Characterizing the astrometric precision limit
for moving targets observed with digital-array detectors^*

In statistics, the CRLB gives the lower bound for the variance of an estimated parameter: it can be represented as the inverse of the Fisher information, which characterizes the amount of information about this parameter contained in an observable random variable. MSL13 established the expression of the CRLB for the astrometric precision for a linear array detector, with the measurement noise driven by a Poisson distribution (the adoption of this probabilistic model is common in contemporary astrometry (e.g., in Gaia, see Lindegren, (2008))). The most generic expression for the CRLB given in MSL13 (their Eq. (11)) can be written as follows:

$\sigma_{CR}^{2}$ is the CRLB for the variance of the PSF center $x_{c}$ of a source observed with a one-dimensional array. The subscript $i$ allows identifying quantities relative to the pixel of index $i$: $\tilde{F}_{i}\left(x_{c}\right)$ represents the flux in pixel $i$ (in photo-$e^{-}$ on the detector), whereas the background flux in the same pixel is denoted by $\tilde{B}_{i}$ and includes contributions from the detector such as dark-current and read-out noise ($RON$), as well as contributions from the sky background (see Eqs. (22) and (37)). Throughout this paper, we assume for simplicity that $\tilde{B}_{i}$ is uniform (constant) under the source (and equal to $\tilde{B}$ in one pixel).

The function $\tilde{F}_{i}\left(x_{c}\right)$ involved in Eq. (14) can be expressed by the integral over pixel $i$ of the one-dimensional PSF ($\Phi$) of the source centered at $x_{c}$ as follows:

Where the integrals bounds $x_{i}^{-}$ and $x_{i}^{+}$ equal $x_{i}-\Delta x/2$ and $x_{i}+\Delta x/2,$ respectively (with $x_{i}$ the coordinate of the center of pixel $i$ and $\Delta x$ the pixel width). The other parameters have the same definition as in Sect. 2: $\tilde{F}$ is the source total flux and $\overline{\Phi}$ the normalized PSF. We note that our function $\overline{\Phi}_{S}$ that we defined in the previous section is the same as the function adopted by MSL13 for the normalized PSF to express $\tilde{F}_{i}$ in Eq. (14).

In a similar way, we can obtain the CRLB expression for a moving object by substituting the function $\overline{\Phi}$ in Eq. (15) by the one-dimensional expression of the spread function $\overline{\Phi}_{M}$ of a moving point source given by Eq. (9). The resulting expression for the CRLB in this case is the following:

where

and where all the parameters and functions have been defined in Sect. 2.

In this subsection, we study the behavior of Eq. (16) when the background in each pixel follows Eq. (22), which corresponds to a realistic expression for the background on a CCD during an astronomical observation,

where $b$ is the sky-brightness component of the background (in units of $e^{-}$/arcsec), while $D$ and $RON$ are the dark current (mostly negligible for modern optical semi-conductor detectors) and the standard deviation of the read-out noise of the detector (in $e^{-}$), respectively. We note that the requirement of a realistic estimator for the noise variance and a noise following a Poisson distribution to apply the CRLB expression (16) leads us to assume that the $RON$ component of the background also follows a Poisson distribution.

The three solid lines (thin, normal, and bold) in Fig. 2 show the square root of the CRLB as a function of detector pixel size for a non-moving source with a Gaussian PSF centered on a given pixel and for an FWHM of $0.5$, $1.0$ and $1.5$”, respectively. The dashed lines correspond to the same curves, but for a slow-moving source (with drifting parameter $L$ equal to the FWHM) whose PSF is also centered on a given pixel.

Figure 2 shows that for a given image quality, the CRLB minimum is understandably degraded by the motion of the source. We also point out that the shapes of the dashed curves (for a slow-moving source) and the solid curves (for a non-moving source) are similar. In particular, we can distinguish three regimes for the behavior of the CRLB according to the pixel size for a slow-moving source with a constant flux $\tilde{F}$ and a constant drifting parameter $L$, as described below.

• $\bullet$

  The oversampled phase: for pixel sizes considerably smaller than the image quality FWHM (in this case, $\Delta x<0.1$ arcsec), the background flux becomes preponderant. In this part of the curve, while the pixel size decreases, the part of flux that is due to the background increases compared to the flux of the source (since a part of the background flux is independent of the pixel size), and this leads to an increase in the CRLB (lower precision).

• $\bullet$

  The undersampled phase: for pixel sizes larger than the image quality FWHM, the main part of the flux from the source is contained in very few pixels and the CRLB increases quickly. In the same way as for a non-moving source, the value of pixel size for which this degradation occurs largely depends on the centering of the spread function inside the pixel (see below).

• $\bullet$

  The well-sampled phase between the two areas described above ($\Delta x\sim\frac{1}{2}$ FWHM), where the CRLB reaches a quasi-constant floor. We note that when the contribution that is due to the source of the total flux decreases, the size of the well-sampled area also decreases (e.g., compare the shape of the lower dashed line with the upper crossed line, which is computed for a source ten times fainter, all else being equal). The reason is that as the background increases, the oversampled area extends to larger pixel sizes, while the undersampled area begins at an unmodified pixel size, close to the FWHM. Then, for a total flux that is largely dominated by the background, the “well-sampled floor” will collapse into a unique point corresponding to an optimal pixel size in which the CRLB reaches its minimum.

We now consider the behavior of the CRLB when the PSF is not centered on a given pixel. Figure 3 shows a set of curves corresponding to the CRLB versus pixel size for an FWHM of $1.0$”, for three different values of the source drift, and for different values of the PSF center inside one pixel. The lowest curves of this figure correspond - like the intermediate dashed lines in Fig. 2 - to the CRLB of a slow-moving source with $L$ = FWHM. As in the case of the stationary source (see MSL13), we see that the decentering effect on the CRLB of a slow-moving source is almost negligible in the oversampled and well-sampled domains, but it plays a leading role in the undersampled case when the pixel size exceeds the FWHM.

We now focus on the behavior of the CRLB for faster sources (i.e., for sources with a drifting parameter $L$ larger than the FWHM). As intuitively expected, we see in Fig. 3 that the larger the drift of the source, the larger the degradation of the CRLB. We also observe a number of significant oscillations of the CRLB values according to pixel size that are not present for stationary or slow-moving sources. The number and amplitudes of these oscillations increase with increasing speed of the source. Their amplitudes form an insignificant part of the CRLB value itself in the case of small pixel sizes (for oversampled and well-sampled sources this effect is negligible), but the contribution becomes important in the undersampled “intermediate” domain that is bounded by pixel sizes between the image quality FWHM and the elongation of the source drift. In the undersampled domain that is defined by pixel sizes larger than the drifting parameter, the effect that is due to oscillations disappears and the decentering effect dominates the behavior of the CRLB (as for stationary or slow-moving sources).

The oscillations of the CRLB that are observed for fast-moving sources are due to the numerical discretization of the source PSF (by the detector array) for which some resonances occur when the ratio between the pixel size and the drifting parameter reaches some specific values. Notably, we observe some peculiar localizations (pixel size values) for which the decentering effect is almost suppressed when the source is affected by a specific fast drift, while this effect is preponderant when the same source is stationary or affected by a slow drift (e.g., compare the behavior of the CRLB for sources with drift parameters of $1.0$” and $2.0$” at a pixel size of $1.5$” in Fig. 3).

These peculiar locations can be of particular significance when performing accurate astrometry of fast objects (e.g., near-Earth objects (NEOs), artificial satellites, space debris, or meteors) even with small telescopes whose pixel size can be substantially larger than the typical seeing. When the speed of the source is known, it is indeed possible (and advantageous) to adapt the exposure time to increase or decrease the drifting parameter such that the decentering effect is suppressed or minimized given the pixel size of the camera in use.

For undersampled sources, it is not possible to simplify the expression given by Eq. (16) of the CRLB (the same is true for stationary sources), since no continuous approximation is practicable to avoid the large effect of the detector-array discretization. Fortunately, the minimum value of the CRLB is not in the undersampled domain, and the effect due to oscillations and decentering can be neglected in the other two phases of the curves, and especially in the oversampled domain, which is further analyzed in the next subsection.

In the oversampled case, as explained in MSL13, the CRLB can be simplified: the pixel width $\Delta x$ is considerably smaller than the FWHM of the PSF, and then Eq. (15) is well approximated by $\tilde{F}_{i}\left(x_{c}\right)=\tilde{F}\leavevmode\nobreak\ \overline{\Phi}\left(x_{i}-x_{c}\right)\Delta x$; in addition, the sum over all pixels involved in Eq. (14) can be approximated by a continuous integral over the interval $]-\infty,+\infty[$. We can then distinguish two limiting situations: the case when the background flux per pixel $\tilde{B}$ is clearly higher than the total flux of the source $\tilde{F}$, in which case Eq. (14) becomes

And the case when the background flux per pixel $\tilde{B}$ is clearly lower than the total flux of the source $\tilde{F}$, in which case Eq. (14) becomes

We note that even though these approximations have been developed by assuming a very small pixel size, the estimates and results deduced from them hereafter remain true in the well-sampled case with less accuracy, of course. This is especially visible in the next figures of this section, where the pixel size is consciously chosen to be relatively large (close to one-third of the FWHM).

To approximate the integrals involved in Eqs. (23) and (24) for a moving source, we first replace the normalized PSF $\overline{\Phi}$ by the 1D MoG function $\overline{\Phi}_{M}$ , and then we consider two distinct cases corresponding to slow- and fast-moving sources.

For a stationary source, the relation between exposure time and astrometric precision (according to the CRLB estimators given by the constant terms in Eqs. (3.3.1) and (26)) is trivial. For a non-moving source, the longer the exposure time, the higher the integrated flux of the source, which leads to an improved astrometric precision (the only limit is the saturation threshold of the detector). However, for a moving source, this is no longer true. Here, the longer the exposure time, the higher the integrated flux of the source, but - and this is the main difference to the stationary case - the larger the detector-array area covered by the source (because the distribution of the source flux drifts). As a result, as we increase the exposure time of a moving source, we add a comparatively larger noise that is due to the areal increase in the background.

In this subsection, we use the previously developed CRLB expressions for a moving source to determine the optimum exposure time that allows reaching the best astrometric precision. We note that instead of the CRLB exact expression (16) itself, we often use its approximations in the oversampled regime here because the CRLB minimum value is reached for pixel size values for which these simplified expressions still give a valid approximation of the CRLB (see Sect. 3.3). As a cautionary note, the estimate of this optimum exposure time is not correct when the pixel size falls into the intermediate or undersampled regimes.

In this scenario, the background is time dependent, and its expression becomes

where $b_{1}\leavevmode\nobreak\ T_{e}$ is the background part depending on exposure time $T_{e}$ , while $b_{0}$ is the time-independent part. $b_{1}$ depends on $b_{s}$, which is the sky component (in units of $e^{-}$/arcsec/sec) and $d,$ which is the dark-current component (in units of $e^{-}$/sec). $b_{0}$ depends on $RON,$ which is assumed to be independent of exposure time. We recall that the relation between the source total flux and exposure time is $\tilde{F}=f_{s}\leavevmode\nobreak\ T_{e}$ , while the relation between source speed and its drifting parameter is given by $L=V_{x}\leavevmode\nobreak\ T_{e}$. With these new notations, we compute the CRLB as a function of exposure time for four sources with a peak S/N bewteen $3$ and $90$ and a similar speed of $2.0$”/min. The pixel size, image FWHM, and background flux (whose RON component is negligible) are kept constant (see Fig. 6).

First we observe that the behavior of all CRLB curves is similar: with the increase in exposure time, the CRLB first decreases, then reaches a quasi-constant floor, and finally, unlike in the case of stationary sources, it reaches a turning point (where its minimum value is attained) and then starts to increase. When the RON is negligible (as for Fig. 6), the turning point is at an exposure time close to $30$ seconds, which corresponds to an elongation of the source drift close to the FWHM (equal to $1.0$” in the figure). When the RON component of the background increases, the turning point is located at a slightly longer exposure time.

We also note that the brighter the source, the longer the exposure time that is required to reach the minimum value of the CRLB (see the strong source case in Fig. 6). For the theoretical limiting case when the background flux is zero, this exposure time is infinity. An estimate of the corresponding minimum value of the CRLB in this theoretical case can be given accurately by expression (30), which is the limit when the exposure time approaches infinity in Eq. (28), which is valid for a high value of the drifting parameter,

Expression (30) is of particular significance. It gives a simple way to estimate an absolute limit for the astrometric precision of all moving sources with the knowledge of only three parameters, which are the FWHM, the flux of the source in electrons per second, and the source speed in arcseconds per second.

By contrast, the fainter the source, the smaller the size of the CRLB floor, and the more the optimum exposure time converges toward a lower limit that is close to the exposure time that corresponds to a drifting parameter that is equal to the image quality FWHM. We call $T_{o}$ the lower limit of the optimum exposure time (see Fig. 6) and $L_{o}$ the corresponding drifting parameter $L_{o}=V_{x}\leavevmode\nobreak\ T_{o}$. We can calculate $T_{o}$ as a power series by considering the roots of the derivative of expression (23) with respect to the exposure time when $\overline{\Phi}$ is replaced by $\overline{\Phi}_{M}$. Expression (23) in the oversampled area yields the limit of the expression (16) of the CRLB when the background dominates the total flux, and Fig. 6 shows that it is precisely in this regime that a lower limit of the optimum exposure time is achieved. Then, we can express $T_{o}$ as follows (see Appendix B for more details):

where $T_{s}$ is the exposure time corresponding to a drifting parameter that is equal to the image quality FWHM ($T_{s}=\text{FWHM}/V_{x}$), and $\mu_{b}$ measures the ratio between the time-independent and the time-dependent parts of the background flux for an exposure time equal to $T_{s}$ ($\mu_{b}=b_{0}/(b_{1}\cdot T_{s})$). The numerical coefficients in Eq. (31) have been confirmed by an estimate of $T_{o}$ based on a binary search algorithm applied on the exact expression (16): for $\mu_{b}$ below $0.2$, the agreement is better than one percent, but for $\mu_{b}$ larger than $0.4,$ this expression of $T_{o}$ is no longer valid. In particular, this means that when the RON component is an important part of the background, the lower limit $T_{o}$ of the optimum exposure time cannot be given by Eq. (31) and has to be determined numerically from the Eq. (16).

To obtain an estimate of the CRLB value for an exposure time equal to $T_{o}$, we replace the parameter $\overline{L}$ by $T_{o}V_{x}/(2\sigma)$ in expressions (3.3.1) and (26) for the case when the total flux is dominated by the background, and the case when the total flux is dominated by the source, respectively (these CRLB estimators can be used here since we have shown that they give correct results even for a drifting parameter $\overline{L}$ equal to one and a half times the FWHM). We obtain the following two expressions:

When $T_{o}$ is used as exposure time and when the RON is a negligible part of the total background, we can deduce from Eqs. (32) and (33) that the degradation of $\sigma_{CR}$ for a moving source is between $19\%$ and $33\%$ of the $\sigma_{CR}$ of the same source with no motion observed during the same exposure time. When the RON is not a negligible part of the total background ($\mu_{b}$ different from zero), the CRLB degradation that is due to the motion of the source is slightly larger (since $T_{o}$ is longer).

When the total flux is dominated by the source (as for the strong sources shown in Fig. 6), the CRLB of a moving source can reach a value slightly below the value given by expression (33). From the comparison between this expression and Eq. (30), however, we deduce that the maximum possible improvement for $\sigma_{o}$ is $7\%$, and only for a theoretical measurement without background, and with an exposure time equal to infinity.

In order to check the validity range of the previous expressions, we compute with the help of a binary search algorithm applied to Eq. (16) the exact minimum CRLB for moving sources with a speed of $2$”/min and with fluxes varying between $f_{s}=30$ $e^{-}$/min and $f_{s}=500000$ $e^{-}$/min. These sources are observed with a CCD pixel size of $0.3$”, an image quality FWHM of $1.0$”, and with a sky-background component $b_{s}=2000$ $e^{-}$/arcsec/min (the dark and the RON background components are equal to zero). Depending on the source brightness, the peak S/N varies between $0.7$ and $250$. Then, we plot in Fig. 7 for each S/N value the exact minimum CRLB value obtained by this numerical method (bold line) and their estimates with expression (32) (dots), with expression (33) (thin dashed line), and with expression (30) (thin solid line).

For weak sources (left part of Fig. 7) the exact determination and the estimate made with expression (32) agree well. When the source is stronger (right part of the figure), the agreement is better with the estimate deduced from expression (33), and except for the strongest source of this example ($S/N=250$), the value provided by this expression is below the exact minimum CRLB value. For the strongest source the estimator given by Eq. (33) is just above the exact value (less than $5\%$ higher). The unreachable limit given by expression (30) is always below the exact value and all the other estimates, as expected.

Optimum exposure time for a set of observations

Most of the time, the observation of a moving source consists of $N$ consecutive images of the source taken with a similar exposure time $T_{e}$. If during $T_{a}$ (the duration of the whole data set) the meteorology, the instrumental conditions, and the linear motion of the target remain unchanged, then the lower bound of the final variance of the astrometric precision of the whole data set is given by $\sigma_{i}^{2}/N$ (where $\sigma_{i}^{2}$ is the variance of the astrometric precision of one independent image). Note that it is beyond the scope of this paper to develop a method to combine the single measurements. Hereafter we only assume that an ideal method exists and deduce a lower bound for the astrometric precision of the ”mean measurement”.

The CRLB formalism can help us to estimate the optimum exposure time of a single image, which maximizes the astrometric precision of a whole data set. Let $T_{N}$ be this optimum exposure time, the number of frames $N(T_{N})$ contained in the data set are $(T_{a}+\Delta_{T})/(T_{N}+\Delta_{T})$, where $\Delta_{T}$ is the time delay between consecutive exposures. Then, the CRLB for the whole data set $\sigma_{a}^{2}(T_{N})$ equals $\sigma_{i}^{2}(T_{N})/N(T_{N})$ (where $\sigma_{i}^{2}(T_{N})$ is now the CRLB of one image).

In what follows we develop a method to graphically locate the optimum exposure time $T_{N}$ for a given source, when its characteristics ($V_{x}$ and $f_{s}$) as well as the observational conditions (FWHM, $T_{a}$, $\Delta x$, $\Delta_{T}$, $b_{s}$, $d,$ and RON) are assumed to be known. We note that finding the optimum exposure time $T_{N}$ that minimizes $\sigma_{a}$ is equivalent to finding the value of $T_{N}$ that maximizes the ratio of $\sigma_{a}^{2}(T_{s})$ to $\sigma_{a}^{2}(T_{N})$, where $T_{s}$, defined previously, is the exposure time corresponding to a drifting elongation equal to the FWHM. With the above notation, we can write this ratio as follows:

We note that the choice of the exposure time $T_{s}$ for the numerator of this ratio is arbitrary and that the same method can be applied with any fixed exposure time. The advantage of considering this ratio instead of $\sigma_{a}$ itself is that this ratio is independent of the total duration $T_{a}$ of the data set. In return, with this method, $N(T_{N})$ is not an integer a priori, and $T_{N}$ will need to be slightly increased or decreased to reach the closest integer $N$ that minimizes $\sigma_{a}$.

We compute and plot in Fig. 8 the CRLB ratio of Eq. (34) (where $\sigma_{i}$ is replaced by its exact expression (16)) according to the exposure time for a moving source with a drifting speed of $2$”/min observed with an FWHM of $1.0$”. The vertical lines at $T_{e}=30$ seconds correspond to $T_{s}$ (which is close to $T_{o}$ , but independent of RON, in contrast to $T_{o}$). In the left panel of the figure, we assume that RON is zero while the time delay $\Delta_{T}$ varies, and conversely, in the right panel of the figure, we assume that $\Delta_{T}$ is zero while RON varies. The star symbols correspond to the maximum of the CRLB ratio, and the abscissa gives the values of $T_{N}$ corresponding to each condition.

We observe that the maximum of the limiting curve when RON and $\Delta_{T}$ tend to zero (the two identical bold curves in the left and right panels) is reached for an exposure time equal to zero, which makes no sense. Another criterion has indeed to be taken into account: the optimum exposure time (according to the CRLB estimator) has to be long enough to allow the detection of the source. As a trivial example, we consider that a source is detected if the peak S/N is higher than a fixed threshold characterizing the reduction pipeline capacity (fixed to $3.5$ hereafter).

For instance, with the parameters of Fig. 8, the minimum exposure time needed to reach an S/N higher than $3.5$ is $13$ seconds when the RON is negligible (left panel), and it increases when the RON increases (for a RON larger than $21$ $e^{-}/pix$, the S/N is always below the threshold). Then, by adding this information concerning the detectability of the source in Fig. 8 (hatched areas), we can easily determine for each observational constraint a “realistic” exposure time $T_{N}$ for the whole data set by restricting the search of the CRLB ratio maximum to these hatched areas.

We see that $T_{N}$ increases quickly with the rise of $\Delta_{T}$ (left panel) or with the rise of RON (right panel): with $\Delta_{T}$ equal to only one-thirtieth of $T_{s}$, $T_{N}$ already equals one-third of $T_{s}$. We also observe that this increase seems bounded by $T_{s}$. Indeed, $T_{N}$ is logically shorter than the optimum exposure time of a single image (minimizing $\sigma_{i}$), and we have shown that even though this exposure time can be longer than $T_{o}$ (or $T_{s}$), the value of $\sigma_{i}^{2}(T_{o})$ is already close to the minimum of $\sigma_{i}$. We note that this bound does not exist for a stationary sources.

We do not show a power series for $T_{N}$ (as done for $T_{o}$ in Eq. (31)) because the location of the maximum of Eq. (34) is the result of a complex interaction between several parameters. However, the simplest and more straightforward numerical approach - an estimate of $T_{N}$ that is based on a binary search algorithm applied to the search of the maximum of the CRLB ratio - allows us to compute $T_{N}$ and then to deduce the number of images $N$ that will optimize the CRLB of a whole set of observation of duration $T_{a}$. This is particularly important when planning astrometric observations of moving targets. We note that This method can also be used without any modification to compute $T_{N}$ when the $N$ images are stacked before the reduction process (but the minimum exposure time for allowing detections is shorter since the S/N is multiplied by $\sqrt{N}$).

A two-dimensional array detector of $n_{x}$ columns and $n_{y}$ rows can be considered as a linear-array detector of $n_{x}\times n_{y}$ pixels. Then, by keeping the same formalism as in the previous section, we can deduce from Eq. (14) the expression below which this corresponds to the CRLB for the astrometric precision along the $X$-axis in the case of a two-dimensional detector array:

$\sigma_{CR_{X}}^{2}$ is the CRLB for the variance of the coordinate $x_{c}$ of the PSF center for a source observed with a two-dimensional array. The subscripts $i$ and $j$ now allow us to identify quantities relative to the pixel in column $i$ and row $j$. The function $\tilde{F}_{i,j}\left(x_{c},y_{c}\right)$ represents the flux contained in the pixel that is located in column $i$ and row $j$, and can be calculated through the integral of the normalized two-dimensional PSF ($\overline{\Phi}$) of the source centered at coordinates $(x_{c},y_{c})$. The expression for $\tilde{F}_{i,j}$ is

where $x_{i}^{-}=x_{i}-\Delta x/2$, $x_{i}^{+}=x_{i}+\Delta x/2$, $y_{j}^{-}=y_{j}-\Delta y/2$ and $y_{j}^{+}=y_{j}+\Delta y/2$ (with $\Delta x$ and $\Delta y$ the pixel width along $X$ and $Y,$ respectively). The background flux in the same pixel is called $\tilde{B}_{i,j}$. As previously, the background is uniform under the source (equal to $\tilde{B}$ in one pixel) and can be divided into three components: the $RON$ and the dark current ($D$), which are independent of the pixel size, and the sky-brightness component $b,$ which was proportional to the pixel size in the one-dimensional case (Eq. (22)) and which is proportional to the pixel area now in the two-dimensional case (Eq (37)),

CRLB approximations for the oversampled case

Similar to the case of the one-dimensional detector-array, when the pixel sizes in X and Y are small compared to the image quality FWHM, the flux in one pixel given by expression (36) is well approximated by $\tilde{F}_{i,j}=\tilde{F}\leavevmode\nobreak\ \overline{\Phi}\left(x_{i}-x_{c},y_{j}-y_{c}\right)\Delta x\Delta y$ and the sums over all pixels present in CRLB expression (35) can be replaced by continuous integrals over the whole space. We can then distinguish two extreme scenarios: the case when the background flux per pixel $\tilde{B}$ is clearly higher than the total flux of the source $\tilde{F}$ (faint sources), in which case Eq. (35) becomes

And the case when the background flux per pixel $\tilde{B}$ is clearly lower than the total flux of the source $\tilde{F}$ (bright sources), when Eq. (35) becomes

More generally, hereafter, we denote $\sigma_{CR_{U}}$ as the CRLB for the astrometric precision along any direction $U$ in the reference plane of the CCD. The expression for $\sigma_{CR_{U}}$ would be given by Eqs. (35), (38), or (39) (depending on the case involved), with the derivative of $\tilde{F}_{i,j}$ (or $\overline{\Phi}$) with respect to $x_{c}$ replaced by the derivative with respect to $u_{c}$ (where $u_{c}$ is the component of the instantaneous PSF center along the $U$-axis). For expressions (38) and (39), the integral over the whole space can also be performed with any coordinate system $(u,v)$ instead of the $(x,y)$ system adopted here since the pixel sizes are smaller than the FWHM.

The expression of the normalized PSF $\overline{\Phi}_{S^{2}}$ of a stationary source that is observed with a two-dimensional detector array is given by Eq. (2) of Sect. 2. Then, by using this normalized PSF in Eq. (36) to compute $\tilde{F}_{i,j}$ and its derivative and by substituting them into Eq. (35), we obtain the following expression for the CRLB along the $X$-axis for a stationary source that is observed with a two-dimensional detector array:

where

Hereafter, we assume that the pixel sizes along $X$ and $Y$ axes are equal ($\Delta x=\Delta y$).

The three solid lines in Fig. 10 (thin, normal, and bold) computed with Eq. (40) show the square root of the CRLB as a function of detector pixel size for a non-moving source centered on a given pixel and observed with a two-dimensional detector for an FWHM of $0.5$”, $1.0$”, and $1.5$”, respectively. We note that the overall behavior of these three lines is very similar to the three solid curves plotted in Fig. 2 for a non-moving source that is observed with a one-dimensional detector and based on Eq. (21) of MSL13. In particular, the three distinct regimes (oversampled, well-sampled, and undersampled) also exist in the two-dimensional case. We can observe, however, that in the two-dimensional case the oversampled regime is slightly more extended, while the well-sampled area begins for pixel sizes that are slightly larger.

Oversampled case approximations:

The CRLB expressions (38) and (39) with $\overline{\Phi}_{S^{2}}$ as normalized PSF allow us to produce simplified expressions for the CRLB of a stationary source that is observed with a two-dimensional array detector (the values of the reciprocals of the integrals involved in Eqs (38) and (39) for $\overline{\Phi}_{S^{2}}$ are given in Appendix A.1 by the functions $I_{S_{1}^{2}}$ and $I_{S_{2}^{2}}$ , respectively). Thus, the expression (40) is well approximated by expression (46) when the background dominates the total flux, and by expression (47) when the source dominates the total flux,

where $\sigma_{CR_{1D}^{B}}$ and $\sigma_{CR_{1D}^{F}}$ correspond to the CRLB expressions given by MSL13 in their Eqs. (39) and (42), respectively, for a stationary source that is observed with a one-dimensional detector. When the background dominates the total flux, we observe that the CRLB in the two-dimensional case is equal to the CRLB in the one-dimensional case multiplied by a factor that is inversely proportional to the pixel size and larger than 1 when $\Delta x<$FWHM. This factor explains that the degradation of CRLB in an oversampled regime occurs for a larger pixel size in the two-dimensional case than in the one-dimensional case. When the source dominates the total flux, the CRLB expression remains unchanged.

We also note that even though the above expressions (40), (46), and (47) are obtained for the CRLB along the $X$-axis, they remain rigorous and unchanged for the CRLB along any direction, since the PSF has been assumed to be circularly symmetric.

Finally, we present in this subsection the CRLB behavior for a moving source observed with a two-dimensional detector array. As a PSF of the moving source, we use $\overline{\Phi}_{M^{2}}$, the 2D MoG function given by Eq. (3) and presented in Sect. 2. This function is not circularly symmetric, and therefore the CRLB is not the same along all directions. However, compared to the stationary case, the main modification of the CRLB behavior is, of course, in the drifting direction. Thus we first study the CRLB along the drifting direction and show the similarities and differences with the results obtained in Sect. 3 for a linear detector. Then, we study the CRLB along the direction normal to the motion of the object - which is of course not present in the linear case.