survey comparisons, Sam-pling, Survey design.I NTRODUCTI ONInterpretersof gravityandgravitygradient datainvariablystudy gridded forms of survey measurements so that a varietyoffilteringcanbeeasilyappliedtoemphasizeanomaliesofdifferent sizes and depths. In general, it is known that filteringto longer wavelengths improves the detectability of anomaliesbut sacrifices overall resolution. If data are over-filtered, thendue to the loss of resolution, positional accuracy is compro-mised and multiple isolated anomalies can even be mistakenas a single entity. On the other hand; without adequate filter-ing, the noise level can be too intrusive and mask the signalcompletely. The noise level therefore dictates the useable res-olution for interpretation and consequently ultimately deter-mines the positional accuracies of any identifiable anomalies.Knowledge of the expected noise on gridded data is thereforecrucial when planning a survey to ensure that the geologicalfeatures of interest can be detected and resolved. To aid thisunderstanding, we estimate and analyse the noise from land,marineandairbornegravimeters andgravitygradiometersshowinghow,givenasetofsurveyparameters,noisefromthe raw acquisition system impinges on the final gridded data.Once in gridded form, the impact of the noise from the dif-E-mail: gary.barnes@arkex.comferent survey systems can be directly compared showing theirrelativebenefitsasafunctionofthefilteringappliedtothegriddeddata.Theformulaederivedhereareintendedtobeused as a general rough guide for survey planning and instru-mentselection.Relyingonvariousapproximationsoutlinedbelow,theyshouldbeusedappropriatelywithinlimitsandonly as a precursor to more comprehensive numerical feasi-bility modelling.SURVEYDATANOI SETofacilitatetheanalysisforbothgradiometerandconven-tional gravimeter surveys, when viewed in the space domain,the noise from the measurement system is assumed to be un-correlated. This canbe partlyjustifiedbyconsideringthetwodomainsof interest; thetimedomainof themeasure-ment systemandthespacedomainforthesurveyandthesignal. Low-frequency instrument noise appears correlated inthe time domain whereas the signal appears correlated in thespace domain. When survey noise is viewed in the space do-main,thetotalnoisepowerisunchangedbutthespectrumbecomes whiter (less correlated) since the ordering of the datais partially randomized.Thechangingofdomainsisgenerallynotenoughtoex-clude low-frequencynoise fromthe space domainandso176 C 2010 European Association of Geoscientists & EngineersComparing gravity and gravity gradient surveys 177various methods are employed to remove the noise from therawdata. Land gravity surveys return to common stations andrepeat measurements allowing one to estimate and remove thedrift variation that occurred in-between. Marine and airbornesurveys achieve similar corrections through the use of tie lineswheredrift canbecorrectedfor byminimizingthediffer-ences at the intersection points. The heterodyne nature of thefull tensor gradiometer measurement system eliminates muchof the low-frequency noise in the first place (Metzger 1977).In well designed surveys, especially those measuring multiplecomponents, much of the low-frequency noise can be removedby advanced methods that attempt to model the drift variationwith time (ARKeX Ltd 2008).Thefollowingsectionswill attempttoestimatethenoiselevels on the raw measurements for each survey system underconsideration.NOI SEPERMEASUREMENTPOI NTAirborne full tensor gradiometerThefull tensorgradiometerconsistsof threedistinct mod-ulesknownasgravitygradientinstrumentseachmeasuringcomponents of differential curvature. For example, a modulealignedinthehorizontal planewouldhavein-line output(Gxx Gyy) / 2 and cross-line output Gxy (note that the fulltensor gradiometer actually adopts a tilted coordinate systemwhere each module axis makes the same angle to the vertical).Byhavingthreeorthogonal modulesinthefull tensorgra-diometer one can sum the three in-line outputs and the signalcancels, (Gxx Gyy) +(Gyy Gzz) +(Gzz Gxx) =0. Whenconstructing this quantity, known as the in-line sum, what isleft forms a useful estimate of the overall full tensor gradiome-ter noise. It relies on an accurate orientation and calibrationof each module so that the real gravity gradient signals can-cel exactly; any remaining signal causes over-estimates in thenoise levels. To mitigate this possibility, the survey chosen fornoise analysis was based on its generally low signal levels dueto very little terrain relief in the area. Often, noise analysis isperformed on long survey lines flown straight at high altitudethusprovidingideal conditionsandattenuatedsignal. We,however, chose to analyse a complete survey to give a betterpicture of what to expect during a real situation.Figure 1 shows actual noise power spectral densities (PSDs)estimated from a survey consisting of almost 250 lines parti-tioned into three equally sized groups according to their av-erage turbulence level. The three series therefore correspondtothelow,mediumandhighturbulenceconditionsexperi-encedduringthesurvey.Theturbulencemeasureisdefinedas the vertical acceleration power over a 5 seconds windowand can vary considerably along the line. Only lines that metthe quality control criteria where used in this analysis, leav-ing those with excessive noise levels (often due to sections ofhigh turbulence) as rejected. Such lines, being out-of-spec, aregenerally reflown in a commercial survey. The statistics builtFigure 1Averaged output channel noisefromfull tensor gradiometer survey dataforthreelevelsofturbulenceasshowninthelegendquantifiedas (meanlevel, me-dianlevel). Blacklineshowsanempiricalnoise model fitted to the data.C 2010 European Association of Geoscientists & Engineers, Geophysical Prospecting, 59, 176187178 G. Barnes et al.Figure2Noiselevel onGzz(blackcurve)calculatedfromanequivalentsourcemodel invertingonfull tensorgradiometermeasurementsreplaced with white noise (red curve).up in Fig. 1 therefore represent the noise levels in a selectedfavourable subset of all the lines flown during a survey. Theoverall turbulence level of these lines is generally less than theaveragesurveybutrepresentativeofatroublefree surveywhere the terrain is flat and the weather is fairly benign. Theblack line is a model of the noise level derived by fitting allthe data to an empirical relation,PSD( f ) = w +a

fc

fcb

ef/ fc

,(1)with fitted parameters: w=12, a =2.33, b =8.56, c =0.375,fc = 4 103.This relation is accurate over the range 1300 mHz wheresufficientdatawereavailabletofitthemodel.Noise atfre-quencies greater than 300 mHz is normally outside the mea-surement bandwidth of interest and below1 mHz the unavail-ability of long enough lines in this survey (having line lengthsup to 65 km) prohibited accurate estimates. Figure 1 showsthat the full tensor gradiometer noise is largely uncorrelatedeven in the raw measurements before tie-line levelling or driftcorrections and full tensor processing.During full tensor processing, all the six measurement chan-nels of the full tensor gradiometer, each having noise charac-teristics as shown in Fig. 1, are combined together to produceanenhancedestimateofthegravitygradient. Wefindthatthisismostsuccessfullyperformedusingequivalent sourcetechniques that invert the measurements into a single densitymodel (Dampney 1969; Li 2001). The resulting density modelcan be used to predict a variety of gravity gradient and gravitycomponents either back at the original measurement points,or projected onto level grids. By forward calculating Gzz backto the original locations of the survey, one can consider (forthepurposesofthisanalysis)thefulltensorgradiometerasan equivalent Gzz gradiometer that measured this data series.To estimate the noise on this equivalent Gzz, the differentialcurvature components on a survey are replaced with randomnumbersmakinguppurewhitenoisewith1E/Hzpowerspectral density. Using this data, the equivalent source inver-sion is performed in exactly the same way as in the real datacase (a procedure that has been verified to honour genuine ge-ological signals) to yield an equivalent source density model ofthe noise. In this process, noise from different measurementsbecome averaged together and attenuated on the basis of non-harmonic variation making the inverted model less affected bymeasurement noise. Forward calculating Gzz from this modeland analysing the noise power (Fig. 2) shows that the noiseon the equivalent Gzz at low frequency is seen to be reducedby a factor 2.1 when compared to the noise level on the in-dividual measurement channels. At higher frequencies (above0.1 Hz), the equivalent Gzz noise power falls rapidly due tothe rejection by the equivalent source inversion of physicallyimpossiblehighfrequencypower.Suchpowerisattenuatedbytheupwardcontinuationofthepotential fieldfromtheEarth to the aircraft.Figure 3 shows the noise levels on the differential curvaturecomponents and the equivalent source calculated Gzz for thereal data of the survey. This situation is less transparent thanthe previous purely numerical analysis since there is a real sig-nal present in the data. However, by using a noisy survey, ap-plying the terrain correction directly on the measurements andC 2010 European Association of Geoscientists & Engineers, Geophysical Prospecting, 59, 176187Comparing gravity and gravity gradient surveys 179Figure 3Power spectral densities of terrain corrected and spatially detrended full tensor gradiometer data (top series) and equivalent sourcecalculated Gzz (blue series).also spatially detrending each data set with a 6thorder polyno-mial function in x and y, the affect of the signal can be reducedleaving a data set where a lower limit on the noise reductionfactor can be made. With reference to Fig. 3, at frequenciesbetween 0.040.1 Hz, this factor is approximately 1.8. Belowthisrangethenoiseestimateiscompromisedbythesignalpower that dominates the power spectral density and abovethis range the factor appears much larger due to the upwardcontinuation as discussed above. This figure fromthe real datatherefore partially supports the conclusion fromthe numericalanalysis.Consequently, inthefollowinganalysis,wewill considerthe full tensor gradiometer as an equivalent instrument mea-suringGzzwithanoiselevelconsistentwiththisenhancedgravity gradient level. Lumping all the channels into an equiv-alent component obviously ignores one of the key advantagesof the full tensor gradiometer having measurements of thefull tensorbutitconsiderablysimplifiesthenoisecompari-son with the gravity systems. Figure 4 shows the cumulativenoisespectrumfortheenhancedGzzcomponentasafunc-tion of bandwidth estimated from full tensor gradiometer sur-vey data. Typical bandwidths used in practice vary between0.20.3 Hz (equivalent to 325217 m wavelengths along theline at 65 ms1) or up to 0.4 Hz (163 m) for highly detailedsurveys with tight line spacing.Over the measurement bandwidth, the noise fromthefull tensorgradiometerisoftenconvenientlyapproximatedas white recognizing the above factors that combat low-frequency noise both in the acquisition system and the subse-quent processing. For a white noise power spectral density, Pover a bandwidth, fbw, the rms variation on each measurementpoint is given bynfull tensor gradiometer =P

fbw. (2)ThecomparisoninFig. 4showsthat forroughcalcula-tions, a white noise approximation is adequate but tends toover-estimate the total noise power slightly. In the followinganalysis, although the full tensor gradiometer noise is assumedto have a white spectrum, the total noise power is evaluatedusing the more accurate model in equation (1).Marine full tensor gradiometerThe time domain full tensor gradiometer noise amplitude fora marine survey is assumed to be the same as in the airbornecase. There will be an equivalent set of noise characteristics butthis time driven by sea-state rather than turbulence. Overall,preliminary evidence is suggesting that mapping the averagenoise statistics (equation (1)) from the airborne to the marinesystem is over-estimating the noise but until we have acquiredsufficient marine data to analyse, we will consider this as ac-ceptable. The biggest difference appears in the space domain,where due to the differing acquisition speeds (65 m/s versus2.5 m/s), the marine case has significantly higher resolutionalong the lines. Analysis of the marine full tensor gradiometernoise will be presented elsewhere in due course.C 2010 European Association of Geoscientists & Engineers, Geophysical Prospecting, 59, 176187180 G. Barnes et al.Figure 4Noise as a function of bandwidth for full tensor gradiometers equivalent Gzz measurement. Blue series: full tensor gradiometer noisemodel (Fig. 1); Red series: white noise approximation, 10 E/Hz.Land gravityWhilst several field land gravimeters can measure the value ofgravity with accuracy suitable for microgravity surveys (Seigel1995), for a standard realistic survey constrained by a cost andtime budget, the instruments precision is rarely the limitingfactor. Foracarefullyacquiredandcorrectedlandgravitysurvey, a more typical level of rms noise per station isnoi se = 0.05 mGal. (3)This figure does, however, vary considerably from one sur-vey to another. Due to the proximity, the terrain correctioncanoftenbethedominantfactorinthelandgravityaccu-racy. In a survey over an area covered with sand dunes risingto 100 m, this uncertainty alone was estimated to contribute0.1 mGal to the total error (Hallinan, Gheit and Reid 2002).When interpreting data over a large area, it is often the casethatthelandgravitymeasurementsderivefromanamalga-mation of surveys of different qualities and vintages makingthenoisecharacteristicschangefromplacetoplace. Manyerrors in legacy data derive from inaccurate height measure-mentscausing0.3086mGalpermetreerrorsinthefreeaircorrection and consequently a much larger uncertainty than amodern GPS assisted survey. Due to the large range of qual-ities, when comparing the different survey scenarios, we willsplit the land gravity performance into three categories; legacyTable 1Uncertainties in land gravity surveysCategory Accuracy (mGal)Legacy 0.5Terrain limited 0.1Carefully acquired and corrected 0.05(pre-GPS), terrain correction limited and a carefully acquiredand corrected modern survey, see Table 1.Airborne gravityFor moving gravity surveys, the noise does not approximatewell to a flat spectrum because of the rapidly increasing noisepower with frequency. In this case, either available data, or asuitable theoretical model that is able to predict the noise as afunction of bandwidth must be used. In the case of airbornegravity,wehavefoundthatusingamodelforGPSderivedaccelerationnoiseusedbyvanKann(2004) approximateswell to available survey data (Fig. 5). Airborne gravity noiseisanalysedmorecomprehensivelyfortheSanderAIRGravsystemasafunctionoftimedomainfilteringinElieffandFerguson (2008). Here it is seen that for a repeatedly flowntestline,theGPSlimitover-estimatesthenoiseforfilteringtimes greater thanroughly70seconds andunderestimatesC 2010 European Association of Geoscientists & Engineers, Geophysical Prospecting, 59, 176187Comparing gravity and gravity gradient surveys 181Figure5Modelforairbornegravitynoisebased on theoretical GPS limit. Data pointsextracted from Argyle et al. (2000).for time constants less than 70 seconds. For the purposes ofthis paper, we adopt the GPS model as a convenient approx-imation for a general airborne gravity system relying on GPSderived corrections and assume its validity for time constants14 km) along the survey and tie lines, hasFigure 6Model for marine gravity noise based on mis-tie analysis.C 2010 European Association of Geoscientists & Engineers, Geophysical Prospecting, 59, 176187182 G. Barnes et al.accumulated signal from different parts of the survey. For thenoise model, data in these extreme limits were excluded fromthe parameter fit giving a relation that can be used to predictthe noise over a useful range,noi se(Tc) = 0.3 +0.16e6.3104Tc. (4)Similar but earlier analysis based on the LaCoste andRomberg meter reported larger errors ranging from0.831.16mGal (10 minute filtering) over a variety of weather conditions(Valliant 1983). These figures most likely reflect the lack ofgood navigation data preventing accurate eotvos corrections.In an evaluation of the BGM-3 meter, Robin and Watts (1986)deduced an accuracy of 0.38 mGal (180 second filter) underforce 1 (optimal) conditions. The consensus of the 2002 SEGworkshop on Gravity Noise Workshop, Salt Lake City, Utah,USA concluded repeatability