# An assessment of inversion methods for AEM data applied to environmental studies

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An assessment of inversion methods for AEM data appliedto environmental studiesDavid Beamish*British Geological Survey, Keyworth, Nottingham NG12 5GG, UKReceived 6 September 2001; accepted 19 July 2002AbstractAirborne electromagnetic (AEM) surveys are currently being flown over populated areas and applied to detailed problemsusing high flight line densities. Interpretation information is supplied through a model of the subsurface resistivity distribution.Theoretical and survey data are used here to study the character and reliability of such models. Although the survey data wereobtained using a fixed-wing system, the corresponding associations with helicopter, towed-bird systems are discussed. BothFraser half-space and 1D inversion techniques are considered in relation to their ability to distinguish geological, cultural andenvironmental influences on the survey data. Fraser half-space modelling provides the dual interpretation parameters ofapparent resistivity and apparent depth at each operational frequency. The apparent resistivity was found to be a remarkablystable parameter and appears robust to the presence of a variety of at-surface cultural features. Such features provide bothincorrect altitude data and multidimensional influences. Their influences are observed most strongly in the joint estimate ofapparent depth and this accounts for the stability of the apparent resistivity. Positive apparent depths, in the example data, resultfrom underestimated altitude measurements. It is demonstrated that increasingly negative apparent depths are associated withincreasing misfits between a 1D model and the data. Centroid depth calculations, which are a transform of the Fraser half-spaceparameters, provide an example of the detection of non-1D influences on data obtained above a populated area. 1D inversion ofboth theoretical and survey data is examined. The simplest use of the 1D inversion method is in providing an estimate of a half-space resistivity. This can be undertaken prior to multilayer inversion as an initial assessment. Underestimated altitudemeasurements also enter the problem and, in keeping with the Fraser pseudo-layer concept, an at-surface highly resistive layerof variable thickness can be usefully introduced as a constrained parameter. It is clearly difficult to ascribe levels of significanceto a measure of misfit contained in a negative apparent depth with the dimensions of metres. The reliability of 1D models isbetter assessed using a formal misfit parameter. With the misfit parameter in place, the example data suggest that the 1Dinversion methods provide reliable apparent resistivity values with a higher resolution than the equivalent information from theFraser half-space estimates.D 2002 Elsevier Science B.V. All rights reserved.Keywords: Airborne geophysics; Electromagnetic methods; Inversion; Assessment of environmental influences1. IntroductionIncreasing use is being made of airborne electro-magnetic (AEM) surveying for assessing infrastructure0926-9851/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved.PII: S0926 -9851 (02 )00213 -6* Tel.: +44-115-936-3432; fax: +44-115-936-3261.E-mail address: D.Beamish@bgs.ac.uk (D. Beamish).www.elsevier.com/locate/jappgeoJournal of Applied Geophysics 51 (2002) 7596(Hodges, 1999; Beard and Lutro, 2000), groundwater(Fitterman and Deszcz, 1998; Sengpiel and Siemon,1998), land-use and environmental issues (Puranen etal., 1999; Beamish and Kurimo, 2000). In a number ofthese cases, the resistivity information may be pro-vided at a high lateral resolution scale and in thevicinity of populated areas. The information is pro-vided through a model and it is the character andreliability of the model that is considered here usingboth theoretical and survey data.AEM surveying can employ frequency-domain ortime-domain techniques, both of which are similar toground-based counterparts (Collett, 1986). The fre-quency-domain systems, discussed here, exist astowed-bird configurations (typically Helicopter HEMsystems) and as fixed-wing (wing-tip sensor) config-urations. The frequencies employed in the twoembodiments are similar; the main difference lies inthe transmitterreceiver coil separation. HEM birdlengths are typically < 7 m and fixed-wing systemsnecessarily exceed this by a factor of about 3. Afurther difference lies in operational height aboveground. Typically, HEM operates the towed sensorbird about 30 m above ground level while fixed-wingsystems (with larger dipole moments) may be flownmuch higher. Early HEM and fixed-wing systemsused one or two simultaneous operating frequenciesand these have been extended both in number andbandwidth.EM induction by elevated magnetic dipoles is gov-erned both by the frequency/bandwidth and the geo-metrical attributes of the system. Transmitterreceivercoil combinations include coplanar, coaxial and verti-cal and horizontal dipole configurations. The under-lying coilcoil orientation issue has been discussed bya number of authors (Sengpiel, 1983; Liu and Becker,1990; Kovacs et al., 1995). The dual frequency fixed-wing system, discussed here, uses a pair of verticalcoplanar coils. In contrast to horizontal coplanar sys-tems, the vertical coplanar coils provide an asymmet-rical and smaller footprint when operated at the sameheight and frequency (e.g. Kovacs et al., 1995).The fixed-wing AEM system developed and oper-ated by the Geological Survey of Finland system wasused in a series of trials in the UK to acquire detaileddata sets in addition to magnetic gradiometer andradiometric information. Four areas in the East Mid-lands were surveyed. Three areas were surveyed atlow elevation (40 m) using 50 m spaced flight lines.Particular targets for the EM data included environ-mentally sensitive zones around conurbations.It is worth noting that, according to Holladay andLo (1997), the Geological Survey of Finland is theonly known operator of a frequency-domain, fixed-wing system and, as such, the current state-of-the-artis confined to two frequencies. A fixed-wing systemoffers operational and cost effectiveness over HEMsystems when the intended use is in relation toproviding systematic (i.e. year-on-year) multipara-meter, regional mapping. Many recent discussions ofairborne EM data have involved quite specific targetssuch as hazardous waste-sites (Doll et al., 2000), thereare only a few examples of AEM applied to generalland-quality issues of the type considered here.When AEM surveys are carried out routinely overpopulated areas, a number of issues are raised. Theinfluence of buildings, services and forest cover be-come important. Although the problem of buildings issometimes mentioned, there are no specific case exam-ples in the literature. A practical example is providedhere.A data acquisition rate of 4 Hz results in an alongline sampling of < 15 m. When data are acquiredusing a 50-m flight line separation and a wing-span of21 m the coverage becomes quasi-continuous (Beam-ish, submitted for publication). With these parameters,information is afforded at the site investigation scale.When used in geological and/or mineral explorationusing, say, 200 m flight line separations, there is scopefor the rejection of problem data. At the site inves-tigation scale, all data points require interpretation andthe issues of at-surface and near-surface features andmultidimensional effects have to be addressed.Although the rejection of problem data is sometimesmentioned (Sengpiel and Siemon, 1998) there are nocase examples in the literature. A practical example isprovided here.Fraser half-space and 1D inversion procedures areexamined using one of the acquired data sets. Thestudy considers data obtained across the 4 1.5 kmLangar area (Fig. 1). The area has a distinctive (andconductive) geological background superimposed onwhich are a number of anomalies. The area containsone closed and one active domestic landfill, both ofwhich give rise to conductive features. The environ-mental assessment is challenging in that it involvesD. Beamish / Journal of Applied Geophysics 51 (2002) 759676Fig. 1. Location map of the Langar example survey area. Cross symbol denotes the position of the closed landfill; the open circle indicates the central position of the active landfill.Flight line 420 is indicated (along Northing 335 km). Central 11 km square is centred on a cement works (labelled Works) and the active landfill. The heavy dash lines indicatethree main geological units, from NW to SE they comprise the Mercia Mudstone Group, a Lower Lias Limestone and a further clay rich member of the Lower Lias. The base map isreproduced from the OS map by British Geological Survey with the permission of Ordnance Survey on behalf of The Controller of Her Majestys Stationery Office, n Crowncopyright. All rights reserved.D.Beamish/JournalofAppliedGeophysics51(2002)759677the detection of conductive pore fluids in conductive(5 V m) clay-rich host rocks.The interpretation of AEM data proceeds using onedimensional (1D) resistivity models. This study firstconsiders the Fraser half-space parameters typicallyused in the interpretation of AEM data sets. Commonprocedures largely developed and described by Fraser(1978, 1986) comprise the modelling of the observedcoupling ratios by a half-space algorithm. Couplingratios are here defined as the secondary to primaryfield ratio multiplied by 106 for both the in-phase andin-quadrature components. The algorithm providesdual interpretation parameters (apparent resistivityand apparent depth) at each frequency. The twoparameters are usually assessed jointly. Apparentdepth can be negative, zero or positive and Fraser(1978) indicates that a negative apparent depthimplies the model is a thin conductor over a resistor.Fraser (1978, p. 155) also describes a negative appa-rent depth as being not meaningful physicallybecause the absolute value of the negative depth isnot a measure of the thickness of the conductive upperlayer. The parameter is rarely presented in the liter-ature but is used in the calculation of a wide variety ofcentroid depth estimates (Siemon, 2001).The dual parameters obtained are routinely used ina variety of approximate depth transforms. Theseprocedures attempt to provide an effective (centroid)depth (Sengpiel, 1988) that can be associated with theresistivity value at a particular frequency. The simpli-fication of the dipole source term used in the estima-tion of the dual parameters and centroid depth requiresa coil separation (r) to height (h) ratio of r < 0.3 h. Intheory, the same procedures cannot be applied to ourfixed-wing data collected at an elevation of 40 myielding r/h = 0.53. A study of the behaviour of thecentroid calculation reveals the degree to which theexisting procedures can be used directly or, wherenecessary, a correction applied. Fraser half-spaceparameters and centroid depths are presented in detailto examine their ability to differentiate geological,cultural (individual buildings and villages) and envi-ronmental (landfill) components.Discrimination of invalid (e.g. non-1D) data addsto the reliability of an interpretation. 1D inversion ofAEM data is becoming more widespread (Patersonand Redford, 1986; Sengpiel and Siemon, 1998) andthe misfit parameter is useful with regard to modelvalidity. Theoretical examples of 1D inversion appliedto a two-layer earth are presented. Fraser half-spaceand 1D inversion resistivity models are comparedover a 11 km area containing a cement works andan active landfill (Fig. 1). The apparent depth infor-mation provided by the Fraser half-space algorithm isalso considered. It is demonstrated that negativeapparent depths are associated with increased misfitlevels obtained by the 1D inversion procedure.2. The AEM survey dataThe first high resolution airborne EM surveys toaddress specific environmental issues in the UK werecarried out jointly by the British (BGS) and Finnish(GTK) Geological Surveys in 1999. The dual fre-quency, fixed-wing EM system operated by GTK wasused in a series of trials to acquire detailed EM datasets in addition to magnetic gradiometer and radio-metric information. Only the EM data are discussedhere. Four areas in the English East Midlands weresurveyed. Targets for the surveys included collieryspoil tips and domestic and industrial landfills (bothactive and closed).The GTK fixed-wing airborne EM system used inthe surveys is described in detail by Poikonen et al.(1998). Jokinen and Lanne (1996) describe environ-mental applications of the system in Finland. Use ofthe system in relation to the mapping of surficialdeposits is described by Puranen et al. (1999). Thecoils are wing-tip mounted (separation of 21.4 m) andare vertical coplanar. Coupling ratios at two frequen-cies (3.1 and 14.4 kHz) are recorded simultaneously at4 Hz. The 3.1 kHz data is referred to here as lowfrequency (LF) and the 14.4 kHz data is referred to asthe high frequency (HF) data. Sampling along theflight direction is typically between 10 and 15 m.Elevation information is provided by a radar altimeter.The calibration of the system is described by Poiko-nen et al. (1998).The airborne data are being assessed for theirpotential relevance to a number of land-use issuesincluding waste planning and pollution control. Someof these issues require quite detailed, local scale ( scale levels of performance. When airborne data areacquired over populated areas, coupling from both at-and near-surface cultural artefacts (e.g. buildings,pipelines, power lines, etc) may occur. The surveydata contain examples of the many influences (geo-logical, cultural and environmental) that pose a chal-lenge to valid data interpretation.One of the four test areas (Langar, Fig. 1) is a4 1.5 km area that was flown at an elevation of 40 musing 50 m EW flight lines. The area was chosen fora study of Fraser half-space and 1D inversion methodssince the cultural features are relatively sparse and canbe clearly identified. There is also very little in theway of tree cover. Since the detail in the data isimportant, results along a single profile (Line 420,Fig. 1) are used to illustrate the results. The profiletraverses three roads, a cement works and an activedomestic landfill.3. Fraser half-space processingThe most common interpretation parametersobtained from frequency domain airborne EM surveysare those developed and described by Fraser (1978).Beard (2000) provides a recent comparison of thesetechniques. The concepts developed relate to theapparent half-space resistivity (qa) that can be deducedfrom the in-phase (P) and in-quadrature (Q) couplingratios measured by airborne systems. One of themodelling concepts is referred to as the pseudolayerhalf-space resistivity. In this model, the sensor altitudeabove ground level (as measured by a laser or radaraltimeter), which may be in error, is not used in thecalculation. The parameters obtained by the pseudo-layer half-space model are shown in Fig. 2.Using the measured in-phase and in-quadraturecoupling ratios at a single frequency, a curve matching(nomogram) algorithm is used to obtain the apparentresistivity of a half-space. The curve is actually twopoints and the procedure is exemplified in Beard(2000, Fig. 1). As a second parameter, the algorithmalso returns an estimate of apparent distance (Da inFig. 2) to the ground surface. Since sensor altitude (h)is also measured, an apparent depth (da) may bedetermined as:da Da h 1When da = 0 (Da = h), the model is ideal in thesense that there are no apparent discrepancies betweenthe uniform half-space model and the data. Whenda>0, it is possible to account for the difference in Daand h either by the presence of a resistive at-surfacelayer or by an incorrect (low) altitude measurement ofh (caused by tree cover or buildings). When da < 0, theparameter suggests that the half-space is overlain by amore conductive, thin layer. The resistivity or thick-ness of the layer is not resolved. A negative value ofda can also result from a magnetic subsurface feature(Beard and Nyquist, 1998). Poor levelling of the datamay also result in negative value of da. The surveydata considered here come from relatively short flightlines and levelling accuracy is judged adequate.The method provides dual interpretation parame-ters (qa,da) at each operating frequency. The statedadvantage of the Fraser pseudo-layer half-spacemethod is that the qa determinations are not influ-enced by errors in the altitude. The dual set ofparameters qa( f),da( f) constitute the Fraser half-spaceinterpretation parameters obtained from an airbornesurvey. The procedure and the parameters obtainedappear very stable and this was probably an importantconsideration in previous decades when sensors andinstrumentation were cruder. The procedure delivers adual parameter set at every point sampled and nomisfit (between model response and the two datapoints) is obtained. This is in contrast to the morecommon numerical 1D inversion schemes in whichthe misfit parameter is an important element in dataand model interpretation.Only a few papers dealing with the interpretationof acquired dual parameter data sets have appeared inthe literature. One paper, which discusses both theo-retical and observational aspects of the behaviour ofapparent depth, is presented by Sengpiel (1983). Thedual parameters are discussed in terms of both layeredhalf-space models and in relation to conductors offinite extent (a thin conducting plate in a homoge-neous half-space). Negative apparent depths can occurin relation to both layered half-spaces and in relationto conductors of finite extent.An example of the dual parameters obtained at twofrequencies along the 3.5 km example profile (line420, Fig. 1) is shown in Fig. 3. Flight (sensor) altitudeis also shown since this has a bearing on the interpre-tation. Fig. 3a shows the apparent resistivity valuesD. Beamish / Journal of Applied Geophysics 51 (2002) 7596 79(logarithmic scale) with two roads, the cement worksand the waste disposal site (see Fig. 1) indicated. Thehalf-space resistivity estimates across the profile arerelatively smooth functions. The background trendsfrom about 30V m in the west to several ohm metre inthe east can be associated with three geological unitsthat traverse the area with a NESW trend. Along Line420 the units comprise the Mercia Mudstone Group(472 to 472.7 km), the Lower Lias Limestone (472.7 to473.7 km) and then a Lower Lias clay series (Fig. 1).The three units provide resistivities of about 25, 10 and5 V m, respectively, and the results therefore offer adegree of geological discrimination. Trends in theapparent depth results (lower frame) are less obvious.The 14.4 kHz data are largely small and positive acrossthe entire profile but the 3.1 kHz data show a distinctnegative level in the west. The apparent depths arelargely confined to the range 5 to 5 m. High wave-number correlations exist between the 3.1 and 14.4kHz apparent depth results. These are probably due toFig. 2. Illustration of parameter relationships involved in the pseudo-layer half-space calculation. The on-board altimeter provides a correctelevation of h or underestimated h* when elevated features are encountered. Apparent distance (to half-space) is Da and formulae give theapparent depth (da) below ground surface.D. Beamish / Journal of Applied Geophysics 51 (2002) 759680the limited accuracy in distance (Da) calculation usedin the curve-matching algorithm and the common useof the height information in the estimates of apparentdepth at the two frequencies.Superimposed on the background trends are pertur-bations that can be associated with man-made features.In the apparent resistivity data, movements to lowervalues are observed in relation to two roads (R), thecement works (C) and within the waste disposal site(W). Flight altitude variations (Fig. 3b) range from 30to 46 m. Within the cement works zone (circled), threesharp movements to low values occur. These areidentified as incorrect (in relation to ground height)low altitude values due to the presence of buildings.Fig. 3. Fraser pseudo-layer half-space parameters along EW flight line 420. (a) Apparent resistivities at 3.1 kHz (LF) and 14.4 kHz (HF).Positions of two roads (R), the cement works (C) and the waste disposal site (W) are indicated. (b) Apparent depth at 3.1 kHz (LF) and 14.4 kHz(HF), together with flight altitude (ALT, right axis). Circle denotes perturbations caused by buildings (B1 and B2) in the cement works area.D. Beamish / Journal of Applied Geophysics 51 (2002) 7596 81The apparent depth results (both frequencies) displayan inverse correlation with the altitude movements andtherefore, in this case, the apparent depth results maybe used to identify the likely influence of buildings onthe survey data. The responses due to two buildingswithin the cement works site are labelled as B1 and B2.In contrast to the behaviour of the apparent depthresults, the apparent resistivity data remain smoothand stable since altitude information is not used in theirdetermination.4. Approximate depth transformsIn order to provide more useful depth informationfrom multifrequency survey data, various approximatetransforms have been suggested. The transforms alltake as their starting point the dual parameters qa (astable parameter) and da (a parameter which may bedifficult to interpret) at single frequencies as discussedabove. Following the work of Mundry (1984), Seng-piel (1988) presented a centroid depth calculation thathas proved very useful in the interpretation of surveydata. The centroid depth calculation is a dipole fieldadaptation of similar concepts that were derived forlow frequency plane-wave (magnetotelluric) investi-gations by Schmucker (1970) and Weidelt (1972).The calculations are based on the complex transferfunction C. For a half-space with resistivity q, the skindepth ( p) in metres is given by:p q=pfl01=2 2where f is frequency (Hz) and l0 = 4p10 7 Vs/Am.For our measuring frequencies, the plane-wave skindepths are: p(14.4 kHz) = 4.19(q)1/2 and p(3.1kHz) = 9.04(q)1/2. In the uniform field case, the com-plex transfer function (Co) is:Co p=2 ip=2 3and has the dimensions of length. Following Weidelt(1972), the real part of the Co transfer function isequivalent to the depth zo of the centroid of theinduced in-phase current system. The imaginary partof Co can be used to determine the apparent resistivity(qo) at the centroid depth zo:qo 4l0pf ImCo2 4withzo ReCo 5By joining qo, zo paired values from a sequence offrequencies, an apparent resistivity depth profile (orsounding curve) is achieved.If the source field can be characterised by a singlewavenumber:k 2pL 6where L is the wavelength of the source, the transferfunction Ck for a half-space becomes:Ck k2 1 i=p21=2 7If the primary field originates from an oscillatingmagnetic dipole, the resulting field components arerepresented by a continuous spectrum of wavelengths.Mundry (1984) used the integral for the normalisedsecondary magnetic field (Z) for a magnetic dipole ata height h above a homogeneous or layered half-spaceof Wait (1982):Z r3Z 40Rof ; k; qzk2exp2khJokrdk 8where r is the separation between transmitter andreceiver, Ro is a reflection coefficient and Jo is theBessel function of order zero. The expression is validfor horizontal coplanar coils; in the case of verticalcoplanar coils, the expression must be multiplied by0.5. The complex reflection factor is a function of thelayered resistivity distribution q(z) and the exponen-tial function governs the propagation through air.Mundry (1984) derived a simplified expression forZ under the condition:h > 3:3r 9i.e. the sensor altitude exceeds the sensor separationby a factor greater than 3.3. Under this condition, theBessel function Jo may be replaced by unity and theparameter r no longer appears under the integral.Condition (9) facilitates the determination of qa andDa (Mundry, 1984).Sengpiel (1988) took the simplified Mundry inte-gral and derived a relatively simple set of equationsrelating a complex transfer function C applicable todipole induction and the in-phase and in-quadratureD. Beamish / Journal of Applied Geophysics 51 (2002) 759682components of the secondary field (Sengpiel, 1988;equations 22 and 23). The plane wave (Co) anddipolar (C) functions were compared with the remark-able result that the imaginary part of the transferfunction for dipole induction was found to be virtuallyidentical to that for induction by a plane wave source.The real part of the C transfer function shows thestrongest deviation from Co when the height (h) is lowin relation to the skin-depth and dipolar geometricaleffects become significant.Thus, under the survey height condition (9),Sengpiel (1988) defined a centroid depth (z*) ofthe in-phase current system for dipole induction byanalogy with Schmuckers zo for a uniform sourcefield:z* hReC 10The properties of the centroid depth have beeninvestigated in a number of studies including 1Dassessments (Sengpiel, 1988; Huang and Fraser,1996) and in relation to multidimensional resistivitydistributions (Sengpiel and Siemon, 1998, 2000).The procedures involve taking the dual parametersqa, da and using qa as the resistivity estimate (i.e.q* = qa) at a particular frequency. Replacing h in Eq.(10) by Da leads to:zs* DaReC 11where the centroid depth zs is measured from thesurface of the equivalent half-space. Since we canassume that there are no currents flowing above theequivalent half-space, we can also measure the cent-roid depth from the true surface of the ground:z* da DaReC 12where da is apparent depth. More recently, Siemon(2001) found that by inserting q = qa into the skin-depth Eq. (2), the centroid depth can be simplycalculated as:zp* da pa=2 13where pa is the apparent skin-depth. A comparison of1D theoretical sounding curves using qa(z*) andqa(zp*) is provided by Sengpiel and Siemon (2000).5. Data evaluationHEM systems typically operate with r < 7 m anduse sensor (towed-bird) elevations of greater than 21 mso that condition (9) is upheld. The fixed-wing surveydata have a sensor separation of 21.34 m and the twomain flight elevations were 40 and 90 m. For the 90 melevation data, condition (9) is satisfied; however, forFig. 4. Centroid depths calculated for three half-space resistivitiesand for elevations in the range 20 to 120 m using a sensor separation(r) of 21.34 m. (a) 14.4 kHz (HF). (b) 3.1 kHz (LF).D. Beamish / Journal of Applied Geophysics 51 (2002) 7596 83the 40 m elevation data, the condition is not met anddipolar geometrical effects may be significant.The significance of the approximation on thecentroid depth calculation was studied using Eq.(10) and the expressions given in Sengpiel (1988).Forward 1D numerical modelling was used to obtainestimates of z* for flight elevations in the range 20 to120 m for a series of homogenous half-spaces. Fig. 4shows the results obtained at high frequency (Fig. 4a)and at low frequency (Fig. 4b). The sensor separation(r) used was 21.34 m and the vertical dashed linesindicate the altitude locations of 2r and 3.3r (con-dition (9)). At both frequencies, departures from thecentroid values obtained at high elevation increasewith increasing resistivity. At the high frequency,there is very little change in the estimated value ofthe centroid depth values across the elevation rangeshown. At the low frequency, the effect is morepronounced and in resistive environments, the cent-roid depth is likely to be underestimated when usinglow altitude survey data from fixed-wing systems.The results of Fig. 4 indicate that a centroid depthcorrection procedure can be applied, when necessary,to low altitude survey data. It is probably not worthattempting to obtain an already approximate centroiddepth to better than 1 m. The centroid value differencesfor 40 m elevation at high frequencies when comparedwith values at 3.3r (Fig. 4) are all less than unity (forresistivities < 100Vm). At low frequencies, the differ-ences are 0.81 m (10 V m), 1.5 m (50 V m) and 5.8 m(100 V m).The pseudo-layer half-space resistivities in theexample data are all < 50 V m and, following theabove discussion, no corrections to the centroid depthestimate given by Eq. (13) are required. The centroiddepths along Line 420 are shown in Fig. 5. Theinfluence of buildings (B1 and B2) is identified.Obviously, in this conductive environment, centroiddepths are small being typically less than 15 m at bothfrequencies. Low frequency values increase in thewest due to a geological transition to higher resistivityvalues. Elsewhere, the separation between the two setsof estimates is about 5 m. In this conductive environ-ment, the two frequencies are essentially sounding thesame vertical profile and one would have to use amuch wider frequency separation to detect any sig-nificant layering (if present). The centroid depths are atransform of information contained in the dual qa, daparameters; they inevitably reflect the geological andman-made features already discussed in relation to theFig. 5. Centroid depths along EW flight line 420 at 3.1 kHz (LF) and 14.4 kHz (HF). The influence of buildings B1 and B2 is indicated.D. Beamish / Journal of Applied Geophysics 51 (2002) 759684dual parameters. The main utility of the centroid depthestimate is to provide an association between the qaestimates and depth below ground surface.A further utility of the centroid depth lies in thediscrimination of problem or multidimensional data.As discussed by Schmucker (1987), for a 1D layeredearth, Weidelts second inequality (Weidelt, 1972;equation 2.31) ensures that z* increases monotoni-cally with decreasing frequency. According to Seng-piel (1988) it is also valid for a dipole source if heightremains constant. For data to be consistent with a 1Dassumption, we have z*( f1) < z*( f2) < z*( f3). . .etc.,where f1>f2>f3. . .etc. In the example data of Fig. 5,the condition is satisfied even across the problemarea associated with the cement works. The utility ofthe centroid depth is better illustrated using thecomplete survey data for the area. Fig. 6a shows thesurvey flight lines across the area. Flight elevationswere largely in the range from 30 to 50 m but thepoints with elevations >50 m have been identifiedwith bold symbols. Three high-fly zones are identifiedas (1) the village of Barnstone in the north, (2) thevillage of Langar in the west (not overflown due to alocal aerodrome) and (3) a zone of flight heightadjustments in the east. In Fig. 6b, the data pointswhich represent z*( f= 14.4 kHz)>z*( f= 3.1 kHz) andwhich are therefore inconsistent with a 1D assump-tion, have been identified by the bold symbols. In thenorth, there is an apparent association between thehigh-fly zone above the village of Barnstone andinconsistent data. It is important to realise, however,that the response data from the high-fly zones areadequate in terms of signal/noise (they would notnormally be rejected on the basis of elevation). Thecentroid depth calculation allows a more rigorousidentification of some, but not all, inconsistent datapoints. Such an identification allows the data to berejected from the interpretation.Fig. 6. Results shown as classified symbol values along flight lines for the Langar survey area. (a) Elevation, values from 26 to 50 m shown aslight symbols, values >50 m shown as bold circles. The two ellipses denote the villages of Barnstone (B) and Langar (L). (b) Bold symbolsdenote points where the centroid depth z*(14.4 kHz)>z*(3.1 kHz).D. Beamish / Journal of Applied Geophysics 51 (2002) 7596 85The approximate dipolar depth transforms can befurther extended, again using an analogy with equiv-alent plane-wave magnetotelluric procedures, as dis-cussed by Huang and Fraser (1996) and Sengpiel andSiemon (1998). The extended procedures are based onthe NiblettBostick transform (Schmucker, 1987) andrely on the estimation of the gradient of the resistivity/frequency information (i.e. dqa/df) (Siemon, 2001).When the number of frequency estimates is greaterthan 2, a sounding curve is defined whose gradientmay be estimated. When, as here, the number offrequencies is 2, the curve is limited to a straight line.While such procedures may be applied in this case,the model is constrained to be a linear variation ofresistivity with depth.6. 1D inversion1D inversion procedures have been applied to bothground-based (meaning non-airborne) and airbornecoilcoil systems. The weighted least-squares formal-ism is usually adopted and nonlinearity is addressedas part of the iterative procedure. One of the mostcommon methods is the ridge-regression procedure(Inman, 1975) with stabilisation incorporated using aFig. 7. 1D inversion results for theoretical data from a two-layer case with a 40-V m cover above a 10-V m half-space, with interface depthsranging from 0 (a 10-V m half-space) to 80 m. The inversion attempts to fit a half-space model for each of the two frequencies (LF = 3.1 kHz,HF= 14.4 kHz). (a) Apparent resistivities (V m). (b) Misfit (%) calculated according to Eq. (14).D. Beamish / Journal of Applied Geophysics 51 (2002) 759686Marquardt (1970) approach. Paterson and Redford(1986) provide a description and examples of theproblems associated with the 1D multilayer inversionof HEM data. One of the key points is that dualfrequency airborne data provide an underdeterminedcase of model estimation, i.e. the number of modelparameters is greater than the number of observations.Ellis (1998) provides further insight into the inherentproblem of non-uniqueness and incomplete band-width that typifies 1D inversion of airborne data.Sengpiel and Siemon (1998, 2000) discuss a Mar-quardt multilayer inversion applied to modern fivefrequencies HEM data.The correct choice of the number of layers (thatcan be resolved by the data) is a difficult problem. Thenumber of layers is an unknown for each multilayerinversion problem. The correct choice is both sitespecific (it depends on the resistivity structure encoun-tered) and system specific (it depends on the numberof frequencies, their separation and their bandwidth).It is always possible to increase the number of layersbeyond those present and, typically, reduce themisfit. However, the problems of parameter equiva-lence and resolution of the additional layers areknown and have been discussed widely (e.g. Ellis,1998).Fig. 8. 1D inversion results for theoretical data from a two-layer case with a 40-V m cover above a 10-V m half-space, with interface depthsranging from 0 (a 10-V m half-space) to 80 m. The inversion attempts to fit a two-layer model using both frequencies. (a) Resistivity of Layer 1(V m). (b) Resistivity of Layer 2 (V m). (c) Depth of interface (m).D. Beamish / Journal of Applied Geophysics 51 (2002) 7596 87A theoretical example of the problem that arises inthe case of the two frequency fixed-wing system isnow considered. Forward modelling of a two-layercase was used to obtain a series of theoretical data.The elevation adopted was 40 m and the coil spacingwas 21.34 m. The two layer model comprises a firstlayer with a resistivity of 40 V m above a 10-V mhalf-space. The interface depth was allowed to varybetween 0 (i.e. a 10-V m half space) and 80 m(effectively a 40-V m half space). A Marquardt non-linear least-squares inversion algorithm, adapted forairborne EM, was applied to the theoretical data.For the small numbers of observations used in theinversion (e.g. 2), an L1 norm (based on the sum ofdifferences) rather than an L2 norm (based on the sumof squared differences) misfit parameter has proveduseful. The issues involved are those of robust sta-tistics (Hampel et al., 1986). Here the L1 normpercentage error is defined as:L1% 100:0Xni1jOi Cij=Ci 14where Oi is the ith observation (an in-phase or in-quadrature coupling ratio) and Ci is the correspondingcalculated value. L1 norms are typically greater thanL2 norms (e.g. an rms misfit) by a factor of 10. Thisdefinition of misfit is used throughout.The results of constructing a half-space solution (1layer) to each of the two frequency data sets byinverse modelling are shown in Fig. 7. The modelhalf-space resistivities are shown in Fig. 7a and thepercentage misfit between data and model is shown inFig. 7b. Apart from the first data point (an interfacedepth of zero indicates a half-space of 10 V m), theinitial set of half-space resistivities (interface depthsfrom 1 to 20 m) returned by the inversion are failedattempts to provide a single half-space resistivity tothe two-layer data. The model resistivities are, how-ever, contained within the correct range from 10 to 40V m. As the thickness of the first layer increases, thehigher frequency data approach a correct layer valuewhen the interface depth is 18 m. The lower frequencyresults, with a larger electromagnetic scale length,continue to mix the two resistivities present untilan interface depth of about 40 m. The misfit valuesreflect the behaviour observed in the resistivity esti-mates. The misfit obtained for the higher frequencyestimation peaks more rapidly, then decays morerapidly than that of the lower frequency inversion.The above example relates to the pitfalls of apply-ing an incorrect model within a fixed number of layersinversion procedure. When the number of layers isappropriate, multilayer inversion schemes can provideextremely accurate results. A two-layer inversionprocedure applied to the joint two frequency theoret-ical data set provided the model results shown in Fig.8. In this case, the estimated model parameters are theresistivities of layer 1 (resistivity 40 V m) and layer 2(resistivity 10 V m) together with their calculatedinterface depth. The misfits obtained in this case areall less than 0.001.7. 1D inversion: application to survey dataThe number of degrees of freedom (input data)obtained with the fixed-wing system is 4 (2 data pointsat 2 frequencies). In theory, the estimation of a two-layer model would involve a slightly overdeterminedinversion procedure since the number of model param-eters in this case is 3. In practice, this is only correctwhen the altitude measurement is a reliable estimate ofheight above ground level. The radar altimeter used hasa resolution of 0.1 m and an accuracy of 0.5 m. Theinstrumental accuracy is not an issue. Barometricheight, although recorded, was found to be unreliable.The real issue is that, when surveying in populatedareas, the height above ground level is often incorrectlydetermined (biased to incorrect low values) due to avariety of elevated features. The most obvious featuresare well-defined forest and copse zones together withdomestic, commercial and agricultural buildings.In order to maintain detailed inversion model accu-racy it has been necessary, in general, to introduce aninitial very high resistivity first layer. This procedureis, in effect, a formalism in keeping with the pseudo-layer half space solution discussed previously. Thefirst layer resistivity (e.g. 106 V m) is a constrainedparameter but its thickness is allowed to vary. Thesimplest inversion strategy is thus to provide a fixedlayer above a half-space model for each of the recordedfrequencies. The procedure is then equivalent to that ofthe pseudo-layer half space model (Fig. 2). The twoparameters estimated are the thickness (of layer 1) andthe underlying half-space resistivity.D. Beamish / Journal of Applied Geophysics 51 (2002) 759688Fig. 9. Half-space model results obtained along the central 1 km profile of EW flight line 420. (a) LF (3.1 kHz) apparent resistivity. Fraserpseudo-layer half-space (symbols), 1D inversion (line). (b) Thickness of resistive at-surface layer (1D inversion), with influence of buildings(B1, B2) noted. (c) Misfit (%) calculated according to Eq. (14).D. Beamish / Journal of Applied Geophysics 51 (2002) 7596 89The results obtained, along example Line 420, forthe low frequency data are shown in Fig. 9. In thiscase, the baseline has been restricted to a 1-km profilelength across the cement works and waste disposalsite. The resistivity, shown as a solid line in Fig. 9a, iscompared with that obtained from the pseudo-layerFig. 10. Half-space model results obtained along the central 1 km profile of EW flight line 420. (a) Comparison of LF (3.1 kHz) coupling ratios,observed (symbol) and modelled (line), in-phase (P) and in-quadrature (Q) components. (b) Comparison of HF (14.4 kHz) coupling ratios,observed (symbol) and modelled (line), in-phase (P) and in-quadrature (Q) components. (c) LF and HF apparent depth estimates obtained fromFraser pseudo-layer half-space algorithm, with influence of buildings (B1, B2) noted.D. Beamish / Journal of Applied Geophysics 51 (2002) 759690algorithm (symbols). A strong overall level of corre-spondence between the two sets of estimates isobserved. In detail, it is clear that the Fraser pseudo-layer estimates are spatially smooth while the 1Dinversion estimates contain a number of higher wave-number components. The thickness of the at-surfaceresistive layer is shown in Fig. 9b. The influence ofthe buildings (B1 and B2) is again observed. Themisfit error (Fig. 9c) displays no correlation with thethickness estimates. This suggests that the influence ofthe buildings is restricted to an incorrect elevationerror rather than, say, their multidimensional effects.Towards the centre of the profile, the misfit estimatesexceed 10%, and display some correspondence withfluctuations in the resistivity estimates.The observed (symbols) and two-layer inversionmodelled (line) data at both frequencies are comparedin Fig. 10a,b. These are the data from which themisfits are derived however the plots serve to dem-onstrate the detailed behaviour of the misfit in relationto the observations in both P and Q components. Forthose not familiar with the raw coupling data fromairborne systems, the overall level of coupling (e.g.10000 ppm) is determined by the subsurface resistiv-ity. The large-scale movements are determined byvariations in flight altitude (see Fig. 3). Any lateralchanges in the resistivity structure may be detected atthe tens of ppm level and would only be observed inFig. 10 by slight changes in gradient. It is the highwavenumber fluctuations in misfit that are of interest.The apparent depth estimates, obtained from thepseudo-layer algorithm, are shown in Fig. 10c. Thebuilding response features (B1, B2) are evident. Thehigh wavenumber features in the data elsewhere,however, display a correspondence with fluctuationsin the modelled response estimates that provide theFig. 11. 1D inversion misfits shown as classified symbol values along flight lines for the Langar survey area. Data have been pre-processedaccording toWeidelts inequality condition applied to centroid depth. (a) LF (3.1 kHz) misfit, values from 0 to 10% shown as light symbols, values>10% shown as bold circles. (b) HF (14.4 kHz) misfit, values from 0% to 5% shown as light symbols, values >5% shown as bold circles.D. Beamish / Journal of Applied Geophysics 51 (2002) 7596 91largest misfits. It appears that, although the pseudo-layer algorithm provides no misfit values, some levelof misfit information is contained within the estimatesof apparent depth.Misfit values obtained from 1D inversion proce-dures are of great utility in both the large scale anddetailed interpretation of the applicability of a givenmodel. Fig. 11a shows the low frequency misfit valuesover the whole survey area classified into two sets ofsymbols at the 10% error level. The data set shownhas first been winnowed following application ofWeidelts second inequality applied to centroid depths(see Fig. 6). Points with a misfit >10% account for18% of the total. If the same level is applied to theFig. 12. Comparison of (a) Fraser half-space and (b) 1D inversion, LF (3.1 kHz) half-space resistivity results across the central 11 km areacontaining cement works and land-fill. The base map is reproduced from the OS map by British Geological Survey with the permission ofOrdnance Survey on behalf of The Controller of Her Majestys Stationery Office, n Crown copyright. All rights reserved.D. Beamish / Journal of Applied Geophysics 51 (2002) 759692Fig. 13. Comparison of (a) negative apparent depth and (b) 1D inversion misfit for LF (3.1 kHz) half-space results across the central 11 kmarea containing cement works and land-fill. (a) Only negative apparent depths (0 to 6 m) are contoured. (b) Only misfits >7% are contoured.Background shows data sampling density of EW flight lines.D. Beamish / Journal of Applied Geophysics 51 (2002) 7596 93high frequency data, less than 2% are included. Thehigh frequency misfit values in Fig. 11b have there-fore been classified at the 5% error level.Apparent in the NW corner of the area is acontinuous zone that displays higher levels of errorat both frequencies. This is interpreted as an area inwhich a half-space solution is not completely appro-priate. The behaviour implies that multilayer (possiblytwo-layer) inversion of the joint two frequency datawould be required to reduce the misfit. The zone,which has higher resistivities (see Fig. 3), correlateswith the known geology and comprises an area ofalluvial cover over Mercia Mudstone trending NESW. These are the only alluvial deposits within thesurvey area. Somewhat surprisingly, the linear road(or features associated with the road), most evident atlow frequencies, provides a lower misfit than thesurrounding area. Elsewhere, high error levels tendto occur as small clusters or individual data points.The ability of the apparent resistivity data sets todiscriminate features at the site investigation scale isnow examined. The 11 km area centred on thecement works and active waste disposal site, shownin Fig. 1, is considered using the low frequency data. Asstated previously, the two survey frequencies providenearly equivalent information in this conductive envi-ronment. The apparent resistivities obtained from theFraser pseudo-layer half-space method are shown inFig. 12awith an overlay of anOrdnance Survey (1:25 k)map. Two highly conductive zones, already observedalong Line 420, are resolved. Line 420 is located alongthe Northing of 335 km. The most conductive feature(about 1 V m) is highly localised and is centred on thecement works. A more diffuse anomaly (2 to 4V m) iscentred on the landfill. Overall, the apparent resistivitydata appear smooth and stable.The apparent resistivities obtained from the 1Dinversion, including rejection of data points at the10% error level, are shown in Fig. 12b. The gridding(kriging) and contour levels selected are the same as inthe previous case. The two resistivity maps show abroad level of agreement, however, the 1D inversionresults display a set of higher wavenumber features thatappear realistic when the data density is considered (seebelow). If we were to define a resistivity level of < 4Vm as anomalous and of interest in relation to enhancedconcentrations of pore fluids, then the two sets ofresults would provide different conclusions (in detail).The Fraser half-space data would indicate two highlycompact zones centred on the works and landfill. The1D inversion results would indicate the same sourcecentres but with a more complex and extensive subsur-face volumetric contribution. Without geochemicalborehole sampling information, the degree of appro-priateness of each model cannot be assessed. Atpresent, the Fraser half-space results are treated as acorrect but conservativemodel; the 1D inversion resultsare treated as a correct guide to ground-truth studies.The apparent depth information provided by theFraser pseudo-layer algorithm should also be consid-ered in relation to the interpretation of the resistivitydata. These data are shown in Fig. 13a in conjunctionwith the flight line data points sampled by the con-toured information. Values in the range 0 to 2 mpredominate and only negative values (0 to 6 m)are shown. Fig. 13b shows the misfit error level (norejection) obtained from 1D inversion. Only misfitvalues of >7% are contoured. Although the dynamicranges of the two parameters are very different, it isclear that there is a strong correspondence betweenincreasingly negative apparent depth and increasingmisfit. The results suggest that model fitting errors inthe Fraser pseudo-layer algorithm are accounted forby the introduction of negative apparent depths. Ashas been demonstrated, positive apparent depths doindeed account for incorrect altitude estimates or mayindicate at-surface resistive zones. The further obser-vation that misfit is also contained in the parameterserves to explain the fact that the jointly estimatedapparent resistivity is a stable and conservative firstestimate of the resistivity distribution.Finally, it is worth noting that the negative apparentdepths in the example are small. Other data setscontain negative apparent depths that exceed thesurvey altitude. The units of negative apparent depthare metres and a level of significance has to beestablished if a rejection criterion is applied. The 1Dinversion methods can provide misfit estimates thathave well-established credentials for testing the sig-nificance of the modelled parameters.8. ConclusionsThe present study has considered the applicability ofFraser half-space and 1D inversion methods applied toD. Beamish / Journal of Applied Geophysics 51 (2002) 759694high-resolution AEM data. It is recognised that anumber of interpretation and modelling issues arisedue to the imprecise measurement of sensor altitudewith respect to ground level. These issues becomeacute when surveys are conducted in populated areas.Two possible improvements to aid modelling would befor the on-board avionics to record accurate (sub-metric) barometric height (e.g. Sengpiel and Siemon,2000) in addition to radar height or to establish equiv-alent accuracies in the elevation information providedby differential global positioning systems already usedfor position information.Modern fixed-wing AEM data can be obtained athigh spatial resolution and used for assessments ofenvironmental influences on the near-surface resistiv-ity distribution. The ability of the data and processingmethods to differentiate geological, cultural and envi-ronmental influences is then an important factor. Thearea chosen for the study contains examples of allthree influences with the latter two occurring inrelatively sparse and identifiable fashion.Fraser half-space modelling provides the dualinterpretation parameters of apparent resistivity andapparent depth at each operational frequency. Theapparent resistivity is a remarkably stable parameterand appears robust to the presence of a variety of at-surface cultural features. Such features provide bothincorrect altitude and multidimensional influences.Their influences are partly incorporated into the jointestimate of apparent depth and this accounts for thestability of the estimated apparent resistivity. Positiveapparent depths, in the example data, can result fromunderestimated altitude measurements. Increasinglynegative apparent depths are associated with increas-ing misfits between the 1D half-space model responseand the observed data.The estimate of apparent depth can be used, inconjunction with apparent skin-depth (incorporatingapparent resistivity), as the basis for an approximatedepth transform. This transform, a dipolar adaptationof plane wave techniques, provides a centroid depthunder the condition that the sensor altitude exceedsthe sensor separation by a factor >3.3. Since thiscondition is not satisfied by low altitude, fixed-wingsurvey data the theory of the centroid depth calcula-tion was examined. The study reveals the degree towhich the existing procedures can be used directly, orwhere necessary, a correction applied. Correctionsbecome increasingly necessary at low frequenciesand in resistive environments. The conditions underwhich existing depth transforms can be applied tofixed-wing surveys have not previously been des-cribed.In a 1D environment centroid depths must increasewith decreasing frequency. This condition can be usedto flag or reject inappropriate data and models. In thefeatured example survey, a swathe of such inappro-priate data is associated with measurements obtainedabove a village. In contrast, data obtained aboveisolated and individual buildings, although influenced,were still classified as appropriate using centroiddepths.1D least-squares inversion of both theoretical andsurvey data has been examined. The issue of depthresolution and number of layers used by 1D methodsis always related to the number of available frequen-cies and their bandwidth. The simplest use of the 1Dinversion methods is in providing an estimate of ahalf-space resistivity. This can be undertaken prior tomultilayer inversion as an initial assessment. Errone-ous altitude measurements also enter the problem and,in keeping with the Fraser pseudo-layer concept, anat-surface highly resistive layer of variable thickness,can be usefully introduced as a constrained parameter.Multilayer inversion of the dual frequency data hasalso been conducted. As has already been stated, inthis conductive environment, the two frequencies areessentially sounding the same vertical profile and onewould have to use a much wider frequency separationto detect any significant layering (if present).A detailed comparison of Fraser half-space andequivalent 1D inversion model parameters has led to abetter understanding of the Fraser half-space parame-ters, particularly that of apparent depth. It is clearlydifficult to ascribe levels of significance to a meas-ure of misfit contained in a negative apparent depthwith the dimensions of metres. The reliability of 1Dmodels is better assessed using a formal misfit param-eter. With the misfit parameter in place, the exampledata suggest that the 1D inversion methods providereliable resistivity models with a higher resolutionthan the equivalent information from Fraser half-spaceestimates. At present, the pseudo-layer results aretreated as a simplified conservative model; the 1Dinversion results are treated as a correct guide toground-truth studies.D. Beamish / Journal of Applied Geophysics 51 (2002) 7596 95AcknowledgementsThe survey data discussed here were obtained as acollaborative venture between the British and FinnishGeological Surveys (BGS and GTK). The BGSwishes to thank Maija Kurimo and the GTK teamfor their dedication and professionalism during thesurvey operations. We are grateful to the Departmentof the Environment, Transport and the Regions andthe Environment Agency for their co-sponsorship ofthe acquisition of the survey data. This report ispublished with the permission of the ExecutiveDirector, British Geological Survey (NERC).ReferencesBeamish, D., 2002. Airborne EM footprints. Geophys. Prospect.(submitted for publication).Beamish, D., Kurimo, M., 2000. Trial airborne EM surveys to as-sess minewater pollution in the UK. 62nd EAGE Conferenceand Technical Exhibition, Glasgow, UK. Expanded Abstracts,vol. 1, D-22.Beard, L.P., 2000. Comparison of methods for estimating earthresistivity from airborne electromagnetic measurements. J.Appl. Geophys. 45, 239259.Beard, L.P., Lutro, O., 2000. Airborne geophysics and infrastructureplanninga case study. J. Environ. Eng. Geophys. 5, 110.Beard, L.P., Nyquist, J.E., 1998. Simultaneous inversion of airborneelectromagnetic data for resistivity and magnetic permeability.Geophysics 63, 15561564.Collett, L.S., 1986. Development of the airborne electromagnetictechnique. In: Palacky, G.J. 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Anomalies of Geomagnetic Variations in theSouthwestern United States. Bull. Scripps Inst. Oceanogr. 13,1165 (La Jolla, CA).Schmucker, U., 1987. Substitute conductors for electromagneticresponse estimates. PAGEOPH 125, 341367.Sengpiel, K.-P., 1983. Resistivity/depth mapping with airborne elec-tromagnetic survey data. Geophysics 48, 181196.Sengpiel, K.-P., 1988. Approximate inversion of airborne EM datafrom a multi-layered ground. Geophys. Prospect. 36, 446459.Sengpiel, K.-P., Siemon, B., 1998. Examples of 1-D inversion ofmultifrequency HEM data from 3-D resistivity distribution. Ex-plor. Geophys. 29, 133141.Sengpiel, K.-P., Siemon, B., 2000. Advanced inversion methodsfor airborne electromagnetic exploration. Geophysics 65,19831992.Siemon, B., 2001. Improved and new resistivitydepth profiles forhelicopter electromagnetic data. J. Appl. Geophys. 46, 6576.Wait, J.R., 1982. Geo-Electromagnetism Academic Press, NewYork.Weidelt, P., 1972. The inverse problem of geomagnetic induction. Z.Geophys. 38, 257289.D. Beamish / Journal of Applied Geophysics 51 (2002) 759696IntroductionThe AEM survey dataFraser half-space processingApproximate depth transformsData evaluation1D inversion1D inversion: application to survey dataConclusionsAcknowledgementsReferences

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