**Contents**

V.3.b.ii.1. Review of the Stationary Model

V.3.b.ii.2. Addition of Motion Estimation

V.3.b.ii.3. Gradient-Descent Algorithm

V.3.b.ii.4. Effect of Noise on the Motion Solution

Source position estimation was augmented by adding linear-in-time motion to the observational model. The previous model continues to be used in what is now called the "stationary fit." The augmented model is called the "PM fit," where "PM" originally referred to proper motion, but in fact many moving sources are close enough for parallax to contribute significantly to timedependent apparent sky position, especially because of the WISE observational mode of scanning very close to 90 degrees off of the sun vector (see section II.6.d.i). The tag "PM" has become ingrained, so the phrase "proper motion" continues to be used but should be considered only a loose-sense tag, not a literal reference to true proper motion.

The use of both fits is called the "dual solution." The stationary fit is performed first, and the full set of photometric parameters is computed. The PM fit begins with the stationary position as the initial position estimate at a fiducial time which is computed for each source as a flux-weighted mean observational epoch. Because of the effects of noise on the nonlinear chi-square minimization algorithm, it is sensitive to the initial estimates used to start the search for the minimum, and of several ways to weight the epoch averaging, flux weighting based on the stationary-fit single-frame flux solutions was found to be the most effective in reproducing motions of known moving objects. The set of photometric parameters computed for the PM fit is a small subset of that for the stationary fit (see section II.1.a.)

**V.3.b.ii.1. Review of the Stationary Model **

The All-Sky observational model is

Eq. 1 |

where *ρ _{λi}* is the flux in the

The reduced chi-square figure of merit is defined as

Eq. 2 |

where *N _{obs}* is the total number of usable pixels, and the circumflexes on

Once the minimum chi-square has been found, the primary component is output and the other solutions are discarded. The only exception to this involves active deblending. The source ensemble starts out composed of a primary component and nearby sources that were detected on the co-adds. The solution for the ensemble is called *passive deblending*. If the chi-square for that solution is too high, an attempt is made (subject to controllable thresholds) to find another source in the neighborhood that was not detected on the co-adds but is blended with the other ensemble sources. The details of this processing are discussed in section IV.4.c.iii of the All-Sky Release Explanatory Supplement. The bottom line is that if the reduced chi-square for the ensemble can be made smaller (by another controllable threshold) by including the new source, then it is kept and called an *actively deblended* source. Since this source is not in the list of coadd detections, it will not come up later as a primary component, and so it is output immediately after the primary component. Although it is theoretically possible to have more than one actively deblended component, in practice only one per blend group was permitted, and the number of secondary passive-deblend components was limited to two.

**V.3.b.ii.2 Addition of Motion Estimation**

The change made to the stationary observational model to produce the PM model in the AllWISE version of WPHOT is to replace **s**_{n} for the primary component with a frame-dependent position that is a linear function of time. Since the primary component is usually component number 1 in the ensemble, for notational convenience we will just assign it *n* = 1 herein. Then

Eq. 3 |

where **s**_{10} is the position of the primary component at a fiducial time *t _{0}*,

The initial estimates for the components of ** μ** are zero. These will evolve to non-zero values if this reduces chi-square. The influence of the motion on chi-square is through its displacements of the PSF in the frames of the co-add stack. So the expression for the reduced chi-square becomes

Eq. 4 |

The inclusion of the 2-vector ** μ** enlarges

The uncertainty estimates for motion are obtained in a manner corresponding to the parameter uncertainty estimation of previous versions of WPHOT. The only difference is the inclusion of the motion terms in the model and chi-square. The Fisher information matrix G is computed, and the error covariance matrix γ expressing the parameter uncertainties is the inverse of G:

Eq. 5 |

where **ρ** is the vector of all pixels used for the calculation, with *N _{obs}* elements, and

The dependence of γ on ** μ** is via the dependence of

**V.3.b.ii.3. Gradient-Descent Algorithm**

Section IV.4.c.iii of the Explanatory Supplement to the WISE All-Sky Data Release Products defines the figure of merit used to estimate point-source fluxes and position. This is summarized in section V.b.3.ii.1 above, and the addition of source motion is described in section V.b.3.ii.2 above.

Both the stationary and the PM models are implemented in AllWISE. In each case the best solution is taken to be that which minimizes the model's chi-square. Because the models are nonlinear in the parameters to be estimated, the solutions cannot be obtained in one step by matrix inversion, and an iterative method is required. As with all iterative methods for solving nonlinear systems, starting estimates are needed. The stationary solution is obtained first, with starting estimates provided by the MDET positions and the aperture fluxes derived from the measurements at that position. Then the PM solution begins at the stationary-solution positions with initial motion estimates of zero. From the starting estimates, improved estimates (i.e., which reduce chi-square) are sought via the gradient-descent method. That is, the gradient of chi-square in the parameter hyperspace is computed, and chi-square is evaluated at successive points in the negative direction of the gradient until a minimum is detected.

The stationary model parameter hyperspace is 6-dimensional in AllWISE for a single source, namely two position parameters (one per axis) and four fluxes. The PM model adds two motion parameters (one per axis) for the primary component of a passive-blend group. In passive-blend groups containing more sources than just the primary component, another six dimensions are added for each additional source in both models. Given estimates for the position(s), which are time-dependent for the primary component in the PM model, the fluxes can be computed directly from the frame data by fitting to the PSF template. This last step is a linear problem. Because of PSF uncertainty, the flux uncertainties do depend on the flux, which makes the problem nonlinear in principle, but whenever the model is evaluated, fluxes from a previous evaluation are used to compute the PSF component of uncertainty, starting with the initial aperture fluxes. Once new estimates of the fluxes are available, chi-square can be computed. So the problem becomes one of efficiently finding position and motion values for which the corresponding flux estimates minimize chi-square.

Since the stationary model is a subset of the PM model, we will focus on the latter, with the understanding that references to motion do not apply to the stationary model. The position and motion model parameters are represented formally by a vector denoted * P*, and the algorithmic description is the same for both models except for the length of the

The location of the chi-square minimum in the * P* space is found by searching for it at discrete locations. A step size

andgC_{0}are calculated at the base location, and the gradient magnitude || is computed; if |g| is zero, the minimum has been found, and the search ends, otherwise the following actions are taken.g

- A step Δ
P = S/|is computed, the step counterg|nis initialized at 0, and then the following operations are performed:

- The components of the
P_{n+1}vector are computed:P_{n+1}(i) =(P_{n}i)-ΔP(gi) for all 2N+2 componentsiof the vectors;

Cis evaluated at_{n+1}P_{n+1};

- if
Cor_{n+1}> C_{n}n+1 ≥ n(where_{max}nis 100 for the stationary solution and 250 for the PM solution), then terminate this part of the search, otherwise increment_{max}nand resume at step B1 above.

- Take one step back:
P_{n}(i) =P_{n+1}(i) + ΔP(gi), increment the iteration counter.

- Compute Δ
C= 2|C_{0}-C| and_{n}C_{min}= 10^{-3}|C_{0}+C_{n}+ 10^{-10}|.

- If
C≥_{n}C_{0}then go to step F, otherwise if ((ΔC≤C_{min}) or (number of iterations is 100)) then end the search withP_{n}, otherwiseP_{0}⇐P_{n}, go back to step A.

- If (Δ
C≤ 0.01C_{min}) or (number of iterations is 100) then end the search with, otherwise continue.P_{n}

- The search has overshot the solution;
P_{0}⇐,P_{n}S⇐S/2; go back to step A.

Subtracting Δ*P g*(

In the stationary solution, when there is only one source in the passive-blend group, the * P* vector has a length

When the stationary solution is obtained for a blend group containing more than the primary source, then in general all sources have significantly non-zero components of * U* on their position axes, and no individual source gets a radial step of 0.0275 arcsec, so no quantization is conspicuous. The same is true if step G is reached, because the step size is cut in half one or more times. If one knows to look, some quantization can occasionally be detected at 0.0275

The sharing of the ** U** vector's components over all

Figure 1 - A plot of RA and Dec motion measurements for detections in a typical Tile that illustrates the motion quantization. |

This manifestation of quantization is the one in which the phenomenon was discovered, after which the cause for quantization in stationary position and PM motion was reconstructed. Since the PM solution has motion parameters only for the primary source in a blend group, the presence of other sources in that group does not significantly disturb the motion quantization in most cases.

**V.3.b.ii.4. Effect of Noise on the Motion Solution**

Chi-square minimization is one of the most powerful methods for estimating model parameters from measurements subject to noise, but the mere presence of noise in the measurements virtually assures the presence of some error in the estimated parameters. This can be quite noticeable when the correct value of a parameter is essentially zero (i.e., extremely small), since any non-zero result is purely the product of noise.

When the parameter being estimated is source motion on the sky, the fact that most sources detectable by WISE have true motions over the observational time span below the WISE spatial resolution leads inevitably to an abundance of pure-noise estimates. In most cases, these noise-produced motions are typically of the order of the motion uncertainty, i.e., not statistically significant. For faint low-S/N sources, however, it sometimes happens that chi-square can be reduced by fitting photometric noise variations in the single-frame observations. It is actually rather rare not to be able to achieve some chi-square reduction by finding some spurious motion that seems partially to explain the photometric variations that are actually just due to noise. This is a common hazard of nonlinear models with a large number of fitting parameters.

In some cases, this stretching of the truth proceeds to rather large motion estimates that may appear significant relative to their uncertainties, because the formal uncertainties have nothing to do with goodness-of-fit (only chi-square can indicate that, and it can fail in these cases because it is what is specifically being minimized), they depend only on the prior uncertainties due to measurement noise, which need not be particularly large to remain compatible with large flat regions in the chi-square hyperspace.

Even chi-square can be unreliable when some implausibly large motion allows the model to be fit better than the stationary model, but the formal motion uncertainties can definitely be misleading, because what they indicate is not goodness-of-fit but rather the uncertainty in whether the model parameters are the best possible compared to other values of the model parameters. In other words, a linear fit to low-noise parabolic data may have very small parameter uncertainties despite a terrible chi-square goodness-of-fit value, because the parameter uncertainties reflect only how good that linear fit is compared to all other possible linear fits. When the model doesn't fit the data, the parameter uncertainties have little to do with anything, and when photometric noise fluctuations prevent a source's true non-moving nature from being obvious, very large spurious motion estimates can result. Chi-square, having been minimized, is more a metric of how the measurements don't rule out large motion than an argument that the motion must be large.

These situations do have a rather characteristic behavior in the PM solution. Because the chi-square hypersurface has no crystal-clear minimum, the search for a minimum over the noisy terrain (see section V.3.b.ii.3 above) can drift like a feather in the wind. But the gradient-descent algorithm has arbitrary surrender points when it seems that it is getting nowhere, and one of these is when it runs off the end of the current iteration's maximum number of steps. This corresponds to the algorithm step B.3 in the previous section; the PM solution uses nmax = 250, and that many steps of size 0.0275 arcsec/yr amounts to 6.875, which is where subsequent steps in the algorithm decide to punt. Thus there is a "pile-up" of motion estimates at or near 6.875 arcsec/yr, and these should essentially all be considered pure-noise solutions.

Figure 2 shows this conspicuous ring for a typical tile. The quantization shown in Figure 1 in section V.3.b.ii.3 is also present but cannot be resolved at this scale. The use of nmax = 250 in the PM solution was chosen specifically to force this "pile-up" to occur at an implausibly large radial-motion value.

Figure 2 - The Noise-Solution "Pile-Up" at 6.875 arcsec/year |

Thus there is a "pile-up" of motion estimates at or near 6.875 arcsec/yr, and these should essentially all be considered pure-noise solutions.

Last update: 19 November 2013