Automatic source localization and spectra generation from sparse beamforming maps
PPREPRINT: Automatic source localization andspectra generation from sparse beamforming maps
A. Goudarzi, a C. Spehr, b and S. Herbold c German Aerospace Center (DLR), Germany Karlsruhe Institute of Technology (KIT), Germany
This paper is part of a special issue on Machine Learning in Acoustics.Beamforming is an imaging tool for the investigation of aeroacoustic phenomena andresults in high dimensional data that is broken down to spectra by integrating spatialRegions Of Interest. This paper presents two methods which enable the automatedidentification of aeroacoustic sources in sparse beamforming maps and the extraction oftheir corresponding spectra to overcome the manual definition of Regions Of Interest.The methods are evaluated on two scaled airframe half-model wind tunnel measurements.The first relies on the spatial normal distribution of aeroacoustic broadband sources insparse beamforming maps. The second uses hierarchical clustering methods. Both methodsare robust to statistical noise and predict the existence, location and spatial probabilityestimation for sources based on which Regions Of Interests are automatically determined. © [https://doi.org(DOI number)][XYZ] Pages: 1–10 I. INTRODUCTION
Multiple noise-generating phenomena and mech-anisms exist in aeroacoustics . Expert domainknowledge and a detailed study of measurements arenecessary to identify these phenomena in measurements.For the localization and investigation of aeroacousticsources, microphone array beamforming is a reliablestandard method . Beamforming measurements usuallyresult in 2D or 3D beamforming maps for each observedfrequency and are often varied over Mach number,angle of attack of the flow, and geometrical parameterof the observed model. The level of the beamform-ing map entries indicates a sound source emissionpower, usually described by the Power Spectral Density(PSD( (cid:126)x, f, M, . . . ), but can also result from backgroundnoise, spurious noise sources, and beamforming artifacts.Additionally, the localization can be disturbed by soundreflections, scattering, and refraction. Consequently,the resulting beamforming maps have to be analyzed toextract the desired source information. For this process,it is useful to integrate the high dimensional PSD overspatial regions of the map to obtain low-dimensionaldata. Ideally, the process only includes the locations ofthe respective source of interest while rejecting locationsof other sound sources. This is aggravated by the fact,that the source location may vary over the frequencyand Mach number due to the flow-dependent nature ofthe sources itself or due to the aforementioned scatteringand refraction within the sound propagation from the a [email protected] b [email protected] c steff[email protected] source to the array microphones.A common way to handle this source identificationis the spatial integration of resulting beamforming mapsover so-called Regions Of Interest (ROI). This resultsin low-dimensional data such as spectra which canbe interpreted by human experts. There exist threeapproaches for the manual definition of ROIs. First,the whole beamforming map is integrated into a singlespectrum which is then analyzed for prominent features,such as tones or peaks. Then, the beamforming mapat these frequencies or frequency bands is observed todetermine the origin of these sources, and ROIs aredefined to account for these. Second, the beamformingmaps are observed at a variety of chosen frequencyintervals, and ROIs are defined based on the consistentappearance of sources at multiple frequencies, intuition,and experience. Third, ROIs are defined based on thestudied geometry. A challenge for these methods isthe distinction between beamforming artifacts and realsources; the correct separation of close and overlappingsources; the detection of sources with a low PSD andsmall-band sources; and the detection of sources thatappear only at some of the measurement variations de-scribed above. The definition of the ROI may thereforenot only depend on the wind tunnel model but on thearray resolution as well as the signal to noise ratio andthe methods used to process the beamforming maps . Awrong or insufficient ROI definition results in degradedor wrong spectra which is especially problematic sincemost of the following aeroacoustic analysis is based onthese. J. Acoust. Soc. Am. / 2 March 2021 Source identification in sparse beamforming maps 1 a r X i v : . [ c s . S D ] F e b he important task of defining ROIs is performed bythe expert manually and takes typically from hours up todays from our experience, depending on the complexity ofthe beamforming maps and the studied model. Machinelearning proved to be a promising tool in acoustics andGaussian Mixture Models were already deployed to trackspeaker sources in space-time . Consequently, this paperpresents two approaches to overcome the difficulties anddrawbacks of this process by automatically identifyingstationary, aeroacoustic sources in sparse beamformingmaps and obtaining their correct spectra using unsuper-vised learning. The scaled air-frame models of a Dornier728 (Do728) and an Airbus A320 are presented to de-rive these methods, discuss their usefulness, and specifya proof-of-concept implementation. II. DATASETS
The data presented in this paper consists of beam-forming measurements of two closed-section wind tunnelmodels: one is of a Do728 and one is of an A320 . Forthe Do728 dataset, values of α i = 1 ° ,3 ° ,5 ° ,6 ° ,7 ° ,8 ° , ° , ° are chosen for angle of attack α and M i = 0 . . . . . .
250 for Mach number M . Themean Reynolds number is (cid:104) Re (cid:105) M = 1 . × based onthe mean aerodynamic cord length D = 0 .
353 m andambient temperature of T = 300 K at an ambient pres-sure p = 1 × Pa. The array consists of 144 micro-phones at an aperture of 1 .
756 m × . f S = 120 kHz. The A320 set con-tains α i = 3 ° ,7 ° ,7 . ° ,9 ° , M i = 0 . . .
225 ata mean Reynolds number of (cid:104) Re (cid:105) M = 1 . × basedon D = 0 .
308 m, T = 300 K, p = 1 × Pa. Thearray consisted of 96 microphones at an aperture of1 .
06 m × . f S = 150 kHz. Thus, the Do728consists of 48 measurements, the A320 dataset con-sists of 12 measurements. Cross-Spectral density Ma-trices (CSM) are calculated using Welch’s method witha block size of 1024 samples for the Do728 and 512 sam-ples for the A320 with 50 % overlap. The beamform-ing is performed using conventional beamforming andCLEAN-SC deconvolution with a focus point resolutionof ∆ x = ∆ x = 5 × − m. III. SOURCE IDENTIFICATION
A general problem concerning beamforming is thatat long wavelengths the localization of acoustic sourcesis difficult. Furthermore, imaging artifacts may occurdue to the sparse spatial distribution of the microphonearray. These artifacts result from background noise, thearray’s Point Spread Function, and aliasing or insufficientWelch estimations . In this part of the paper, we dis-cuss two ideas on how to identify sources from beamform-ing maps contaminated with noise and obtain their spec-trum. The ideas are based on sparse beamforming mapswhich can be achieved by inverse beamforming methodsor conventional beamforming in combination with whatis known in the aeroacoustic beamforming community as FIG. 1. (Color online) A320, CLEAN-SC result on 2D-planeusing conventional beamforming, the z-axis displays the fre-quency. The color represents the normalized PSD in decibelat M = 0 . α = 3 ° . “deconvolution”, such as CLEAN-SC or DAMAS . Forthis paper, we choose conventional beamforming with di-agonal removal in combination with CLEAN-SC overDAMAS, because of the huge number of computed beam-forming maps and the high spatial resolution of the maps.CLEAN-SC assumes point-like sources and then sub-tracts coherent portions of the dirty beamforming map .This removes most of the Point Spread Function but willalso result in a single PSD( (cid:126)x , f ) representation of spa-tially distributed sources. We make this an advantageas this results in extremely sparse representations of thesource map. For the terminology, we call every recon-structed PSD( (cid:126)x , f ) a source-part s . Thus, the result-ing sparse beamforming maps are a list of source-partvectors s i = [ (cid:126)x i , f i , α i , M i , PSD i ]. Figure 1 displays thesource-parts of the CLEAN-SC result on a 2D-focus gridfor the A320. On the z-axis, the frequency is displayed,the color represents the normalized PSD. We can iden-tify multiple vertical pillars of source-parts s which, spa-tially integrated, represent a source spectrum PSD( f ).However, we also observe pillars that suddenly split withincreasing frequency (e.g., at the flap side edge) or pil-lars with higher density embedded in blobs with lowerdensity (e.g., the slat track at the inner slat). Up tonow, large ROIs were defined manually as integration ar-eas to obtain spectra such as the whole slat and flapregion. This partly contradicts the beamforming idea,as we often do not know where sources are located andwhether all sources within the integration region are ofthe same source type. In the following part, we introducetwo methods on how to estimate the existence and posi-tions of individual sources in sparse beamforming mapsand how to correctly assign the corresponding source-parts to them. A. Source Identification based on spatial Normal Distribu-tions (SIND)
Figure 2 a ) shows the normalized Overall SoundPressure Level (OASPL) over the spatial location (cid:126)x for [m] x [ m ] a) x [m]b)40 30 20 10 0 norm OASPL [dB] log (n+1) FIG. 2. (Color online) A320, section of the CLEAN-SC mapat M = 0 . , α = 3 ° . a ) shows the normalized OASPL, b )shows a log-histogram of the source-parts s per focus point (cid:126)x . the A320. The OASPL is the integration of the source-parts Sound Pressure Level (SPL) over frequency. Weobserve that sources cannot be easily distinguished basedon the OASPL because the sound carries most energy atlong wavelengths. Due to the array resolution, beam-forming is not able to localize sources well at these wave-lengths (see Figure 1). However, ignoring the SPL andsimply counting how often a source-part s was recon-structed by CLEAN-SC at every location (cid:126)x provides abetter grasp on individual source distributions, which areshown in the logarithmic histogram in Figure 2 b ). Wesee mostly distinguishable blobs with maxima in theircenter that probably represent aeroacoustic sources, asthe blob’s positions coincide with the location of the slattracks, the slat side edge, and the flap side edge. Whilethe blobs in the log-histogram resemble normal distri-butions, statistical tests such as the Shapiro-Wilk or theAnderson-Darling test do not determine that data as nor-mal. The reason for this is the discrete spatial sampling,the overlapping of sources, as well as the large populationof source-parts. Instead, we use a soft normal distribu-tion test to verify the normality assumption. First, wefit a normal distribution to the log-distribution of theappearance of source-parts by minimizing the differencebetween the source-part position histogram and the esti-mated Probability Density Function (PDF). Then, wecompare the estimated distribution with the observeddata. The normal distribution in 2D is calculated witheq. 1 . For practical applications, we recommend opti-mizing for the normal distribution’s amplitude A , thestandard deviations σ x i , and the distribution rotation θ, x i, by using a bounded optimization method withequations 2. The histograms maximum determines thestarting values for A, x i, , the bounds A ± ε A , (cid:126)x ± ε (cid:126)x pre-vent the optimizer from wandering off to a completelydifferent source. x [m] x [ m ] a) 0.10.30.50.70.9 0.05 0.00 0.05 x i n o r m P D F b) hist( x )hist( x )pdf( x )pdf( x ) FIG. 3. (Color online) Do728, flap side edge region. a ) showsthe isocontour lines of the by A normalized distribution (dot-ted lines) and its fitted PDF (full lines). b ) shows the nor-malized distribution and PDF on its principal axis ˆ x andˆ x which are achieved by rotating the coordinate system by θ ≈ − ° . f ( x , x ) = A exp (cid:18) − (cid:0) a ( x − x , ) + 2 b ( x − x , )( x − x , )) + c ( x − x , ) (cid:1)(cid:19) (1) a = cos θ σ x + sin θ σ x (2a) b = − sin 2 θ σ x + sin 2 θ σ x (2b) c = sin θ σ x + cos θ σ x (2c)Figure 3 a ) shows the normalized log-distribution ofthe source-parts (dotted lines) for the Do728 flap sideedge region. We can determine two overlapping blobsin this region, a major one upstream and a minor onedownstream. Using eq. 1, a 2D normal distribution isfitted to minimize the major source-part blob (full lines).We introduce two principal axes ˆ x and ˆ x for which thenormal distributions standard deviations σ x i are inde-pendent. They are obtained by simply rotating the coor-dinate system by θ ≈ − ° . Figure 3 b ) shows the com-parison of the actual and fitted distributions along theseaxes. To find and fit all sources in the beamforming map,we introduce the distance metric d , see eq. 3, to mea-sure and minimize the distance of the estimated normaldistributions and real distributions. With the set X S i containing all grid points (cid:126)x i that belong to a source S i ,we want to minimize d S i for all assumed sources S i ∈ S in the beamforming map, so that the difference of thesource-part histogram and the superposition of all fittednormal distributions achieves a minimum. d S i = (cid:88) (cid:126)x i ∈ X Si (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) hist( (cid:126)x i ) − (cid:88) S i ∈ S (PDF S i ( (cid:126)x i )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3) J. Acoust. Soc. Am. / 2 March 2021 Source identification in sparse beamforming maps 3 [m] x [ m ]
12 34 56 789 1011 121314 151617181920 2122232425 26 2728 ( n FIG. 4. (Color online) A320. The SIND solution for t I = 20 isshown. The source numbers correspond to the order of foundsources via the maxima in the histogram. The ellipses aroundthe sources represent the PDF functions at 1 − σ . The coloredpoints are the log-histogram values from all source-parts. Using this metric we can implement a greedyalgorithm that minimizes d S i by iteration. First, wefind the maximum in the histogram of the source-parts;second, we fit a normal distribution that minimizes thehistogram; and third, we subtract the fitted distributionfrom the histogram and repeat the process until theremaining histogram-maximum drops below a thresh-old t I . This threshold represents a lower significancebound and prevents endless fitting iterations. Aftersubtracting a PDF from the histogram for a source,occasionally single (cid:126)x i with a high source-part countremain due to fitting inaccuracies (single beamform-ing map pixels are then fitted with a distributionin the next iteration). We integrate the total PDFarea A S i = (cid:82) x (cid:82) x PDF( S i ) dx dx for each sourceto account for these artifacts. If A S i drops below athreshold t A we can reject it as a fitting artifact. A S i ofartifacts is orders of magnitude below A S i of real sources.Figure 4 shows the result of the procedure for theA320. The threshold t I for the iterating process is t I = 20 source-parts. No sources are rejected as fittingartifacts. The crosses mark the determined sources, thenumbers correspond to the order in which they are iden-tified. The ellipses around the marked sources represent x [m] x [ m ]
12 3 45 6 78 9101112 13141516 171819 2021 2223 242526 2728 ( n FIG. 5. (Color online) Do728. The SIND solution for t I = 30is shown. The source numbers correspond to the order offound sources via the maxima in the histogram. The ellipsesaround the sources represent the PDF functions at 1 − σ . Thecolored points are the log-histogram values from all source-parts. PDF S ( (cid:126)x ) = 1 − σ of the fitted normal distributions.Figure 5 shows the result of the procedure for the Do728for t I = 30, no sources are rejected as fitting artifacts.Finally, we calculate for all source-parts the prob-ability of belonging to each source-cluster using PDF S and assign them to the source with the highest proba-bility. Then we drop all source-parts with a PDF valuebelow a threshold t σ . Figure 6 shows the source-partsthat were assigned to the upstream flap side edge regionwithin t σ = 1 − σ confidence at M = 0 . α = 9 ° ,Figure 7 shows the the downstream flap side edge region.The color encodes the corresponding normalized PDFvalue, which can be interpreted as the confidence thatthe source-parts belong to the assigned source. Weobserve three rows of points with similar shape over fre-quency. We assume that the two rows at a low SPL areartifacts from the CLEAN-SC process, as CLEAN-SCfailed to remove these source-parts from the dirty mapwithout residue. After integrating all source-parts overthe frequency, we obtain a mostly smooth spectrum, [ m ] x [ m ] f [ H z ] f [Hz]204060 P S D [ d B / H z ] P ( s S i ) FIG. 6. (Color online) The source-parts of the A320 upstreamflap side edge region (source number 1 in Figure 4) at M =0 . α = 9 ° from the SIND-solution that lie within 1 − σ confidence. The confidence (normalized PDF-value) that thesource-part belongs to the source is displayed in color. Theblack line represents the integrated spectrum from all source-parts. x [ m ] x [ m ] f [ H z ] f [Hz]204060 P S D [ d B / H z ] P ( s S i ) FIG. 7. (Color online) The source-parts of the A320 down-stream flap side edge region (source number 7 in Figure 4) at M = 0 . α = 9 ° from the SIND-solution that lie within1 − σ confidence. The confidence (normalized PDF-value)that the source-part belongs to the source is displayed incolor. The black line represents the integrated spectrum fromall source-parts. indicated by the black line. Figure 8 and Figure 9 showthe same for the Do728 at M = 0 . α = 7 ° . Fig-ure 10 shows an exemplary Do728 slat / slat track source.SIND assumes that the source positions do not fun-damentally change in the beamforming map over M or α (considering a focus grid that rotates and moves with α ) so that the source-parts of different measurement con-figurations can be simply stacked and fitted at once toobtain global source positions and distributions. How-ever, beamforming can suffer from the approximation ofGreens Functions in complex medium flows to calculatethe sound propagation from the source position to themicrophone array or errors in the position of the focalplane . The first results in a shift or stretch of the beam- x [ m ] x [ m ] f [ H z ] f [Hz]204060 P S D [ d B / H z ] P ( s S i ) FIG. 8. (Color online) The source-parts of the Do728 up-stream flap side edge region (source number 4 in Figure 5) at M = 0 . α = 6 ° from the SIND-solution that lie within1 − σ confidence. The confidence (normalized PDF-value)that the source-part belongs to the source is displayed incolor. The black line represents the integrated spectrum fromall source-parts. x [ m ] x [ m ] f [ H z ] f [Hz]204060 P S D [ d B / H z ] P ( s S i ) FIG. 9. (Color online) The source-parts of the Do728 down-stream flap side edge region (source number 13 in Figure 5)at M = 0 . α = 6 ° from the SIND-solution that lie within1 − σ confidence. The confidence P (normalized PDF-value)that the source-part belongs to the source is displayed in color.The black line represents the integrated spectrum from allsource-parts. forming maps, the second results in a source that movesthrough the map with increasing angle α because of theprojection error (the strakes of the Do728 in Figure 5show this behavior). The first problem can be fixed byaligning the beamforming maps prior to fitting the nor-mal distributions. To do so, the source-part histogramof each individual configuration is calculated, then a his-togram is chosen as a reference. All remaining histogrampositions are then linearly modified with f ( x i ) = a i x i + b i (4)to achieve a maximum correlation with the reference his-togram using standard optimization methods. Eq. 4 canthen be used to modify the source-parts positions x i . Fig-ure 11 shows the obtained parameters a i , b i for the A320. J. Acoust. Soc. Am. / 2 March 2021 Source identification in sparse beamforming maps 5 [ m ] x [ m ] f [ H z ] f [Hz]204060 P S D [ d B / H z ] P ( s S i ) FIG. 10. (Color online) The source-parts at the DO728 slattrack (source number 5 in Figure 5) at M = 0 . α = 1 ° from the SIND-solution that lie within 1 − σ confidence. Theconfidence P (normalized PDF-value) that the source-part s belongs to the source S i is displayed in color. The black linerepresents the integrated spectrum from all source-parts. a a =3.0°=7.0° =7.2°=9.0° 0.02 0.00 0.02 b b M=0.175M=0.200 M=0.225
FIG. 11. (Color online) A320, beamforming map alignmentstretch a i and shift parameters b i for the source-part positions x i relative to the reference beamforming map at M = 0 . α = 3 ° . While the stretch factors a i are small, the shift factors b i show a clear trend. The beamforming maps wanderslightly with increasing angle of attack and substantiallywith increasing Mach number downstream (more than b ≥ x ). B. Source Identification based on Hierarchical Clustering(SIHC)
A second approach to identifying sources and assign-ing the corresponding source-parts is clustering methodswhich can automatically group source-parts in a multi-dimensional space. Since we do not know the numberof expected clusters and their distribution beforehand,we choose Hierarchical Density-Based Spatial Cluster-ing of Applications with Noise (HDBSCAN). Similarto SIND, HDBSCAN requires a threshold t below whicha cluster is rejected as noise. The threshold has a great FIG. 12. (Color online) A320. Resulting clusters from HDB-SCAN at t = 105, using an euclidean distance metric. Thecluster midpoints are marked, the corresponding source-partsare displayed in the same color. The transparency level dis-plays the probability of belonging to the cluster. Grey source-parts were rejected as noise. effect on the resulting clusters and has to be determinedwith the expert in the loop. We cluster the source-parts based on their normalized location (cid:126)x i , normalizedStrouhal number St i and Mach scaled, normalized PSDlevel (normalized to the range [0 , (cid:91) PSD = PSD −
10 log M n (5)with n ≈ . .Figure 12 shows the result of HDBSCAN for theA320 at t = 105 and Figure 13 for the Do728 at t = 500.The crosses mark the cluster midpoints of the corre-sponding source-parts, displayed in the same color. Greysource-parts are rejected as noise as their confidenceof belonging to any source is below t σ = 1 − σ . Thecolor intensity displays the classification confidence.Figure 14 shows the A320 source-parts that wereassigned to the flap side edge region at M = 0 . α = 9 ° within t σ = 1 − σ confidence, Figure 15 shows IG. 13. (Color online) Do728. Resulting clusters from HDB-SCAN at t = 500, using an euclidean distance metric. Thecluster midpoints are marked, the corresponding source-partsare displayed in the same color. The transparency level dis-plays the probability of belonging to the cluster. Grey source-parts were rejected as noise. the same for the Do728. Figure 17 shows the same slattrack source from the SIND solution in Figure 10 andFigure 16 shows the upper part of the corresponding slat. C. Comparison of SIND and SIHC
Both methods yield comparable ROIs and areable to identify the prominent source locations suchas the flap side edge, slat tracks, wing tip or strakes.SIND often separates individual sources in dense andoverlapping source regions that are clustered together bySIHC, especially at the inner slat or the flap side edgeregion. SIHC finds additional source regions that arenot well localized and spread over the map, especiallysources that are not located on the wing. We observethat SIND and SIHC assume different underlying sourcedistributions when comparing the resulting flap sideedge spectra in Figure 6, Figure 7 and Figure 14 forthe A320 and Figure 8, Figure 9 and Figure 15 for theDo728. For the A320 flap side edge, a detailed spectrumanalysis shows that the up- and downstream separationof SIND is reasonable, see Figure 18 and Figure 19. Thespectra’s SPL are Mach scaled with eq. 5 to reveal theirself-similarity. While the low-frequency peak scales overStrouhal number, the high-frequency peaks scale over x [ m ] x [ m ] f [ H z ] f [Hz]204060 P S D [ d B / H z ] P ( s S i ) FIG. 14. (Color online) The source-parts at the A320 flap sideedge (source number 5 in Figure 12) at M = 0 . α = 9 ° from the SIHC-solution that lie within 1 − σ confidence. Theconfidence P (normalized PDF-value) that the source-part s belongs to the source S i is displayed in color. The black linerepresents the integrated spectrum from all source-parts. x [ m ] x [ m ] f [ H z ] f [Hz]204060 P S D [ d B / H z ] P ( s S i ) FIG. 15. (Color online) The source-parts at the Do728 flapside edge (source number 2 in Figure 13) at M = 0 . α = 7 ° from the SIHC-solution that lie within 1 − σ confidence. Theconfidence P that the source-part belongs to the source isdisplayed in color. The black line represents the integratedspectrum from all source-parts. Helmholtz number which suggests different aeroacousticsource mechanisms and justifies the spatial separation.While SIND and SIHC separate most slat from slattrack ROIs, SIHC reconstructs more smooth spectra byidentifying the corresponding source-parts than SIND.Figure 10 shows, that the low-frequency slat tones arenot well localized and scattered around the slat area.SIHC not only separates the Strouhal number scaling slattones, see Figure 20, from the Helmholtz number scalingslat track source, see Figure 21, it assigns the source-parts mostly correct to the corresponding source spectra.Performance-wise SIHC’s computation time scalesaround O ( n log n ) for the number n of source-parts .Since SIND does not cluster the points directly, the com-putation time is independent of the number of points,which is a huge advantage for large datasets. The total J. Acoust. Soc. Am. / 2 March 2021 Source identification in sparse beamforming maps 7 [ m ] x [ m ] f [ H z ] f [Hz]204060 P S D [ d B / H z ] P ( s S i ) FIG. 16. (Color online) The source-parts at the DO728 slat(source number 20 in Figure 13) at M = 0 . α = 1 ° fromthe SIHC-solution that lie within 1 − σ confidence. Theconfidence P that the source-part s belongs to the source S i is displayed in color. The black line represents the integratedspectrum from all source-parts. x [ m ] x [ m ] f [ H z ] f [Hz]204060 P S D [ d B / H z ] P ( s S i ) FIG. 17. (Color online) The source-parts at the DO728 slat(source number 29 in Figure 13) at M = 0 . α = 1 ° fromthe SIHC-solution that lie within 1 − σ confidence. Theconfidence P that the source-part s belongs to the source S i is displayed in color. The black line represents the integratedspectrum from all source-parts. St [ ]405060708090 P S D [ d B / H z ] , s c a l e d b y n = . a) 10 He [ ]60708090100110 P S D [ d B / H z ] , s c a l e d b y n = . b) M=0.175M=0.200M=0.225 FIG. 18. (Color online) The A320 spectra of the upstreamflap side edge region at α = 9 ° over a ) Strouhal number and b ) Helmholtz number. The spectra are Mach scaled with thescaling exponent n , see eq. 5. St [ ]405060708090 P S D [ d B / H z ] , s c a l e d b y n = . a) 10 He [ ]60708090100110 P S D [ d B / H z ] , s c a l e d b y n = . b) M=0.175M=0.200M=0.225 FIG. 19. (Color online) The A320 spectra of the SIND down-stream flap side edge source (number 7 in Figure 4) at α = 9 ° over a ) Strouhal number and b ) Helmholtz number. The spec-tra are Mach scaled with the scaling exponent n , see eq. 5. St [ ]6080100120 P S D [ d B / H z ] , s c a l e d b y n = . a) M=0.125M=0.150M=0.175M=0.200M=0.225M=0.250 He [ ]6080100120 P S D [ d B / H z ] , s c a l e d b y n = . b) FIG. 20. (Color online) The Do728 spectra of the SIHC slatsource (number 20 in Figure 13) at α = 1 ° over a ) Strouhalnumber and b ) Helmholtz number. The spectra are Machscaled with the scaling exponent n , see eq. 5. number of source-parts in the Do728 dataset is around n = 10 , which SIND processes within seconds and SIHCwithin an hour on a standard laptop. Both methodsprocess the A320 dataset within seconds, which containsaround n = 10 source-parts. IV. DISCUSSION
We presented two methods on how to detect sourcesand extract their spectra from sparse beamformingmaps. Instead of using generic data, we evaluated themethods on real measured wind tunnel data. The reasonfor this choice was that aeroacoustic experts only needsupport in identifying sources in beamforming maps ofcomplex, ambiguous data. The drawback of this choiceis the lack of a ground truth to quantify the results witha related metric. Thus, the results can only be discussedqualitatively by comparing them to each other, theirconsistency, and to the expectation of the aeroacousticexperts.SIND was based on the idea that the source-part’spositions of real acoustic sources at different frequencies St [ ]6080100120 P S D [ d B / H z ] , s c a l e d b y n = . a) M=0.125M=0.150M=0.175M=0.200M=0.225M=0.250 He [ ]6080100120 P S D [ d B / H z ] , s c a l e d b y n = . b) FIG. 21. (Color online) The Do728 spectra of the SIHC slattrack source (number 29 in Figure 13) at α = 1 ° over a )Strouhal number and b ) Helmholtz number. The spectra areMach scaled with the scaling exponent n , see eq. 5. appear spatially normal distributed in sparse beam-forming maps. Thus, it yielded good results in findingpoint-like sources such as slat tracks, strakes, flap tracks,or the wing tip in both datasets. SIND was also able toidentify dense, overlapping sources like the flap side edgeor point-like sources that were embedded in distributedsources such as the nacelle and the slat tracks in the in-ner slat region. It profited from stacking the histogramsof multiple measurements at different Mach numbersand angles of attack to increase the histogram statistics,yet failed to recognize sparse source blobs with no clearmidpoint. Wind tunnel noise was a prominent examplefor this, as this source was projected on different partsof the image with increasing angle of attack α due tothe mismatched focal plane. SIND’s results are robustagainst variations of the introduced parameters andthus, were consistent with the expert out of the loop.The source positions on the two similar airframe modelsare consistent and based on the underlying source-parthistogram we assume they are correct. The identificationof line-like sources, such as the slat, is difficult for thisapproach. The slat noise and its resulting low-frequencyslat tones that scale over Strouhal number is visible inFigure 1 between the slat tracks. SIND tends to identifyit as multiple point-like sources due to its distributionassumption in combination with CLEAN-SC processingor included these in the slat track sources.SIHC was based on the hierarchical clusteringmethod HDBSCAN and thus did not assume a pre-defined source distribution. The source-parts wereclustered directly in space, frequency, and SPL withthe expert in the loop, as the results depend stronglyon the set threshold. This means the correct thresholdhas to be determined manually to give accurate results.Because of the additional frequency and SPL informationSIHC has the potential to separate spatially overlappingsources, such as slat tracks and slats. On the one hand,it clustered the inner slat and the flap side edge tosingle sources for which we assume the SIND solutionto be more precise. On the other hand, it was able to identify sources containing source-parts that were toofar scattered around the map to be identified by SIND,such as spurious noise sources that were not located onthe wing.Despite the similar identified source regions, SIND’sseparation of individual sources is more refined comparedto the SIHC solution. While both methods identifiedthe individual slat tracks (except for the A320 innerslat, where we assume the existence of two slat tracks,embedded in a distributed high-frequency noise source,see Figure 1), the strakes and the wing tip on the Do728,SIHC missed the flap track closest to the wing tip on theDo728 and A320. It also clustered the inner slat regionof the Do728 to a single ROI, as well as the nacelleregion of the A320 and Do728 and the outer slat tip ofthe A320 and Do728.The Do728 and A320 flap side edge, as well as anA320 slat source, were shown in detail to evaluate theROI quality. While the source-parts of the flap sideedge form two overlapping normal distributions, SIHCidentified a single source-distribution that featuredmultiple regions of high confidence in space, frequency,and SPL. We expect the flap side edge to be a line-sourcewith a frequency-dependent source location and showedthat the spectrum is governed by two different sourcemechanisms. Thus, we favor the SIND result over theSIHC result. The example Do728 slat source showedthat the Strouhal number scaling tones are a distributedline source that is superimposed with point-like slattrack sources which scale over Helmholtz number. WhileSIND identified most of the slat sources as point-likesources between the slat tracks, it was not able to assignthe low-frequency source-parts to the slat that werelocated at the slat track positions. Since SIHC hasthe additional SPL and frequency information of eachsource-part and had no prior assumption of the sourcedistribution it was able to assign the source-parts ofoverlapping sources to the correct sources in this case.Thus, we favor the SIHC result for the slat sources.Both methods proved useful with different advan-tages and disadvantages to the real-world airframedatasets. SIHC works well for small datasets (e.g., asingle angle of attack and few Mach variations) with lit-tle statistical noise. It is advantageous for exploring thedataset because a single threshold drastically changes theROI outcome. Generally, density-based clustering meth-ods tend to fail separate clusters when too much noiseis present that connects the clusters, so-called bridgepoints. Consequently, SIHC yields better results whendecreasing the Welch block size, which increases the num-ber of FFT averages and results in less statistical noisebut also a lower frequency resolution. SIND works wellfor noisy datasets with high-resolution PSDs (large Welchblock sizes) and yields stable results, that are mostly in-dependent of the selected thresholds and profits fromlarge datasets. Since its thresholds only limit the pro- J. Acoust. Soc. Am. / 2 March 2021 Source identification in sparse beamforming maps 9 essing time and drop sources after the identification, in-creasing or decreasing these values will not change theoutcome of the remaining sources. Thus, SIHC is well-suited for an iterative process with the expert in the loopthat can be fine-tuned to a desired outcome, while SINDrequires no tuning to generate stable results and can beemployed autonomously. The overall quality of SIND’sresults decreases with smaller data sets, as the source-part histogram statistics decreases, while SIHC’s resultsimprove, as it has to handle less statistical noise and viceversa. In specific cases, when two sources overlap spa-tially but can be distinguished based on their SPL( f ),such as slat sources, the SIHC method has a clear ad-vantage over SIND, which can not detect this behavior.While dense source-distributions with bridge-points areproblematic for SIHC, it is able to detect sparse source-distributions without a clear midpoint, which SIND can-not detect. It is possible to combine both methods byfirst employing SIND to extract the high-density clus-ters and then performing SIHC on the remaining source-parts. Performance-wise SIND is superior to SIHC andcan be employed on datasets of any size. Together, bothmethods cover the automatic source identification andspectrum generation from single, sparse low-resolutionFFT beamforming maps to high-resolution FFT beam-forming maps including multiple parameter variationswith speed and accuracy that is unmatched by humanexperts. V. CONCLUSION
We presented the two methods “SIND” and “SIHC”,which automatically detect aeroacoustic sources in de-convolved beamforming maps. They identify underlyingsource-distributions and thus, allow for the automatic de-termination of Regions Of Interest. To the best of ourknowledge, these are the first automated approaches thatcan identify sources and generate corresponding spectrafrom sparse beamforming maps without prior informa-tion about the source locations. Both methods togethercover a variety of real-world scenario used-cases, fromsingle measurements with sparse source distributions tohigh-dimensional datasets with parameter variations andcan be combined. Implementation details and resultswere discussed on scaled airframe half-model measure-ments. In particular, the resulting Regions Of Interestand spectra of the flap side edge and a slat track werepresented and showed that SIND is superior in separatingdense, overlapping source regions, while SIHC is superiorin assigning the source-parts to the correct sources whichresults in an improved reconstruction of spectra at lowfrequencies. For Future work, SIND should be extendedwith a spectrum continuity criterion that ensures thatthe scattered low-frequency source-parts are assigned tothe correct sources.
ACKNOWLEDGMENTS
We want to thank the experts of the aeroacousticgroup G¨ottingen, especially Dr. Thomas Ahlefeldt, forthe helpful discussions on the analyzed beamforming re-sults. The authors also acknowledge the DLR, Instituteof Aerodynamics and Flow Technology, Department ofExperimental Methods (contact: Carsten Spehr) for pro-viding the SAGAS software which generated the beam-forming and CLEAN-SC results for this paper. Abramowitz, M. ( ). Handbook of Mathematical Functions,With Formulas, Graphs, and Mathematical Tables, (Dover Pub-lications, Inc.). Ahlefeldt, T. ( ). “Aeroacoustic measurements of a scaledhalf-model at high reynolds numbers,” AIAA Journal (12),2783–2791, doi: . Ahlefeldt, T. ( ). “Microphone array measurement in euro-pean transonic wind tunnel at flight reynolds numbers,” AIAAJournal (1), 36–48, doi: . Bahr, C. J., Humphreys, W. M., Ernst, D., Ahlefeldt, T., Spehr,C., Pereira, A., Lecl`ere, Q., Picard, C., Porteous, R., Moreau, D.,Fischer, J. R., and Doolan, C. J. ( ). “A comparison of mi-crophone phased array methods applied to the study of airframenoise in wind tunnel testing,” in , doi: . Bianco, M. J., Gerstoft, P., Traer, J., Ozanich, E., Roch,M. A., Gannot, S., and Deledalle, C. A. ( ). “Machinelearning in acoustics: Theory and applications,” The Journalof the Acoustical Society of America (5), 3590–3628, doi: . Dobrzynski, W., and Pott-Pollenske, M. ( ). “Slat noisesource studies for farfield noise prediction,” in , Vol. 5805, doi: . Guo, Y. P., and Joshi, M. C. ( ). “Noise characteristics ofaircraft high lift systems,” AIAA Journal (7), 1247–1256, doi: . Howe, M. S. ( ). “On the generation of side-edge flap noise,”Journal of Sound and Vibration (4), 555 – 573, doi: . Howe, M. S. ( ). Hydrodynamics and Sound (CambridgeUniversity Press). McInnes, L., Healy, J., and Astels, S. ( ). “hdbscan: Hier-archical density based clustering,” The Journal of Open SourceSoftware (11), doi: . Merino-Mart´ınez, R., Sijtsma, P., Rubio Carpio, A., Zamponi,R., Luesutthiviboon, S., Malgoezar, A., Snellen, M., Schram,C., and Simons, D. ( ). “Integration methods for dis-tributed sound sources,” International Journal of Aeroacoustics , 1475472X1985294, doi: . Merino-Mart´ınez, R., Sijtsma, P., Snellen, M., Ahlefeldt, T., An-toni, J., Bahr, C. J., Blacodon, D., Ernst, D., Finez, A., Funke,S., Geyer, T. F., Haxter, S., Herold, G., Huang, X., Humphreys,W. M., Lecl`ere, Q., Malgoezar, A., Michel, U., Padois, T.,Pereira, A., Picard, C., Sarradj, E., Siller, H., Simons, D. G.,and Spehr, C. ( ). “A review of acoustic imaging methodsusing phased microphone arrays,” CEAS Aeronautical Journal (1), 197–230, doi: . M¨uller, E.-A., ed. ( ). IUTAM Symposia
Mechanics of SoundGeneration in Flows (Springer-Verlag Berlin Heidelberg). Sijtsma, P. ( ). “Experimental techniques for identificationand characterisation of noise sources,” Technical Report. Sijtsma, P. ( ). “Clean based on spatial source coherence.international journal of aeroacoustics,” International Journal ofAeroacoustics , doi: ..