# Daytime sky polarization calibration limitations

**David M. Harrington**

National Solar Observatory, 3665 Discovery Drive, Boulder, Colorado 80303, United States

Kiepenheuer-Institut für Sonnenphysik, Schöneckstr. 6, D-79104 Freiburg, Germany

University of Hawaii, Institute for Astronomy, 2680 Woodlawn Drive, Honolulu, Hawaii 96822, United States

**Jeffrey R. Kuhn**

University of Hawaii, Institute for Astronomy Maui, 34 Ohia Ku Street, Pukalani, Hawaii 96768, United States

**Arturo López Ariste**

IRAP CNRS, UMR 5277, 14, Avenue E Belin, Toulouse, France

*J. Astron. Telesc. Instrum. Syst*. 3(1), 018001 (Jan 24, 2017). doi:10.1117/1.JATIS.3.1.018001

#### Open Access

## Abstract

**Abstract.**
The daytime sky has recently been demonstrated as a useful calibration tool for deriving polarization cross-talk properties of large astronomical telescopes. The Daniel K. Inouye Solar Telescope and other large telescopes under construction can benefit from precise polarimetric calibration of large mirrors. Several atmospheric phenomena and instrumental errors potentially limit the technique’s accuracy. At the 3.67-m AEOS telescope on Haleakala, we performed a large observing campaign with the HiVIS spectropolarimeter to identify limitations and develop algorithms for extracting consistent calibrations. Effective sampling of the telescope optical configurations and filtering of data for several derived parameters provide robustness to the derived Mueller matrix calibrations. Second-order scattering models of the sky show that this method is relatively insensitive to multiple-scattering in the sky, provided calibration observations are done in regions of high polarization degree. The technique is also insensitive to assumptions about telescope-induced polarization, provided the mirror coatings are highly reflective. Zemax-derived polarization models show agreement between the functional dependence of polarization predictions and the corresponding on-sky calibrations.

Polarization calibration of large telescopes and modern instruments is often limited by the availability of suitable sources for calibration. Several calibration techniques exist using stars or the sun, internal optical systems, or *a priori* knowledge of the expected signals, but each technique has limitations. For altitude–azimuth telescopes, coudé or Nasmyth instruments, or telescopes with off-axis primaries, the polarization calibration usually requires bright, highly polarized sources available over a wide range of wavelengths, altitude–azimuth pointings. In night-time astronomy, polarized standard stars are commonly used, but they provide very limited altitude–azimuth coverage are faint, and have low polarization amplitudes (typically $<5%$^{1}^{–}^{3}). Unpolarized standard stars also exist, but they are also faint and provide limited altitude–azimuth coverage. Solar telescopes can use solardisk-center as a bright, zero-polarization target, provided there is no magnetic field activity. Solar observations often lack bright, significantly polarized targets of known properties. Smaller telescopes can use fixed polarizing filters placed over the telescope aperture to provide known input states that are detected and yield terms of the Mueller matrix as for the Dunn Solar Telescope.^{4}^{–}^{8} Symmetries of spectropolarimetric signatures from the Zeeman effect have been used in solar physics to determine terms in the Mueller matrix.^{9}^{,}^{10} Many studies have either measured and calibrated telescopes, measured mirror properties, or attempted to design instruments with minimal polarimetric defects (cf. Refs. ^{11}^{12}^{13}^{14}^{15}^{16}–^{17}). Space-based polarimetric instruments such as Hinode also undergo detailed polarization calibration and characterization.^{18}^{,}^{19}

Many telescopes use calibration optics such as large polarizers, polarizer mosaic masks, polarization state generators, or optical injection systems at locations in the beam after the primary or secondary mirror. For some systems, a major limitation is the ability to calibrate the primary mirror and optics upstream of the calibration system. These systems include telescopes with large primary mirrors, systems without accessible intermediate foci, or many-mirror systems without convenient locations for calibration optics. System calibrations are subject to model degeneracies, coherent polarization effects in the point spread function, and other complex issues such as fringes or seeing-induced artifacts.^{4}^{,}^{5}^{,}^{20}^{–}^{22} Modern instrumentation is often behind adaptive optics systems requiring detailed consideration of active performance on polarization artifacts in addition to deconvolution techniques and error budgeting.^{23}^{–}^{26} Modeling telescope polarization is typically done either with simple single-ray traces using assumed mirror refractive indices or with ray-tracing programs such as Zemax.^{11}^{,}^{24}^{,}^{27}

Every major observatory addresses a diversity of scientific cases. Often, cross-talk from the optics limits the polarization calibration to levels of 0.1% to $>1%$ in polarization orientation (e.g., ESPaDOnS at CFHT, LRISp at Keck, and SPINOR at DST^{4}^{,}^{28}^{–}^{30}). Artifacts from the instruments limit the absolute degree of polarization (DoP) measurements from backgrounds or zero-point offsets. Hinode and the Daniel K Inouye Solar Telescope (DKIST) project outline attempts to create error budgets, calling for correction of these artifacts to small fractions of a percent. The calibration techniques presented here aim to calibrate the cross-talk elements of the Mueller matrix to levels of roughly 1% of the element amplitudes, consistent with internal instrument errors. We also show that the limitation of the method is not the model for the polarization patterns of the sky but the other instrumental and observational issues.

The High-resolution Visible and Infrared Spectrograph (HiVIS) is a coudé instrument for the 3.67-m AEOS telescope on Haleakala, Hawaii. The visible arm of HiVIS has a spectropolarimeter, which we recently upgraded to include charge shuffling synchronized with polarization modulation using tunable nematic liquid crystals.^{31} In Ref. ^{31}, hereafter called H15, we outline the coudé path of the AEOS telescope and details of the HiVIS polarimeter.

The DKIST is a next-generation solar telescope with a 4-m-diameter off-axis primary mirror and a many-mirror folded coudé path.^{32}^{–}^{34} This altitude–azimuth system uses seven mirrors to feed light to the coudé lab.^{32}^{,}^{35}^{,}^{36} Its stated scientific goals require very stringent polarization calibration. Operations involve four polarimetric instruments spanning the 380- to 5000-nm wavelength range with changing configuration and simultaneous operation of three polarimetric instruments covering 380 to 1800 nm.^{6}^{,}^{35}^{–}^{37} Complex modulation and calibration strategies are required for such a multi-instrument system.^{35}^{,}^{36}^{,}^{38}^{–}^{41} With a large off-axis primary mirror, calibration of DKIST instruments requires external (solar, sky, and stellar) sources. The planned 4-m European Solar Telescope, though on-axis, will also require similar calibration considerations.^{42}^{–}^{45}

The following discussion of polarization formalism closely follows Refs. ^{46} and ^{47}. In the Stokes formalism, the polarization state of light is denoted as a four-vector: $Si=[I,Q,U,V]T$. In this formalism, $I$ represents the total intensity, $Q$ and $U$ are the linearly polarized intensities along polarization position angles 0 deg and 45 deg in the plane perpendicular to the light beam, respectively, and $V$ is the right-handed circularly polarized intensity. The intensity-normalized Stokes parameters are usually denoted as $[1,q,u,v]T=[I,Q,U,V]T/I$. The DoP is the fraction of polarized light in the beam: $DoP=Q2+U2+V2I=q2+u2+v2$. For this work, we adopt a term angle of polarization (AoP) from the references on daytime sky polarimetry, which defines the angle of linear polarization (AoP) as $ATAN(Q/U)/2$. The Mueller matrix is a $4\xd74$ set of transfer coefficients that describes how an optic changes the input Stokes vector ($Siinput$) to the output Stokes vector ($Sioutput$): $Sioutput=MijSiinput$. If the Mueller matrix for a system is known, then one inverts the matrix to recover the input Stokes vector. One can represent the individual Mueller matrix terms as describing how one incident polarization state transfers to another. In this paper, we will use the notation

The daytime sky is a bright, highly linearly polarized source that illuminates the telescope optics similar to distant targets (sun, stars, satellites, and planets) starting with the primary mirror. A single-scattering Rayleigh calculation is often adequate to describe the sky polarization to varying precision levels and is introduced in great detail in several text books (e.g., Refs. ^{48} and ^{49}).

There are many atmospheric and geometric considerations that change the skylight polarization pattern. The linear polarization amplitude and angle can depend on the solar elevation, atmospheric aerosol content, aerosol vertical distribution, aerosol scattering phase function, wavelength of the observation, and secondary sources of illumination such as reflections off oceans, clouds, or multiple scattering.^{50}^{–}^{66} Anisotropic scattered sunlight from reflections off land or water can be highly polarized and temporally variable.^{67}^{–}^{72} Aerosol particle optical properties and vertical distributions also vary.^{58}^{,}^{73}^{–}^{82} The polarization can change across atmospheric absorption bands or can be influenced by other scattering mechanisms.^{83}^{–}^{87} Deviations from a single-scattering Rayleigh model grow as the aerosol, cloud, ground, or sea-surface scattering sources affect the telescope line-of-sight. Clear, cloudless, low-aerosol conditions should yield high linear polarization amplitudes and small deviations in the polarization direction from a Rayleigh model. Observations generally support this conclusion.^{88}^{–}^{96} Conditions at twilight with low solar elevations can present some spectral differences.^{79}^{–}^{82}^{,}^{97}

An all-sky imaging polarimeter deployed on Haleakala also shows that a single-scattering sky model is a reasonable approximation for DKIST and AEOS observatories.^{95}^{,}^{96} The preliminary results from this instrument showed that the AoP agreed with single-scattering models to better than 1 deg in regions of the sky more than 20% polarized. We will show later how to filter data sets based on several measures of the daytime sky properties to ensure that second-order effects are minimized. The daytime sky DoP was much more variable, but as shown in later sections, the DoP variability has minimal impact on our calibration method. More detailed models include multiple scattering and aerosol scattering and are also available using industry standard atmospheric radiative transfer software such as MODTRAN.^{98}^{,}^{99} However, the recent studies on Haleakala applied measurements and modeling techniques to the DKIST site and found that the AoP was very well described by the single-scattering model for regions of the sky with DoP greater than 15%.^{95}^{,}^{96} The behavior of the DoP was much more complex and did not consistently match the single-scattering approximation. The technique we developed uses only the AoP.

Sky polarization modeling is well represented by simple single-scattering models with a few free parameters. The simplest Rayleigh sky model includes single scattering with polarization perpendicular to the scattering plane. A single scale factor for the maximum degree of linear polarization ($\delta max$) scales the polarization pattern across the sky.

The all-sky model requires knowing the solar location and the scale factor ($\delta max$) to compute the DoP and the AoP projected onto the sky. The geometry of the Rayleigh sky model is shown in Fig. 1. The geometrical parameters are the observer’s location (latitude, longitude, and elevation) and the time. The solar location and relevant angles from the telescope pointing are computed from the spherical geometry in Fig. 1. The maximum DoP ($\delta max$) in this model occurs at a scattering angle ($\gamma $) of 90 deg. The Rayleigh sky model predicts the DoP ($\delta $) at any telescope pointing (azimuth and elevation) as

**F1 :**

The celestial triangle representing the geometry for the sky polarization computations at any telescope pointing. $\gamma $ is the angular distance between the telescope pointing and the sun. $\theta s$ is the solar angular distance from the zenith. $\theta $ is the angular distance between the telescope pointing and the zenith. $\phi $ is the angle between the zenith direction and the solar direction at the telescope pointing. The angle $\psi $ is the difference in azimuth angle between the telescope pointing and the solar direction. The input $qu$ components are derived as sin and cos of $\psi $, respectively. The law of cosines is used to solve for any angles needed to compute DoP and position AoP.

**F2 :**

Various Rayleigh sky model parameters computed in a range of projections for mid-morning on January 27th, 2010, on Haleakala when the sun was at an elevation of 50.5 deg and an azimuth of 172 deg. The single-scattering model was scaled to a maximum DoP of 100% ($\delta max$). (a) The model DoP with white at 100% and black as 0% for all altitudes and azimuths plotted on a Cartesian rectangular grid. (b) The same DoP model data but in an orthographic projection with North up and East left. (a, b) show equivalent DoP data just with differing projections. (d) The scattering angle in an orthographic projects. This scattering angle shows the input linear polarization angle in the reference frame of the telescope, which always has the +elevation axis point. (e, f) $q$ and $u$ in an orthographic projection where white is $+1$ and black is $\u22121$. The coordinate system for $qu$ was chosen to be $+1$ in the $+altitude$ direction of an altitude–azimuth system. This system has a singularity at the zenith where the telescope optics can degenerately point to the zenith with any azimuth. The $+altitude=+q$ system is referenced to the optical train through the orientation of the primary mirror mount against the sky. (g, h, i) Properties of the sky polarization model on the horizon. (g) The DoP with peaks in the East and West. The scattering angle in (h) shows sign changes at the solar azimuth of 172 deg. (i) shows $q$ as the solid black line and $u$ as the dashed line. Stokes $u$ changes sign at the solar azimuth of 172 deg.

We model the $3\xd73$ cross-talk elements ($quv$ to $quv$ terms) of the Mueller matrix as a rotation matrix.^{100} This method makes the assumption that a telescope with weakly polarizing optics can have a Mueller matrix that is well represented by a rotation matrix. We find cross-talk of 100%, but the induced polarization and depolarization terms are less than 5%. A rotation matrix has been a good fit to our past data, is predicted by our Zemax modeling, and is easily described with three Euler angles to produce the nine terms of the cross-talk matrix.^{31}^{,}^{101} We also perform a sensitivity analysis in later sections to show that this approximation is reasonable. We find in the appendices that we can neglect the first row and column of the Mueller matrix as the correction to the inner $quv$ to $quv$ terms is second-order in these neglected terms.

For our procedure, all Stokes vectors are scaled to unit length (projected onto the Poincaré sphere) by dividing the Stokes vector by the measured DoP. This removes the residual effects from changes in the sky DoP, telescope-induced polarization, and depolarization. Since we ignore the induced polarization and depolarization, we consider only the $3\xd73$ cross-talk elements as representing the telescope Mueller matrix. We denote the three Euler angles as ($\alpha ,\beta ,\gamma $) and use a short-hand notation where $cos(\gamma )$ is shortened to $c\gamma $. We specify the rotation matrix ($Rij$) using the $ZXZ$ convention for Euler angles as

When using this rotation matrix approximation for the telescope Mueller matrix, the Rayleigh sky input Stokes parameters multiply each term of the rotation matrix to give a system of equations for the three Euler angles ($\alpha $, $\beta $, $\gamma $). This system of equations can be solved using a normal nonlinear least-squares minimization by searching the ($\alpha $, $\beta $, $\gamma $) space for minima in squared error. This direct solution of this set of equations using standard minimization routines is subject to several ambiguities that affect convergence using standard minimization routines. The details of our methods for deriving Euler angles and an example of how one could plan sky calibration observations are outlined in the Appendix of Ref. ^{100}.

Equating Mueller matrix elements to rotation matrix elements, we can write the system of equations for the three Euler angles. This system of equations can be solved using a normal nonlinear least-squares minimization by searching the ($\alpha $, $\beta $, $\gamma $) space for minima in squared error. With the measured Stokes vector ($Si$), $i=(1,2,3)$, the Rayleigh sky input vector ($Rj$), $j=(1,2)$, and a rotation matrix ($Rij$), we define the error ($\epsilon $) as $\epsilon 2(\alpha ,\beta ,\gamma )=\u2211i=13\u2211j=12[Si\u2212RiRij(\alpha ,\beta ,\gamma )]2$. For $n$ measurements, this gives us $3\xd7n$ terms. This solution is easily solvable in principle but has ambiguities. An alternative method for the direct least-squares solution for Euler angles is done in two steps. First, we directly solve a system of equations for the Mueller matrix elements that are not subject to rotational ambiguity. With the estimated Mueller matrix elements in hand, we can then perform a rotation matrix fit to the derived Mueller matrix element estimates. This two-step process allows us to use accurate starting values to speed up the minimization process and to resolve Euler angle ambiguities. When deriving the Mueller matrix elements of the telescope, one must take care that the actual derived matrices are physical. For instance, there are various matrix properties and quantities one can derive to test the physicality of the matrix.^{102}^{–}^{105} Noise and systematic errors might give overpolarizing or unphysical Mueller matrices. By fitting a rotation matrix, we avoid unphysical matrices.

The normal solution for Mueller matrix elements can be computed via the normal least-squares method. We can rearrange the time-varying Rayleigh sky inputs to ($Rij$) for $i$ independent observations and $j$ input Stokes parameters. The measured Stokes parameters ($Si$) become individual column vectors. The unknown Mueller matrix elements are arranged as a column vector by output Stokes parameter ($Mj$). If we write measured Stokes parameters as ($qmi,umi,vmi$) and the Rayleigh input Stokes parameters as ($qri,uri$), we can explicitly write a set of equations for two Mueller matrix elements

This solution is stable provided a diverse range of input states are observed to give a well-conditioned inversion. The noise properties and inversion characteristics of this equation can be calculated in advance of observations and optimized. We can write the matrix $A$ with an implied sum over $i$ observations for each term. As an example for a single element, if we compute the inverse of $A$ and multiply out $A\u22121$ for the $QQ$ term, we can write

The solution to the equations for the three sets of Mueller matrix elements is outlined in the Appendix of Ref. ^{100}. In this manner, we can easily implement the usual matrix formalism with a time-series of daytime sky observations to measure six Mueller matrix elements.

The assumption of a single-scattering model for computing the daytime sky polarization is incorrect under some circumstances. Multiple scattering, contributions from multiple light sources (upwelling, cloud reflections, and ocean reflections) all complicate the computation of the DoP and associated linear polarization angle. In this section, we outline a second-order scattering model and show how this model can be used to choose calibration observations to avoid such issues. By planning observations in regions of the sky where multiple scattering issues are minimized, this calibration technique can be efficiently used with a simple single-scattering model.

We show here that the common two-component multiple-scattering model imparts minimal changes to the AoP in wide regions of the sky. Several additions to the single-scattering model are possible but behave similarly. Along any line of sight in the sky, there are contributions from the single-scattered sunlight along with multiply scattered light off a range of airborne- and ground-based sources as well as extinction. Contributions from Mie scattering of water droplets, ice crystals, or large aerosols modify the models in complex ways. As an example of the variations between the single-scattered Rayleigh model and a simple multiple-scattering model, we follow the mathematical formalism of Ref. ^{66} to derive general properties of the polarization imprinted from the most common multiple scattering source.

In their notation, they use $\zeta $ to denote the location of a point on a stereographic projection of the sky. In Cartesian geometry, $\zeta =x+iy$. In polar coordinates, $\zeta =r\u2009ei\phi $. In Ref. ^{66}, they used the term $w$ to represent the polarization pattern across the sky. By breaking the exponential equation into an amplitude term $|w|$ and a complex orientation term $\gamma (\zeta )$, they represent the stereographic projection for the sky polarization pattern as $w(\zeta )=|w|e2i\gamma (\zeta )$. For the single-scattering case, this simple relation behaves as $\zeta 2$ and can be scaled to an amplitude of 1 and written in polar coordinates ($r,\phi $) as $w(\zeta )\u223c\zeta 2=r2e2i(\phi \u2212\pi 2)$.

To add multiple scattering to this equation, we must consider the shift of the zero polarization points away from the solar and antisolar locations. These zero points are Brewster and Babinet points near the sun as well as the Arago and second Brewster point near the antisolar location. Several empirical results show that the singularities are found above and below the sun along the solar meridian. This generally follows from the empirical result that double scattering is the dominant contribution to multiple scattering in the typical locations surveyed. This double-scattering contribution is generally polarized in the vertical direction as it represents the light scattered into the line of sight from the integrated skylight incident on all points along the line of sight. When the sun is low in the horizon, the low DoP regions of the sky are also low on the horizon. This double-scattering contribution is of the same amplitude as the single-scattered light when the single-scattered light is weak and horizontally polarized, which occurs above and below the sun at low solar elevations during sunrise and sunset.

The simplest perturbation to the model is to add a constant that represents a small additional polarization of assumed constant orientation, denoted as $A$. Following Ref. ^{66}, the zero polarization singularities fall at the locations of $\zeta =\xb1iA$, which corresponds to a Cartesian $y$ value of $\xb1A$. To make the singularities at the antisun location, the equation was generalized to $w(\zeta )\u223c(\zeta 2+A2)(\zeta 2+1A2)$.

A simple example of this two-term scattering model is shown in Fig. 3. The stereographic projection convention has been used. In this case, we put the sun on the horizontal axis to match the North = up convention of Fig. 2. However, in this formalism, the Stokes $qu$ parameters are not referenced to the altitude–azimuth frame and there is no singularity at the zenith. An angular splitting of 27 deg was chosen, and the sun is at an elevation of 89 deg. This solar elevation puts the sun in the center of the image with the horizon projected on the edge of the circle.

**F3 :**

The multiple-scattering model with a splitting constant of $\delta =4ATAN(A)=27\u2009\u2009deg$. All projections are stereographic with North up and East left. The sun is at the Zenith. (a) Stokes $q$ and (b) Stokes $u$ in the altitude–azimuth frame. The gray-scale corresponds to $white=1$ and $black=\u22121$. (c) The DoP with black as 0 and white as 100%. The AoP computed as 0.5ATAN($q,u$) is shown in (d). The two polarization zero points are seen as the split singularities near the zenith in the center of the AoP image. The AoP is linearly scaled from black to white from 0 deg to 180 deg.

The calibration method we have pursued is based on the assumption that the AoP of the sky polarization pattern is known as a modeled input parameter with a high degree of accuracy. Variations between the single Rayleigh scattering model and the real input Stokes vector can cause errors in our calibration methodology.

Figure 4 shows the AoP variations between the simple single-scattering model and the multiple-scattering model, considering the double scattering term in a stereographic projection for a range of multiple-scattering models. In the regions of highest DoP at scattering angles of 90 deg, the difference between this second-order model and the simple Rayleigh model is less than 0.001 deg, as can be seen in Fig. 5. The band of high DoP following the 90-deg scattering angle arc shows similar agreement in AoP. Regions near the neutral points show strong angular variation. This is in agreement with the all-sky imaging polarimetry on Haleakala.^{95}^{,}^{96}

**F4 :**

The angular differences (in deg) between the single-scattering Rayleigh model and the multiple-scattering model outlined above with the double-scattering term. We use a log color scale and chose a separation of $\delta =4ATAN(A)=12\u2009\u2009deg$ and the sun is at 10 deg elevation. A region within 20 deg of the sun has been masked and shows up as a black circle. The scale bar on the right shows the color scheme with white as 0.1 deg AoP angular difference and black as 0.001 deg AoP difference. The line of symmetry between the sun and the antisolar location is a region of minimal difference as is the 90-deg scattering plane shown as the curved black arc in this stereographic projection.

**F5 :**

The AoP variation between the single-scattering Rayleigh model and the multiple-scattering model here considering double scattering. For this figure, the sun was placed on the horizon at an elevation of $\alpha =0\u2009\u2009deg$. Each curve shows a trace from horizon to the zenith (elevation 90 deg along the 90 deg scattering plane for maximum DoP). The different colors correspond to different splitting angles $\delta =4ATAN(A)$ of 3 deg up to 27 deg in steps of 3 deg. As the double-scattered term grows stronger and the splitting angle increases, the AoP variations between single Rayleigh scattering and this multiple-scattering model increases from 0.005 deg up to and approaching 0.4 deg. However, at the 90-deg scattering location, the angular differences between single- and double-scattering models drop significantly.

In this section, we outlined a second-order scattering model that included two components contributing to the polarization pattern of the sky. We showed that by choosing regions of the sky with high DoP, one can avoid several contaminations of the AoP to a small fraction of a degree as shown in Fig. 5. Choose points near the 90 deg scattering plane and away from the horizon with high airmass to avoid multiple scattering contamination when using this calibration technique.

This technique requires a diversity of input polarization angles to minimize noise propagation when deriving telescope Mueller matrices. There is an analogy between the time-dependent Rayleigh sky input polarization and the retardances chosen to create an efficient modulation scheme for polarization measurements. By choosing telescope pointings and observing times such that the solution for Mueller matrix elements is well conditioned (efficient modulation by the daytime sky), a good calibration can be derived. Polarimeters typically produce intensity modulations by changing the incident polarization state with retardance amplitude and orientation changes. This retardance modulation translated into varying intensities using an analyzer such as a polarizer, polarizing beam splitter, or crystal blocks such as Wollaston prisms or Savart plates. These modulation schemes can vary widely for various optimizations and schemes to maximize or balance polarimetric efficiency over user-chosen Stokes parameters, wavelengths, and instrumentation systems.^{106}^{–}^{111} There have been many implementations of achromatic and polychromatic designs in both stellar and solar communities.^{40}^{,}^{41}^{,}^{112}^{–}^{118} In the notation of these studies, the instrument modulates the incoming polarization information into a series of measured intensities ($Ii$) for $i$ independent observations via the modulation matrix ($Oij$) for $j$ input Stokes parameters ($Sj$): $Ii=OijSj$. This is analogous to our situation in which we changed the matrix indices to be $i$ independent Stokes parameter measurements for $j$ different sky input Stokes parameters: $Si=RijMj$.

In most night-time polarimeters, instruments choose a modulation matrix that separates and measures individual parameters of the Stokes vector, typically called a Stokes definition modulation sequence

Other instruments choose a wide range of modulation schemes to balance the efficiencies over a number of exposures. One recovers the input Stokes vector from a series of intensity measurements by inverting the modulation matrix ($O$) via the normal least squares formalism: $S=OTIOTO$. The demodulation matrix is typically defined as $Dij=[OTO]\u22121OT$.

In our daytime sky technique, the Rayleigh sky input parameters become the modulation matrix ($Oij=Rij$) and the formalism for noise propagation developed in many studies such as Refs. ^{107} and ^{108} apply. If each measurement has the same statistical noise level $\sigma $ and there are $n$ total measurements, then the noise on each demodulated parameter ($\sigma i$) becomes $\sigma i2=n\sigma 2\u2211j=1nDij2$. The efficiency of the observation becomes $ei=(n\u2211j=1nDij2)\u221212$.

One must take care with this technique to build up observations over a wide range of solar locations, so the inversion is well conditioned, as outlined in the Appendices of Ref. ^{100}. The path of the sun throughout the day will create regions of little input sky Stokes vector rotation causing a poorly constrained inversion with high condition number. For instance, at our location in the tropics, the sun rises and sets without changing azimuth until it rises quite high in the sky. We are constrained to observing in early morning and late evening with the dome walls raised since we may not expose the telescope to the sun. This causes input vectors at East-West pointings to be mostly $q$ oriented with little rotation over many hours. Observations at other times of the year or at higher solar elevations are required to have a well-conditioned inversion. One can easily build up the expected sky input polarizations at a given observing site with the Rayleigh sky polarization equations. Then, determining the modulation matrix and noise propagation for a planned observing sequence to ensure a well-measured telescope matrix with good signal-to-noise (SNR) is straightforward.