Using the sky model

If you elect to model the sky, you will see some graphs showing the progress of the fitting procedure. First, if some bright stars were observed, you see a plot of sky data (for a given band and night) as a function of the adjacent star's brightness. A crude linear fit, indicated by + signs, shows the initial estimate of the star contribution to ``sky'' data. Second, if there are enough dark-sky data to model, you will see a plot of the dark-sky data (corrected for star light) as a function of airmass, with the fitted term drawn in as + signs. Both these plots show instrumental intensity units on the vertical scale.

Next, if there are enough data with the Moon above the horizon to fit the term, you will see a plot of the moonlit sky, corrected for both the stellar and the dark-sky terms, as a function of elongation from the Moon. To show the importance of the lunar aureole (whose presence is an indicator of considerable aerosol scattering, and hence probably non-photometric conditions), this plot is normalized by multiplication by the factor in square brackets (cf. the equation for in the previous subsection) and division by the airmass; that is, it is simply a plot of the E-dependent factor in parentheses. Again, the fit is shown with + signs. If the subtraction of the stellar and dark-sky terms produced some apparently negative intensities, the calculated zero level for the term will be drawn as a horizontal line.

If the sky is good, this plot will be nearly horizontal. Usually, it bends up near 0 and 3 radians, with a minimum near 1.7; your data may not cover this whole range, so pay attention to the numbers on the horizontal scale. The vertical scale is chosen to make the range of most measurements visible, so the zero level may be suppressed; look at the numbers on the vertical scale.

It often happens that the range of the independent variables in these fits is inadequate to allow a full fit of all the parameters. Reasonable starting values will be used for the indeterminate parameters. The fitting strategy is to adjust the best-determined parameters first, and release parameters in turn until nothing more can be done. At each stage, the results of a fit are examined to see whether the values determined are reasonable. For example, most of the parameters must be positive; and the dimensionless ones should not be much larger than unity.

You will be informed of the progress of these fitting attempts. Do not be alarmed by ``error messages'' like DLSQ FAILED or SINGULAR MATRIX or Solution abandoned. It is quite common for solutions to fail when several parameters are being adjusted, if the data are not well distributed over the sky. Sometimes, when a solution fails, the fitting subroutine will check to make sure the partial derivatives were computed correctly. This is a safety feature built into this subroutine, and you should always find that the error in the derivatives is or less. The program should then comment that the model and data are incompatible, because we are usually trying to fit more parameters than the data can support. Remember that the program will adopt the best solution it can get; so watch for messages like 3-parameter fit accepted, and don't worry about the failures.

Pay more attention to the graphs that compare the fits to the data. Are there regions where they become widely separated? If so, the fit is poor, and you can forget about the model. If the fit looks good, the model is useful for detecting bad sky data, and may even be useful for interpolating missing sky measurements.

When the fitting is complete, the program will print the terms used in the fit. Then, three summary graphs display the quality of the fit. The moonlit data are marked M in each of these three plots. The first shows the observed sky brightness as a function of the model value. This plot should be a straight line.

The second diagnostic plot shows the residuals of the sky fit as a function of the adjacent star brightness. These points should be clustered about the axis indicated by dashes. If you tend to measure sky farther from bright stars than from faint stars, as some beginners tend to do, the points will show a downward trend toward the right. That's a sign that you need to be more careful in choosing sky positions. Likewise, a large scatter in this plot probably means you have been careless in choosing sky positions, sometimes measuring closer to the star and sometimes farther away. (Here, ``large'' means large compared to the typical sky values on previous plots.)

The final plot in this group shows the ratio of the observed to the computed (modelled) sky brightness, as a function of time. If the airglow changes with time, you will see waves and wiggles in the dark-sky portion. Likewise, if the aerosol is varying with time, you will see coherent variations in the moonlit portion. The upper and lower limits of this plot are fixed in absolute value, so the scatter visible here is a direct indication of the quality of the overall fit.

Finally, if there are aberrant points that do not fit the model, they will be tabulated. If the data are not well represented by the model, there will be many entries in this table. Pay particular attention to the last column, which gives the ratio plotted in the last diagnostic graph. If the fits were generally satisfactory, the few sky measurements tabulated here may be in error, and may indicate a instrumental or procedural problem. They should be examined carefully to determine the cause of the problem.

After all this information has been presented, you will be asked whether you want to subtract the modelled sky from the stellar data. You can reply Yes or No, or enter R to repeat the whole process, or H to ask for help. If you reply Yes, the model values will be subtracted from all the stellar data that were not taken during twilight. However, as twilight is not modelled, the nearest-neighbor method will be used to correct stars observed during twilight. If you reply No, you will be given the option of subtracting the nearest neighbor in all cases.

The sky models do not work well if only a few observations are made with the Moon either above or below the horizon. They do not handle solar or lunar twilight. They can have difficulties if the observations are not well distributed over the sky. In these, as in some other cases discussed above, one should choose some other method instead of using a model for sky subtraction.