Application of Convolutional Neural Networks to Identify Stellar Feedback Bubbles in CO Emission
Duo Xu, Stella S. R. Offner, Robert Gutermuth, Colin Van Oort
DDraft version January 15, 2020
Typeset using L A TEX preprint2 style in AASTeX62
Application of Convolutional Neural Networks to Identify Stellar Feedback Bubbles in CO Emission
Duo Xu, Stella S. R. Offner, Robert Gutermuth, and Colin Van Oort Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA; Department of Astronomy, University of Massachusetts, Amherst, MA 01003, USA; University of Vermont, Burlington, VT 05405, USA;
ABSTRACTWe adopt a deep learning method casi (Convolutional Approach to Shell Identi-fication) and extend it to 3D ( casi-3d ) to identify signatures of stellar feedback inmolecular line spectra, such as CO. We adopt magneto-hydrodynamics simulationsthat study the impact of stellar winds in a turbulent molecular cloud as an input togenerate synthetic observations. We apply the 3D radiation transfer code radmc-3d to model CO ( J =1-0) line emission from the simulated clouds. We train two casi-3d models: ME1 is trained to predict only the position of feedback, while MF is trained topredict the fraction of the mass coming from feedback in each voxel. We adopt 75% ofthe synthetic observations as the training set and assess the accuracy of the two modelswith the remaining data. We demonstrate that model ME1 identifies bubbles in simu-lated data with 95% accuracy, and model MF predicts the bubble mass within 4% of thetrue value. We test the two models on bubbles that were previously visually identifiedin Taurus in CO. We show our models perform well on the highest confidence bubblesthat have a clear ring morphology and contain one or more sources. We apply our twomodels on the full 98 deg FCRAO CO survey of the Taurus cloud. Models ME1 andMF predict feedback gas mass of 2894 M (cid:12) and 302 M (cid:12) , respectively. When includinga correction factor for missing energy due to the limited velocity range of the COdata cube, model ME1 predicts feedback kinetic energies of 4.0 × ergs and 1.5 × ergs with/without subtracting the cloud velocity gradient. Model MF predicts feedbackkinetic energy of 9.6 × ergs and 2.8 × ergs with/without subtracting the cloudvelocity gradient. Model ME1 predicts bubble locations and properties consistent withprevious visually identified bubbles. However, model MF demonstrates that feedbackproperties computed based on visual identifications are significantly over-estimated dueto line of sight confusion and contamination from background and foreground gas. Keywords:
ISM: bubbles – ISM: clouds – methods: data analysis – stars: formation [email protected]ff[email protected] INTRODUCTIONStellar winds driven by young stars create dis-tinct features in molecular clouds. The ejectedmass compresses and heats the ambient gas, a r X i v : . [ a s t r o - ph . GA ] J a n producing shocks (Hollenbach & Tielens 1999).In the case of more massive stars, stellar windscombined with radiation create bubbles contain-ing luminous H ii regions. Observational sur-veys find that signatures of such bubbles areubiquitous. For example, Churchwell et al.(2006, 2007) visually identified numerous H ii regions in the Spitzer Galactic Legacy InfraredMid-Plane Survey Extraordinaire (GLIMPSE)data. They concluded these bubbles have a sig-nificant impact on the dynamics and star forma-tion of molecular clouds, and they found 12%of the shells are associated with young sources,which may have been triggered by shell expan-sion. Over 50% of the bubbles identified are notspherically symmetric due to fluctuations in lo-cal gas density and/or anisotropic stellar windsand radiation fields. These complications makebubble identification more challenging.A variety of groups have investigated the im-pact of stellar feedback bubbles due to stel-lar winds within molecular clouds. Nakamuraet al. (2012) found several parsec-scale bubblesexpanding and compressing the ambient gas,which they proposed has contributed to the for-mation of several dense filaments. They sug-gested one of these dense filaments is converg-ing with another filament, triggering recent starformation in the cloud. Quillen et al. (2005)mapped the expanding cavities in NGC 1333,a sub-region in the Perseus molecular cloud.They found the kinetic energy of these cavitiesis sufficient to power the turbulence in this re-gion. Arce et al. (2011) identified stellar feed-back bubbles in a full map of the Perseus molec-ular cloud and drew similar conclusions aboutthe energy budget. However, Li et al. (2015)found the energy injected from bubbles in theTaurus molecular cloud is only 29% of the tur-bulent energy of Taurus.A variety of theoretical work has also investi-gated the impact and signatures of stellar feed-back. Boyden et al. (2016) studied statistical signatures of stellar winds in synthetic obser-vations. They found that the covariance ma-trices of the velocity channels are sensitive tothe existence of stellar feedback. Offner, & Liu(2018) found that the slope of the velocity powerspectrum becomes steeper in simulations withfeedback compared to those without feedback.However, stellar feedback might also indirectlyadd energy to the ambient gas where there isno injected feedback mass (Offner, & Liu 2018),making quantitative statistical study of feed-back more challenging in observational data.Historically, bubbles, such as those found inthe above studies, have been identified “byeye ‘’ . However, given the exponentially in-creasing amount of data, visual identification isnot scalable, i.e. it is almost impossible for hu-mans to look through all the data by eye (Moli-nari et al. 2010; Peek et al. 2011). Moreover, tostudy the dynamics of bubbles, it is necessary toswitch from two dimensional images (Beaumontet al. 2014) to three dimensional data cubes(Arce et al. 2011; Li et al. 2015) those containinformation about the gas motion. An extra di-mension makes it much more time intensive toidentify the bubble features. Moreover, it is im-possible for humans to consistently identify orclassify without bias. However, systematic andrepeatable identification is possible with the aidof machine learning approaches (Beaumont etal. 2011, 2014; Van Oort et al. 2019).Several machine learning algorithms havebeen applied to identify stellar feedback fea-tures (Beaumont et al. 2011, 2014). Beaumontet al. (2011) applied Support Vector Machines(SVM) to distinguish a supernova remnant fromthe ambient gas in CO J = 3 − “ Random Forests, ” toidentify bubbles in dust emission. To trainBrut, they adopted bubble identification re-sults from over 35,000 participants in the MilkyWay Project, a citizen science project based onGLIMPSE data. Xu & Offner (2017) expandedon this work by supplementing the Brut train-ing set with synthetic observations of bubblesin simulated clouds. After retraining on the en-hanced training set,
Brut more efficiently iden-tified ultra-compact and compact H ii regionsgenerated by B-type stars. Leveraging bothobservational data (e.g., Simpson et al. 2012;Jayasinghe et al. 2019) and synthetic observa-tions can significantly enhance the performanceof machine learning algorithms. However, Brut requires the bubble to be centered in the imagefor it to be accurately identified. This makesit computationally expensive to identify bub-bles in a large sky survey map because the datamust be cropped into small chunks centered atdifferent positions with different sizes to ensurethe target bubbles are centered in at least oneimage.Due to the evolution of high performance com-puting and the power of GPUs (Graphics Pro-cessing Units), deep learning is gaining popular-ity thanks to its general applicability and highaccuracy. Recently developed deep learningmethods are more powerful in image recogni-tion than earlier methods, like
Brut . Ntampakaet al. (2019) developed a deep machine learningtool based on Convolutional Neural Networks(CNNs) to estimate the mass of galaxy clus-ters in X-ray emission. The CNN is not sen-sitive to the position of galaxy clusters, mak-ing it straightforward to apply to large sky sur-vey maps. Van Oort et al. (2019) developedan “Encoder-Decoder” Convolutional Approachto Shell Identification ( casi ) to identify stel-lar wind bubbles in density slices and 2D COemission. Once trained, casi can identify struc-tures in minutes and achieves a 98% pixel-levelaccuracy. However, one caveat of these CNNmodels is that they are limited to 2D integratedintensity maps or individual velocity channels.These algorithms do not take radial velocity in-formation into consideration, which may lead to a high false detection rate when applied to3D data. In other words, this technique mayidentify a clear ring structure as a bubble eventhough this structure is caused by a turbulentpattern without any evidence of expansion inthe spectra. Alternatively, it may miss struc-tures with coherent expansion across a range ofchannels but which do not have a bubble mor-phology in most channels.In this paper, we adopt the deep learningmethod casi (Van Oort et al. 2019) and ex-tend it to 3D ( casi-3d ) in order to exploit thefull 3D CO spectral information to identify bub-bles. We develop two deep machine learningtasks. Task I predicts the position of feedback.Task II predicts the fraction of the mass in thevoxels that is coming from feedback. We de-scribe our deep machine learning algorithm ar-chitecture and how we generate the training setfrom synthetic observations in Section 2. In Sec-tion 3, we present the performance of the CNNmodel in identifying bubbles in both syntheticdata and observational data. We summarize ourresults and conclusions in Section 4. METHOD2.1. casi-3d
Architecture
In this section we give a brief overview of the casi architecture and describe our implementa-tion in 3D. casi-3d is available on GitLab .Van Oort et al. (2019) developed the casi architecture with residual networks (He et al.2016) and “U-net” (Ronneberger er al. 2015).The residual networks exponentially increasethe complexity of the networks to prevent over-fitting (He et al. 2016; Veit et al. 2016). “U-net” adds cross connections between differentlayers, which enhance the performance in con-structing the output image (Ronneberger er al.2015). casi utilizes a widely-used optimizationmethod, stochastic gradient descent (SGD) with https://gitlab.com/casi-project/casi-3d momentum, in the training. Momentum indi-cates that the SGD takes the past time stepinto consideration when conducting the opti-mization. It helps accelerate SGD during train-ing. casi is trained on simulated molecularcloud density 2D slices or CO integrated in-tensity maps. It learns to predict the “tracerfield,” which is an orion field (Li et al. 2012)that tracks the fraction of gas in each cell that islaunched in the stellar winds. A more detaileddescription of casi can be found in Van Oort etal. (2019).We modify the casi architecture and replacethe 2D convolutions with 3D convolutions, asshown in Figure 1. The “encoder” part extractsthe features from the input data, then the “de-coder” part translates these features into an-other image, e.g., the tracer field. In Van Oortet al. (2019), the input data is a 256 ×
256 array.The image goes through four down-sampling(max pooling) layers. Given the extra dimen-sion in our model, we cannot maintain the samespatial resolution in our 3D model due to thelimited memory on GPU. We reform the inputdata to an array shape of 64 × ×
32 in position-position-velocity (PPV). The data cube under-goes three down-sampling layers. We apply7 × × × × Training Task
We develop two training tasks to identify stel-lar feedback bubbles. Task I involves trainingto predict the position of feedback, which re-produces the morphology of bubbles. Task IIinvolves training to predict the fraction of themass that comes from stellar feedback, whichgives a better mass estimation of the bubbles.2.2.1.
Training Task I: Predicting the Position ofFeedback
In Task I, we aim to predict the position offeedback on a pixel by pixel basis. We use themean squared error (MSE) as the loss functionin the training. The loss function is a metricto quantify the performance of the model pre-dictions. The mean square error is defined as:
M SE = 1 n n (cid:88) i =1 ( Y pred − Y tracer ) , (1)where Y pred represents the prediction from themodel, Y tracer represents the “ground truth” asdescribed by the tracer field, and n indicatesthe number of samples. In this model, Y tracer is the emission of CO at the specific positionswhere there is feedback gas. The CO intensityis proportional to the mass of the gas in theoptically thin regime.Observational data includes noise so we mustadopt a prescription to define feedback that ac-counts for noise limits. We adopt MSE in partbecause it is less affected by the noise in the datacube than other loss functions. We set a thresh-old based on the input data noise level, 0.2 K,to binarize the prediction map, i.e., a pixel isassigned a 1 if the predicted emission is above0.2 K, and a pixel is assigned a 0 if the predictedemission is below 0.2 K. The binarized predic-tion map indicates the location of the bubblesin PPV space. The final loss indicates the errorof the prediction, which is related to the massestimation uncertainty.To explore the performance of different mod-els with different hyper-parameters, we trainmodels with different learning rates, differentepochs, different optimization methods and dif-ferent loss functions. We easily rule out otherloss functions and optimization methods basedon the performance on the training set, andadopt MSE as the loss function and SGD as theoptimization method. We list different modelswith different learning rates and with differentepochs in Table 1. We additionally explore thecompleteness of the training set on the perfor- mance. More discussion can be found in Sec-tion 2.3.4.2.2.2.
Training Task II: Predictions for theFraction of Feedback Mass
Task I basically classifies pixels individuallyas belonging to feedback or not belonging tofeedback. However, it does not take into ac-count that many pixels contain some feedbackand some non-feedback gas. To address this,we train another model to predict the fractionof the mass that comes from stellar feedback.We adopt the same learning rate, epoch andoptimization method as the best model in Sec-tion 2.5. We define the new training data tobe PPV cubes containing the fraction of mass,rather than the emission, that comes from thefeedback. We describe how these cubes are con-structed in more detail in Section 2.3.2.We test the MSE, Intersection over Union(IoU) and a combination of MSE and IoU as theloss function and compare their performance inAppendix A.2. The IoU is a metric to evaluatethe overlap fraction between the prediction andthe tracer field, which is defined as:
IoU = Y pred ∩ Y tracer Y pred ∪ Y tracer . (2)The combination of MSE and IoU is defined as: L new = ω × M SE + IoU, (3)where ω is the weight of the MSE in the newloss function. Here we set ω = 100. Since thefraction is between 0 and 1, the MSE is not sen-sitive to small values, while the IoU is stronglysensitive to small values. We find the combina-tion loss function performs the best in predict-ing the fraction of the mass that comes fromstellar feedback.2.3. Training Sets
Synthetic CO Observations
Convolutional LayersConvolutional Layers X F (X) F (X)+X selu Residual BlockInput64x64x32 R e s i d u a l B l o c k x x x A v e r a g e P oo l i n g R e s i d u a l B l o c k * R e s i d u a l B l o c k * x x x R e s i d u a l B l o c k * x x x Residual Block*2 U p s a m p l i n g R e s i d u a l B l o c k * U p s a m p l i n g R e s i d u a l B l o c k * x x x U p s a m p l i n g R e s i d u a l B l o c k * x x x U - n e t U - n e t U - n e t Output64x64x32
EncoderDecoder A v e r a g e P oo l i n g A v e r a g e P oo l i n g Figure 1.
The architecture of the U-net CNN model with residual functions.
Table 1.
Training Model ParameterModel Task Training Set Learning Rate EpochFiducial Resolution Negative Set High Resolution(5 pc × × (cid:88) (cid:88) (cid:88) adaptive 277ME2 I (cid:88) (cid:88) X adaptive 223ME3 I (cid:88) (cid:88) (cid:88) fixed 60ME4 I (cid:88) (cid:88) (cid:88) adaptive 60ME5 I (cid:88) X (cid:88) adaptive 189ME6 I (cid:88) X (cid:88) adaptive 260ME7 I X (cid:88) (cid:88) adaptive 260MF a II (cid:88) (cid:88) (cid:88) adaptive 277 a : The training set and the hyper-parameters of model MF is the same as those of ME1. The onlydifference is the training set. Model MF adopts the fraction of the feedback mass as the target. Theother models adopt the intensity of the feedback emission in the training. To train casi-3d , we adopt magneto-hydrodynamic(MHD) simulations that model sources launch-ing stellar winds in a piece of a turbulent molec-ular cloud (Offner & Arce 2015). The simula-tion box is 5 × × , with a total massof 3762 M (cid:12) and mean density of ∼
500 cm − .Offner & Arce (2015) conduct different simula-tion runs with different mass loss rates, differentmagnetic fields, different turbulent patterns and different evolutionary stages to study the im-pact of stellar winds on the ambient gas. Moredetails about the simulations can be found inOffner & Arce (2015).We apply the publicly available radiationtransfer code radmc-3d (Dullemond et al.2012) to model the CO ( J =1-0) line emissionof the simulation gas. We use CO emissionrather than CO since CO is optically thick atthe average simulation column density. To con-struct the synthetic observations, we adopt thesimulation density, temperature and velocitydistribution for the radmc-3d inputs. In theradiative transfer, H is assumed to be the onlycollisional partner with CO. In general, weassume the CO abundance is constant where[ CO/H ]=1 . × − . However, when gas con-ditions are likely to result in full dissociationof CO (
T > n (H ) <
50 cm − ),we set the abundance to zero. In addition, wealso set the CO abundance to zero in condi-tions where it would freeze out onto dust grains( n (H ) < cm − ) or where it would be disso-ciated by strong shocks ( | v | >
10 km/s).We increase the training set by also consider-ing thin clouds. Qian et al. (2015) study thethickness of molecular clouds from the core ve-locity dispersion (CVD) and find the line-of-sight thickness of Taurus, Perseus and Ophi-uchus molecular clouds are ≤ of their length.To recreate the distinct circular structures ofthe observed stellar feedback, we crop the datacube into thinner slices, including widths of 2 pcand 0.9 pc. We show the emission from COwith different cloud thicknesses in Appendix B.1and discuss why we adopt CO in favor of COin our analysis.2.3.2.
Training Target: Tracer Field for Task I
To construct the training set for task I, wefirst define the position of the bubbles usingthe tracer field that indicates the fraction of gascoming from stellar feedback at each position.A given voxel is assigned to be part of a bub-ble structure where more than 2% of the masscomes from stellar feedback. Conversely, a voxelis assigned to be pristine gas where less than2% of the mass comes from stellar feedback. Tobetter capture the morphology of the bubbles,we define the position of a bubble by the gastemperature, where T ≥
12 K. We discuss dif-ferent definitions of bubbles in Appendix B.2,including a criterion using the gas velocity. We mask all the positions of pristine gas. Weset the CO abundance to be 0 in the maskedregion and compute the radiative transfer toobtain the CO emission that is only comingfrom the stellar feedback, which we refer to asthe CO feedback map. Figure 2 shows an ex-ample of synthetic CO observations and the CO feedback map. In PPV space, the feed-back map is the CO emission that comes fromstellar feedback, which allows us to distinguishbetween the feedback and diffuse emission fromthe host molecular cloud. The wind tracer is thepositive signal that casi-3d learns to pick outfrom the messy CO emission of the molecularcloud.In PPV space, the CO feedback map emis-sion is only a fraction of the total emission ineach voxel, due to the foreground and back-ground emission along the line of sight. As aresult, we built the target in Task I by fillingthe voxels where there is feedback with the cor-responding CO emission in PPV space. Thisclosely emulates what astronomers do in obser-vational data (Arce et al. 2011; Li et al. 2015).The model in Task I adopts the raw CO datacube and returns a data cube of the same shapethat only has the CO intensity in the feedbackregions.In the final step, we compute the bubble prop-erties from the output prediction. The predic-tion is the reconstructed CO emission in vox-els associated with feedback in PPV space. Weapply a threshold based on the noise level ofthe input data to mask out noise-induced falsebubble identifications. We then sum over all thevoxels to calculate the mass, momentum and en-ergy of the identified bubbles.2.3.3.
Training Target: Mass Fraction for Task II
To attempt to improve mass, momentum andenergy estimation in the feedback identification,we adopt the fraction of feedback mass as thetarget in Task II. To build the target, we firstconvert the raw density from position-position-position (PPP) space to PPV space, just likethe CO data cube. Next, we convert the den-sity cubes that only include the feedback gas, toPPV space. We take the ratio of the two con-verted cubes to get the fraction of feedback massin PPV space. The fraction ranges from 0 to 1and the fraction value is not necessarily propor-tional to the actual CO intensity. If the COemission is optically thin, its column density isproportional to the emission intensity. Know-ing the fraction of the feedback in each positionallows us to calculate the actual feedback mass.Note that the emission predicted by Task Idoes not exclusively come from the feedbackgas. Pristine gas that is not associated withfeedback also contributes to the emission. Thus,Task I overestimates the mass coming from feed-back. Consequently, Task II is a more advancedapproach to estimate the physical properties ofthe feedback bubbles.2.3.4.
Data Augmentation
We adopt simulations with different mass lossrates, different magnetic fields, different turbu-lent patterns and different evolutionary stagesto create synthetic observations. The simula-tions have a physical scale of 5 pc on a side.To enhance the diversity of the training set, weconduct radiative transfer from three differentangular views and rotate the images every 15 ◦ from 0 ◦ to 360 ◦ . Figure 2 shows an exampleof the integrated intensity and the tracer fieldof CO with different model outputs. We alsoconstruct a “zoomed in” synthetic observationon bubbles with an image size of 2.5 pc × casi distinguish feedback bubblesfrom shell-like structures produced from super-sonic turbulence in the molecular cloud, we alsoconduct synthetic observations on purely tur-bulent simulations including noise, which donot contain feedback sources. We adopt the non-feedback cloud emission data as a nega-tive training set. This negative training set isessential because it trains the algorithm to ig-nore large (e.g., 0.5-2 pc) arc-like and shell-likefeatures that ambient turbulent motions maygenerate despite no recent stellar feedback. Inthe Taurus survey, for instance, there are areasthat are considerably larger than our adoptedpostage-stamp view-port size that lack any feed-back sources while curving, filamentary struc-tures are visually apparent. The model mustperform well in those areas or else may pro-vide false detections. Table 1 lists the propertiesof the training sets adopted by seven differentmodels.To make the synthetic cubes closer to the realobservational data in Section 2.4, we convolvethem with a telescope beam of 50” and addnoise. We assume the synthetic images are at adistance of either 140 pc or 250 pc and are ob-served by Five College Radio Astronomy Obser-vatory (FCRAO) (Ridge et al. 2006). Figure 3shows a bubble before and after we convolve theimage with the beam and add noise. The noiselevel, 0.125 K, is the same as the RMS noisein the Taurus CO observational data. More-over, we randomly shift the central velocity ofthe cubes between -1 to 1 km s − to increasethe diversity of the training set.In total, we generate 7821 synthetic datacubes: 3910 have a field of view (FoV) of5 pc × × Taurus Data
The Taurus CO J = 1 − C O I n t e g r a t e d I n t e n s i t y Different Turbulence Late Evolutionary Stage Strong Wind Weak Magnetic Field C O t r a c e r f i e l d K km/s
K km/s
K km/s
K km/s
Figure 2.
Upper row: CO integrated intensity including emission from the full cloud. Bottom row:integrated intensity of the full CO (upper row) masked by a CO synthetic observation of the tracer fieldto obtain the pixel locations of the feedback in PPV space (see Section 2.3.2). First column: syntheticobservations corresponding to model W2 T2 t0 in Offner & Arce (2015). Second Column: synthetic obser-vations corresponding to model W2 T2 t1. Third column: synthetic observations corresponding to modelW1 T2 t0. Fourth column: synthetic observations corresponding to model W2 T3 t0. CO without noise1 pc CO with noise TMB 8
Figure 3.
Integrated intensity of CO ( J =1-0). Left: CO integrated intensity of an simulated bubblewithout noise. Middle: CO integrated intensity of a simulated bubble convolved with a beam of size 50”and with 0.125 K noise. Right: CO integrated intensity of an observed bubble, TMB 8, identified by Li etal. (2015).
The map covers an area of ∼
98 deg with abeam size of 50 (cid:48)(cid:48) . The data have a mean RMSantenna temperature of 0.125 K. We combinethis with the Class III YSO catalog of Kraus etal. (2017) as a reference to determine the po-tential driving sources of the bubbles. Kraus et al. (2017) reexamined 396 candidate membersfrom previous surveys in the literature covering3 h m < α < h m and 14 ◦ < δ < ◦ . Theyconcluded 218 YSOs are confirmed or likelyTaurus members, but 160 candidates are con-0firmed or likely interlopers, and the remaining18 objects are uncertain.Li et al. (2015) visually identified 37 bubblesin the Taurus molecular cloud from the COemission, and we adopt these as an observa-tional test sample for our models. We dividethese 37 bubbles into three categories based ontheir morphology and likely driving source. Thethree ranked categories of bubbles are:A: An A bubble contains at least one YSO in-side the bubble and has clear circle/arc mor-phology.B: An B bubble contains no YSOs inside butcontains at least one YSO on the bubble rimor near the bubble boundary.C: An C bubble contains no known YSOsin/around the bubbles.Among the 37 bubbles, 7 are Rank A, 16 areRank B and 14 are Rank C.To make the observational data suitable forthe algorithm, we first down-sample the Taurus CO data cube by a factor of 3, so that it has asimilar resolution ( ∼ (cid:48) ) to that of the trainingset. We shift the down-sampled cube’s meanvelocity to 0, and then crop the velocity rangefrom -4 km s − to 4 km s − . We crop the cube to2.7 pc × × ×
32. See Appendix D for more detail.However, investigating only the 37 previouslyidentified bubbles is limiting. casi-3d doesnot require the bubbles to be centered in theimage, and the algorithm is able to rapidlysearch the entire Taurus map if it is dividedinto smaller data cubes. Furthermore, compar-ing the casi-3d identification of the previouslydefined, cropped bubbles to those identified inthe full map in an unbiased search allows us toverify that the algorithm is translation invariantand insensitive to the position of the bubble. casi-3d requires input data that has the samedimensions as the training data. When applyingthe CNN model to the full map, we decomposethe Taurus data into a series of 64 × × ∼ (cid:48) )resulting in 92% overlap with adjacent steps.We begin cropping from the northeast corner ofthe full map. Then we move to the next positionalong the RA direction with a step size of 5pixels. When we finish the sampling procedurealong the RA direction at fixed declination, wemove 5 pixels in declination and then repeat theprocess again.2.5. Model Selection
Validation
After training, we find all models in Task Iconverge to a MSE below 0.1, and a combina-tion of MSE and IoU in Task II (model MF)converges to unity. Figure 4 shows the trainingand validation errors of model ME1. After 277epochs, this model converges to a MSE of 0.06.The number of epochs used in the training, 277,is set by the maximum job run-time permittedon our computing resources. We show the per-formance of seven CNN models on a test set ofsynthetic observations in Figure 5. All modelsin Task I and II capture the shell features pro-duced by stellar feedback clearly.2.5.2.
Mean Opinion Score: Visually Assessingthe Model Performance
To visually assess the performance of the CNNmodels on observational data, we apply eightCNN models as listed in Table 1 to the Taurus CO bubble data. All test bubbles have a clearcircular or arc-like structure across a range ofvelocity channels, which provides an appropri-ate test sample. However, we do not quantita-tively know the true bubble boundaries, so weuse visual identification and assessment to eval-uate the performance of the models. We intro-duce the mean opinion score (MOS) to visuallyquantify the performance of the Task I models1 M S E L o ss TrainingValidation
Figure 4.
Training and validation errors of modelME1 during training. on the Rank A and B bubbles in Taurus. TheMOS is expressed as an integer ranging from1 to 5, where 1 is poor performance, and 5 isexcellent performance. We create a rubric out-lining the characteristics of each score to ensurevisual ranking between assessors is as uniformas possible. The rubric is as follows:5: excellent performance. The prediction cov-ers the full rim structure of the bubble, withless than 10% extra emission, i.e., a minimalamount of false positive pixels.4: good performance. The prediction covers75% of the rim structure of the bubble, withless than 25% extra emission.3: average performance. The prediction covershalf the rim structure of the bubble, with lessthan 50% extra prediction.2: fair performance. The prediction covers onethird of the rim structure of the bubble, withless than 70% extra prediction.1: poor performance. The prediction coversless than one third of the rim structure of thebubble, with more than 70% extra prediction.Figure 6 and 7 show the integrated intensityof example Rank A and B bubbles and comparethe predictions from the CNN models. We conduct a blind rating of the performanceof the seven models in Task I, where each co-author assessed the quality of each bubble pre-diction. To judge the prediction, the co-authorslooked at the integrated intensity map and thechannel by channel prediction of each model foreach bubble. Figure 8 shows the MOS of eachmodel as rated by the four co-authors. Five ofthe seven models have an overall MOS that isabove average. We easily rule out models ME6and ME7, which have lower MOSs. ME1 ex-hibits decent performance on the Taurus bub-bles, and it is trained using the most completetraining set that includes both negative exam-ples (no bubbles) and higher resolution bubbles.We adopt ME1 as the fiducial model in the fol-lowing analysis.For Task II, we maintain the same trainingset and hyper-parameters as those adopted forME1 and simply replace the training target datawith the fraction of mass coming from feedbackin the training set as discussed in Section 2.1. RESULTS3.1.
Assessing Model Accuracy UsingSynthetic Observations
In this section we use the synthetic images toassess how accurately physical properties can bedetermined from the identified bubbles. We ap-ply all the models to the synthetic observationsof the bubbles in the test set as shown in Fig-ure 5. We mask the prediction cubes at 0.2 K,which is consistent with the input cubes’ noiselevel. We then calculate the mass of the bubblesby assuming that the CO emission line is op-tically thin and has an excitation temperatureof 25 K (Narayanan et al. 2012; Li et al. 2015).We examine the uncertainty of the bubble massestimation in terms of the choice of excitationtemperatures in Appendix C. We take 1 . × − as the abundance ratio between CO and H (Narayanan et al. 2012; Li et al. 2015). Finally,2 CO1 pc CO tracer ME1 ME2 ME3ME4 ME5 ME6 ME7 MF
K km/s
Figure 5.
The integrated intensity of the CO (top left), the integrated intensity of the full CO maskedby a CO synthetic observation of the tracer field to obtain the pixel locations of the feedback in PPV space(second from top left), and the prediction of eight CNN models for the bubble emission. COTMB12 1 pc ME1 ME2 ME3 ME4ME5 ME6 ME7 MF K k m / s Figure 6.
The results of the eight models applied to Rank A bubble TMB12. First row, first column panel:integrated intensity of CO overlaid with a yellow ring indicating the position and thickness of the bubble.Star symbols show the location of the Class III YSOs from Kraus et al. (2017). The remaining panels: thepredicted intensity integrated along the velocity axis for the eight CNN models. we compute the total mass by summing over thebubble volume.Figure 9 shows the mass estimated from thetwo models, ME1 and MF. We also plot thetrue feedback mass, which we estimate directlyby adding the mass contained in all cells with T ≥
12 K and a tracer fraction ≥
2% (see § COTMB37 1 pc ME1 ME2 ME3 ME4ME5 ME6 ME7 MF K k m / s Figure 7.
The same as Figure 6 but depicting the Rank B bubble TMB37.
ME1 ME2 ME3 ME4 ME5 ME6 ME7Model Name2.42.62.83.03.23.4 M e a n O p i n i o n S c o r e Figure 8.
The overall mean opinion score (MOSfor each model, averaged over all 23 Rank A andRank B bubbles and the visual rankings by the fourhuman judges. small, and they do not expand enough to breakout of the cloud such that more gas along theline of sight contributes to the CO emission.Their velocities are also small, yielding moresurrounding gas in the velocity channels wherethe feedback is. On the other hand, high-massbubbles are usually large with more gas comingfrom the driving YSOs and have larger expand-ing velocities. The gas along the line of sight ofthe high-mass bubbles occupies a smaller frac- tion (but still a large portion) in each velocitychannel compared to that of low-mass bubbles.We compare the 1D line of sight momentumbetween the model prediction and the true sim-ulation feedback in Figure 10. We define the1D momentum as the sum of the gas mass ineach channel multiplied by the channel velocity,where we have shifted the mean cloud velocityto zero. Model ME1 overestimates the 1D mo-mentum by a factor of 2.8. In contrast, modelMF is able to correctly predict the 1D momen-tum within 10% error.Under the assumption of isotropic expansion,the 3D momentum would be expected to be afactor of √ | v | > − . Mean-while, 20% of the momentum and 44% of the en-ergy are missed due to the limited CO velocityrange. In observations, the amount of missingmass, momentum and energy will depend bothon the observation spectral range and the sourcemasses, which are often not well constrained. Inthe following sections, we correct the totals forthe missing mass, momentum and energy.The different accuracies achieved by the ME1and MF models can be understood as follows.Model ME1 is trained using the tracer intensityand thus can only predict the feedback position.We find the training set for ME1, namely thefull CO emission masked by the position ofthe tracer field in PPV space, is not a good in-dicator of the fraction of feedback mass in eachvoxel, because the emission at the predicted po-sition does not exclusively come from feedbackgas. Instead, gas along the line of sight thatis not associated with feedback contributes tothe emission and can dominate the total. Thismatches the current state of the art in humanidentification, since visual identification of feed-back cannot disentangle the feedback from thenon-feedback gas within a given velocity rangeand voxel. This is why our model ME1 pre-dicts total mass, momentum and energy similarto that estimated by Li et al. (2015) as we willshow in Section 3.2. However, model MF adoptsthe fraction of mass coming from feedback asthe training target, which allows the model notonly to predict the position of feedback but alsoto predict the fraction of the mass coming fromfeedback in each voxel. This allows a signifi-cantly more accurate determination of the mass,momentum and energy.3.2.
Physical Properties of the IndividualTaurus Bubbles M )050100150200250 C NN P r e d i c t e d M a ss ( M ) ME1MF 10:13:11:1
Figure 9.
The relation between the CNN pre-dicted bubble mass and the true feedback mass fordifferent bubbles. The filled symbols indicate themass calculated from model MF. The open symbolsrepresent the mass calculated from model ME1.The black dashed line indicates where the CNN cor-rectly predicts the true mass as determined by thetracer field and gas temperature. The blue dashedline has a slope of 3 and the red dashed line has aslope of 10. The physical parameters of the simu-lations with different labels are listed in Offner &Arce (2015).
Next, we estimate the masses of the bubblesin Taurus identified by models ME1 and MFand compare them with the previous observa-tional estimates. For the purpose of compari-son with the prior visual identifications, we an-alyze postage stamps centered on each of the 37Rank A, B and C bubbles. We calculate thebubble mass and momentum for each model asdescribed in Section 3.1. The observational ap-proach to calculate the observed bubble mass,momentum and energy is as follows. Li et al.(2015) adopts an annulus as a mask for eachbubble rim region, where the inner and outerradii of the annuli are determined by visual in-spection, and then adds up the emission of COin the masked region to calculate the mass. Thebubble velocity extent along the line of sight isalso determined by eye.5 M km s )050100150200 C NN P r e d i c t e d D M o m e n t u m ( M k m s ) ME1MFME1MF
Figure 10.
The relation between the CNN pre-dicted bubble momentum and the true feedback1D momentum from different bubbles. The filledsymbols indicate the momentum calculation frommodel MF. The open symbols represent the mo-mentum calculation from model ME1. The dashedline indicates where the CNN correctly predicts thetrue momentum.
Figure 13 compares the bubble mass calcu-lated from the two CNN models and that fromthe observational approach for all the bubblesidentified by Li et al. (2015). The mass esti-mated from ME1 shows an approximately lineartrend with that from the observational approachwithin a factor of 2. Figure 13 also comparesthe mass calculated from MF with the mass es-timated from the observational approach. Wefind the observational approach overestimatesthe bubble mass by an order of magnitude.Figures 14 and 15 compare the momentumand energy calculated from the CNN modelswith those from the observational approach.The momentum and energy estimated fromME1 both show approximately linear trendscompared to those from the observational ap-proach and are within a factor of 2. However,when considering the fraction of mass comingfrom feedback, both model ME1 and the obser-vational approach overestimate the momentumand energy by an order of magnitude. M km s )050100150200 C NN P r e d i c t e d D M o m e n t u m ( M k m s ) ME1MF 1:13:1
Figure 11.
The relation between the CNN pre-dicted bubble momentum and the true feedback3D momentum from different bubbles. The filledsymbols indicate the momentum calculation frommodel ME1. The open symbols represent the mo-mentum calculation from model ME1. The blackdashed line has a slope of 1 and the blue dashed linehas a slope of 1/ √
3, which indicates that velocitysymmetry is a reasonable assumption to estimatethe true 3D momentum. C NN P r e d i c t e d D E n e r g y ( e r g s ) Figure 12.
The relation between the CNN pre-dicted bubble energy and the true feedback energyfrom different bubbles. The filled symbols indicatethe energy calculation from MF. The open sym-bols represent the mass calculation from ME1. Theblack dashed line has a slope of 1 and the bluedashed line has a slope of 1/3. casi-3d models may alsooverestimate the total bubble properties sincethere may be more than one bubble identifiedin each postage stamp. To address this, we setthe postage stamp size to minimize this effect.Several different effects may cause errors in es-timating the mass, momentum and energy fromthe CO emission. We find that the choice ofexcitation temperature could cause a factor oftwo error in mass estimation, but it cannot ac-count for a factor of ten (see Appendix C formore detail). Likewise, the assumption of LTEhas a small affect on the mass estimation. Weconclude the line of sight gas contamination isthe main uncertainty in mass estimation. Aswe discussed in Section 3.1, low-mass bubblesare overestimated by a larger factor (a factor often) compared to high-mass bubbles (a factor ofthree) due to the line of sight gas contamination.For low-mass bubbles, the line of sight contam-ination is the dominant factor overestimatingthe mass, but for high-mass bubbles, the un-certainty that comes from line of sight contam-ination is similar to the uncertainty that comesfrom assuming a fixed excitation temperature.It is also worth considering the estimates froma physical perspective. The mass associatedwith feedback based on our understanding of the launching velocities of feedback is usuallycomparable to the young stars’ mass. It isphysically impossible for a ∼ few M (cid:12) youngstar to drive a 50-60 M (cid:12) bubble. Arce et al.(2011) pointed out that these high mass bub-bles could be produced if v wind =200 km/s with˙ m wind = 10 − − − M (cid:12) /yr. However, thesemass-loss rates are orders of magnitude higherthan those that can be explained by stellarwinds, outflows (considering outflows are colli-mated, Bally 2016), ionization or radiation pres-sure from the stars observed in Taurus (Smith2014). The mass directly launched by youngstars in both theoretical work and observationsis small, ∼ − M (cid:12) /yr (e.g., Shu et al. 1994;Hartigan et al. 1995). Numerical simulationssuggest the entrained gas can contribute threetimes more mass than the direct mass loss fromyoung stars (Offner, & Chaban 2017). In ob-servations, the mass associated with feedbackis included in the estimate of the entrained gas.Model ME1 does the same thing, i.e., it predictsthe gas associated with feedback (including theentrained gas) but cannot disentangle the line ofsight contamination. Model MF goes one stepfurther to predict the fraction of gas mass as-sociated with feedback (including the entrainedgas). Although we include the entrained gas inthe bubble mass estimation, the result is sig-nificantly less than 10-100 M (cid:12) . Reducing thebubble mass by excluding extra gas along theline of sight brings the estimates closer in linewith both empirical and theoretical models forfeedback.Table 2 lists the physical parameters of all theTaurus bubbles. It includes the estimates fromLi et al. (2015) and our models ME1 and MF.Since Li et al. (2015) do not consider the cor-rection factors for bubble mass, momentum andenergy due to the limited CO velocity range,to make a fair comparison, we do not apply cor-rection factors to the predictions from modelsME1 and MF.7
Table 2.
Physical Parameters of Taurus BubblesBubble Rank Li+ (2015) ME1 MFID Mass Momentum Energy Mass Momentum Energy Mass Momentum Energy(M (cid:12) ) (M (cid:12) km/s) (10 ergs) (M (cid:12) ) (M (cid:12) km/s) (10 ergs) (M (cid:12) ) (M (cid:12) km/s) (10 ergs)TMS 1 B 59 65 7 15 39 11 1.9 6.2 2.25TMS 2 B 31 34 4 43 32 4 9.1 8.2 1.29TMS 3 B 129 207 33 120 169 35 10.7 15.1 3.33TMS 4 B 56 62 7 42 65 14 3.6 5.4 1.14TMS 5 B 32 52 8 56 45 5 3.6 3.5 0.54TMS 6 B 33 37 4 10 9 1 2.7 3.2 0.64TMS 7 C 91 191 40 95 73 9 10.5 9.7 1.50TMS 8 B 46 97 20 33 48 12 2.8 4.5 1.12TMS 9 C 22 17 1 39 34 4 5.3 4.3 0.58TMS 10 C 217 282 37 228 272 57 24.8 35.8 8.26TMS 11 A 78 149 28 94 70 9 7.1 5.9 0.81TMS 12 A 45 108 26 57 43 5 3.7 3.0 0.39TMS 13 B 84 176 37 112 88 11 8.1 8.6 1.46TMS 14 C 18 33 6 13 44 17 2.8 9.0 3.71TMS 15 A 98 107 12 109 90 12 6.0 7.6 1.68TMS 16 A 57 92 15 75 60 8 1.5 2.4 0.64TMS 17 C 10 8 1 15 19 3 1.4 2.4 0.67TMS 18 C 54 103 19 72 63 9 2.5 3.0 0.57TMS 19 A 41 78 15 66 77 13 3.5 4.0 0.68TMS 20 B 12 13 1 21 28 5 1.8 2.3 0.42TMS 21 B 11 24 5 3 15 9 1.1 4.6 2.38TMS 22 B 119 131 14 134 168 33 12.3 18.4 3.92TMS 23 C 10 11 1 24 21 3 0.1 0.1 0.02TMS 24 C 20 22 2 29 52 10 1.8 2.9 0.56TMS 25 C 9 26 8 7 11 2 0.2 0.6 0.22TMS 26 B 11 23 5 8 29 12 3.0 10.6 4.71TMS 27 C 46 111 26 24 82 33 9.4 32.3 13.87TMS 28 C 205 656 209 293 365 68 14.2 18.4 3.50TMS 29 A 91 119 15 120 145 27 7.0 7.3 1.19TMS 30 C 420 672 107 576 636 106 23.2 26.3 4.65TMS 31 C 62 81 10 43 89 26 6.6 23.9 10.33TMS 32 C 12 22 4 7 29 14 2.6 12.6 6.74TMS 33 B 92 74 6 143 153 25 5.6 5.9 0.96TMS 34 C 67 182 49 36 39 7 4.7 5.3 1.51TMS 35 B 30 39 5 22 27 4 4.4 3.8 0.64TMS 36 A 29 47 7 16 25 6 3.7 4.2 1.00TMS 37 B 25 53 11 19 40 12 7.0 17.6 6.96 Mass obs (M )10 M a ss C NN ( M ) rank A (ME1)rank B (ME1)rank C (ME1)rank A (MF)rank B (MF)rank C (MF) Figure 13.
Bubble mass estimated from the casi-3D model predictions and the observational massestimate from Li et al. (2015). The grey dashedline indicates the trend for equal mass, while thepurple dashed line is scaled down by 10. The bluesymbols indicate the mass calculated from modelME1. The red symbols represent the mass calcu-lated from model MF. Momentum obs (M km s )10 M o m e n t u m C NN ( M k m s ) rank A (ME1)rank B (ME1)rank C (ME1)rank A (MF)rank B (MF)rank C (MF) Figure 14.
Bubble momentum estimated from the casi-3D model predictions and the observationalmomentum estimate from Li et al. (2015). The greydashed line indicates the trend for equal momen-tum, while the purple dashed line is scaled downby 10. The blue symbols indicate the momentumcalculated from model ME1. The red symbols rep-resent the momentum calculated from model MF.
Assessing the Global Impact of Feedback:Full Taurus Map Energy obs (ergs)10 E n e r g y C NN ( e r g s ) rank A (ME1)rank B (ME1)rank C (ME1)rank A (MF)rank B (MF)rank C (MF) Figure 15.
Bubble energy estimated from the casi-3D model predictions and the observationalenergy estimate from Li et al. (2015). The greydashed line indicates the trend for equal energy,while the purple dashed line is scaled down by 10.The blue symbols indicate the energy calculatedfrom model ME1. The red symbols represent theenergy calculated from model MF.
Feedback Features Identified in the FullMap
We apply the casi-3d models to the completeTaurus map to predict all the emission associ-ated with feedback. We divide the Taurus mapinto smaller cubes as discussed in Section 2.4.To create the full prediction map, we adopt thelargest value from the overlapping predictionsat each pixel. Note that the 5x5 pixel regionsin the map corners have only one cube predic-tion for each pixel.To check the accuracy of this method, wecompare the model predictions of the postagestamps and those from the large map. Figure 16shows that the large map prediction capturesthe bubble rims better than the single postagestamp predictions.Figure 17 and 18 show the predictions frommodels ME1 and MF for the whole Taurus map.The casi-3d model predictions cover almost allthe previously identified bubble regions and pre-dict additional feedback regions in the Taurusmap. The new predictions are correlated with9the locations of Class III YSOs as shown on themap. Figure 17 shows that most predictions areclose to several groups of YSOs. For example,new bubble N3, which was not previously iden-tified, seems to enclose a large group of YSOs.This suggests that the YSOs are shaping thesurrounding clouds through their feedback andcreating a wind signature in the CO spectra.We discuss the newly detected feedback regionsfurther below.We identify three types of bubbles in ourmodel predictions: high-confidence bubblesthat were identified by the previous observa-tional survey (red boxes), high-confidence bub-bles that we believe are real bubbles that weremissed in the previous survey (yellow boxes),and low-confidence bubbles that are new bub-bles found by our models but we believe areless certain (white boxes). The first categoryof high-confidence bubbles correspond to “truepositives.” The second category of missing high-confidence bubbles corresponds to “true nega-tives”, and the final category of low-confidencebubbles may represent “false positives.”First, we discuss the high-confidence bubbles(true positives) that are consistent with theprevious human identifications. These bubbleshave a clear ring or arc-like structure and haveat least one YSO inside. Bubbles H1, H2,H3 and H4 correspond to TMB 37, TMB 29,TMB12 and TMB7 in Li et al. (2015), respec-tively. These bubbles are identified by bothmodel ME1 and MF, although the extent of theemission in model MF may be smaller if thefraction of the mass coming from feedback ispredicted to be low.Next, we discuss the high-confidence bubblesthat were not included in Li et al. (2015). Thesebubbles have a clear bubble rim morphologyand have YSOs nearby if not directly within thebubble center. For example, in the yellow boxN1 we see the bubble rim and the cavity. More-over, one Class III YSO is centered in the cavity, which is likely to be the driving source of thebubble. The predictions from both ME1 andMF for N1 highlight the bubble rim. In anotherexample, N3, we can easily identify the bubblerim in Figure 17. Supporting the bubble’s exis-tence is a group of Class III YSOs inside its rim.However, when we look at the prediction frommodel MF for N3 in Figure 18, we cannot seethe bubble rim prediction. This suggests thatthere is likely a small amount of mass comingfrom the feedback.Finally, we discuss the low-confidence bub-bles. These bubbles tend not to be associatedwith any YSOs. In addition to the Class IIIYSOs identified in Kraus et al. (2017), we checkall types of YSOs in Taurus that were identifiedby Rebull et al. (2010). These are plotted inFigures 31 and 32 in Appendix E. We highlightfour such bubbles in white boxes in Figure 17.We note a number of the bubbles identified byLi et al. (2015) do not contain any Class IIIYSOs, such as L4. In these cases, the drivingsource may have moved out of the bubble, or theYSO census may be incomplete. Another pos-sible explanation is that although the morphol-ogy is circular or arc-like, they are caused bycloud turbulence, which causes coherent motionacross several velocity channels. It is difficultto distinguish the bubble structure from turbu-lent patterns when a circular or arc-like patternshows across multiple channels. However, webelieve this last explanation is unlikely, since weinclude pure turbulence snapshots in the train-ing set as negative training images. Thus, casi-3d should not be prone to misidentify turbulentpatterns as bubbles.Overall, we conclude that the two CNNmodels perform as well or better than “by-eye ” visual identifications of bubbles. Theyappear to reasonably predict both the bubbleposition and the fraction of mass coming fromfeedback.0 K km/s
Figure 16.
Comparison of the prediction of a postage stamp and that from the large map on bubbleTMB29. Left: integrated CO intensity. The red stars indicate Class III YSOs from the Kraus et al. (2017)catalog. Middle: integrated prediction from model ME1 run on the postage stamp shown in the left panel.Right: integrated ME1 prediction from the full map prediction, reconstructed from overlapping postagestamps.
Mass, Momentum and Energy of theFeedback Identified in the Full TaurusCloud
We now calculate the feedback mass, momen-tum and energy in Taurus based on the predic-tions from models ME1 and MF. Table 3 liststhe feedback properties calculated in this workand those calculated in Li et al. (2015).Model ME1 predicts 2630 M (cid:12) of gas associ-ated with feedback, which is consistent within afactor of two with the feedback mass calculatedin (Li et al. 2015). However, model MF predictsthat only 275 M (cid:12) of gas is associated with feed-back, which is an order of magnitude smallerthan the previous calculations. The smalleramount of feedback mass is also consistent withthe total stellar mass in Taurus, which is esti-mated to be on the order of 200 M (cid:12) in Krauset al. (2017). The feedback mass predicted bymodel ME1 and that calculated from Li et al.(2015) are 10 times the stellar mass, which isinconsistent with the expect amount of gas en-trained by feedback (Offner, & Chaban 2017).Despite the detailed machine learning iden-tification, we must still confront the challengeof how to disentangle feedback from the bulk cloud motion. For example, Taurus has a veloc-ity gradient that stretches from the south-eastto north-west. Not accounting for this gradi-ent may artificially enhance the feedback total.The most accurate way to account for the bulkmotion is not clear; thus, we present two ap-proaches to calculate the feedback momentumand energy. The first way treats the molecularcloud as a whole, with the same fixed centralvelocity. We shift the central velocity to zeroand calculate the 1D momentum and 1D energychannel by channel as described in Section 3.1.The second approach is similar but treats themolecular cloud locally, which means the clouddoes not have a fixed central velocity but has acentral velocity gradient across the entire cloud.We subtract the central velocity pixel by pixeland then calculate the momentum and energychannel by channel. To convert the 1D line ofsight estimates to 3D, we make the assumptionof isotropic expansion to calculate the 3D mo-mentum and 3D energy. Finally, in Section 3.1,we assessed the model accuracy using syntheticobservations and found that 20% of the momen-tum and 44% of the energy are missed due to thelimited CO velocity coverage. Considering thelimited velocity range of the CO data cube,1 h m m m m D e c H1 H2 H3 H4 H5N1 N2 N3N4 N5 N6N7L1 L2L3 L4 L5L6
Figure 17.
The CO integrated intensity of Taurus molecular cloud overlaid with the integrated predictionof feedback position from ME1 along velocity channels in red color. The arcs in yellow indicate the positionof previously identified bubbles in Li et al. (2015). The star symbol demonstrates the location of the ClassIII YSOs from Kraus et al. (2017). we apply these correction factors for the miss-ing momentum and energy here.The momentum estimate without the velocitygradient treatment from model ME1 is close tothat from Li et al. (2015). The momentum esti-mate with the velocity gradient treatment frommodel ME1 is 38% smaller than the calculationin Li et al. (2015). Once corrected for the ex-tra CO emission from the foreground or back-ground, the momentum (with and without thevelocity gradient treatment) predicted by modelMF is an order of magnitude smaller than thecalculation in Li et al. (2015). Both energy estimates (with and without thevelocity gradient treatment) from model ME1are within a factor of two compared to the en-ergy calculated in Li et al. (2015). In contrast,model MF implicitly corrects for the extra COemission coming from the foreground or back-ground, such that the predicted energy is an or-der of magnitude smaller than that calculatedin Li et al. (2015). We discuss the implicationsin the following section.3.3.3.
Assessing the Relative Energies ofTurbulence and Feedback
In this section we compare the total energy as-sociated with feedback and the total cloud tur-2 T a b l e . P r o p e r t i e s o f F ee db a c k i n t h e T a u r u s M o l ec u l a r C l o ud ∗ M o d e l W i t h o u t s ub t r a c t i n g t h e v e l o c i t y g r a d i e n t Sub t r a c t i n g t h e v e l o c i t y g r a d i e n t M a P D E D b ˙ E c M a P D E D b ˙ E c ( M (cid:12) )( M (cid:12) k m / s )( × e r g s )( × e r g s / s )( M (cid:12) )( M (cid:12) k m / s )( × e r g s )( × e r g s / s ) L i + ( ) ( . % ) . ( . % ) . ( . % ) ---- M E ( . % ) ( . % ) ( % ) ( . % ) . ( . % ) . ( . % ) M F ( . % ) . ( . % ) . ( . % ) ( . % ) . ( . % ) . ( . % ) ∗ : M o d e l n a m e , f ee db a c k bubb l e m a ss , D f ee db a c k m o m e n t u m , D f ee db a c k e n e r g y , e n e r g y i n j ec t i o n r a t e f r o m f ee db a c k bubb l e s . T h e nu m b e r s i n t h e t a b l ec o n s i d e r t h ec o rr ec t i o n f a c t o r s du e t o t h e li m i t e d v e l o c i t y r a n g e o f t h e C O d a t a c ub e . a : T h e nu m b e r i n t h e p a r e n t h e s e s i nd i c a t e s t h e p e r ce n t ag e o ff ee db a c k m a ss c o m p a r e d t o t h e w h o l e m o l ec u l a r c l o ud m a ss ( P i n e d a e t a l. ) . b : T h e nu m b e r i n t h e p a r e n t h e s e s i nd i c a t e s t h e p e r ce n t ag e o ff ee db a c k e n e r g y c o m p a r e d t o t h e w h o l e m o l ec u l a r c l o ud t u r bu l e n t e n e r g y ( L i e t a l. ) . c : T h e nu m b e r i n t h e p a r e n t h e s e s i nd i c a t e s t h e p e r ce n t ag e o f e n e r g y i n j ec t i o n r a t e f r o m f ee db a c k bubb l e s c o m p a r e d t o t h e t u r bu l e n t d i ss i p a t i o n r a t e o f t h ec l o ud . T h e t u r bu l e n t d i ss i p a t i o n r a t e a d o p t e dh e r e i s L t u r b = . × e r g s − , w h i c h a ss u m e s a m e a n c l o udd e n s i t y o f n = c m − . T h i s t u r bu l e n t d i ss i p a t i o n r a t e i s a b o u tt w o t i m e s h i g h e r t h a n t h a t f r o m L i e t a l. ( ) , w h i c h a ss u m e s a l o w e r m e a n c l o udd e n s i t y o f n = c m − . h m m m m D e c H1 H2 H3 H4 H5N1 N2 N3N4 N5 N6N7L1 L2L3 L4 L5L6
Figure 18.
The same as Figure 17 but predicted by MF. L turb = E turb t diss , (4)where t diss is the turbulent dissipation time.The method to estimate the turbulent dissipa-tion time in Li et al. (2015) is from Mac Low(1999), t diss ∼ ( 0 . κ M rms ) t ff , (5)where t ff is the free-fall timescale, M rms is theMach number of the turbulence, and κ is theratio of the driving length to the Jean’s lengthof the cloud. For M rms = 5 and a free-falltime t ff = 7 × yr, which assumes a meancloud number density of n = 20 cm − , the tur-bulent dissipation rate is 3 . × erg s − .However, Taurus is not a uniform sphere, themean number density of n = 20 cm − adoptedby Li et al. (2015) is too low. The typical meannumber density of a molecular cloud is around n = 100 cm − , which gives t ff = 3 . × yr and L turb = 6 . × erg s − . Arce et al. (2010) andNarayanan et al. (2012) also adopt this methodto calculate the turbulent dissipation rates inPerseus and Taurus, respectively.One caveat here is that the equation to cal-culate the turbulent dissipation rate is obtainedfrom simulations, which depend on the initialconditions and the way turbulence is driven.The energy injection rate is defined as L bubble = E bubble /t kinetic , where E bubble is the kinetic en-ergy of the bubble and t kinetic is the kinetictimescale of the bubble. The kinetic timescale ofthe bubble can be calculated as t kinetic = R/V exp ,where R is the radius of the bubble and V exp is the expansion velocity of the bubble. Wefind the energy injection rate from bubbles in ME1 is L turb , ME1 = 1 . × erg s − , whichis slightly larger that the turbulent dissipationrate of the cloud. If we subtract the velocitygradient, the energy injection rate from bub-bles is L turb , ME1 , G = 2 . × erg s − , which isabout half of the turbulent dissipation rate ofthe cloud. In summary, like Li et al. (2015), weconclude that feedback is sufficient to maintainthe current level of cloud turbulence.However, we have shown that model ME1overestimates the energy because excess fore-ground and background material is included inthe calculation. Consequently, we find that af-ter recalculating the feedback energy using themore accurate model MF prediction, the kineticenergy from the feedback decreases by an orderof magnitude, which means the energy injec-tion rate from stars is smaller by an order ofmagnitude: L turb , MF = 2 . × erg s − . Un-der this circumstance, the energy injection ratefrom feedback is 29% of the turbulent dissipa-tion rate of the cloud. If we subtract the veloc-ity gradient, the energy injection rate from feed-back is L turb , MF , G = 6 . × erg s − , which isan order of magnitude smaller than the turbu-lent dissipation rate. This indicates that someadditional energy is needed to drive turbulencein the Taurus molecular cloud, which could beprovided by outflows for example. Feedbackfrom bubbles may not be sufficient to maintainthe cloud turbulence over long timescales.The Taurus molecular cloud is host to an olderpopulation of stars ( τ ∼ −
20 Myr), which in-dicates the lifetime of the cloud is at least 10-20million years (Kraus et al. 2017). However, thislife time is much longer than the gravitationalcollapse free-fall time of the Taurus molecularcloud estimated from CO, which is 3.3 millionyears. This suggests that there must be energyinjected to support the cloud against gravita-tional collapse, which suggest feedback is play-ing some role in driving turbulence but is notdominant.53.3.4.
Quantifying the Impact of Feedback withTurbulent Statistics
With an accurate prediction of the positionof feedback in hand, we compute multiple as-trostatistics to study the different propertiesbetween regions with and without feedbackin Taurus. We adopt the statistical analysispackage, turbustat , to conduct the statisticalanalysis (Koch et al. 2017, 2019). turbustat contains 15 different statistics, but here we con-sider only the spatial power spectrum (SPS) andthe covariance matrix used to compute princi-ple component analysis (PCA). We adopt thesestatistics since they have previously been shownto be sensitive to feedback as discussed in theintroduction. The SPS is defined as the squareof the 2D Fourier transform of an image: P ( k ) = (cid:88) | (cid:126)k | = k |M ( (cid:126)k ) | = | (cid:90) ∞−∞ (cid:90) ∞−∞ M ( (cid:126)x ) e − πj(cid:126)k(cid:126)x d(cid:126)x | . (6)It is applied to the integrated intensity map.The covariance matrix is defined as: C jk = 1 n n (cid:88) i =1 X ij X ik , (7)where X ij = T ( r i , v j ) − [ n (cid:88) k =1 T ( r k , v j )] /n, (8)in which T ( r i , v j ) is the spectral cube, where r i = ( x i , y i ) is the position on the sky and v j indicates the spectral velocity channel. It pro-vides information about velocity correlations.In addition to these two statistics, we also con-sider the distribution of linewidths of the feed-back and non-feedback gas as well as the dis-tance between YSOs and pixels associated withfeedback.Figure 19 shows the SPS of the full CO in-tegrated intensity map of Taurus, the SPS of the region where the emission is above 0.2 K(i.e., excluding noise) and the SPS of the modelME1 and MF predicted feedback regions. Fig-ure 19 shows that the slope of the SPS is flat-tened over the feedback injection region. If theemission is optically thin and the temperatureis roughly constant, this indicates mass or en-ergy has been injected into smaller scales by thefeedback. Here, the CO is mostly opticallythin with the exception of dense cores.Next, in Figure 20 we present the covariancematrices of the velocity channels for the full CO integrated intensity map of Taurus, thehigh signal-to-noise region and the predictionof the models ME1 and MF. For comparison,Figure 20 also shows the covariance matricescalculated using the synthetic data. The co-variance matrices of the predicted feedback re-gions clearly show off-diagonal velocity features,which indicate coherent motions at these veloc-ities. These features can be characteristic ofthe expansion of bubbles or high-velocity gas(Boyden et al. 2016), but it may also representcoherent cloud motions (e.g., Feddersen et al.2019). In either case, the clear differences be-tween the identified feedback and non-feedbackgas underscore that casi-3d is indeed identify-ing statistically distinct regions.Next, we assess the relative distance to theYSO locations, which provide additional evi-dence that our regions are associated with feed-back. Figure 21 shows the distribution of theprojected distances between the YSOs and theemitting gas, and the distribution of the pro-jected distances between YSOs and the feed-back gas predicted by ME1 and MF. The me-dian value of the projected distance between theYSOs and the feedback gas is closer than thatbetween the YSOs and all the emitting gas. Thetypical distance between the YSOs and the feed-back gas is 0.7 pc, which is also the typical sizeof the bubbles.6Finally, we expect feedback regions to havelarger velocity dispersions. Figure 22 shows thedistribution of the full width at half maximum(FWHM) of the high signal-to-noise emission re-gion and the FWHM of the ME1 and MF pre-dicted feedback regions. The median values ofthe FWHM of the feedback regions are indeedlarger than that of the FWHM of the full map.The higher FWHM indicates larger velocities inthe spectrum associated with feedback. CONCLUSIONSWe adopt a deep learning method, casi , andextend it to 3D ( casi-3d ) to identify stellarfeedback features in 3D CO spectral cubes. Bycreating different training sets, we develop twodeep machine learning tasks. Task I predicts theposition of feedback. Task II predicts the frac-tion of the mass coming from feedback. Ourmain findings are the following:1. casi-3d is a powerful method to iden-tify bubbles. casi-3d performs well onsynthetic test data and recovers feedbackwith an accuracy of 4% on a pixel level.2. casi-3d successfully infers/predicts hid-den information, e.g., the fraction of masscoming from feedback.3. We apply casi-3d to the CO observa-tions of the Taurus molecular cloud andshow that casi-3d successfully identifiespreviously known, visually identified bub-bles.4. We find that training Task I reproducesthe mass, momentum, and energy of indi-vidual bubbles inferred by human visualidentifications. In contrast, Task II, whichis trained on the feedback mass fraction,indicates that the true mass, momentumand energy are an order of magnitudelower.5. casi-3d suggests previous studies overes-timate feedback mass and energy in the Taurus molecular cloud. The feedbackmass is overestimated by a factor of five.The feedback energy is overestimated by afactor of five compared to that calculatedwithout subtracting the velocity gradientover the full map, and it is overestimatedby a factor of ten compared to that calcu-lated with subtracting the velocity gradi-ent over the full map.6. We carry out an analysis of the spatialpower spectrum to quantify the turbu-lence properties in the feedback and non-feedback regions. We show that feed-back flattens the slope of the spatial powerspectrum of the full CO integrated in-tensity map of Taurus, indicating thatmass and/or energy has been injected atsmaller scales by feedback.7. We calculate the covariance matrix andshow that the presence of feedback ap-pears as off-diagonal peaks in the covari-ance matrices.8. The median value of the projected dis-tance between YSOs and the feedback gas(0.64 pc predicted by model ME1 and 0.75pc predicted by model MF) is closer thanthat between YSOs and all the emittinggas (0.84pc). The median value of thefull-width at half maximum (FWHM) ofthe feedback regions (1.2 km s − predictedby model ME1 and 1.1 km s − predictedby model MF) is larger than that of theFWHM of the full emitting regions (0.9km s − ).In future work, we plan to apply casi-3d toother star-forming regions and other types offeedback, such as protostellar outflows (Arce etal. 2010).D.X., S.S.R.O., R.A.G. and C.V.O. were sup-ported by NSF grant AST-1812747. S.S.R.O.also acknowledges support from NSF Career7 l o g P ( K ) R e s i d u a l s Full Map -2.95±0.02 l o g P ( K ) R e s i d u a l s Map Excluding Noise -3.06±0.03 l o g P ( K ) R e s i d u a l s ME1 -2.78±0.04 l o g P ( K ) R e s i d u a l s MF -2.53±0.03 Figure 19.
The spatial power spectrum (SPS) of the full CO integrated intensity map of Taurus and theSPS of the emission regions (excluding noise regions) where the emission is above 0.2 K and the SPS of theME1 and MF predicted feedback regions grant AST-1650486. The authors acknowl-edge the Texas Advanced Computing Center(TACC) at The University of Texas at Austin for providing HPC resources that have con-tributed to the research results reported withinthis paper.APPENDIX A. CASI-3D PARAMETERSA.1.
Down-sampling Methods
We test two widely used down-sampling methods to reduce the size of the data: max poolingand average pooling. Max pooling picks out the largest value to replace its adjacent pixels. Maxpooling can extract the most important features, but it is not proficient in dealing with different noisebackgrounds. Since all large-map sky surveys are conducted through substantial observing periodswith different weather conditions and with different baselines, the noise level is different in differentpatches of the large map. When applying max pooling to down sample the data, the boundarybetween patches distinctly appears, which makes the data inconsistent across the map. On the otherhand, average pooling extracts features smoothly and it preserves the overall value during down8
Full Map Map Excluding Noise
Velocity (km/s)
ME1 MF
Full Map Map Excluding Noise
Velocity (km/s)
ME1 MF
Figure 20.
The covariance matrices of the velocity channels on the full CO integrated intensity map ofTaurus and the covariance matrices of the emission regions and the covariance matrices of the ME1 and MFpredicted feedback regions. Left panel: the covariance matrices of the velocity channels on synthetic data.Right panel: the covariance matrices of the velocity channels on Taurus CO data. N u m b e r Map Excluding Noise, d med =0.84pcMF, d med =0.75pcME1, d med =0.64pc Figure 21.
The distribution of the projected dis-tance between YSOs and the emitting gas, andthe distribution of the projected distance betweenYSOs and the feedback gas predicted by ME1 andMF. sampling., Figure 23 shows an example of the two different down-sampling methods tested on COTaurus molecular cloud data. A.2.
Loss Function
We test three types of loss functions – mean squared error (MSE), intersection over union (IoU) anda combination of MSE and IoU – to predict the fraction of the mass that comes from stellar feedback.9 N u m b e r Map Excluding Noise, med =0.9km/sMF, med =1.1km/sME1, med =1.2km/s
Figure 22.
The distribution of the FWHM of theemission regions and the FWHM of the ME1 andMF predicted feedback regions.
MaxPooling
AveragePooling
K km/s
K km/s
Figure 23.
Comparison of two different down-sampling methods tested on CO Taurus molecular clouddata.
Figure 24 shows the performance of the model using different loss functions on a test bubble. Themodel adopting IoU as the loss function can capture the morphology of the bubble clearly but missesthe value information. The IoU model predicts almost unity at the feedback position but does notreflect the actual fraction value. The MSE model is able to capture the position of larger feedbackvalues but underestimate the smaller values which is useful to predict the emission but not thefraction. The model adopting both MSE and IoU as the loss function performs the best. This modelnot only captures the distinct bubble morphology but also returns reasonable fraction values.0 CO Slice1 pc True Feedback IoU MSE IoU+MSE f (%)
T (K)
Figure 24.
The performance of model MF adopting different loss functions to predict the fraction of themass that comes from stellar feedback on a test bubble. Left: integrated CO intensity. Second from left:integrated true feedback fraction. Right panels: models using the IoU, MSE, and IoU+MSE loss functions,respectively. B.
TRAINING SETSB.1.
Comparison of CO Emission with Different Cloud Thicknesses
Figures 25 shows the difference in the synthetic observations between the whole cube and thecropped data for CO. The CO bubble rim is embedded in the diffuse gas emission and the bubblecavity is not clear in the integrated intensity map when the thickness of the cloud is 5 pc. Since COis even more optically thick, CO is not an appropriate proxy to trace stellar feedback winds (e.g.,Arce et al. 2011; Li et al. 2015). When the thickness of the cloud becomes smaller, the bubble rimand its cavity are recognizable in the CO integrated intensity map. Although some bubble rims ortheir cavities are not distinct in the integrated intensity map of CO, these feedback features becomerecognizable in PPV space.B.2.
Different Definitions for the Bubble Extents
In this section we assess the impact of different choices for the bubble definition on the results.In addition to the bubble definition described in Section 2.3.2, we also examine the tracer field inthe simulation data. The gas adjacent to the tracer gas has a velocity vector going outwards, whichindicates the feedback gas compresses the ambient gas without direct contact. Although the fractionof feedback gas compared to the entire amount of gas contained in these voxels is almost zero, theadjacent layer contributes to the momentum and the energy of the cloud. Consequently, we define thetracer field with the velocity vector going outwards from the central stars, which increases the massof the feedback bubble by a factor of 3. We furthermore test a temperature cut at T ≥
12 K near thetracer gas to calculate the bubble mass. The simulation data cubes have an average temperature of 10K. The temperature drops quickly from the bubble rim to the ambient gas. We compare the differenttracer definitions in Figure 26. Both the velocity cut plus the tracer field and the temperature cutplus the tracer field are slightly larger in area than the original tracer field. Both of these definitionsyield bubbles that are similar in shape. Since there are five individual stars in the simulation box,the bubbles generated by theses stars are easily connected to each other during the expansion. Thisaffects the gas velocities, which makes it difficult for us to define the gas flow direction and determinewhich expansion is part of the shell. Under this circumstance, the temperature cut is a better optionto define the bubble boundary. We compare the bubble mass calculated from the velocity-basedbubble definition and the the temperature-based bubble definition in Figure 27. Larger bubbles aremore likely to overlap during the the expansion, which makes the velocity-based bubble definition1 C O I n t e g r a t e d I n t e n s i t y D=5 pc D=2 pc D=0.9 pc C O t r a c e r f i e l d K km/s
Figure 25.
Integrated intensity of CO ( J =1-0). Upper left: integrated intensity of CO generatedusing the whole data cube. Upper middle and upper right: integrated intensity of CO generated usingthe cropped data cube. Bottom left: integrated intensity of CO wind tracer generated using the wholedata cube. Bottom middle and bottom right: integrated intensity of CO wind tracer generated using thecropped data cube. ”D” is the line of sight thickness of the the data cube. mass slightly smaller than the the temperature-based bubble definition mass. Overall, we concludethe temperature-based bubble definition is the most appropriate definition of the bubble boundary.B.3.
Comparison of the Training and Observed Bubble Mass Distributions
In this section, we examine the distribution of the bubble masses in the training set. Figure 28shows the maximum bubble mass in the training set is ∼ M (cid:12) , and it spans the range of thebubble masses in the test samples in Section 3.1. Moreover, to extend the range of bubble masses,we have included “zoomed-in ” synthetic observations. In these 64 ×
64 postage-stamps, the originalbubble is enlarged by a factor of two in both length and width, which indicates the bubble areaand the mass both increase by a factor of 4. Since casi-3d takes postage-stamp cubes as inputs,regardless of the actual physical size of the cubes, this means the training set spans bubble massesup to ∼ × M (cid:12) , and it spans the range of the individual bubble masses in observations inSection 3.2. In some cases, a single 64 ×
64 postage-stamp cannot cover an entire bubble. Onlypart of the bubble appears within the input window, such as an arch or a half circle. These casesin the training set are consistent with the cases of larger bubbles that are contained in the full mapprediction in Taurus. Thus, to obtain the masses of the largest bubbles in Taurus, we combine a2 CO1 pc CO tracer T cut V cut
K km/s
Figure 26.
Different definitions for a synthetic bubble. First panel: the CO integrated intensity map.Second panel: the original definition of the bubble using the orion tracer field integrated over the velocitychannels. Third panel: the temperature-based definition (
T >
12 K plus the tracer field) integrated over thevelocity channels. Fourth panel: the velocity-based definition (gas with expanding velocities plus the tracerfield) integrated over the velocity channels. M )01020304050607080 V e l o c i t y C u t T r a c e r M a ss ( M ) W2_T3_B1W2_T3_B2W1_T1_B1W1_T1_B2W2_T2_B1W2_T1_B1W2_T1_B2W2_T3_B1W2_T3_B2W1_T1_B1W1_T1_B2W2_T2_B1W2_T1_B1W2_T1_B2
Figure 27.
Comparison between the velocity cut tracer mass and the temperature cut tracer mass. stack of postage-stamps that cover different parts of each bubble to get the full prediction and thencalculate the bubble mass as described in Section 3.2. C. EXCITATION TEMPERATURE SELECTION AND IMPACTIn this section, we explore the uncertainty in the bubble masses due to the choice ofexcitation temperature. We find that 25 K (e.g., Narayanan et al. 2012; Li et al. 2015) isthe most appropriate choice to convert CO emission to column density in the syntheticobservations. We show the ratio between the mass estimated from CO assuming a 25 Kexcitation temperature and the true mass calculated from the simulations in Figure 29.The ratio is within a factor of two of unity when assuming LTE and a 25 K excitationtemperature, which in turn demonstrates that both LTE and the choice of 25 K arereasonable. Mass ( M ) F r a c t i o n Bubble Mass Distribution in The Training Set
Figure 28.
The distribution of bubble masses in the training set. M ( T ex = 25 K )/ M ( True )0.00.20.40.60.81.01.2 F r a c t i o n Figure 29.
The ratio between the mass estimated from CO assuming a 25 K excitation temperature andthe true mass calculated from the simulations.
Under the assumption of LTE, the mass estimation goes linearly with the excitationtemperature. From previous feedback mass estimates (e.g., Arce et al. 2011; Li etal. 2015), the choice of excitation temperature ranges from 10 K to 50 K. This couldintroduce a factor of two uncertainty in the mass estimation, but it cannot account fora factor of ten. D. ASSESSING THE SENSITIVITY OF THE DATA WINDOWIn this section we check the sensitivity of the data window to feedback as a function of voxellocation. As discussed in Section 2.2.1, the casi-3d models predict the full Taurus map feedbackusing a stack of 64 × ×
32 cubes, in total 11,340 cubes. We examine the “response” of each voxelin a 64 × ×
32 cube. We define the response as the fraction of the stacked voxels over the 11,340cubes that are detections in a 64 × ×
32 cube. A detection is defined as a voxel above 90% of themaximum prediction value for all overlapping voxels at the corresponding full map location. Figure 30shows the response integrated over the velocity channels. The central region of the postage stamp4is the most sensitive region, where a higher fraction of the stacked voxels fall above the predictionthreshold. The boundary region of the postage stamp is less sensitive to features and detection is lessefficient. This figure illustrates that choosing the appropriate cube offset is important to achievingthe best sensitivity. To ensure that all data are covered by the highly sensitive part of the window,the maximum step size should not exceed 16 pixels. In our stacked prediction, we adopt a step sizeof 5 pixels, which is smaller than the required step size, to crop the full Taurus map.We note that the range of the window sensitivity likely depends on the training data and the targetfeature size. Here, we aim to find bubbles that have typical sizes greater than ∼
16 pixels or a quarterof the the cube length. We recommend that other users of casi-3d check the window sensitivity fortheir problem to determine the appropriate offset size when applying casi-3d to large data maps.
Figure 30.
The integrated response of a voxel in the cube, i.e., the fraction of voxels predicted to beassociated with feedback in the same position in the stack of cubes summed over all velocity channels.E.
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