Small Near-Earth Asteroids in the Palomar Transient Factory Survey: A Real-Time Streak-detection System

A near-Earth asteroid (NEA) is by definition any asteroid with perihelion q < 1.3 au and aphelion Q > 0.983 au. From the largest NEA (of diameter D ≈ 30 km) down to D ≈ 0.5 km in size—for which the known population is largely complete—the cumulative size-frequency distribution (Figure 1) goes roughly as N(D) ∝ D⁻², where N(0.5 km) ≈ 10⁴. Harris (2008, 2013) presents these statistics and describes how the original "Spaceguard" goal to catalog 90% of all D > 1 km NEAs was achieved by the mid-2000s, while the current congressional mandate is to find 90% of all D > 140 m NEAs by 2020. See Harris & D'Abramo (2015) for a review of estimates of the size‐frequency distribution of near earth asteroids.

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The incrementally decreasing target size in the NEA census has been mostly motivated by risk mitigation. Over the quarter century that began with our realization of an asteroid's role in the dinosaurs' extinction (e.g., Alvarez et al. 1980) through to our fulfillment of the 1-km Spaceguard goal, the estimated risk of an individual's death from asteroid impact—initially believed to be comparable with that of a commercial airplane accident—dropped by an order of magnitude. Surveying to the currently recommended D > 140 m can decrease this risk by yet another order of magnitude (Harris 2008). Hence, discovery of D < 100 m NEAs will contribute only minimally to any further significant reduction in the risk of death to any individual. However, events like the Tunguska and Chelyabinsk airbursts (Brown et al. 2013) could have caused a significant numbers of deaths if the impact parameters had been different and did cause environmental or other damage. This suggests that impacts from 10–100 m objects qualify as "natural disasters" that merit advance warning, and possibly prevention via space-based manipulation of hazardous NEAs.

However, the size-frequency distribution informing these estimates is uncertain across orders of magnitude in impactor size and constrained on the small end (D ≲ 10 m) by infrasound detections of bolide fluxes (Silber et al. 2009). Besides impact mitigation (e.g., Ahrens & Harris 1992; Lu & Love 2005), other space-based activities benefiting from small NEA discoveries include in situ compositional studies (Mueller et al. 2011) and resource utilization (Elvis 2014). NEAs have also been declared a major component of NASA's manned spaceflight program (Obama 2010). NEA rendezvous feasibility critically depends on mission duration and fuel requirements; these in turn are functions of the NEA's orbit and relative velocity (Δv) with respect to Earth (Shoemaker & Helin 1978; Elvis et al. 2011). Robotic missions that have been proposed may facilitate or complement the manned program. One example was the proposed Asteroid Retrieval Mission (ARM; Brophy et al. 2012), which evolved into the NASA Asteroid Redirect Mission. At the time of the writing of this paper, NASA has chosen to retrieve a boulder from a larger asteroid rather than retrieve an entire small asteroid. Natural temporary capture of meter-scale NEAs into Earth-centric orbits, if confirmed via the discovery of "mini-moons" (Granvik et al. 2012; Bolin et al. 2014), would present another appealing class of targets.

Cleary, the discovery of 1–100 m sized NEAs is motivated by different (and more diverse) applications than those that have driven the census of larger NEAs. The discovery method often likewise differs. Most large NEAs were found via the "tracklet" method of linking several serendipitously observed positions within a night or across several nights. This is the basis of 'MOPS'-like detection software (e.g., Denneau et al. 2013), which in its present state is most efficient at detecting NEAs moving slower than ∼5 deg/day (Jedicke et al. 2013). Below this rate, an NEA's individual detections are nearly point-like for typical survey exposure times (e.g., 30–60s) and sufficiently localized on the sky given typical intra-night pointing cadences (e.g., 15–45 minutes). Hazardous NEAs occupy a range of orbits with moderate eccentricities, so they spend most of their time far from the Earth and Sun where their sufficiently slow apparent motions allow them to be easily detected with this technique. Searches for NEAs with DECam (Allen et al. 2013 & 2014) using conventional MOPS techniques on a large telescope appear to be particularly promising for finding small NEAs. Recent searches using NEOWISE (Mainzer et al. 2014) also yielded interesting new detection of small NEAs.

In contrast, the method of streak detection enables the discovery of much smaller and closer (i.e., brighter and faster-moving) NEAs. Whereas slower-moving NEAs can be mistaken for main-belt asteroids, streaked asteroids having an angular rate of larger than 1 deg/day are likely to be NEAs. Unlike the tracklet method, discovery via streak detection is possible on the basis of a single exposure via recognition of the streak morphology, meaning repeat visits to the same patch of sky are unnecessary and more area can be searched. Lastly, NEAs that are detected as streaks are typically closer and therefore are two to three times brighter than those found by the tracklet method when they are tracked non-sidereally, making them more convenient for follow-up from dedicated (including amateur-class) facilities once an approximate orbit is determined.

Survey-scale application of the streak-detection method for NEA discovery was pioneered by Helin & Shoemaker (1979) using photographic plates on the Palomar 18-inch Schmidt telescope in the 1970s. Rabinowitz (1991) was the first to apply this method with CCD detectors in near real time with the Spacewatch survey. Combining Spacewatch's streaked NEA detections (e.g., Scotti et al. 1991) with its tracklet-detected NEAs (Jedicke 1995) produced a debiased NEA number-size distribution (Rabinowitz et al. 2000) spanning four orders of magnitude in size (10 km > D > 1 m). In 2005, Spacewatch initiated a public Fast-Moving Object (FMO) program (McMillan et al. 2005) that allowed public access to the survey's imagery on the internet so that members of the public could scan images and report detections of streaks.

Figure 2 breaks down the number of streaked NEA discoveries as a function of time and survey, from 1991 through 2014-Oct. Here, "streaked" is taken to mean any detection wherein the length of the imaged streak is greater than 10 seeing widths. The counts in Figure 2 were compiled by first retrieving all NEA discovery observations from the Minor Planet Center (MPC) database and then using JPL's HORIZONS service (Giorgini et al. 1996) to compute the on-sky motion at the discovery epoch. These rates were then converted into streak lengths in units of seeing widths, where the continental surveys all have assumed 2'' seeing and PanSTARRS has assumed 1'' seeing. The assumed exposure times come mostly from a table in Larson (2007), except for PTF and PanSTARRS, which have assumed exposure times of 60s and 45s, respectively.

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Before 2005, Spacewatch was the only contributor of significantly streaked NEA discoveries and it is also the most prolific streaked-NEA discoverer overall. There are two likely reasons for this: (1) Spacewatch's relatively long 120s exposure time and (2) the active role of a human screener ("observer") during data collection, as documented by Rabinowitz (1991). The Catalina Sky Survey also has a dedicated human operator to scan candidates and conduct same-night follow-up (Larson 2007), which explains its similarly consistent contribution of streaked discoveries. Some major NEA surveys of the past two decades not contributing to the streaked discoveries in Figure 2 include LINEAR (Stokes et al. 2000)—likely because of its short 8s exposures, as well as NEAT (Pravdo et al. 1999) and LONEOS (Stokes et al. 2002)—which to our knowledge lacked real-time human interaction with their respective data flows.

The years 2013 and 2014 marked a clear upturn in the discovery of streaked NEAs. The purpose of this paper is to document a new streak-discovery pipeline that has contributed in part to this increased discovery rate.

2.1. Technical and Operational Characteristics

2.2. Previous Solar System Science with PTF

2.3. Real-time Data Reduction at IPAC

2.4. Detections of Known Streaking NEAs

Early in the development of our streak detection system, we sought to extract all observations of known fast-moving NEAs from existing PTF data. We used the table of all predicted PTF sightings of all known asteroids compiled by Waszczak et al. (2013), updated to include data through early 2014.

There is a total of 539 predicted sightings (of 158 unique objects) for which the predicted motion was faster than 10''/minute and the predicted magnitude was brighter than V = 20. For objects having predicted positional uncertainties greater than 10'', the images were visually inspected around the predicted location for the presence of a streak. Because the V < 20 brightness criterion is based upon HORIZONS-predicted magnitudes, which have a typical accuracy of ∼0.5 mag, in certain cases the actual magnitude was almost certainly fainter than V = 20. These particular predicted sightings (having good positional localization but possibly too faint for detection) are still included as long as the predicted (point source) magnitude is brighter than V = 20.

Figure 3 shows some examples of visually confirmed PTF streak detections from this set of predicted sightings. Qualitative variations in morphology due to a differences in magnitude, streak length, and seeing are apparent.

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As described below, these 539 serendipitous sightings constituted the initial test bed for development of our streak detection algorithm. At the time of this paper, PTF acquired ∼90 new additional detections including unconfirmed PTF discoveries, confirmed PTF discoveries, and PTF-observed discoveries from other surveys in 2014.

The principal steps of the streak-detection process, referenced to component products in Figure 4, are

• 1.  
  Image processing and subtraction of a reference image to produce a differenced image (A&B);
• 2.  
  Detection of candidate streaks as regions of contiguous pixels on the differenced image (C&D);
• 3.  
  Measurement of a set of morphological features describing each candidate streak (E);
• 4.  
  Filtering of likely non-real detections on the basis of their computed features (F); and
• 5.  
  Human recognition of real streaks by reviewing images of the filtered candidates (G).

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The above five steps comprise the discovery phase and entail the creation of the data products labeled (A) through (G) in Figure 4. Upon discovery of a real streak, data products (H) through (J) are created as part of the follow-up phase, which we discuss later (Section 4).

Initial image processing and reference-image subtraction (the first of the above-enumerated steps) are described by Laher et al. (2014) and Masci (2017), respectively. Step 2 involves identifying the pixels on the differenced image belonging to candidate streaks. Whereas pixel-level data for point-source transients (e.g., supernovae or slow-moving asteroids) can be efficiently extracted with commonly used software such as Source Extractor (Bertin & Arnouts 1996), streaked detections require a distinct approach as their image footprints contain many more pixels, often with much lower signal-to noise per pixel. To meet this need, we developed an original piece of software called findStreaks.

3.1. Object Detection with findStreaks

3.1.1. Algorithm

The findStreaks software is derived from code originally created for the IPAC processing pipeline to identify and mask very long tracks in PTF exposures due to satellites and aircraft (Laher et al. 2014). findStreaks was developed in the C programming language to maximize computing speed. The software first thresholds the image pixels above a local background noise level, then groups contiguous pixels into objects or "blobs" (i.e., candidate streaks), and lastly computes morphological features for each object.

The findStreaks code is a single-threaded application. Parts of the software could be straightforwardly made to process with multiple threads using the pthread library, but the overall speed-up that may result is yet to be quantified. The algorithm is likely amenable to GPU parallelism.

Real-time performance is currently achieved on the iPTF system by running as many as four pipelines simultaneously on each 24-CPU machine with 2.4-GHz CPUs and 16 GBytes total memory (parts of the pipeline other than findStreaks are parallel processes). In this configuration, the findStreaks code typically runs in 13–21 seconds per CCD image for moderately crowded fields, requiring roughly one wall-clock second of run time per 1500 sources extracted from the image by the Source Extractor software.

The differenced image's local median background values are computed on a coarse grid with 64-pixel grid spacings and 129-pixel windows, with bilinear interpolation used to fill in the pixel values between the grid points. These median values are used to threshold the positive difference image at 1σ above the local median. All below-threshold pixels are discarded and only above-threshold data are considered further (e.g., Figure 4 item C). Image-edge pixels are ignored (to avoid artifacts along CCD edges). Ignoring image-edge pixels does introduce some selection effects, however we do not attempt to estimate absolute rates of NEA detections in this paper.

The findStreaks module arranges all contiguous blobs of pixels, each blob in one or more segments of computer memory, where adjacent pixels in the cardinal and diagonal positions are considered to be connected. For efficient memory management, the module is configured to handle up to one million memory segments and up to 1000 pixels per segment. The sky-background-subtracted blob flux and instrumental magnitude are computed, along with their respective uncertainties. The median and dispersion of the pixel-blob intensity data are computed and subsequent morphological analysis is done only on pixels with intensities that are within ±3σ of the median, where σ is given by half the difference between the 84th and 16th percentiles. A line is fit to all pixel positions in each blob and the slope and y-intercept are obtained, as well as the linear correlation coefficient

$$\begin{eqnarray}&&r=\displaystyle \frac{{\sum }_{i}({x}_{i}-\bar{x})({y}_{i}-\bar{y})}{\sqrt{({\sum }_{i}{({x}_{i}-\bar{x})}^{2})({\sum }_{i}{({y}_{i}-\bar{y})}^{2})}}.\end{eqnarray} \tag{ 1 }$$

Perpendicular distances from the linear model to constituent pixels are used to find the blob half-width, defined as half the difference between the 84th and 16th percentiles of these distances. With size and shape parameters now in hand, several hard filters are used to eliminate blobs that are not considered to be streaks. Blobs containing more than 400,000 pixels, having long axes shorter than nine pixels (i.e., 3.6 deg/day motion), or having half-widths larger than 16 pixels are discarded. Blobs for which the absolute value of the linear correlation coefficient is less than 0.5 are also discarded.

The findStreaks module outputs a table of streak metadata, where each table row corresponds to a streak detection. The real-time pipeline augments this table with additional columns including the proximity of the candidate to the nearest reference image (stationary) object, as well as the brightness of this nearest reference-image object. Table 1 lists the 15 features currently retained for each candidate, and used in the classification stage that follows. This list of features will be updated to include additional morphological metrics in future versions of this software, but the results of this paper only include analysis of the above-described 15 features.

Table 1.  Morphological and Other Features Saved for Streak Candidates

Feature	Description
Pixels	Number of pixels associated with detected object
Length	Long axis length
hwidth	Half-width
dMax	Perp. distance of maximum-flux pixel from longest axis
Angle	Proper angle (in RA, Dec coords)
Median	Median pixel flux
Scale	1σ variation in pixel flux
Slope	Slope (dy/dx in image coordinates) of fitted line
Correl	Correlation coefficient of fitted line
Flux	Total flux of object
refDist	Distance from midpoint to nearest object in reference image
refMag	Magnitude of nearest object in reference image
Epoch	Epoch (modified Julian date)
ra	Right ascension of object midpoint
decl.	Declination of object midpoint

Download table as:  ASCIITypeset image

3.1.2. Completeness and Contamination

To ascertain findStreaks's completeness and the number and nature of false positives it detects, we tested the software on a set of images containing both known real asteroid streaks (the 539 predicted sightings described in Section 2.4) and a large number of injected synthetic (simulated) streaks.

To generate each synthetic streak "stamp," we first considered a 2D-Gaussian point-spread function of flux f, full-width-at-half-maximum (FWHM) θ, and center at (x₀, y₀)

$$\begin{eqnarray}&&\mathrm{PSF}(x,y,{x}_{0},{y}_{0},f,\theta )=f\times \displaystyle \frac{4\mathrm{ln}2}{\pi {\theta }^{2}}\\ &&\quad \times \exp \left(-\displaystyle \frac{{(x-{x}_{0})}^{2}+{(y-{y}_{0})}^{2}}{{\theta }^{2}}\times 4\mathrm{ln}2\right).\end{eqnarray} \tag{ 2 }$$

In terms of Equation (1), a simulated asteroid streak of length L oriented at angle ϕ is given by

$$\begin{eqnarray}&&\mathrm{Streak}(x,y,{x}_{0},{y}_{0},f,\theta ,L,\phi )\\ &&\quad =\displaystyle \frac{1}{L}{\displaystyle \int }_{t=0}^{t=L}\mathrm{PSF}(x-t\cos \phi ,y-t\sin \phi ,{x}_{0},{y}_{0},f,\theta )\ {dt}.\end{eqnarray} \tag{ 3 }$$

Vereš et al. (2012) present a similar streak model, albeit with a slightly different analytical expression.

We evaluate the integral in Equation (2) numerically over a grid with spacings Δx = Δy = 005. Assuming the physical units of x and y are PTF image pixels (=10), and assuming a typical PTF seeing value of θ ≈ 2'' (though we randomly vary θ along with other parameters; see below), the 0.05'' grid spacing ensures the simulated streak is initially oversampled (by a factor of several tens) relative to the final (coarsened) image of the streak.

For each synthetic streak, the various model parameters in Equation (2) are randomly drawn from flat distributions on the following intervals:

$$\begin{eqnarray}0\lt {x}_{0}\lt 1 & 0\lt {y}_{0}\lt 1\\ 1\buildrel{\prime\prime}\over{.} 4\lt \theta \lt 3^{\prime\prime} & 10^{\prime\prime} \lt L\lt 60^{\prime\prime} \\ 0^\circ \lt \phi \lt 180^\circ & \quad {\rm{1800}}\,{\rm{counts}}\lt f\lt {\rm{7200}}\,{\rm{counts}}.\end{eqnarray} \tag{ 4 }$$

A synthetic streak's flux f relates to its apparent magnitude m according to $m={m}_{0}-2.5\,{\mathrm{log}}_{10}\,f$, where m₀ is the zeropoint of the streak's host image. As the insertion of synthetic streaks into host PTF images is random (see below), it follows that the apparent magnitude m is not sampled from a uniform distribution, unlike the parameters f, θ, ϕ, and L. The counts for f prescribed above roughly simulate 15 mag < V < 21 mag for typical PTF zeropoints (given normal variations in sky background, extinction, etc.).

To coarsen each synthetic streak (prior to injection into an image), we evaluate the mean flux value in each 1'' × 1'' bin, equivalent to downsampling the initial simulated image by a factor 20. We round the counts in each resulting pixel to the nearest integer and crop the streak image to a rectangular "stamp" including all non-zero pixels. Lastly, to simulate shot noise, we replace the value of each non-zero pixel with a random integer sampled from a Poisson distribution whose mean is equal to the original pixel value.

We generated a set of 5000 synthetic streak image stamps following the above process and then inserted these at random locations into the 539 PTF images containing each of the 539 predicted known streak sightings (Section 2.4). In particular, for each image the number of streaks injected was determined by drawing from a Poisson distribution with mean equal to 5. Our results are not highly sensitive to the number of injections. A set of that number of synthetic streaks was then randomly drawn (with replacement) from the pool of 5000 and stamped into the image at randomly chosen (x, y) coordinates. The total number of injected streaks was 2631. We then processed each image with an offline version of the IPAC real-time image-differencing pipeline (Section 2.3) and ran findStreaks on the differenced images.

In the real-time streak detection pipeline, the output of findStreaks is subsequently subjected to machine-learned classification and human vetting. However, in the interest of initially assessing the completeness and reliability of findStreaks as an isolated module, we here simply (albeit arbitrarily) define a "successful detection" (i.e., a true-positive detection) as any case wherein findStreaks found an object whose measured center lies within a 15'' radius of the streak's true center, and the measured length minus the true length is less than four times the streak's measured width. The successful and failed detections according to these criteria are plotted in Figure 5.

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Two trends evident from the synthetic streaks are limiting magnitude-versus-length (Figure 5 top row) and lack of sensitivity to near-vertical or near-horizontal streaks (Figure 5, bottom row). In general, the completeness drops sharply at a certain limiting magnitude; this limiting magnitude brightens from ∼19 mag at 20 pixels to ∼18 mag at 60 pixels. Streaks oriented very near to either 0° = 180° or 90° are much less reliably detected by findStreaks (at all magnitudes): this is due the imposed hard limit on correlation ($| r| \gt 0.5$), a criterion which both near-vertical and near-horizontal streaks fail to satisfy. These are due, for instance, to diffraction spikes and CCD imperfections. See the additional discussion below in Section 3.2.

The total number of candidate streaks returned by findStreaks in this test was 21,783, or an average of ∼40 per image, most being false-positive detections (also referred to as "bogus" detections later in this paper). Figure 6 details the distribution of false positives in magnitude, length, and orientation space. These plots indicate that the most common type of false-positive detections are faint and short, consistent with these contaminants being mostly star/galaxy subtraction artifacts and segments of extended, low-surface brightness objects like optical ghosts, space debris trails, and bright-star halos. Figure 7 presents a gallery of examples of successful, failed, and contaminant detections.

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Among the 539 predicted sightings of real streaking asteroids (see Section 2.4) in the test images, a total of 240 were successfully detected by findStreaks. The left-side plots of Figure 5 show successful and failed detections in the same feature subspaces in which the synthetics are also plotted in Figure 5. A distinction between the y-axis-plotted "magnitude" for the reals and that of the synthetics is that the magnitude of the reals is again the predicted brightness, accurate to ∼0.5 mag, whereas the synthetic magnitudes are more precisely known (even for the non-detected synthetics, as they still have a known flux and well-defined image zeropoint).

While we have characterized the completeness and contamination of the findStreaks algorithm, we have not undertaken a detailed optimization of the algorithm to maximize the number of real streaking asteroids detected by the algorithm. Future upgrades of findStreaks are planned, particularly for the implementation of a small asteroid detection pipeline for the Zwicky Transient Facility (see Section 6 for a discussion of ZTF).

3.2. Machine-learned Classification

3.2.1. Overview

As described above, a typical PTF image (single CCD) may contain several tens of false-positive streak candidates so that a full night of PTF observations—consisting of several thousand such images—may typically produce an order 10⁵ raw candidate objects. This is far too many to screen by eye. This is especially important for small NEAs with high angular rates that can move quickly enough that they may not be in the observed field during the second observation of a night. As discussed in Section 4.1, if a streaked NEA candidate is identified in a single image, "target of opportunity" observations are made later in the night of those fields that may contain the NEA based on streak length and orientation.

In addition to reducing the number of candidates for human vetting, there is a second advantage to machine-learned classification. Imposing simple filters on the measured morphological features (Table 1) can eliminate large subsets of false positives, but these hard cuts generally come at the cost of decreased completeness. A good example is the filter on the linear correlation coefficient condition ($| r| \gt 0.5$) discussed above, and the resulting insensitivity to near-vertical and near-horizontal streaks.

To address this issue, we trained and implemented a machine-learned classifier to discriminate real streaks from false positives. We adopt a supervised ensemble-method approach for classification, originally popularized by Breiman et al. (1984), specifically the random forest (RF) method (Breiman 2001). RF classification has extensive and diverse applications in many fields (e.g., economics, bioinformatics, and sociology). Within astronomy in particular, RF classification is one of the more widely employed methods of machine-learning, though many alternatives exist. For example, Masci et al. (2014) use the RF method for variable-star lightcurve classification, while others have approached this problem via the use of, e.g., support vector machines (Woźniak et al. 2004), Kohonen self-organizing maps (Brett et al. 2004; Masters et al. 2015), Bayesian networks and mixture models (Mahabal et al. 2008), principal component analysis (Deb & Singh 2009), multivariate Bayesian and Gaussian mixture models (Blomme et al. 2011), and thick-pen transform methods (Park et al. 2013).

For general descriptions of RF training and classification, we refer the reader to Breiman (2001), Breiman & Cutler (2004), and the many references cited by Masci et al. (2014). Our use of an RF classifier is particularly motivated by its already proven application to the discovery and classification of astrophysical transients in the same PTF survey data (Bloom et al. 2012).

Streak candidates in PTF images are cast into a vector of quantitative morphological and contextual features, namely the 15 features listed in Table 1. Given a large set of such candidates, these metrics define a multi-dimensional space that can be hierarchically divided into subspaces called nodes. The smallest node—also known as a leaf—is simply an individual candidate. Given a set of leaves with class labels, i.e., a training set, one can build an ensemble of decision trees (called a forest) with each tree representing a different, randomly generated partitioning of the feature space with respect to a subset of the total training sample (and a subset of the total list of features). The forest allows one to assign a probability that a given vector of features belongs to a given class. For the PTF candidates, we are interested in a binary classification, i.e., whether the candidate is real or "bogus". Bloom et al. (2012) and Brink et al. (2013) coined the term "realBogus" to describe this binary classification probability. In the present work, we are essentially adapting Bloom et al.'s realBogus concept to the problem of streaking asteroid discovery.

3.2.2. Implementation and Training

We employ a Python-based RF classifier included as a part of the scikit-learn Python package.⁸ Specifically, we use the ExtraTreesClassifier class in the sklearn.ensemble module. This particular code is an implementation of the "extremely randomized trees" method (Geurts et al. 2006), a variant of the Random Forest method containing an added layer of randomness in the way node splitting is performed. Specifically, ExtraTrees chooses thresholds randomly for each feature and picks the best of those as the splitting rule as opposed to the standard RF, which picks thresholds that appear most discriminative. The additional randomization tends to improve generalization over the standard RF algorithm; this was verified empirically for our streak data.

Our training data consists of all candidate streak detections from the 539-image synthetic-injection test described in Section 3.1.2. This includes 240 real detections (out of the 539 predicted sightings from Section 2.4), 1285 synthetic detections (out of the 2631 total injected) and 20,072 bogus detections. Various examples of these bogus (false-positive) detections are shown in the right column of Figure 7, while their distributions in magnitude-versus-length and magnitude-versus orientation space are shown in Figure 6.

Among the 15 features (Table 1) describing the streak candidates, several of them exhibit some level of correlation, as shown in Figure 8. Most correlations are reasonable as they express the relationship between geometrically similar quantities: the length of a streak is generally correlated with the number of pixels and the fitted linear slope correlates with the proper angle. A strong correlation between median and scale (measures of flux signal and noise, respectively) is simply a expression of the Poisson noise associated with photon counting. Assessing the correlation between features aids in the interpretation of relative feature importances (Figure 9) derived during the training process (described below). In particular, among the top four most discriminative features (according to Figure 9), three are significantly correlated (pixels, hwidth, and length).

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The classifier evaluation consists of a ten-fold bootstrapping process, wherein we split the data (reals, synthetics, and boguses) into 10 disjoint sets using stratified random sampling. Then, in each cross-validation fold, we train using nine of the sets and test on the remaining one; however, we exclude the synthetics from this test sample. In each of the ten cross-validation trials, the classifier outputs a classification probability for each object in the test sample and we track the true-positive rate (TPR; fraction of real streaks accurately classified as reals) as a function of the false-positive rate (FPR; fraction of bogus streaks inaccurately classified as reals). In astrostatisics, TPR is also commonly called completeness while FPR is equivalently one minus the reliability. The results of the separate trials, as well as the averaged result, are shown in Figure 10. By tuning the minimum classification probability (i.e., the realBogus score) used to threshold the classifier's output, one effectively moves along the hyperbola-shaped locus of points in TPR-versus-FPR space seen in the plot.

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Several parameters can be adjusted or tuned when working with a random forest classifier. First is the number of decision trees generated during the learning stage. Classification accuracy typically increases with the number of trees and eventually plateaus. Most applications employ hundreds to thousands of trees; here we found that 300 trees provide sufficient performance. Increasing the size of the forest to beyond 300 trees did not produce substantially more accurate results. Another tunable parameter is the number of randomly selected features (out of the 15 total here considered) with respect to which nodes are split in building the decision trees. Breiman (2001) recommends using the square root of the number of features; however, here we found optimal accuracy when splitting with respect to all  15 features. Other parameters that can be tweaked are the maximum depth of a tree, the minimum number of samples per leaf, the minimum number of samples used in a split, and the maximum number of leaf nodes. We do not constrain any of these parameters, meaning that we allow trees of any depth with any number of leaf nodes, leaf nodes consisting of a single sample, and splits based on the minimum of two samples.

3.2.3. Post-training Performance

In addition to tracking the classifier's performance during the training cross-validation trials, after training we subjected the classifier to a new sample of ∼400 synthetics. These newly generated synthetics were injected into the same 539 test images using the same procedure described in Section 3.1.2. Given the distinct random numbers used in this run, these synthetics are distinct from those that were used in training, and appear at different locations on the PTF images.

As was done in cross-validation, the purpose of this post-training trial was to ascertain the detection completeness, though this time using synthetics (which were used previously for training but not testing). Another difference is that we now consider completeness for a fixed classification probability threshold (p > 0.4) and do so as a function of magnitude and length (similar to the analysis done for findStreaks in Section 3.1.2).

The top plots of Figure 11 show the same information as was shown in Figure 5, albeit for this new sample of synthetics (and at slightly coarser resolution). Namely, we first examine the completeness delivered by findStreaks alone and again see the limiting magnitude-versus-length trend. In the bottom plots of Figure 11, we show detection completeness for the same sample only this time for the combined findStreaks plus machine classifier system. In other words, all the blue data points in the lower left Figure 11 plot were both successfully detected by findStreaks and were subsequently classified as real with a probability p > 0.4.

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In Figure 12, we again show data from the same synthetics sample, this time plotting the loss in absolute detection completeness due solely to the application of the machine classifier. In the top plot of Figure 12, green data points were successfully detected by findStreaks but did not score high enough (p > 0.4) in the classification stage. The 2D histogram below it shows that the most significant loss in completeness occurs for short faint streaks. Likely not coincidentally, this region suffers from the largest number of bogus findStreaks detections, as indicated by Figure 6. Integrating over all bins in this magnitude-versus-length histogram, we observe an average completeness drop of ∼0.15, consistent with Figure 10 for a true-positive rate of ∼85% accompanying a false-positive rate of ∼5%.

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3.3. Web-based Screening Interface

The final component of the discovery portion of the PTF streak-detection pipeline consists of a web page for human vetting of image stamps of streak candidates to which the classifier has assigned a high probability of being real. Figure 4 includes a screenshot of this web page. Given the ∼5% false-positive rate quoted in the preceding paragraph and the ∼10⁵ detected candidates accumulated in a typical night (cf. Section 3.2.1), this web page displays on average several thousand candidates per night.

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Including operations on Palomar Mountain, the data transfer from Palomar to Caltech, and the IPAC real-time processing pipeline (Section 2.3), a typical lag-time of ∼30 minutes (approx. ±10 minutes) elapses between the acquisition of exposures with the PTF camera and the posting of streak candidates from said exposure to the scanning web page. The image stamps have fields of view of 200'' × 200'' with linear contrast scaling from −0.5σ to 7σ (as in Figure 3). Undifferenced images are reviewed as opposed to the differenced images to better provide context to the scanner, enabling visual assessment of the observing conditions (i.e., the density and image quality of background stars).

The kinds of false positives commonly encountered on the scanning web page include all of those shown on the right-hand side of Figure 7. Image stamps are viewed in chronological order so that candidates from a common image appear consecutively on the scanning page. This enables rapid recognition of false positives of a common origin. For example, multiple segments of a long satellite trail, large optical ghost, or artifacts from a poorly subtracted or high stellar density image will appear together and are thus easily dismissed (see Figure 13). Artifacts that do not appear in groups, such as cosmic ray hits, background sky noise, and poorly subtracted galaxies, are rapidly visually dismissed as well. A full night's set of candidates (several thousand) can be reliably reviewed by a trained scanner in 5–10 minutes, though the reviewing time is distributed over the during observing session, as the web page is refreshed every 20–30 minutes.

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Clicking on the image stamp of a candidate streak presents another web page with more detailed information including astrometry; photometry; realBogus score; image stamps of the differenced and reference images; and a larger FOV around the detection. Certain types of false positives are more easily identified using this additional information, including portions of optical ghosts and periodically glinting space debris. This summary page also contains information for real-time follow-up, as discussed in the next section.

Once a real streak is discovered in PTF via the steps outlined in the previous section, we trigger real-time follow-up with the same telescope. Its wide FOV (Section 2.1) makes the PTF camera particularly well suited for recovering fast-moving NEAs within a few hours of an initial detection. As described below, the follow-up process effectively interrupts the nominal robotic survey by injecting high-priority exposure requests into the queue. The final step involves reporting observations to the Minor Planet Center to facilitate subsequent confirmation and follow-up worldwide.

4.1. Target-of-opportunity (ToO) Requests

4.2. Initial NEA Discoveries

The full streak-discovery system, incorporating the IPAC real-time data products, findStreaks and the trained machine classifier, began real-time operations 2014 May 1. About a week later, the first PTF streaking NEA discovery was made (2014 JG₅₅). Passing at one-third of a lunar distance, this object is also the smallest and closest-approaching NEA yet discovered by PTF.

The largest streaking NEA discovered by PTF to date is 2014 WK₇, at H = 22.4 mag (D ≈ 166 m), while the PTF discovery having an orbit with the lowest Δv with respect to Earth and most accessible by a robotic space mission (e.g., see Jedicke et al. 2013) is 2014 ST₂₂₃.

Table 2 details the 11 total streaking-NEA discoveries made by PTF as of 2014 December 1 (see also Figure 14). Nearly all of these (the one exception being 2014 LL₂₆) were followed-up and confirmed by multiple observatories within 24 hours. A total of 25 different observatories have provided follow-up observations within 24 hours of at least one of the NEA discoveries listed in Table 2. After the Sun has risen in California, most short-term follow-up of PTF discoveries occurs from Japan and Europe (and occasionally Australia), as most longitudes west of Palomar fall in the Pacific.

Table 2.  PTF Discoveries of Streaked NEAs (between 2014 May 1 and 2014 December 1)

Name	Date	# PTF	Diameter (m)	Inclination	Eccentricity	Semimajor	Min. Earth Orbit	Δv w.r.t.	Speed	V
		Detections	(7% albedo)	(Deg)		Axis (au)	Intersect. (LD)	Earth (km s−1)	(''/min)	Magnitude
2014 WS7	Nov 19	3	17.4	8.687	0.47655	1.8918	4.36	6.1	25.7	18.7
2014 WK7	Nov 18	4	166	23.91	0.36071	1.5605	3.38	8.4	39.8	16.7
2014 UL191	Oct 30	2	66.2	2.070	0.65017	1.7553	3.73	7.8	66.9	17.1
2014 ST223	Sep 23	4	15.2	5.875	0.17045	1.0532	1.56	5.2	31.8	18.4
2014 SE145	Sep 23a	2	18.6	8.406	0.54698	2.2354	4.03	6.6	62.3	19.1
2014 SC145	Sep 23	3	36.4	20.17	0.20651	1.2216	6.61	7.8	28.4	19.0
2014 SE	Sep 16	2	43.4	20.02	0.17187	1.2428	13.5	7.9	24.7	18.7
2014 RJ	Sep 02	2	41.8	19.57	0.27083	1.4086	7.39	7.4	37.2	17.9
2014 LL26	Jun 09	2	43.7	9.182	0.10177	1.1427	5.25	5.4	29.1	17.2
2014 KD	May 17	4	66.2	5.238	0.54605	2.1519	7.74	6.2	30.0	17.5
2014 JG55	May 10	4	7.26	8.739	0.41257	1.5843	0.336	5.8	60.0	18.1

Note.

^aDesignated by MPC as "First observed at Palomar Mountain–PTF on 2014 September 23", but MPC also cites PanSTARRS 1 observations on September 22.

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Figure 15 shows the discovery position of the NEAs in Table 1 relative to opposition. A majority of the objects were found within 40° of opposition. An outlier is 2014 WK₇, which was discovered 73° from opposition (phase angle 71°), though this NEA is also an outlier in the sample in terms of its size.

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Table 3 lists the various sub-surveys (also known as "iPTF experiments"; see Section 2.1) to which the NEA discovery exposures belong. Here "TILU" stands for Transients in the Local universe'. A key point here is that nearly all of PTF's streaked NEA discoveries to date have been made in images originally purposed for non-solar-system science. A dedicated iPTF experiment designed to maximize the area covered around opposition was carried out for several nights in Fall 2014, though only one exposure from said program produced a discovery, 2014 ST₂₂₃.

Table 3.  iPTF Sub-surveys Containing Streaked-NEA Discovery Exposures

Name	Sub-survey in which NEA was Discovered	Filter	Degrees from Opposition
2014 WS7	Permanent Local Galaxies	R	28
2014 WK7	TILU K2 Campaign	g	73
2014 UL191	TILU Fall 2014	g	47
2014 ST223	Opposition NEA search	g	18
2014 SC145	RR Lyrae	R	32
2014 SE	RR Lyrae	R	21
2014 RJ	TILU Fall 2014	g	35
2014 LL26	Star-forming low-cadence	R	9
2014 KD	TILU Spring 2014	R	49
2014 JG55	iPTF14yb follow-up	R	34

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All follow-up was unsolicited apart from having posted the discoveries on the NEOCP and attests to the dedication of the worldwide NEA follow-up community. We note, however, that while they are on the NEOCP, PTF-discovered streaking NEAs are consistently the brightest on the list—all were V ≤ 19 mag—whereas most of the 50+ objects typically found on the NEOCP have V ≥ 20 mag. It is therefore not surprising that more follow-up facilities are able and willing to recover these bright objects as compared with the typical faint and slow NEOCP candidates.

4.3. Blind Real-time Recovery of known NEAs

4.4. Unconfirmed Discoveries

A total of five PTF objects posted to the NEOCP (between May 1 and December 1) did not receive external follow-up, meaning they never obtained confident orbit solutions and thus were not assigned provisional designations by the MPC (Table 4 and Figure 16). For four of these unconfirmed objects, PTF had submitted only two observations to the NEOCP. We note that five out of the 11 confirmed objects (Table 2) also were reported with only two observations, from which we naively conclude that a two-observation discovery has only a 50% probability of being successfully followed-up (for three- and four-observation discoveries the recovered fraction increases to 66% and 100%, respectively).

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Table 4.  Unconfirmed PTF Streak Discoveries (from 2014 May 1 to 2014 December 1)

NEOCP	Date	Num.	Speed	V	Notes
Name	Found	Obs.	(''/min)	(mag)
PTF5i5	May 4	2	46.8	19.5
PTF9i2	Jul 8	2	36.9	17.9	85% moon, near dawn
PTF3k8	Sep 23	2	64.9	18.0	Likely a satellite
PTF8k2	Sep 25	3	27.6	18.9
PTF7l3	Oct 25	2	30.1	17.5

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While in reality the recovery probability depends also on the temporal spacing of the observations, the object's speed and magnitude, and the availability of follow-up resources (e.g., less facilities operate around full moon), the number of observations seems to be a useful indicator of the recovery likelihood. Users of the PTF real-time scanning and ToO system attempt to obtain at least three observations for discoveries, though this is not always possible, e.g., for discoveries made early in the night in the western sky, or just before dawn. Occasional technical issues with the real-time processing and/or ToO system also can hinder PTF self-follow-up.

4.5. Artificial Satellites

Many distant Earth-orbiting artificial satellites can, at certain parts of their orbit, appear consistent with an Earth-approaching NEA. Our streak-recognition pipeline has on several occasions detected such satellites. Figure 17 shows some examples, including one of the THEMIS mission spacecraft studying the Earth's magnetosphere (Angelopoulos 2008) and a Titan IIIC rocket body. The MPC's automated observation-ingestion processes outputs known artificial satellite matches to NEOCP submissions (as was the case for the three in Figure 17), though in some cases the object will be posted to the NEOCP and remain on the list for some time prior to its recognition as artificial. Three examples of the latter were PTF7i2, PTF8i6, and PTF0n2.

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While we see the same value in blind reporting of artificial satellites as we do blind reporting of known NEAs (Section 4.3), some high-orbit satellites have geosynchronous orbits and can therefore appear in the same area of sky for many consecutive nights. An example is the THEMIS spacecraft, whose apogee was coincident with opposition, causing it to be repeatedly observed by PTF in autumn 2014. For routine identification of known satellites, we therefore adopted the useful software tool sat_id by Project Pluto.¹²

The right panel of Figure 15 shows the distribution of streaked NEA detections (including confirmed and unconfirmed discoveries as well blind recoveries). In this section, we use this sample of detections and the distribution of PTF exposures with respect to opposition to derive the debiased streaked-NEA detection rate as a function of radial distance from opposition. Figure 18 shows the same data as in Figure 15, removing the azimuthal information to only show the one-dimensional radial distributions.

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We seek to estimate the frequency f of streaked NEA detections per unit area of sky per unit time (equivalently, per survey image). The posterior probability distribution of f (assuming a constant prior) is given by an appropriately normalized Poisson distribution,

$$\begin{eqnarray}&&P(f)=\displaystyle \frac{{({NC})}^{n+1}}{{\rm{\Gamma }}(n+1)}{f}^{n}\exp (-{NCf}),\end{eqnarray} \tag{ 5 }$$

where N is the total number of images searched for streaked NEAs, n is the number of detected streaked NEAs, C is the completeness (true-positive rate) of the PTF streak detection system as a whole, and Γ(...) is the gamma function (which contributes to the normalization of the distribution).

Figures 5 and 11 indicate that the completeness C depends on which volume in magnitude, length, and orientation space under consideration, as well as the separate efficincies of sub-components like findStreaks and the machine classifier. For simplicity, in the following analysis we evaluate two separate values for C (0.5 and 0.7), but the most accurate values for C would in principle come from direct application of the completeness data in Figures 5, 11 and 12.

We apply Equation (5) to the image count N and streak count n within each of the 13 bins in Figure 18. In particular, by numerical integration we compute the 16th and 84th percentiles of the resulting Poisson distributions, and plot these bounds as a function of distance from opposition in Figure 19. The estimates are 1–3 streaked NEA detections per 10⁴ deg² of sky near opposition, dropping to about 1 or less beyond 40–50 deg from opposition. The images acquired by PTF from 2014 May 1 through 2014 December 1 represent 191,435 deg², and a total of 19 streaked NEAs (10 confirmed, four blindly recovered, and five unconfirmed) were detected in these data. If the areal density of the streaks were independent of distance from opposition, this would correspond to a coarse estimate of ∼1 detected streak per 10⁴ deg², in agreement with the radially-binned rates multiplied by the actually radial distribution of images (which are mostly 40 deg or more from opposition).

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We here derive a quantitative FoM proportional to the average number of streaked asteroids detectable per unit time by a survey. The FoM will depend on a number of survey specifications including the

• field of view Ω in deg²,
• seeing width θ_PSF in arcseconds,
• limiting magnitude of a point source m_lim, for a given exposure time τ, and
• the time between exposures τ_tot, which includes readout and telescope slew time.

Assume that the density of asteroids and their velocity distribution is independent of distance. The volume of streaked asteroids detectable at any given time goes as ${\rm{\Omega }}{D}_{\mathrm{strk}}^{3}$, where D_strk is the maximum distance at which an asteroid can be detected as a streak. The figure of merit (asteroids detectable per unit time) therefore scales as

$$\begin{eqnarray}&&\mathrm{FoM}\propto \displaystyle \frac{{\rm{\Omega }}{D}_{\mathrm{strk}}^{3}}{{\tau }_{\mathrm{tot}}}.\end{eqnarray} \tag{ 6 }$$

For a point source, ${D}_{\mathrm{pnt}}^{2}\propto {10}^{0.4({m}_{\mathrm{lim}}-{m}_{0})}$, where m₀ is a normalization and consequently we define the point source FoM to be

$$\begin{eqnarray}&&{\mathrm{FoM}}_{\mathrm{pnt}}\equiv \displaystyle \frac{{\rm{\Omega }}{10}^{0.6({m}_{\mathrm{lim}}-{m}_{0})}}{{\tau }_{\mathrm{tot}}}.\end{eqnarray} \tag{ 7 }$$

A principal result, shown in the Appendix, is that the FoM for long streaks is simply

$$\begin{eqnarray}&&{\mathrm{FoM}}_{\mathrm{strk}}={\mathrm{FoM}}_{\mathrm{pnt}}\left(\displaystyle \frac{{\theta }_{\mathrm{PSF}}}{{\theta }_{\mathrm{strk}}}\right),\end{eqnarray} \tag{ 8 }$$

where θ_strk is the angular length of the asteroid streak for an asteroid with a given velocity perpendicular to the line of sight at the limiting point source detection distance for the asteroid.

Streak detection thus differs significantly from the more conventional analysis that was designed for detection of asteroids with point-like appearance in individual images. It is therefore of interest to estimate the capabilities of various current and future surveys assuming that those surveys implemented a streak detection capability of the type described in this paper.

For deep surveys with larger aperture telescopes (in particular DECam, SST, and LSST), many small NEAs will be detected as point sources because they can be detected far from Earth where the apparent angular velocity will be small. Consequently, for these surveys, streak detection will provide a significant improvement only for the smallest fast-moving asteroids. For shallower surveys, such as PTF and ZTF, streak detection can significantly improve the capability for discovery of small NEAs.

In the Appendix, we derive an analytical expression for FoM, proportional to the average number of streaked asteroids detectable per unit time. As defined, the FoM is proportional to the volume per unit time surveyed by a telescope for an asteroid of given normalized brightness and given velocity (direction and magnitude). We discuss the analytical form of a generalized FoM that is applicable to both point source detection and streaked asteroid detection and transitions smoothly between the two regimes for any survey. This computation can be done for any desired representative class of small NEAs, specified by their brightness (H magnitude) and asteroid velocity perpendicular to the line of sight (v_⊥). We also discuss the numerical integration of the generalized FoM over an isotropic distribution of angles to obtain an FoM for asteroids of a given H magnitude and velocity relative to Earth (v_ast), taking into account the relative numbers of objects with a given perpendicular velocity to the line of sight.

Table 5 summarizes the results for various surveys. The choice of surveys is selective, with emphasis on current and near-future capabilities. PTF with streak detection is used as the reference in all cases. Two FoMs are given for each survey and each case of brightness and velocity of the NEA. The first number in each case is the FoM for streak detection, relative to the streak detection FoM for PTF. The second number (in parentheses) is the FoM for "peak" detection relative to the streak detection FoM for PTF. The FoM for peak detection is described in the Appendix and corresponds to the MOPS approach used by many asteroid surveys, namely, identification of a candidate asteroid when the flux integrated over the typical point-spread function is above a given threshold.

Table 5.  Comparison of Relative FoM for Detection of Small Near-Earth Asteroids. The H Magnitudes Correspond to 10 m, 50 m, 100 m, and 200 m Asteroids (for H = 28.5, 25.0, 23.5, and 22.0, respectively, Assuming an Albedo of 0.07). The Two FoMs in each Case are with and without Implementation of Streak Detection, the Latter Case (Peak Detection) is in Parenthesis. Variables are Further Defined in the Text and the Appendix. The FoMs are Averaged Over an Isotropic Distribution of Velocities of Magnitude, v, Computing the Asteroid Velocity, v_⊥, Perpendicular to the Line of Sight for Each Direction

Telescope	Ω	mlim	${\theta }_{\mathrm{PSF}}$	τ	${\tau }_{{tot}}$	$\langle \mathrm{FoM}\rangle$		$\langle \mathrm{FoM}\rangle$		$\langle \mathrm{FoM}\rangle$		$\langle \mathrm{FoM}\rangle$
	(deg2)	(mag)	(arcsec)	(s)	(s)	(H = 28.5)		(H = 25.0)		(H = 23.5)		(H = 22.0)
						(v = 20 km s−1)		(v = 20 km s−1)		(v = 20 km s−1)		(v = 5 km s−1)
						Streak	Peak	Streak	Peak	Streak	Peak	Streak	Peak
PTF	7.25	20.7	2.0	60	105	$\equiv$ 1	$\equiv$ 1	$\equiv$ 1	$\equiv$ 1	$\equiv$ 1	$\equiv$ 1	$\equiv$ 1	$\equiv$ 1
DECam	3.2a	23	0.8	40	60	31	(1.4)	28	(6)	26	(11)	20	(18)
PS1	7	21.8	1.1	30	40	20	(1.0)	18	(4.4)	17	(7)	13	(11)
ZTF	47	20.4	2	30	45	17	(0.8)	15	(3.5)	14	(6)	11	(9)
ATLAS	60	19.9	2.5	30	35	14	(0.6)	13	(2.8)	12	(5)	9	(8)
CSS/MLS-II	5.	21.5	1.3	20	35	17	(1.2)	14	(5)	12	(6)	7	(6)
CSS-II	19.4	19.5b	1.5	30	45	1.0	(0.02)	1.1	(0.1)	1.1	(0.2)	1.2	(0.9)
SST/Lincoln	6.	20.5	1.	2.	6.	96	(33)	44	(33)	29	(25)	14	(13)
LSST	9.6	24.4	0.7	30	39	2000	(200)	1500	(650)	1200	(700)	700	(650)

Notes.

General notes: Ω is the field of view. m_lim is the 5σ median limiting r-band magnitude. θ_PSF is the width of the telescope point-spread function. τ is the exposure time and τ_tot is the time between exposures.

PTF: m_lim is the measured PTF value.

DECam: Ω, m_lim and τ from Allen et al. (2015). Note ^a: the value of Ω quoted by Allen et al. (2015) for NEO observations is less than the 3.2 deg² normally quoted. θ_PSF from Shaw (2015), Table 4.1. τ_tot assumes a read time of 20s (from Shaw 2015, Table 4.2).

PS1: Performance is from the 3π survey of PS1. Recent upgrades may have improved performance for PS1/2 (Morgan et al. 2012). Ω from Kaiser (2004). m_lim is median r-band limiting magnitude from Morganson et al. (2012). θ_PSF from Lee et al. (2012).

ZTF: m_lim is the 5σ median limiting r-band magnitude estimated using PTF observing history and accounting for new optics and shorter exposure time.

ATLAS: The dark sky (not median) magnitude limit in r-band is 20.2 (from Tonry 2016). We take m_lim ∼ 19.9 as an estimate of an achievable median performance. Other values from Tonry (2016).

CSS/MLS-II (Mt. Lemmon upgrade): Ω = 5 sq deg from Christensen et al. (2015). θ_PSF from Jedicke et al. (2016). Other parameters from Larson (2007) except for τ_tot which is estimated.

CSS-II (Catalina upgrade): Ω = 19.4 sq deg, Christensen et al. (2015). Other values from Mahabal (2016). Note ^b: CSS-II may be able to detect NEOs at less than 5σ, perhaps as low as 1.2σ as described in Christensen (2014) for CSS. See discussion in the text.

SST/Lincoln: Ω, τ, and τ_tot from Ruprecht et al. (2014). θ_PSF is estimated, including seeing. m_lim from Ruprecht et al. (2015).

LSST: Ω, τ, and τ_tot from LSST Project (2016). m_lim and θ_PSF from Ivezic et al. (2008).

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The estimates of relative FoM are only approximate because of the uncertainty of the values for median performance for each survey. For instance, an error of 0.15 magnitudes in the median limiting magnitude, m_lim, will appear as a 25% error in the FoM. We also note that the FoM is a detection volume and the number of NEAs actually detected in any observation depends critically on many factors: the direction in which the observation is made, e.g., in the ecliptic, at opposition, etc.; the cadence of repeat observations of a given field, as well as other observational parameters such as phase of moon; airmass; etc. Assessment of these factors for a survey typically requires a detailed simulation. The FoM attempts to quantify an instrument's potential for carrying out a small asteroid survey. How a given instrument is used for asteroid observations will ultimately determine its effectiveness. Individual surveys obviously differ significantly in their overall survey strategy. For example, for the PTF observations reported here, the overall survey strategy was largely determined by objectives other than solar system science, such as supernova science, resulting in about 2/3 of the observations being greater than 20 degrees from the ecliptic plane (see e.g., Figure 15).

The H magnitudes shown in Table 5, 28.5, 25.0,23.5, and 22.0 correspond approximately to NEAs of 10 m, 50 m, 100 m, and 200 m in size, for an albedo ∼0.07. A velocity of 20 km s⁻¹ is chosen as typical (Jeffers et al. 2001). The circumstances chosen span a range of size, with the longest streaks expected for the case of H = 28.5 and v_ast = 20 km s⁻¹ and the most point-like objects to be seen in the case of H = 22.0 and v_ast = 5 km s⁻¹. Table 5 shows that for most of the telescopes streak detection is about 20 times more efficient than conventional peak detection for very small fast moving asteroids (H = 28.5, v_ast = 20 km s⁻¹), about four times more efficient for 50 m asteroids (H = 25.0, v_ast = 20 km s⁻¹), about 2.5 times more efficient for 100 m asteroids (H = 23.5, v_ast = 20 km s⁻¹), and, as expected, only slightly more efficient at detecting slower-moving 200 m asteroids (H = 22.0, v_ast = 5 km s⁻¹). The FoM values shown for H = 22.0, v_ast = 5 km s⁻¹ are in fact reasonably well approximated by the simple point source FoM given in Equation (10) in the Appendix.

Taking into account that the estimates of FoM in Table 5 are only approximate, we can make some general statements. For the four cases of asteroid size and velocity shown in Table 5, the capabilities of DECam, PanSTARRS, ZTF, ATLAS, and CSS/MLS-II are similar in terms of performance per unit time if implementing a similar survey strategy. The FoM values for these five surveys are within about a factor two of each other and therefore the total time for which asteroid surveys are carried out and the choice of survey pointing strategy will be the significant factors in determining the relative long-term productivity of each of the surveys for small NEO detection. Clearly, both SST/Lincoln and LSST represent a major improvement in capability for small asteroid detection.

To a large extent, the PTF observations described in this paper were directed towards the detection of ∼10 m asteroids. Although PTF is nominally about 13 times less effective than PanSTARRS for detecting slow-moving 200 m NEAs (entry for H = 22. for PS1 in Table 5), using streak detection it has roughly the same effectiveness per unit time as PanSTARRS for fast moving 10 m asteroids; our estimate is that PanSTARRS would be ∼20 times more effective than PTF if streak detection were implemented (entry for H = 28.5). This is borne out by the statistics of small NEA detection for the period during which PTF carried out its small NEA demonstration, 1 May 2014 to 1 December 2014. The statistics were taken from MPC data on discovered NEOs for that period. During this period PTF discovered one NEA with H > 28, 2014 JG55, while PanSTARRS-1 detected two such NEAs. For H > 27, PTF discovered four NEAs, while PanSTARRS discovered nine. This indicates that PTF was roughly half the effectiveness of PanSTARRS in discovering the smallest NEAs, This is consistent with the predictions of Table 5, particularly after taking into account that the survey strategy of PTF was far from optimal, determined as it was by other science objectives, and probably at least a factor of two less effective than PanSTARRS.

Also interesting are the statistics for other survey programs. For the period 2014 May 1 to December 1, Mt. Lemmon detected 21 NEAs with H > 27 and the CSS detected 13. DECam did most of its observations during other times of the year and had the highest number of detections of the smallest asteroids (H > 28) over the entire year. The FoM for CSS in 2014 should be about 2.4 less than that for CSS-II shown in Table 5, due to its smaller FOV. The relatively high numbers of detections of NEAs with H > 27 during the 2014 May 1 to December 1 period is likely due to the ability of CSS to detect NEAs at less than the nominal 5σ level used to calculate the FoM. Christensen (2014) attributes this to the effectiveness of human scanning and validation of candidate NEAs.

Table 5 also indicates the importance of accurate calibration of the efficiency of detecting small NEOs if population size is to be estimated, particularly using peak detection. For the smallest fast-moving asteroids (H = 28.5, v_ast = 20 km s⁻¹), there is typically a difference of several hundred between the effective detection volume employing streak detection and the point source detection volume, i.e., same size asteroid but with v_⊥ = 0. Calibration of the detection efficiency is therefore critical and can be accomplished using simulated NEO tracks injected into representative survey images (see Allen et al. (2015) as an example). Such a calibration can then be used to estimate the effective detection volume and a corresponding population density of small NEOs.

This work uses data obtained with the 1.2-m Samuel Oschin Telescope at Palomar Observatory as part of the Palomar Transient Factory project, a scientific collaboration among the California Institute of Technology, Columbia University, Las Cumbres Observatory, the Lawrence Berkeley National Laboratory, the National Energy Research Scientific Computing Center, the University of Oxford, and the Weizmann Institute of Science, and the Intermediate Palomar Transient Factory project, a scientific collaboration among the California Institute of Technology, Los Alamos National Laboratory, the University of Wisconsin, Milwaukee, the Oskar Klein Center, the Weizmann Institute of Science, the TANGO Program of the University System of Taiwan, and the Kavli Institute for the Physics and Mathematics of the universe.

The authors thank Robert Jedicke and Quan-Zhi Ye for valuable comments on the paper, as well as the informative comments and suggestions by the referee.

A. Waszczak has been supported in part by the W.M. Keck Institute for Space Studies (KISS) at Caltech and this work was part of his thesis research. The work presented in this paper was in part inspired by a workshop on an Asteroid Retrieval Mission organized and funded by KISS in 2010–2012. This work also was also supported by NASA JPL internal research and technology development funds. Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.

In this appendix, we derive a figure of FoM for the detection of asteroids that are trailed in the image. We call this "streak detection." Elsewhere in the literature, this is sometimes called "trail detection" and the loss in detection sensitivity due to the trailed image is termed "trailing loss." Here, we take a somewhat different approach than some previous treatments (e.g., Milani et al. 1996; Rabinowitz 1991; Jedicke et al. 2002; Vereš et al. 2012). We estimate the detection sensitivity loss in magnitudes and, in addition, use this sensitivity loss to calculate a figure of merit by which to compare instruments with different FOVs, limiting magnitude, exposure time, and readout time, taking into account a distribution of asteroid velocity directions. Assume a sky background limited observation of a field. The FoM calculated here is a quantity proportional to the average number of streaked asteroids detectable per unit time, which for any given pointing direction is proportional to the detection volume for asteroids of a given size and velocity perpendicular to the line of sight. The FoM is defined to be

$$\begin{eqnarray}&&{FoM}\equiv \displaystyle \frac{{\rm{\Omega }}{D}^{3}}{{\tau }_{{tot}}}\end{eqnarray} \tag{ 9 }$$

and will depend on a number of factors including the field of view, Ω; the pixel size (arcsec), θ_PSF; the limiting magnitude for detection of the telescope, m_lim; the integration time of the observation, τ; and the total time for observations, τ_tot, of a given field including number of revisits and dead time due to readout and slewing and the velocity of the asteroid perpendicular to the line of sight, v_⊥. We chose not to include the number of revisits in the definition of FoM since this can vary with time for a given survey.

Appendix. Point Source Detection FoM

The distance, D, to which a point source can be detected is related to its magnitude, m, as

$$\begin{eqnarray*}&&m-{m}_{0}=5\ {\mathrm{log}}_{10}(D)\end{eqnarray*}$$

where m₀ is a normalization. Consequently,

$$\begin{eqnarray*}&&D={10}^{0.2(m-{m}_{0})}.\end{eqnarray*}$$

The FoM for point sources is then

$$\begin{eqnarray}&&{\mathrm{FoM}}_{\mathrm{pnt}}=\displaystyle \frac{{\rm{\Omega }}{10}^{0.6({m}_{{lim}}-{m}_{0})}}{{\tau }_{{tot}}}.\end{eqnarray} \tag{ 10 }$$

Appendix. Streak Detection

Streak SNR

The equations for a streak need to be modified because the signal-to-noise (S/N) saturates when the length of the streak becomes larger than θ_PSF. While the number of pixels containing a signal increases with time, the noise per pixel also increases, with the two effects canceling each other. Increases in exposure time therefore do not increase the effective SNR and the streak becomes more difficult to detect because the surface brightness decreases with time.

The S/N has limiting cases:

$$\begin{eqnarray*}{{SNR}}_{\mathrm{strk}} & \sim & \displaystyle \frac{F/(4\pi {D}^{2})}{\sqrt{B}}\\ & & \times \sqrt{{\tau }_{\mathrm{PSF}}}\ \ \ (\mathrm{for}\ \tau /{\tau }_{\mathrm{PSF}}\gg 1,\ {\rm{i}}.{\rm{e}}.,\ \mathrm{streak})\\ {{SNR}}_{\mathrm{pnt}} & \sim & \displaystyle \frac{F/(4\pi {D}^{2})}{\sqrt{B}}\\ & & \times \sqrt{\tau }\ \ \ (\mathrm{for}\ \tau /{\tau }_{\mathrm{PSF}}\ll 1,\ {\rm{i}}.{\rm{e}}.,\ \mathrm{point}\ \mathrm{source})\end{eqnarray*}$$

where F is the source strength, B is the background flux per unit PSF, τ is the exposure time, ${\tau }_{\mathrm{PSF}}={\theta }_{\mathrm{PSF}}/\dot{\theta }$ is the time required for the asteroid to traverse a length of one PSF, and $\dot{\theta }$ is the angular velocity of the asteroid during the observation.

A useful approximation that has the proper limiting behaviors as well as appropriate intermediate behavior (e.g., SNR_strk < SNR_pnt in all cases) is

$$\begin{eqnarray}{SNR} & \sim & \displaystyle \frac{F/(4\pi {D}^{2})}{\sqrt{B}}\times \displaystyle \frac{\sqrt{\tau }}{\sqrt{1+\tau /{\tau }_{\mathrm{PSF}}}}\\ & = & \displaystyle \frac{F/(4\pi {D}^{2})}{\sqrt{B}}\times \displaystyle \frac{\sqrt{\tau }}{\sqrt{1+\tau \dot{\theta }/{\theta }_{\mathrm{PSF}}}}.\end{eqnarray} \tag{ 11 }$$

Streak FoM

The detection volume for a given type of source goes as ΩD³/3, where D is the detection limit of the telescope for that source. The detection limit is defined for some limiting SNR, say SNR = 5. We assume that the density of asteroids and their velocity distribution is independent of distance (a reasonable assumption as most are detected fairly close to Earth). To achieve an SNR of 5 for a streaked object, the object must be closer than a point source of the same magnitude. Specifically, because SNR² ∝ D⁻⁴, the decrease in distance to achieve the same SNR as a point source implies (using Equation (11))

$$\begin{eqnarray}&&{D}_{\mathrm{strk}}^{4}\sim {D}_{\mathrm{pnt}}^{4}\times \displaystyle \frac{1}{1+\tau \dot{\theta }/{\theta }_{\mathrm{PSF}}}.\end{eqnarray} \tag{ 12 }$$

This yields an expression for the decrease in limiting magnitude due to a streak of

$$\begin{eqnarray}&&\Delta {m}_{\mathrm{lim}}=5\,{\mathrm{log}}_{10}({D}_{\mathrm{pnt}}/{D}_{\mathrm{strk}})\sim 1.25\,{\mathrm{log}}_{10}\left(\displaystyle \frac{1}{1+\tau \dot{\theta }/{\theta }_{\mathrm{PSF}}}\right).\end{eqnarray} \tag{ 13 }$$

Note that the reduction in limiting magnitude in Equation (13) is not the same as the trailing loss often discussed in the literature. The magnitude change in Equation (13) is the decrease in limiting magnitude assuming that a candidate streak has been identified and that the signals in the pixels along the streak have been summed together, i.e., what we call streak detection. This is the same as the quantity "SNR trailing loss" referred to in the context of LSST by Jones et al. (2016). As will be shown in Section A.3, the magnitude change in Equation (13) is half of what is usually called trailing loss. Indeed, the gain in detection rate using streak detection is a direct result of the fact that the magnitude losses are only half that incurred in conventional "peak detection".

Figure 20 shows the estimated magnitude limit versus streak length in PSF units as computed from Equation (13).

Zoom In Zoom Out Reset image size

Returning to the derivation of a figure of merit and dropping the approximate symbols, Equation (12) can be expressed as

$$\begin{eqnarray}&&{D}_{\mathrm{strk}}^{4}={D}_{\mathrm{pnt}}^{4}\times \displaystyle \frac{1}{1+\tau \dot{\theta }/{\theta }_{\mathrm{PSF}}},\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&=\,{D}_{\mathrm{pnt}}^{4}\times \displaystyle \frac{1}{1+\tau ({v}_{\perp }/{D}_{\mathrm{strk}})/{\theta }_{\mathrm{PSF}}},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&\approx \,{D}_{\mathrm{pnt}}^{4}\times \displaystyle \frac{{D}_{\mathrm{strk}}{\theta }_{\mathrm{PSF}}}{\tau {v}_{\perp }}\ \ \ \ \ \ \ (\mathrm{for}\ \tau \dot{\theta }\gg {\theta }_{\mathrm{PSF}}),\end{eqnarray} \tag{ 16 }$$

where v_⊥ is the asteroid velocity perpendicular to the line of sight, and therefore

$$\begin{eqnarray}&&{D}_{\mathrm{strk}}^{3}\approx {D}_{\mathrm{pnt}}^{3}\times \displaystyle \frac{{D}_{\mathrm{pnt}}{\theta }_{\mathrm{PSF}}}{\tau {v}_{\perp }}\ \ \ \ \ \ \ (\mathrm{for}\ \tau \dot{\theta }\gg {\theta }_{\mathrm{PSF}}),\end{eqnarray} \tag{ 17 }$$

Using Equation (17) then yields for the FoM:

$$\begin{eqnarray}{\mathrm{FoM}}_{\mathrm{strk}} & = & \displaystyle \frac{{\rm{\Omega }}{D}_{\mathrm{strk}}^{3}}{{\tau }_{{tot}}}\approx \displaystyle \frac{{\rm{\Omega }}\ {D}_{\mathrm{pnt}}^{3}}{{\tau }_{{tot}}}\left(\displaystyle \frac{{D}_{\mathrm{pnt}}{\theta }_{\mathrm{PSF}}}{\tau {v}_{\perp }}\right)\\ & = & {\mathrm{FoM}}_{\mathrm{pnt}}\left(\displaystyle \frac{{\theta }_{\mathrm{PSF}}}{{\theta }_{\mathrm{strk}}}\right)\ \ \ \ \ \ \ (\mathrm{for}\ \tau \dot{\theta }\gg {\theta }_{\mathrm{PSF}}),\end{eqnarray} \tag{ 18 }$$

where τ_tot is the total time per field including integration time, readout time, and slew time, and θ_strk is the angular extent of the streak at the point source limiting distance for the asteroid. Note the difference between v_⊥ and v_ast as used in Section 6. If we integrate Equation (18) over an isotropic distribution of asteroid velocity directions, we then obtain an additional π/2 ∼ 1.57 in the volume factor.

The approximation $\tau \dot{\theta }\gg {\theta }_{\mathrm{PSF}}$ in Equation (17) is not strictly necessary. The formula for D_strk (Equation (15)) can be solved exactly, although the expression is rather cumbersome. Mathematica was used to derive the exact solution and it is used in the calculations of FoM. The exact expression requires choosing a fiducial v_⊥. An advantage to the exact calculation is that there is a smooth transition between point source behavior and streaked source behavior. The averaging over asteroid velocity directions can be done for the exact calculation, but involves a numerical evaluation. Results of these calculations are given in Table 5.

Appendix. Peak Detection

To estimate the detection volumes for a survey that does not specifically implement streak detection, we estimate the probability that the measured flux exceeds the limiting magnitude at some point along the streak. We call this peak detection. For peak detection within a streak, the surface brightness of the streak decreases like τ/τ_PSF and therefore in analogy to Equation (11):

$$\begin{eqnarray*}&&{SNR}\sim \displaystyle \frac{F/(4\pi {D}^{2})}{\sqrt{B}}\times \displaystyle \frac{\sqrt{\tau }}{1+\tau \dot{\theta }/{\theta }_{\mathrm{PSF}}}.\end{eqnarray*}$$

Consequently, the detection distance for peak and point detection are related by

$$\begin{eqnarray}&&{D}_{\mathrm{peak}}^{2}={D}_{\mathrm{pnt}}^{2}\times \displaystyle \frac{1}{1+\tau \dot{\theta }/{\theta }_{\mathrm{PSF}}}.\end{eqnarray} \tag{ 19 }$$

This yields an expression for the decrease in limiting magnitude due to streak length of

$$\begin{eqnarray}&&\Delta {m}_{{lim}}=5\,{\mathrm{log}}_{10}({D}_{\mathrm{pnt}}/{D}_{\mathrm{peak}})=2.5\,{\mathrm{log}}_{10}\left(\displaystyle \frac{1}{1+\tau \dot{\theta }/{\theta }_{\mathrm{PSF}}}\right),\end{eqnarray} \tag{ 20 }$$

i.e., the decrease in the magnitude limit for simple peak detection is double that of full streak detection. This change in magnitude (Equation (20)) is often called the "trailing loss" in the literature. As examples, Equation (20) accurately describes the measured loss insensitivity shown in Figure 2 of Jedicke & Herron (1997) for the case of trailed images in Spacewatch, and also agrees to within about 20% with the estimate of trailing losses for LSST. See the upper curve in Figure 5 of Jones et al. (2016)). Equations (13) and (20) provide a natural explanation of the factor of two difference between the "SNR trailing losses" and "detection trailing losses", shown in that figure.

Equation (19) can be expanded to yield

$$\begin{eqnarray*}{D}_{\mathrm{peak}}^{2} & = & {D}_{\mathrm{pnt}}^{2}\times \displaystyle \frac{1}{1+\tau \dot{\theta }/{\theta }_{\mathrm{PSF}}}\\ & = & {D}_{\mathrm{pnt}}^{2}\times \displaystyle \frac{1}{1+\tau ({v}_{\perp })/({\theta }_{\mathrm{PSF}}{D}_{\mathrm{peak}})}.\end{eqnarray*}$$

This can be solved exactly to yield

$$\begin{eqnarray}&&{D}_{\mathrm{peak}}=\displaystyle \frac{1}{2}\left(\sqrt{4{D}_{\mathrm{pnt}}^{2}+{\left(\displaystyle \frac{\tau {v}_{\perp }}{{\theta }_{\mathrm{PSF}}}\right)}^{2}}-\displaystyle \frac{\tau {v}_{\perp }}{{\theta }_{\mathrm{PSF}}}\right),\end{eqnarray} \tag{ 21 }$$

which can then be inserted into the FoM

$$\begin{eqnarray}&&{\mathrm{FoM}}_{\mathrm{peak}}\equiv \displaystyle \frac{{\rm{\Omega }}{D}_{\mathrm{peak}}^{3}}{{\tau }_{{tot}}}({10}^{\Delta {m}_{\mathrm{multi}}}),\end{eqnarray} \tag{ 22 }$$

where we account for the small improvement, Δm_multi, in effective limiting magnitude due to the fact that there are multiple possibilities for detection along the streak, thus increasing the probability for exceeding the detection threshold. The averaging over angles can also be done for the peak case, and requires a numerical integration over the effective volume for detection as a function of v_⊥. The results of these numerical computations of the peak detection FoM are provided in Table 5 as numbers in parentheses.

Appendix. Discovery versus Detection