TRACKING ERRORS AND OPTICAL SCATTER IN A SOLAR TRACKER WITH LINEAR REGRESSION ERROR CORRECTION .by: Norman B. Wood and Stephen K. Cox Department of Atmospheric Science Colorado State University LIBRARIES -1 T 2 8 1998 t Fort Collins, CO 80523 COLORADO STATE UNIVE:RSITYJ Funding Agencies: • National Aeronautics and Space Administration (Grant No. Nag 1-1704) • Office of Naval Research (Contract No. N00014-91-J-1422) • Department of Defense Center for Geosciences Phase II (Contract No. DAAH04-94-G-0420) "TRACKING ERRORS AND OPTICAL SCATTER IN A SOLAR TRACKER WITH LINEAR REGRESSION ERROR CORRECTION" Norman B. Wood and Stephen K. Cox Department of Atmospheric Science Colorado State University Fort Collins, CO 80523 Research Supported by National Aeronautics and Space Administration (Grant No. NAG 1-1704) Office of Naval Research (Contract No. N00014-91-J-1422) and Department of Defense Center for Geosciences Phase II (Contract No. DAAH04-94-G-0420) March, 1998 Atmospheric Science Paper 663 U18401 6641775 34 ODJL12.a2 ~i 12198 XU O fJ :13-{ll}--OJ GBC ~i ABSTRACT TRACKING ERRORS AND OPTICAL SCATTER IN A SOLAR TRACKER WITH LINEAR REGRESSION ERROR CORRECTION Tracking errors were assessed for a computer controlled solar tracker. The effects of optical scattering on radiometric measurements performed with the tracker were also evaluated. As the position of the tracker is iteratively corrected over time, linear regression is used to calculate a best-fit correction for tracking error. The performance of the tracker was found to be sensitive to the timing of the iterative corrections and to the errors associated with those corrections. Using an optimized scheme for iterative corrections in a field test, the average tracking error was found to be 0.11 ± 0.05 degrees for 48 hours following the final iterative correction. The solar tracker may be fitted with a mirror which can reflect the image of a target into an instrument. Because the mirror is exposed to multiple sources of illumination (direct sunlight, skylight, and light from surrounding objects) the scattering properties of the mirror are important. The intensity of light scattered from the mirror was compared with the intensity of diffuse skylight. Scatter from the diffuse field incident on the mirror (background scatter) was found to be more significant than scatter from the direct solar beam, and both were significant compared to the intensity of diffuse skylight. Background scatter ranged from 20% to 70% of the total measured signal, depending on 11 t C.C, S5s2 .('=, ~ - C:,b3 P..,\J\{)<; scattering geometry and wavelength. Solar scatter ranged from 1 % to 20%, also depending on scattering geometry and wavelength. The scattering properties of the mirror, as measured by the bidirectional reflectance distribution function, appeared to be anisotropic, possibly because of surface defects. For the wavelengths examined, the scattering properties did not follow the wavelength scaling law predicted by Rayleigh-Rice theory for clean, smooth, front-surface reflectors . Ill ACKNOWLEDGEMENTS Dr. John Davis and Dave Wood assisted us greatly, providing both advice and helping hands with th(? solar tracker. We'd also like to thank Chris Cornwall, who performed much of the original development work on the solar tracker, and Melissa Tucker, who assisted in the preparation of a number of earlier versions of this document. This research was supported by the National Aeronautics and Space Administration under contract NAG 1-1704; the Office of Naval Research under contract N00014-81-J-1422, P00007; and the Department of Defense Center for Geosciences Phase II under contract DAAH04- 94-G-0420. IV TABLE OF CONTENTS 1. Introduction . 1 1.1 Instrumentation Requirements 1 1.2 The Solar Tracking System . 3 1.3 Objectives . 5 2 . Sources of Tracking Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 . Compensation for Misalignment .. 3.1 The Transfonnation Matrix . 3.2 Numerical Stability Testing . 3 .3 Correction Algorithm Testing . 4 . Assessment of Other Sources of Error . 4.1 The Astronomical Algorithm 4.2 Electromechanical Errors . . 5. System Testing and Performance . . 5.1 Field Testing of the System . 6. Methods Used in Mirror Scatter Testing. 6.1 Mirror Scattering Effects . . 6.2 Mirror Scattering Properties . V 11 11 21 25 35 35 37 38 38 41 41 42 6.3 Field Tests of Mirror Scattering .. . ...... . . ...... .. .... 45 7 . Mirror Scattering Test Results . . .. 7 .1 Uncertainty in Measurements . 8 . Discussion of Mirror Scatter Test Results . 8.1 Background Scattering 8.2 Solar Scattering . . . . 9 . Conclusions . .. .......... .. . 9.1 Compensation for Misalignment 9.2 System Performance .... 9.3 Mirror Mode Performance . 53 62 65 65 69 73 73 73 74 10. References ... . . . . . . . .. ... . .... .. .. ... . .. ..... 77 11 . Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 VI 1. Introduction A number of areas of atmospheric radiation research benefit from high-quality solar radiation measurements at the surface, including: • studies of the surface energy budget • studies of the energy budget for the lower atmospheric boundary • verification of radiation algorithms in numerical models, and • remote sensing. While the energy budgets may be evaluated to first order using instruments with hemispheric fields of view, more precise evaluations require observation of the direct component alone; this requires accurate tracking of the sun. The purpose of this research is to develop and test a solar tracking system which will be used in research investigations in the four areas noted above. 1.1 Instrumentation Requirements To accurately measure the directional variation in a radiance field, the orientation of the instrument must be accurately controlled so that it samples the required portion of the field . For example, a normal incidence pyrheliometer must point directly at the sun. A radiometer with a narrow field of view might be used to measure the radiance of a point in a cloud at a fixed angular separation from the sun. In both of these applications, an accurate tracking system is required. When a tracking system is deployed in the field, accurate tracking becomes a problem. Although the sun's position may be predicted with fairly high accuracy relative to the local coordinate system, it is difficult to align a tracker with this coordinate system. Typically, iterative manual corrections are made to the tracker's alignment over the course of a day ' s operation. With luck, these iterative corrections result in an optimum setup and accurate tracking. More commonly, this method results in accurate solar tracking only around the time of day of the most recent adjustment. An additional problem is that an instrument may be too large to mount on a field-deployable tracker. In this case, a mirror may be used to reflect the radiance from the target into the instrument. In this configuration, a mirror mounted on the tracker essentially becomes an external element in the instrument's optical path. However, unlike the other internal elements, the mirror is exposed to extraneous light sources. Since a mirror is not perfectly specular, a portion of this extraneous light may be scattered into the path to the instrument's detector, contaminating the desired signal. A desirable tracker, then, would be portable and easily deployable in the field . It should forgive moderate inaccuracies in its alignment. Its tracking accuracy should be sufficient to insure that targets are kept within the instrument's field of view. It should be usable with large instruments which are too heavy to mount in the tracker directly . In this mode, it should not introduce significant contamination of the desired measurement. Finally, its position should be easily correctable. 2 PC Azimuth & elevation position command and position feedback D D D D Servo amplifier 0 Figure 1: Schematic diagram of solar tracker components 1.2 The Solar Tracking System The tracking system discussed here is described more fully elsewhere (Wood et al., 1996). Relevant excerpts are included in Appendix 1. A short description is included here, insofar as it relates to the work that follows . The tracking system includes a two-axis gimbal optical mount driven with servo motors, servo motor amplifiers, and a . personal computer equipped with a two-axis motor controller card (Figure 1). A two-axis mount was used for two reasons. First, the azimuth-elevation nature of the mount translates easily into azimuth-elevation solar position data. Second, a two-axis mount allows for the two modes of operation described above: a direct mode with instruments mounted directly on the gimbal, and a mirror mode in which a mirror is mounted on the gimbal . 3 Each of the two servo motors used to drive the mount is equipped with an optical encoder for position indication and feedback. The encoders have an absolute resolution of 1,000 counts per encoder revolution and quadrature is used to increase this to 4,000 counts. The gear trains used on the azimuth and elevation stages of the gimbal give a reduction of 54: 1. The final resolution is then 216,000 counts per revolution of the azimuth or elevation axis (600 counts per degree) (Aerotech, Inc., 1991). The motors are controlled using a Motion Engineering MC-200 two-axis controller card installed in an IBM AT personal computer. The computer software includes: • a routine to calculate the true solar coordinates as a function of time and location on the earth, • interfaces to the motor control software libraries, • routines needed to make iterative, manual adjustments to the gimbal position, and • routines to correct for tracking errors. The algorithm to calculate the true solar coordinates was adapted from Meeus (1991) and Sax (1991 ). The motor control libraries were provided by Motion Engineering (1990). To simplify adjustments for tracking errors, a semi-automated correction method was developed. The method applied in this solar tracker is similar to coordinate transformations used for computer graphics applications and other mathematical transforms (Newman and Sproull, 1979). After setup, iterative corrections are made simply by changing the azimuth and elevation angles of the tracker so that it is pointing directly at the sun. These iterative corrections are made through the computer software. After three iterative corrections are made, one can solve for a transformation matrix 4 which will convert the true solar coordinates into the coordinates used by the gimbal. True solar coordinates are calculated using the Meeus/Sax astronomical software, then multiplied by the transformation matrix to obtain the gimbal coordinates. While a minimum of three iterative corrections are required, additional corrections allow averaging in a least-squares sense to minimize the effects of random errors in the corrections. Since the tracker is aimed using explicitly calculated azimuth and elevation angles, a number of tracking modes are possible. In solar mode, the tracker points directly at the sun. In offset mode, the tracker points to a position on the sun's apparent trajectory, but at a fixed temporal offset from the current solar position. For example, for a +5 minute offset, the solar position at (local time+ 5 minutes) is calculated, and this is where the tracker is pointed. In scanning mode, angular offsets can be applied separately to azimuth and elevation angles and can be made to vary sinusoidally as a function of time. 1.3 Objectives A number of issues affect the accuracy of measurements made using the tracker. Chief among these is the accuracy of the tracker itself. If the radiance field being measured varies spatially (as might occur when measuring across the forward scattering peak of a parcel containing cloud particles), small errors in the tracker's position can potentially cause large errors in the measured radiance field . If the tracker is to be left unattended for long periods of time, the magnitude of the tracking error as a function of time becomes an issue. For ease of use, it is desirable that the tracker maintain accurate tracking for a period of several hours. For unattended use, the period might extend to a day or longer. 5 The second, related issue is the numerical behavior of the algorithm used to calculate the transformation matrix. Under some conditions, the algorithm might prove to be excessively sensitive to small errors in the iterative correction points and to the limited precision of the computer's floating point arithmetic. At best, this sensitivity could result in gross errors in the calculation of the transformation matrix. At worst, it could result in numerical errors which might cause the tracking program to crash. The third issue relates to the scattering effects of the mirror. The mirror itself has a hemispherical field of view. Under typical conditions, the mirror is exposed to two distinct radiance fields : an intense, direct field originating from the sun; and a low-level diffuse field consisting of sky light plus radiance emitted or reflected from the ground and surrounding objects. Two scenarios can be considered for operation in mirror mode. In one scenario, an instrument is positioned to receive the direct solar beam reflected by the mirror. In this scenario, the signal received by the instrument is the specular reflection of the direct solar beam plus light from the diffuse field which has been scattered by the mirror into the instrument's field of view. In most cases, the radiance due to the direct solar beam will be much larger than that due to the mirror-scattered diffuse field, and the scattering effects of the mirror likely can be ignored. In the second scenario, an instrument is positioned to measure a diffuse radiance reflected by the mirror. For example, a radiometer might be positioned to view, through the mirror, a part of the sky at a few degrees angular separation from the sun. In this scenario, the signal received by the instrument is the specular reflection of the diffuse sky light, plus light from the remainder of the diffuse field which has been scattered by the mirror into the instrument's field of view, plus light from the direct solar beam which has 6 been similarly scattered into the instrument's field of view. Unlike the first scenario, it is not clear that the desired signal (the specular reflection of the skylight) will be much larger than the other two contributions. The scattering effects of the mirror may not be negligible. The objectives of the work performed here are then threefold. First, assess the performance of the tracking error correction algorithm. This includes evaluating the numerical stability of the calculation of the transformation matrix and insuring that it remains stable under typical operating conditions. Second, evaluate the performance of the tracking system as a whole in terms of tracking error. Third, examine the effects of the mirror's scattering properties on measurements of the diffuse field . 7 2. Sources of Tracking Error Tracking errors can be attributed to a number of causes, primarily: • misalignment of the gimbal, • inaccurate calculation of the true solar position, and • electromechanical errors in the tracking system. Assuming local horizontal coordinates are being used, misalignment can be partitioned into two distinct sources of error, error in meridional alignment and error in zenith ( or level) (Figure 2) . The vertical axis around which the gimbal rotates should be pointed along the local zenith. The horizontal axis around which the gimbal rotates should be +x (gimbal) ~----- --- +x (lhc) t +z (gimbal) I +z (lhc) +y (lhc) - - - Meridional misalignment --- +z (lhc) +z (gimbal) : /+y (gimbal ) I +x (gimbal) ------- - - - +x (lhc) - - - - - - Zenith (level) misalignment Figure 2: Sources of error in alignment. The figure on the left illustrates a rotation about the z-axis. The figure on the right, a rotation about the y-axis. 8 normal to the local meridian when the gimbal is pointed north or south, and normal to the vertical axis. Provided the horizontal and vertical axes of the gimbal are normal to each other (this would be controlled by the assembly of the gimbal and is taken to be true) any real misalignment will consist of some combination of error in meridional alignment and error in zenith. The actual, misaligned orientation of the gimbal can then be related to the ideal orientation by a simple rotation about some axis (Figure 3). If the error were strictly an error in meridional alignment, the rotation axis would be vertical. If the error were strictly an error in zenith, the rotation axis would lie somewhere in the horizontal plane. The coordinate systems used to define the true local horizontal coordinates and the gimbal's coordinates are defined in more detail below. vector representing misalignment rotation L---- ---- --- - - - - +x (south) +z +y (west) Figure 3: Misalignment defined by rotation about an arbitrary axis. 9 Calculation of the true solar position requires that a number of parameters be accurately known. These parameters include the observer' s location on the earth (latitude, longitude and altitude), the date, the time, and the effects of atmospheric refraction. Provided these quantities are accurately known, the accuracy of the calculation of the true solar position depends on the approximations used in the calculation algorithm. In the field, the observer' s location can usually be determined using a global positioning system (GPS) receiver. Synchronization with coordinated universal time (UTC) to within one second can be readily achieved either through the GPS signal or through National Institute of Standards and Technology broadcasts from station WWV. The effects of atmospheric refraction can also be approximately calculated. However, refraction is negligible over most of the sky, ranging from zero degrees when the zenith angle is zero, to 0.1 arcminutes when the zenith angle is 45 degrees, and to 35 arcminutes when the zenith angle is 90 degrees (Meeus, 1991 ). In this work, the effects of atmospheric refraction have been neglected. Two sources of error occur in the electromechanical system (i.e. , the servo motor amplifiers, the servo motors, and the gimbal) . The first, the servo error, arises when the servo motor position deviates from the position commanded by the control software. The servo error is expected to be mainly a function of the motor control parameters (gain, pole, zero) and the load on the gimbal. The second, the mechanical error, occurs when, because of looseness or friction in the drive train, the position of the gimbal does not match that of the servo motors . The mechanical error is expected to be mainly a function of the fit of the drive train and the load on the gimbal. 3. Compensation for Misalignment Assuming one can determine the true solar coordinates with sufficient accuracy, and that the electromechanical errors in the tracker are negligible, tracking errors can be attributed to misalignment. This section describes the geometry and mathematics involved in the misalignment problem and the techniques used to compensate for misalignment. 3.1 The Transformation Matrix As described in Wood et al. (1996), two different coordinate systems must be overlaid: the local horizontal coordinate system, in which the true solar coordinates are calculated using astronomical software, and the gimbal coordinate system, which is defined by the initial setup of the gimbal. Both coordinate systems are represented as three-dimensional cartesian coordinates, sharing a common origin, with axes defined in a left-handed sense. For the local horizontal coordinates, positive xis to the south, positive y is to the west, and positive z is vertical. The gimbal coordi nates have a similar orientation, but are not perfectly aligned with the local horizontal coordinates. Since the two coordinate systems are cartesian and share the same origin, an arbitrary position in the local horizontal coordinates can be transformed to a position in the gimbal coordinates through the use of a single 3 X3 transformation matrix. Thus, once the transformation matrix is known, the true solar coordinates are multiplied by the transformation matrix, giving the sun ' s position in the gimbal coordinate frame. The gimbal can then be moved to the required position . 11 In cartesian coordinates, an object's position in the local horizontal coordinates (/he) can be represented by a vector: Xlhe = [ X /he, Y /he , z /he ] as can the object's position in gimbal coordinates: The transformation from local horizontal coordinates to gimbal coordinates can be accomplished by multiplyingx1he by the 3 X3 transformation matrix, A : (1) (2) (3) Given known values for Xzhe and Xg;m, the equation may be solved for A. If only a single pair of vectors Xzhe and Xg;m is known, the matrix A is under-determined. However, once three or more distinct pairs of Xzhe and Xg;m are known, a solution for A is possible. Let X1he and Xg;m be matrices containing multiple instances of Xzhe and the corresponding Xg;m, respectively : r x,he,1 Y1he ,l I x,1,c ,2 Y11,c ,2 x/he =1 I Lxlhe,N Y1he,N and r xgim,1 Ygim,I I X . Y gim,2 I g,m.- xgim =1 I l xgim,N Ygim,N 12 z,1,c ,1 l z lhe ,2 I I I z/he, N J zgim,I 7 z . I g,m.2 I I I zgim,N J (4) (5) where N is the number of iterative corrections. When N = 3, the transformation equation can be exactly solved for A : (6) 3.1.1 The Effects of Uncertainty In reality, neither Xu.cnor Xgw are known exactly. The elements of Xu.care obtained from calculations of the solar position by the astronomical software and so are subject to any errors in the model used to develop the calculations. The matrix Xgw can also only be approximated. The position of the sun in gimbal coordinates is obtained each time the operator makes an iterative correction to point the tracker at the sun. Once such a correction is made, the gimbal position is read from the optical encoders. Since errors exist in each iterative correction, some errors also exist in the sun's gimbal coordinates. Equation (6) can thus be rewritten as: (7) where A' represents an approximation to the transformation matrix, Eihc is the matrix of errors in the local horizontal coordinates, and Eg;m is the matrix of errors in the gimbal coordinates. As before, when N = 3, (7) can be solved exactly for A' . However, because of uncertainty (primarily the measurement errors in Xg;m), the calculated A' may not be accurate. In particular, if the matrix (Xthc + Eihc) is somewhat ill-conditioned, its inverse may amplify the errors Eg;m when (7) is evaluated, causing A' to be grossly inaccurate. When N> 3, A' is over-determined. The over-determined case probably provides better results, since multiple iterative corrections should reduce the uncertainty in A', provided 13 a least-squares solution technique is applied to the data. Since errors are associated with both Xihc and Xg;m , a total least-squares approach is indicated (Van Huff el and Vandewalle, 1991 ). However, the algorithm used to calculate A' must run quickly, so as not to interfere with the real-time operation of the tracker. Since the errors associated with X ihc (see section 4.1) are small relative to those that might be expected for X g;m, a simpler, classical least-squares technique is used. Ignoring the errors in Xthc and lettingXobs = (Xg;m + Eg;m), the matrix form of the transformation equation is: A typical method is to calculate A' as: A'= (X;hc ·X11,cr1 (X~J(X 0bJ where x r/hc is the transpose of X1hc- The inverse term, (XT/hc • X1hs 1 , may be evaluated using a number of different techniques, but this approach is prone to problems if x rlhc. X thc is ill-conditioned. (8) (9) To avoid this problem, a technique using singular value decomposition (SVD) is applied. Details of the technique are described in Press et al., (1992) and are summarized here. The matrix X1hc, which is of size Nx3, can be decomposed into the product of an Nx3 column orthogonal matrix U, a 3 x 3 diagonal matrix W, and the transpose of a 3 x 3 orthogonal matrix V: 14 j Ul ,I u l.2 u1 3 l I U2,l u2.2 u2.3 l r wl 0 0 l I v;I T v 1\ l I 0 I.IV; Vl,2 ; I 1·1 0 T x,hc =1 W2 J l 2.1 V2,2 Vi3 j I I l 0 0 W3 v;,I T (I 0) v3,2 v3,3 L11N .1 UN ,2 UN ,3J w yr u = u-w-vr The diagonal elements of W, denoted as w;, are the singular values of Xihc• These are the nonnegative square roots of the eigenvalues of XTihc • Xihc (Kincaid and Cheney, 1991 ). In order to find the least-squares solution for A', the pseudoinverse of X1hc, denoted as X-ihc, must be calculated, and is given by: (11) Since Wis diagonal, the inverse w·1 is simply the matrix whose diagonal elements are the reciprocals of w,. Problems occur if some w; = 0, or if some w; are near zero. These small values of w; indicate that Xihc is ill-conditioned. To overcome problems related to the ill-conditioning, the reciprocal 1/w; is replaced by zero ifw; = 0 or ifw; is exceedingly small (Press et al., 1992). Finally, A' is calculated as : A'= x;hc . Xoos (12) 3.1.2 Implementation in the Solar Tracking System Correction points may be collected each time an iterative correction is made to the gimbal' s position. In typical operation, the operator sets up, aligns and starts the tracker. Over time, as tracking errors are detected, the operator may switch the tracking software to manual control, then adjust the azimuth and elevation angles until the gimbal is on target. Once the target (typically the sun) is attained, the operator presses a key to signal 15 the tracking program that the adjustment is complete. The computer records the angular position of the gimbal relative to its home position via the optical encoders mounted on the servo motors. This becomes the observed position. The program then calls the astronomical software to calculate the true solar position in local horizontal coordinates and stores the results. These two positions are stored in a data file, along with any previously collected corrections, as azimuth and elevation angles . Next, the program checks if sufficient data exist to calculate A '. If so, the program recalls each correction, converts the angular positions into cartesian coordinates by assuming a unit sphere, and stores the results in matrices Xobs and X thc• The SVD least-squares technique described above is then performed to obtain the matrix A'. After A' is calculated, it may be used to correct for misalignment. The tracking program calculates the true solar position in local horizontal coordinates, then multiplies by A ' to obtain the applied coordinates. The gimbal is then moved to the applied coordinates . 3.1.3 Sensitivities The performance of the error correction algorithm is expected to be sensitive to a number of vari ables, including: • magnitude and orientation of misalignment • errors in the site data (latitude, longitude, altitude, time) • errors in iterative corrections (i.e., in X obs) • timing and number of iterative corrections • time of year • location (primarily latitude) of the tracker 16 • time lapse between iterative corrections and observations Errors in the site data should have effects similar to misalignment, but on smaller scale. For example, a one degree error in latitude would be similar to a one degree rotation of the tracker about the y axis. A one degree error in longitude would be similar to a one degree rotation of the tracker about the x axis. Small errors in time or altitude also correspond to alignment errors. Since accurate site data can usually be obtained, and since the resulting errors are similar to misalignment, errors in site data were not considered further. 3.1 .4 Tracker Simulation Design A computer simulation of the tracker was developed to test the effects of the remaining variables on the performance of the correction algorithm. Using the variables listed above as inputs, the simulation marched forward in time, calculating the solar position in both the local horizontal coordinates (true solar position) and in the gimbal coordinates (observed solar position). At any chosen time interval , the simulation could be paused and an iterative correction could be made. After three or more iterative corrections, the transformation matrix was calculated and could be applied to the true solar position. Then the transformed true solar position was compared with the observed solar position. The magnitude and orientation of the misalignment of the gimbal coordinates with respect to local horizontal coordinates were controlled by specifying a rotation vector (Figure 3). The direction of the vector represents the axis of rotation, while the magnitude of the vector specifies the magnitude of the rotation. 17 With exception of the numerical stability tests, iterative correction errors were simulated as random deviates normally distributed about a mean of zero so that the mean iterative correction corresponded with the true solar position. A simple central limit theorem approach was used to model the errors (Rade and Westergren, 1990): (13) where e is the simulated iterative correction error, U; is a series of simulated, uniformly distributed random deviates ( obtained from a random number generator), cr is the desired standard deviation for the simulated normal distribution, and µ is the desired mean for the simulated normal distribution. The parameterµ is set to zero and cr is chosen to control the probable size of e. For example, with a true normal distribution, a standard deviation of 0.255 causes the error to fall between ±0.5 with 95% probability. Thus, the typical magnitude of the errors could be increased or decreased by modifying cr. For the numerical stability tests, the iterative correction errors were simulated as uniformly-distributed random deviates. The number and timing of iterative corrections (including the time intervals between corrections) were controlled during operation of the simulation. Iterative corrections could be made at any time during the simulation. The simulation recorded a maximum of thirty iterative corrections and used a minimum time interval of ten minutes. These corrections were stored sequentially and, if more than thirty corrections were made, the earliest corrections were discarded. 18 The elapsed time between the corrections and the observations was also controlled during operation of the program. The simulation ran forward in time from the start date, up to a maximum of thirty days, but only within the start month. The transformation matrix could be applied to the tracker at any time during the simulation. So, for example, iterative corrections could be recorded during day #1 , then several simulated days or weeks later, the transformation matrix could be applied and tracking error measured. 3.1.5 Development of Test Cases A set of test cases was established to use throughout the testing of the correction algorithm. The cases consisted of: • moderate misalignment • varied axes of misaligment • time of year near an equinox (to obtain a typical 12-hour day) • mid-latitude, northern hemisphere The test cases were chosen by running the simulation with no error correction and looking at the tracking error as a function of time. Latitude and longitude were set at 40 degrees N, 105 degrees W, the magnitude of the misalignment was set at 1 degree, and the date was set at 22 September, 1992. The orientation of the rotation axis was varied over elevation angles from O degrees to 90 degrees and over azimuth angles from -180 degrees to + 180 degrees. The resulting tracking errors were plotted as a function of local time and a set of four typical cases was chosen to use in the remainder of the testing (Figure 4). The test cases, along with the orientations of the misalignment vectors, were: • Axis A (Elevation O degrees, azimuth O degrees) - pure zenith misalignment 19 V> Q) l!: 00 Q) "C 1-' e 1- u.J 00 C: u I- I- 2.0 .......----.-----,------,-------,.----r------.---r-----, 1.8 1.0 ................ . 0.8 Case A . . 0.6 ······································ ······'.···· ················.····························· · ·. . · Case B 0.2 ······························································ ········································································································· 0.0 ..J...__----1----+-----t-------,----+-----+----~___J 00:00 4 :00 8:00 12:00 16:00 20:00 24:00 Local Standard Time, hours Figure 4: Tracking errors associated with cases used to test the error correction algorithm • Axis B (Elevation 30 degrees, azimuth -90 degrees) - combined zenith and meridional misalignment • Axis C (Elevation 45 degrees, azimuth -180 degrees) - combined zenith and meridional misalignment • Axis D (Elevation 90 degrees) - pure meridional misalignment 3.2 Numerical Stability Testing Initially, the correction algorithm used Gauss-Jordan elimination (GJE) to solve for A', following the method shown in equation (9) . Stability problems with the algorithm 20 occurred when iterative corrections were made at 5-minute or shorter intervals. It wasn ' t clear whether other situations might create similar stability problems. The GJE version of the correction algorithm was tested by running the simulation for several dates at two-month intervals throughout a year (the 20th day of January, March, May, July, September and November). At each test date, the simulation program was run using test case B with+ 1.0 degrees rotation. During the first twenty-four simulation hours, corrections were made at 0800, 1200 and 1600 local standard time (LST). The simulation program calculated the transformation matrix, then applied the matrix to the tracker position during the second twenty-four simulation hours. Tracking errors were calculated at intervals of ten simulation minutes throughout each test. The transformation matrices and tracking errors were examined to evaluate the performance of the correction algorithm. In addition, to help assess the cause of the stability problems, condition numbers were calculated for the x r,hc · Xihc matrices used in the correction algorithm. Matrix condition numbers were calculated using the column-wise lrnorm (Kincaid and Cheney, 1991). After this testing was completed, the GJE technique was replaced by SVD. The SVD algorithm requires a parameter (which we've called TOL) which measures the smallness of the elements w ; of the diagonal matrix W(see equation IO and the following discussion). lfw ; is smaller than the limit established by this parameter, the reciprocal 1/w; in the inverse w-1 is replaced by zero. A series of tests were run with the SVD simulation to determine an appropriate value for this parameter. First, a benchmark was established by running both the GJE and SVD versions of the simulation to determine an ideal transformation matrix. Test case B was used with+ 1.0 21 degree misalignment, a simulation date of 20 March and no errors in iterative corrections. For the SYD simulation, TOL was set to zero. Then, starting at a simulation time of 0630 LST, approximately 24 iterative corrections were made at half-hour intervals. The transformation matrix was calculated (both SYD and GJE versions gave the same result) and this matrix was used as a comparison to evaluate the performance of the SYD version in subsequent tests with different values of TOL. Next, a series of six SYD simulations was run using the same parameters described above, but with values of TOL ranging from I o-6 to I 0-1 . As above, for each test iterative corrections were made at half-hour intervals starting at 0630 LST. The resulting transformation matrices were compared with that obtained in the benchmark test and an optimum value for TOL was chosen based on the comparisons. This value of TOL was used through all of the subsequent tests. 3.2.1 Results of Numerical Stability Testing The initial testing of the GJE version of the simulation produced transformation matrices for each of the six simulation dates (Table 1). Also shown (Table 2) are the largest tracking errors occurring during the second simulation day of each test and the condition number for the corresponding XTrhc • Xrhc matrix. To choose a suitable value for TOL, a benchmark transformation matrix was calculated, along with transformation matrices obtained from GJE and from SYD with various values of TOL (Table 3). 3.2.2 Discussion of Numerical Stability Test Results Given a small misalignment, as was used to perform the numerical stability tests, the transformation matrix should be very close to an identity matrix. For four of the GJE test 22 Table 1: Numerical stability test results. Transformation matrices calculated using Gauss-Jordan elimination 109913 -0.0085 -0.00957 1-1.4334 1.7222 091481 20 Jan 92: I 1.0016 I 20 Mar 92: l 0.0052 1.0013 I l 0.0015 0.0048 j 0.0057 j 0.0337 -0.0009 09835 2.0404 -1.4399 02289 , 09968 -0.0150 -0.02847 ' 09959 - 0.0112 - 0.02847 20 May 92: l 0.0106 09973 -0.0013 I 20 Jul 92: I 0.0086 09973 - 0.0011 I 0.0125 0.0015 1.0061j l 0.0126 0.0011 1.0060j 1 o.s653 0.1212 -023847 1 rno93 -0.0121 - 0.0060 7 20 Sep 92: I 09973 -0.0010 I 20 Nov 92: l 0.0090 - 09976 - 0.0015 I l 0.0112 1.1849 j 09911J 03665 -0.1033 -0.0055 0.0139 Table 2: Largest tracking errors from numerical stability tests, along with condition numbers for the matrix X!c · X/hc Simulation Date Largest tracking error, degrees Condition Number 20 Jan 92 1.504 77.25 20Mar92 4.800 601,889 20 May 92 0.469 60.68 20 Jul 92 0.406 69.12 20 Sep 92 0.925 29,670 20 Nov 92 2.643 79.06 cases this was true, while in the remaining two cases (March and September) the transformation matrices differed significantly from identity matrices. Thus it appears that the correction algorithm failed to calculate the appropriate transformation matrix. As would be expected if the transformation matrix were inaccurately calculated, the tracking errors for March and September were much larger than in the other cases. 23 Table 3: SVD tuning results. Selected transfonnation matrices for benchmark, Gauss-Jordan elimination and SVD with various values of TOL 109998 -0.0087 -o.01517 1 1.1855 -0.1064 -024957 Benchmark: l 0.0087 1.0000 -0.0001 j GJE Results:l 0.0081 0.9994 I 0.0007 J 0.0151 -0.0001 09999 -0.1429 0.0806 1.1964 1 o.4169 -0.0057 0.4858 7 1 o.8742 0.1622 0.04727 TOL = 10·1 : I 0.0076 09995 I TOL = 10..1 :I 0.0082 1.0007 I lo5001 0.0007 J l 0.1206 0.0007 J -0.0045 05835 -0.1417 09466 105631 0.0487 -0.4954 7 1 0.8124 -0.0760 -0.1919 7 TOL = 10"5 : j 0.0074 1.0002 -0.0007 I TOL=lO~:I 0.0079 0.9998 -0.0009 I loJ789 -0.0471 13993J l 0.1101 0.0564 1.1480 J The cause of the inaccurate transfonnation matrices was revealed when the condition numbers were calculated for the x r lhc • X 11,c matrices. The condition numbers for both March and September were much larger than for the remaining cases, indicating that these matrices had a higher degree of ill-conditioning. Larger condition numbers indicate that the solution to an inverse problem is more sensitive to errors in the input data. This increased sensitivity causes the small iterative correction errors to be magnified, resulting in a grossly-erroneous transfonnation matrix. The timing of the tests (both the March and September tests were near an equinox) appeared to be significant, although this was not explored further. The SVD tests suggested that a value of 10·3 for TOL gave better perfonnance than other values that were tested. This value produced a transfonnation matrix which most closely matched the benchmark matrix, both in fonn and in ability to reproduce transformed coordinates. 24 3.3 Correction Algorithm Testing Once the SVD technique was implemented, four tests were performed to evaluate the sensitivities of the correction algorithm to what were considered to be the most significant (see section 3.1.3) input parameters. The parameters included orientation of misalignment, magnitude of misalignment, magnitude of errors in the iterative corrections, and timing of iterative corrections. Each test was performed by running the simulation for 48 simulation hours using a simulation date of 20 March. Corrections were made during the first simulation day, then the transformation matrix was applied during the second day . Each run was repeated three times (a replicate set) and the average tracking error was calculated at each interval of ten simulation minutes. • Orientation of misalignment Four replicate sets were performed, one with each of the test cases described in section 3.1.5. The magnitude of the misalignment vector was fixed at 5 degrees. Corrections were made at half-hour intervals between 0630 LST and 1800 LST on the first simulation day. • Magnitude of misalignment Three replicate sets were performed. The magnitude of the misalignment was varied from 0.5 degrees to 5 degrees to 15 degrees while the rotation axis (Case C: -180 degrees azimuth, +45 degrees elevation) was kept constant. During the first simulation day, corrections were made at half-hour intervals between 0630 LST and 1800 LST. 25 • Magnitude of iterative correction errors To evaluate the sensitivity to errors in the iterative corrections, three replicate sets were run with varying magnitudes of iterative correction error. The rotation axis (Case C: -180 degrees azimuth, +45 degrees elevation) was kept constant for all three sets, as was the magnitude of the misalignment (5 degrees). During the first simulation day, corrections were made at half-hour intervals between 0630 and 1800 LST. The magnitude of the iterative correction error was controlled by varying the simulation parameter cr . The desired cr for the error model was determined by assuming that the iterative correction errors should fall within a specified range with a specified probability. For example, a 95% probability that the iterative correction errors fall between -0.3 degrees and +0.3 degrees gives a desired cr of 0.153 degrees using a table of the standard normal distribution function (Rade and Westergren, 1990). Similar calculations were made for the cr·values used in these tests ( Table 4). • Timing of iterative corrections Four replicate sets were performed, each with different schemes for the timing of iterative corrections. For all four sets, six iterative corrections were made, beginning at 0630 LST, and no corrections were made during nighttime hours . Iterative corrections were made at approximately 30-minute intervals for the first set, 2-hour intervals for the second set, 6-hour intervals for the third set, and 12-hour intervals for the fourth set. The magnitude and orientation of the misalignment (Case B: -90 degrees azimuth, +30 degrees elevation, +5 degrees rotation) was kept constant for all sets. In addition, once these four timing tests were completed, a further test was performed in which an attempt was made to optimize the timing of the corrections. 26 3.3.1 Results of Correction Algorithm Testing The performance of the correction algorithm was largely insensitive to changes in the magnitude and orientation of misalignment (Figure 5 and Figure 6). The performance of the correction algorithm was sensitive to the magnitude of the errors in iterative corrections (Figure 7). Tracking errors increased as the parameter cr increased (Table 4) . For each case, the resulting tracking error varied from approximately cr/2 to 3cr/2. The performance of the correction algorithm was sensitive to the timing of iterative corrections (Figure 8 through Figure 11 ). With closely-spaced iterative corrections (Figure 8), the tracking error was weakly periodic with a strong upward trend. The standard deviation of the tracking error remained small. With more broadly-spaced iterative corrections (Figure 9 and Figure 10), the average tracking error was more periodic but had less of an upward trend. However, the standard deviation was larger than with closely-spaced iterative corrections. For iterative corrections spaced at approximately 12-hour intervals (Figure 11), the tracking error was strongly periodic and, although the trend seemed to be flat, the tracking error was larger than ,in the previous tests . Table 4: Probable iterative correction error (95% probability) with minimum and maximum tracking errors for various values of sigma PROBABLE RANGE OF ITERATIVE MIN/MAX Q CORRECTION ERROR TRACKING ERROR degrees degrees degrees 0.01 +/- 0.02 0.00557 I 0.01817 0.10 +/- 0.20 0.08613 I 0.23667 1.00 +/- 2.0 0.52960 I 1. 76883 27 0.80 I I I I I I I I 0.75 _ _ Simulation Date: 22 - 23 Sep 92 -- Test 10 C1 (0.5 degrees) Site Location: - - Test 10 C2 (5 degrees! 0.70 -- latitude: 40 N . .. . · Test 10 C3 (1 5 degrees) 0.65 0.60 0.55 Vl 0.50 Q) Q) ... 0.45 tlO Q) '"'C 0.40 ._, 0 0.35 ... ... - t- Longitude: 105W V Elevation: o.oom ---- - Time Zone: MST (UTC - 7) /.t ___./ r-----.. ... 1.~- / "'- ./. ....,/'/. __/,,. / . · --- :-:- -~ ./ ----. . . ... •,.;,,....- ....... _/ .... . .. --..... - ... --·/ w 0.30 ./ 0.25 •/ , 0.20 0.15 0.10 00:00 02 :00 04 :00 06 :00 08 :00 10:00 12 :00 14:00 16:00 18:00 20:00 22 :00 24:00 Local Standard Time, hours Figure 5: Tracking error for various magnitudes of misalignment 0.50 I I I I I I Simulation Date: 20- 21 Mar 92 -- Test 7A 0.45 - - Site Location : - - Test 7 B ... Latitude: 40 N · · · · · Test 7C 0.40 0.35 Vl 0.30 Q) Q) ... 0.25 tlO Q) '"'C ' 0.20 ... 0 ... ... w 0.15 - - Longitude: 105W - Test 7 D I Elevation : o.oo m ·· -r -- .·--Time Zone: MST (UTC - 7) _.- / .... / .: r--. . / / I . .. r· I"-. · , / I/ " ' · / ', ....... ~. -~---·· / I '- .. I _, ... ' ............. ;;/ i.-- - ,- , - ·:--..... ' ' . . '- -_:::, ::;:::::;,,' -· 0.10 0.05 0.00 00:00 02 :00 04:00 06 :00 08:00 10:00 12 :00 14:00 16:00 18:00 20:00 22:00 24 :00 Local Standard Time, hours Figure 6: Tracking error for various orientations of misalignment 28 1./') QJ QJ ... t:IO QJ ""O ... ... 0 ... ... w 2 .00 I I I I - Test 11 Cl (sigma= 0.01 ) . . 1.50 -- Test 11 C2 (sigma = 0.1 0l ... .. Test 11 C3 (sigma= 1.00) . . . . . . . . . . 1.00 .. .. Simulation Date: 20 • 21 Mar 92 0 .50 ___J;ite Location : . . .... , latitude: 40 N 0 .25 L 'td 105W , - ong1 u e: / Elevation: o.oom --0.20 Time Zone: MST (UTC - 7) ./ -- / /' 0.15 0 .10 t-- - -- -- .,,,.,. --.... __ .,,,.,. -- ----- 0.05 0.00 00:00 02 :00 04 :00 06 :00 08 :00 10:00 12 :00 14:00 16:00 18:00 20:00 22 :00 24 :00 Local Standard Time, hours Figure 7: Tracking error for various iterative correction errors Based on the sensitivity of the correction algorithm to the timing of the iterative corrections, a final scenario was tested in which multiple iterative corrections were made in several clusters distributed throughout the day. A pattern consisting of six clusters at approximately 6-hour intervals (again with n (l) 0 0.020 .,.....-------------------------~ 0.016 0 .012 0 .008 0 .004 - Test1A · · · · Test 1B . - - Test 1E · ... . 0.000 ---------,.----~ ----+----------~- ----.J 00:00 04:00 08:00 12:00 16:00 20:00 24:00 Local Standard Time, hours Figure 13 : Error in solar position calculated using astronomical algorithm (compared versus positions calculated using U. S. Naval Observatory data). 36 tests. These errors were small given the fields of view for instruments typically used for atmospheric radiation measurements. For example, at a range of 100 m, an instrument with a 0.5 degree half-angle, circular field of view observes a circular target with a radius of 87.3 cm and an area of23 ,943 cm2 . With a tracking error of0.02 degrees, the observed area is shifted 3.5 cm relative to the target area, resulting in the coverage being shifted by about 611 cm2 (2.5% of the total target area). With an error of0.008 degrees, the coverage is shifted only about 244 cm2 (1% of the total target area). 4.2 Electromechanical Errors Two sources of electromechanical error were noted for the tracking system: servo error and mechanical error. The solar tracker control program measures the servo error by comparing the command position with the position indicated by the optical encoders on the motor shafts. During normal operation of the tracker with a light load in the gimbal (2 - 3 kg), the servo error is typically about 15 encoder counts (0.025 degrees) for each axis . An estimate of the mechanical error is available from manufacturer' s literature for the optical mount (Aerotech, 1991). The accuracy for each axis is specified as 0.05 degrees . 37 5. System Testing and Performance 5.1 Field Testing of the System The previous sections isolated and examined three sources of tracking error: misalignment, the astronomical algorithm and the electromechanical system. The tracking error correction algorithm was tested only for its ability to correct for misalignment; however, during actual operation of the tracker, all three sources of error are present. It was unclear how the tracking error correction algorithm would perform when all three sources of error were present. To evaluate this, a field test was performed on the complete system. For this test, the tracker was set up in the field with an intentional misalignment, a series of iterative corrections were made, then the tracking error was measured over a two-day period. The tracker was mounted on a sturdy aluminum table and the table was rotated about its vertical axis to an azimuth error of about + 1.5 degrees . Then the table was tilted approximately 1.2 degrees (downward to the west). To improve the resolution of the subsequent iterative corrections and measurements, the tracker was zeroed to a point marked on a wall about 30 m south of the tracker. Iterative corrections were made on the first and fourth days of the test. The timing for the iterative corrections was based on a scheme developed from the results of the sensitivity testing described above. Iterative corrections were made in clusters of three, and three clusters were made each day. The clusters were separated by about three hours (four 38 hours on the fourth day) and individual corrections within each cluster were separated by about five minutes. The tracker was operated in mirror mode, so the solar image was projected to the zero point on the wall south of the tracker. Iterative corrections were made so that the solar image was centered on the zero point. On the fifth and sixth day of the test, no corrections were made, and the tracking error was measured at approximately half-hour intervals. The distance from the center of the solar image to the zero point was measured and converted to an angular error. The precision with which this measurement could be made was estimated as 2 cm, which resulted in 0.04 degrees of uncertainty . 5.1.1 Results of System Field Testing Testing of the solar tracker in the field showed tracking errors ranging from 0.06 degrees to 0.18 degrees over a two-day period following the final iterative correction (Figure 14). The tracking error appeared to vary somewhat periodically, much as occurred in the simulations . Averaged over all measurements, the tracking error was 0.11 degrees. 5.1.2 Discussion of System Field Test Results The perfonnance of the tracker when tested in the field was similar to the perfonnance predicted by the tests with the simulation. In the simulation test which used the optimized scheme for iterative corrections, tracking errors were less than 0.2 degrees. For this field test, the average tracking error was 0.11 +/- 0.04 degrees with a maximum error of 0.18 +/- 0.04 degrees . How do these tracking errors compare with the errors in the iterative corrections? When projected to the zero point at a range of about 30 m, the image of the solar disk was a fairly well-defined bright circle with a diameter of 39 Ill Cl) Cl) ,_ 0.0 Cl) ""O ,_, 0 ,_ ,_ UJ 0 .24 0 .20 0 .16 0 .12 0 .08 0 .04 0 .00 00:00 ' ' 6 :00 12 :00 • ,,~. ,, ,,. l•P ' 18:00 00:00 6:00 12:00 18:00 24 :00 Local Standard Time, hours Figure 14: Tracking error for field testing with improved iterative correction scheme approximately 35 cm. The target area was ruled with concentric circles at 15 cm increments of radius. When iterative corrections were made, the position of the solar disk could typically be adjusted so that it was misaligned by no more than about half the spacing between the circles on the target, or about 7.5 cm. This is equal to an uncertainty of about 0.14 degrees in the iterative corrections. Thus, in this field testing, the maximum tracking error was slightly larger than the uncertainty in the iterative corrections, just as was observed in the tests with the simulation. The orientation of misal ignment was similar to axis B used in the simulation tests, with a magnitude on the order of 5 degrees. 40 6. Methods Used in Mirror Scatter Testing 6.1 Mirror Scattering Effects Scattering from optical elements such as mirrors and lenses in an instrument may contaminate the signal produced by the instrument. If light from a non-target object (an object outside the instrument's field of view) is incident on such an element, that light may be scattered into the instruments's field of view and contribute to the measured flux, even though the object lies outside the field of view. If the target object is bright, while the non-target objects are dim, the contamination is likely insignificant. An example would be making radiometric observations of the sun through a clear sky. The signal due to the sun is much stronger than the signal due to the scattering of diffuse skylight into the radiometer' s field of view. The contamination may be significant if the situation is reversed, i.e., observing a dim target like diffuse skylight in the presence of a bright non-target object like the sun. If the direct solar beam strikes an element in the optical path of the instrument and if even a small fraction of the beam is scattered into the field of view, the scattered light may significantly contaminate the measurement. In the solar tracking system, measurements may be made in either direct mode or mirror mode. In mirror mode, contamination may result from scattering from the mirror surface. It may not be possible to shade the mirror adequately to prevent this contamination, especially as the position of the sun changes with time. Then it is necessary to determine whether the contamination is significant compared to the signal being measured. If the contamination is significant, it's important to know if the contamination is predictable or 41 if it is possible to specify a mirror with properties that adequately reduce the contamination. The primary concern, then, is the case when a mirror is used in the solar tracking system with an instrument which is observing diffuse skylight. In that case, the total signal produced by the instrument can be partitioned into signals from three distinct sources: • Target reflection the light exiting the target which is specularly reflected by the mirror into the instrument, • Solar scattering the light from the direct solar beam which is scattered by the mirror into the instrument, and • Background scattering the light from other miscellaneous sources which is scattered by the mirror into the instrument. The first item in the list is the desired signal. The other two items are contaminants. Both of these contaminants depend on the scattering properties of the mirror along with the intensity and spatial distribution of the undesirable light sources which are incident on the mirror. The second item may be predicted if the scattering properties of the mirror are known along with the irradiance and direction of the direct solar beam. The third item appears unpredictable due to the complexity and variability of the incident sources. 6.2 Mirror Scattering Properties The scattering properties of a surface can be described in terms of the bidirectional reflectance distribution function (BRDF) (Stover, 1990; Nicodemus et al., 1977). The BRDF relates the radiance scattered from a point on a surface to the irradiance incident on that point. If the scattering properties of the surface are uniform and isotropic, the BRDF can be expressed as (Nicodemus eta!., 1977): 42 (14) or, alternately, in terms of the incident radiance : wheref,. is the BRDF, dLr is the scattered radiance, E; is the incident irradiance, L; is the incident radiance, and dro; is the infinitesimal solid angle through which L; is incident. The angles (0;, q> ;, 0s, s), which define the directions of the incident and reflected radiances, are defined as shown in Figure 15. The BRDF is closely related to the topography of the reflecting surface. Provided the surface is a clean, optically smooth, front-surface reflector, the BRDF can be related to the power spectral density function (PSD) of the surface profile by using the Rayleigh-Rice vector perturbation theory (Stover, 1990). The PSD gives the square of 0, 0-l Figure 15: Scattering geometry 43 the surface roughness height per unit spatial frequency . The BRDF and PSD are related by: where A is the wavelength of the incident light, Q is a polarization factor which relates the polarization states of the incident and scattered beams, and Sis the PSD, which is a function of the spatial frequencies..fx and_t;,. Thus, given a particular surface profile (i.e., a particular PSD), (16) allows the BRDF to be expressed as a function of wavelength, polarization and incident angle, provided the conditions of Rayleigh-Rice theory are met. If the incident beam is unpolarized, and the receiver is insensitive to polarization, Q is given by: 1[ ] 0 =- 0 +O + +O - 2 - ss -sp Q ps ~ PP (I 7) where Q ss, Qsp, Q ps, and Qpp represent the various combinations of incident and detected polarization states and are approximated for good reflectors by (Stover, 1990): (17a) 0 = (sin(JJ 2 -sp cos(SJ (17b) Q =(sin(s)J 2 sp cos(S;) (17c) 0 = ( cos(J - sin(S; )sin(Ss )J 2 - PP cos(S; )cos(S s ) (17d) The spatial frequenciesfx and};, are derived from the hemispherical grating equations for diffraction and are defined by: 44 I' = sin(S,)cos( Q> , )- sin(S;) Jx "- f = sin(S,)sin(q>,) y "- Assuming the surface is isotropic,.fx andJ;, can be reduced to a single frequency,J,so, where: 6.3 Field Tests of Mirror Scattering (18a) (18b) (18c) To assess the significance of the scattering terms, a series of tests was performed which allowed the individual components of the mirror-mode measurement (target reflection, solar scattering and background scattering) to be calculated and compared with the direct-mode measurement. In addition, the solar scattering was used to calculate the BRDF and PSD for the mirror. A sunphotometer was used to make measurements of the solar beam and of diffuse skylight both directly (i.e. , with the sunphotometer pointed at the sky) and through the mirror (with the sun photometer pointed at the mirror and the mirror positioned to reflect skylight into the sunphotometer). Specifications for the sunphotometer and mirror are shown in Table 6. Over the course of three days, the test was repeated three times for each of two wavelengths (Table 7). The BRDF varies with the incident and scattering directions, so data had to be collected for a range of scattering geometries. Note from Figure 15 that the coordinate system for the scattering geometry is defined by the normal to the mirror surface and the azimuth direction of the reflected beam. Since solar scattering was used for the BRDF calculation, the direction of the reflected solar beam defined the azimuthal coordinates. 45 Then the scattering geometry was varied by changing the position of the diffuse target. With the solar tracker, the position of the diffuse target can be specified by an offset from the current solar position. The tracker expresses offsets in units ohime. To obtain the offset position, the tracker software calculates the solar position at the current local time Table 6: Specifications for the sunphotometer and mirror Sunphotometer: Manufacturer: Telescope: Detector: Filters: Mirror: Design: Flatness: University of Arizona Field of view 2 degrees, full angle Entrance aperture 0.75 inches Photodiode, EG&G model UV-444B Temperature stabilized, photovoltaic operation Ten narrow-band three-cavity interference filters Bandpasses between 7 and 15 nm, FWHM 8-inch, aluminized, front-surface with a protective coating. Unspecified Table 7: Parameters for the mirror scattering tests Test Date & Time Waveleneth, nm Offsets relative to sun, minutes A 1 Oct. 97, am 440.9 +25 to + 180 B 5 Oct. 97, pm 440.9 -35 to -180 C 6 Oct. 97, pm 440.9 -35to-180 D 1 Oct. 97, am 779.0 +25 to +180 E 5 Oct. 97, pm 779.0 -35 to -180 F 6 Oct. 97, pm 779.0 -35 to -180 46 plus the specified offset. This position becomes the target position for the tracker. For these measurements, the offsets ranged in magnitude from 25 minutes to 180 minutes, giving angular offsets (relative to the solar position) ranging approximately from 7 degrees to 45 degrees. Smaller offsets could not be used because of problems with scattering of the solar beam by the objective lens of the sunphotometer. These angular offset positions were later used to calculate the scattering geometry . In order for the measurements at each offset to be useful, the target observed in the direct measurement should be the same as the target observed in the mirror measurement (implying that, ideally, the direct and mirror measurements should be made simultaneously and at precisely the same target). However, since the sunphotometer had to be repositioned to switch from direct mode to mirror mode, simultaneous direct and mirror measurements were not possible. Therefore, it was necessary to make sequential direct and mirror measurements at each off set and to minimize the elapsed time between measurements. Also, measurements were confined to days with minimal cloudiness, so as to minimize the spatial and temporal variability of the illumination from the sky . Two mounting positions were set up for the sunphotometer. For direct measurements, a bracket was attached to the back of the gimbal 's optics ring. This allowed the sunphotometer to be mounted on the gimbal, above and behind the mirror so as not to interfere with the gimbal ' s rotation. For mirror measurements, an instrument stand was clamped to the tracker table at a point approximately south of the tracker. The instrument stand was positioned such that, when the sunphotometer was mounted on the stand, the sunphotometer was approximately aligned so that light from the tracker' s target would be reflected into it's aperture. 47 Prior to taking measurements, the mirror was mounted in the gimbal and the tracker was started following the procedures described in Wood et al. (1996). For direct .measurements, the sunphotometer was first mounted on its bracket on the gimbal. An iterative correction was made so that the sunphotometer was pointed directly at the sun, then the direct measurements were made. Next the sunphotometer was moved from the gimbal to its instrument stand. The tracker was switched to mirror mode, any offset was removed, and, if necessary, an iterative correction was applied so that the reflected solar beam was incident on the sunphotometer. Then the mirror measurements were made. Next the sunphotometer was returned to its bracket on the gimbal , realigned and the . remaining direct measurements were made. The data set taken for each particular offset angle consisted of seven data points: • direct-solar sunphotometer pointed directly at the sun, at solar zenith angle 8101, 1 • direct-offset sunphotometer pointed directly at a patch of diffuse skylight at offset angle a • mirror-solar sunphotometer pointed at the mirror, viewing the sun, at solar zenith angle 8101,2 • mirror-offset-sun sunphotometer pointed at the mirror, viewing a patch of diffuse skylight at offset angle a., with the solar beam incident on the mirror • mirror-offset-nosun sunphotometer pointed at the mirror, viewing a patch of diffuse skylight at offset angle a., with the solar beam obscured from the mirror • direct-solar sunphotometer pointed directly at the sun, at solar zenith angle 8101,3 • direct-offset sun photometer pointed directly at a patch of diffuse skylight at offset angle a 48 The direct-solar and direct-offset measurements were repeated in order to bracket the variation in the signals for the direct solar beam and the diffuse target. The elapsed time from the beginning to end of a single set of measurements was approximately ten minutes. The local standard time, the sunphotometer gain (typically in millivolts of output signal per microamp or nanoamp of photodiode current), and the sunphotometer output (in millivolts) were recorded at each measurement. Since the PSD should be independent of the wavelength at which measurements are made (the PSD describes the physical properties of the mirror), measurements were taken in two bands, one centered at 440.9 nm and the other 779.0 nm. For each measurement, geometric calculations were performed to obtain the offset angle (for offset measurements), the solar zenith angle in local horizontal coordinates, the scattering angles, and the spatial frequency coordinatesfx andJ;,. Also, the sunphotometer output in millivolts was converted to photodiode current in nanoamps, based on the gain recorded at the measurement. As noted above, when the mirror is used to observe diffuse skylight, the radiance exiting the mirror can be considered to be due to three terms: target reflection, solar scattering and background scattering. Then the signal from an instrument observing that radiance can be written as the sum of the signals due to each term : ; -; +i +i mi"or- offset - spec seal ,so/iJr scat ,badcground = R,pec i direct-offset + i,ca1 ,solar + i,cat ,background where imirror-offset is the total signal output by the instrument for a mirror-offset measurement, ispec is the signal due to the specularly reflected target ( = R,pecidirect-offset, where Rspec is the specular reflectance of the mirror and ;direct-offset is a measured 49 (19) direct-offset signal) . iscat.solar is the signal due to scattering of the direct solar beam, and i scat,background is the signal due to scattering of other light incident on the mirror. The data were then analyzed to evaluate the individual terms in ( 19). First, the specular reflectance of the mirror was found by calculating the ratio of the mirror-solar measurement to the direct-solar measurement: R = i mirror - solar spec · I direct - solar (20) Since a direct-solar measurement was not made simultaneously with the mirror-solar measurement, the value for idirect-solar was interpolated from the initial and final direct-solar measurements. This was done by assuming a Beer's Law dependence on zenith angle and interpolating to the solar zenith angle at the time of the mirror-solar measurement. Then the ratio was calculated, giving Rspec as a function of incident angle relati ve to the mirror normal for each of the two wavelengths. The dependence on incident angle was weak, so an arithmetic mean value for each wavelength was used in subsequent calculations. Next, i direct-offsec was determined. Since neither of the two direct-offset measurements were simultaneous with the mirror-offset measurement, the direct-offset measurements were . interpolated to the position of the mirror-offset measurement. In this case, an arithmetic mean of the initial and final direct-offset measurements was used. Next, i scat,solar was expressed as the difference between the measurements mirror-offset-sun and mirror-offset-nosun. These two measurements were typically made within a few seconds of each other and were treated as being simultaneous. 50 Finally, iscat,background was determined from : i =i -R i -i scat ,background mirror- s, and A, thenfx,hand Q can be calculated. Usingfx andJ;,, a value of S(fx,fy) can be interpolated from the data, or, if the spatial frequencies are outside the range of the data, the inverse power law can be used to extrapolate S(fx, fy) to the required spatial frequencies . Once S(fx, fy) is known, the BRDF can be calculated using (16). Given the BRDF and an incident irradiance E0, the scattered radiance Lr can be calculated. Rearranging ( 15) to solve for the reflected radiance gives: 51 L,(0s ,s) = f /,(0;, ;, 0s,;)cos(0;)dro i 0, Equation 23 indicates that, for a non-specular reflector, any incident light, from a direction in which!,. is not zero, contributes to the scattered radiance. If the incident irradiance is from the sun, the direct solar beam may be approximated using the Dirac delta function (Liou, 1980): (23) (24) where Eo is the solar irradiance perpendicular to the direct beam. This approximation allows (24) to be evaluated as: If the incident radiance is from a more general source, such as diffuse skylight, the integral (23) will most likely have to be solved using numerical techniques. 52 (25) 7. Mirror Scattering Test Results The results of the mirror scattering tests are cumbersome to present graphically, because, at a given wavelength, the BRDF depends on four variables: 0;, Q>;, 0s, and cl>s (the zenith and azimuth angles, relative to the mirror normal, for both the incident and scattered beams). Normally, tests would be performed with the direction of the incident beam fixed and with measurements made along a constant cl>s so that only one independent variable remains . However, in these tests, the direction of the incident beam is not fixed, nor is cl>s, due to the changing position of the sun and the mirror normal. The dependencies can be simplified somewhat by assuming that the scattering properties of the mirror are isotropic, so that the BRDF is independent of incident azimuth angle but still dependent on 0;, 0s, and cl>s- Sti ll , these parameters vary considerably over even a single set of measurements. So, to simplify further, the raw measurements of the signals for the specular reflection, solar scatter and background scatter are presented only as functions of offset angle. At both wavelengths, solar scattering and background scattering make measurable contributions to the desired signal. After the measurements were decomposed per (19), the target reflection U spec), the solar scatter ( iscar.solar) and the background scatter Uscat,background) were plotted versus offset angle (Figures 16 and 17). In general , i spec decreases with increasing offset angle. In tests A and D, ispec initially decreases, then increases with increasing offset angle. Values for i scar,background vary only slightly with offset angle. Values for isca,.salar are higher at small offset angles, then decrease with 53 C -~ V) <( C C 00 vi TestA-1 Oct 97 0.025 -.------------------,--------,------, 0.020 •••••••• • ••••••••••••••• 0.015 ················ , ..................... . 0.010 ......... ....... ,. ............. ·············· 0.005 0 .000 -0.005 -0.010 0 10 20 30 40 50 Offset Angle, degrees Test B - 5 Oct 97 0 .025 0 .020 0.D15 0 .010 0 .005 0 .000 -0.005 -0.010 0 ················--·--·······1·······························: i --- Target reflection _._ Scattered solar -...- Sc.attered background ! ............ . ·············· ................ : ......................... ...... : ............................... i ........ ...... .. .............. :., -, .......... ........... . 10 20 30 40 50 Offset Angle, degrees Test C - 6 Oct 97 0.025 -.-------,-------,------,------,------, 0.020 ··········--·······-·····--· . : : ' : ; ! 0 .015 0 .010 0.005 0 .000 -0.005 -0 .010 0 ......................................... : ................................. ; ................................................ .. .. ........ .. ' .. • ••• •• : : ·······························:•······························'.·······························;·······························r ···························•·• 10 20 30 40 50 Offset Angle, degrees Figure I 6: Target reflection, solar scattering and background scattering as a function of offset angle for for A= 440.9 nm 54 - C: .!=fl r.J) C'w C: .!=fl r.J) Test D - 1 Oct 97 0.05 0 .04 0 .03 0 .02 0 .01 0 .00 -0.01 ······························i·······························!······························.l ................... ............ l ............. .... ............. · ••• • •• ••n• •--•• •• •• •••• • • ~•• •• ••••• •• • .. ••••••• •• .. •••••~• .. • •••••• •• •--•••••••••••••••• .:-- ••• •••••••• ••• • •••• •••••••••: •••• •••••• ••• • • •• OHHOOH•••• • - · • •ooo 00 0 0 0 0 000000000 00000 ·······························=·······························=·······························-·························· .... ........ ········•···· : : 0 10 20 30 40 so Offset Angle, degrees Test E - 5 Oct 97 0.05 ~---------~----~----~---- 0.02 0 .00 -..- Target reflection _,._ Scanered solar .......- Scanered background .... . . ·······························i·······························1·······························:············· -0.01 -+-----,-------,------+-------;-------, 0 10 20 30 40 so Offset Angle, degrees Test F - 6 Oct 97 0.05 ....------,-----------------'=-----, 0.0: = <1>: = 0. 56 * * ('ti C .~ t./) TestA-1 Oct 97 60 ~---------~----~----~-----, SO •················ ••••••••••••••••••• ••• -+- Scattered solar · · • · · Scattered baclcground 40 ............. .............. ' ........... .. ............... : ···························< ! i j 30 ······ .... · ............ .. .-..... < ......................... .-... , ..... ......... ,4. .... -- ....... ' ............ · · . . : ,. . 20 ·····························:·····················~ :··.··~···: ...... ....... . •:········:-···········:······· 10 .. .................. . ··············~---: -i·: ·:··· ............ : ..... ..... ~. 0 .. .............. ... . : .. .... +·····························:······························•:••·········································· ..... .. .... .. . -10 -+------+------;-------;-------;---------i 0 10 20 30 40 so Offset Angle, degrees Test B - 5 Oct 97 60 ....-------------------------, so 40 - ·····························i ........................ i ··························'······························'······················· ..... .. l i : 30 ·····························+·····························i······························t···························.·1 :·:·:··.-·.-·:. ·•····•······· 20 ······························'····························:t·. ·~ •·:·::.; ·: ·. ;-:··;·.J, .. . ,., .•..• .. ,.: .. :.: .. t ····························· 10 0 -10 -+------.-------;-------;------~-------i 0 10 20 30 40 so Offset Angle, degrees Test C - 6 Oct 97 60 ,--------------------------, so .... .... .................. ............................. .... ... ..... ....................... : ....... ....... .. .......... .. .. , .... ........... ....... ....... . 40 : : 30 . . .............. .......... ..... ······················:··················••···········:······························ 20 10 : . . i. .. . .. ·• ····························· '··.· ·:·~:·:··.~·:·:·~.··.··:~·:·:·~.··:·:~·.-·:·:·:·:·:~·:·:·:·•···-•=·:.:. ·······:····· ······················· 0 ····························· ........ ................ .. ............. ... .... ... ...... .. ..... , .. ... ............. .... ........ , ....... ..... . -10 0 10 20 30 40 so Offset Angle, degrees Figure 18: Magnitudes of solar scattering and background scattering, as% relative to the total mirror-offset signal, as a function of offset angle, for;\= 440.9 nm 57 TestD-1 Oct97 80 ~----~---------~----~----, 70 ·················"' ·"··· ····:······························:··········· ···········;a; ·············•········"'···· ...... . :. · .. : .. .. . : :.: ... •. :.: :.·. . - . •· 60 ············ ....... .. ······,·· ··············:.-:·;·•·'·· • _ 50 : ....... >'·'•-... :. --t ·· . . . . ,. 4o ······························: .. . 1c··········· .. ····· ! ......... .................. ' --+- Scattered solar 3o ·····················:4·.&:t: ..................... ..... )....................... .. · ·A· · Scattered baclcground ······ . . . . . . 10 . . ························································································· 0+------+-----+----------------1 0 10 20 30 40 so Offset Angle, degrees Test E - 5 Oct 97 80 70 60 .. :::::::::::::::::::::·::::::1::::::::::::::::::::::: :: :::::1:::::::::::::·::::::::::::::::.·::::::::::::::::::::· , , 50 '#.. 40 t'tl 30 C: .~ 20 (/} : ::::::: ::::=::·:::: :r::::::::: : ::::::: ::: :: ;::::::; ········ ....... ··; . .. .. . . .. l-....... .... :::::::::: :::::: :. ::l :: .. . ........ ::. ::::::::r::::::::':~ ....... :.:.: .. ~.:.:.:::::·::~::::~:·: :·::::c::::: .................... . 10 ·························~·+··--·· ··:·················--'····7· .-a::-= --~-.:.:-.=-:-=~.:.:.:: 0+----------+------------~-- 0 10 20 30 40 50 Offset Angle, degrees Test F - 6 Oct 97 80 70 60 :::::::::::: :::::::: ....... ·•······••··:::::::: ::: ::::::::::::::::::::::::::::::::::::::1::::::::::::::::::::::::::::::1:::::::::::::::::::::::::::::: '#.. so ···························································?·•····························~······························=······························ - 40 t'tl C: 30 .~ (/} 20 .... ~JE; :- :~:: r :A 10 0 0 10 20 30 40 so Offset Angle, degrees Figure 19: Magnitudes of solar scattering and background scattering, as% relative to the total mirror-offset signal, as a function of offset angle, for A= 779.0 nm 58 Since <1> : = 0, then from (18b) / : = 0 and: (28) e: is found from (18a) as: (29) Then (30) allowing the B F to be plotted solely as a function of e: (Figures 20 and 21 ). The BRDF values are generally higher at small scattering angles and decay to lower values at large scattering angles. The measurements appear reasonably repeatable among ,.... •,_ Vl _._ 1 Oct97 0.1 :::::::·:·:_·::::::. :::·:. ·::i.·:.·:.·:::.·::::.·::::::::.·::::::::i::.·:::::::::::.:·::::::·:·:···::i-:.·::::::::::::::::::.···:···:.··-:-:::.::·:··::: ·. _,_ 5 Oct 97 ._._._._._._._. ....... -.·.·.·.·.·:.· .. ·::::. ·-.i.-:::.-::.-::::.-::.-.-:.-:::::.-.-.-.-:::.l-::.-::·:.-:::::·:::::::::::::::-.:.-.-:·::: ··:::·::::::::::.-: ·· ····'·········· ······· 6 Oct 97 ... ... ..... ... .... . ... ••••• ·······~····•·· •••••••••• ••• • •••• --- ·~ · -· · · · ························· .. . . ... . . . ...... .... . . ... ........... ...... , 'i · ...................................................... . ............. ..... . . . .. ....... . ............. : ................... . ......... ..... ·:·--.. ... .. . .. .. . ... .. .. ... . . . . .. ·:······ .............. ·········--:············••·•· ........ ······:······················ .. ········~ ..... ...... ................ ··••,•········ .. ······· ..... ...... .. . : : . . . . ...... --~.. . ..... .... ... . . . . . . . . . . ·~ ............................. ··:··.... . ... ... ........... . ·····~······· . •··•··· ............ . . 0 .01 ..... ... ... ................... j .................... .. ....... j .. i . ...... ·····················'··················· .. . ... .... . . , ·························-r· ... ·r ...... .............. . 0 .001 : : ....... .......... ........... .. ... ... .... .. . ··············••·)•······--· .................. . 0 10 20 30 40 50 60 Isotropic, normal incidence 0s, degrees Figure 20: Scaled BRDF for normal incidence, plane ofincidence measurement, A= 440.9 nm 59 -..... II) L.L. ... 0 co ...,._ 1 Oct9 7 _..,_ 50ct97 0.1 .............................. : ... ..... ... .... .. ... ........... , ....... .... .................... :. ........ . .............. .. ...... ...... .. .......... 6 0c 97 :::::::::::::::::::::::::::: ·:t:::::::::::::::::::::::::::::.-:.·::::::.·:::::.·:.·:::::::::::::::;:.·::::::::::::::::.·::.·.·:::::::.r.·:·· ·:::::·:·:::.·:::: ... --- t :::::::::::::::::::::::::: :::: j::::::::::::::::::: ::::::::::::i:::::::: ::::::::::::::::::::::: j::::::::::::::::::::::::: ::: :::;:·:::::::::::::::::::::::::::::::: ........... :::::::::::::::::: ·····························-i--····························r···················· .. ·······-:-······························:-······· .. :::··:::::::::::::::::: .. ··:::::::::::::::::::::::::: i i j : ······················ ·····•·············-········ ; .. ..... .. ....... ... ...... ......... ... .... .......... ... . . 0.01 ---------·-------------· .... :::::····················::::::::::::::··········:·::::::::::::::::::::::::: ·········•············· ······························i····························· ................. .: .. ................. .. ........ j ............ ..... .............. j ... ........................... . . . . . . . . . . . . . . . . . . . . . . . . . . . .. -~. ! ! 0 .001 -+-----;---------;..-----+-----+-----.--'----- 0 10 20 30 40 so 60 Isotropic, normal incidence 05, degrees Figure 21 : Scaled BRDF for nonnal incidence, plane ofincidence measurement, 1. = 779 .0 nm the three replicate tests perfonned at each wavelength. At small scattering angles, the BRDF' s for}.= 440.9 appear to be somewhat larger than those for}..= 779.0. At larger scattering angles, the differences seem less significant. Again, there are some anomalous results at the smallest scattering angles. Power spectral density data were also derived from the measurements and plotted versus /;so (Figures 22 and 23). Following (22), nonlinear regression was performed on the data ( excluding the PSD values for the anomalous solar scatter values already noted) so that values for n and Kn could be detennined (Table 8). The standard errors for each parameter and the correlation coefficient for the regression are also shown in the table. The figures and linear regression results show that the PSD for 779.0 nm is significantly larger than that for 440.9 nm. 60 N E ::$.. N E e +-' Vl tlO C: < 6 N E ::$.. N E 0 .... +-' Vl tlO C: < o' rJ) 0... 105 -,----------.. -••• -: --.- .-. - . -• •• ••• - .-•• -•• - .- . --~_. .--------.---------.,-•• CC •• "'.C •• '."C ••• CC •• "'.C •• '."C • • ------------, - . . .• •• • • ••• •• • •••••••• • . :. ::.: •• ; • • : •• • •• • ::·: :·:~::.:·: . ::::::·.~·-· ; • • : :- : 10• 103 101 10• 103 101 101 :: : : : : :::::::::: :: :: :::::::::::::·::!::: ::::::::: ........ i .... : ............ :::::::.·: ;: ....... -:- ..... --!-- -- .. -i-.... -~ .... ; ....... ..... .. .... .. : ........... ·· •••• •• • •••• • • • • ·::···· --- 1 Oct97 _,._ 50ct97 ··············•··••················l.···················.:.···················•···-i··•·····:·······:······:·····:····;································ ....,...... 6 Oct 97 ....... ........... ..... ............ = .................... =... ··':':': . '.'.':':':':':':':':':'::'J:':':':':':':':.':.!,:.':.':.':.':.·':.':.I.·'.:':.':.':.'.:': .. :,.:':.':.':.':.·':.-l,· ':.' .. :':.':.1,.=:.':.':.':.':.':.':.·':.':.':.·':.'·.:'.:':':':':':':':':':':':':':':':':':':':':':':t:': :·:·: :·: :::-: : : : : : : : : : :: : ... .. ..... . . ··········· · ········-················· • • .. ···············•······································· ·<······ .... ········••··••····················· ................ . . .. . . .............. = .......... . . . •.• .... L ...................... J ...... L ___ _. ... :11, r.lli~;:::'11',..~=- ::::::::::::::::::.::::::::::::i:)::::::::::::::::::i:;i::·:·i:-::::::::,:.:::iiiiJ::::.:::f i :::L ... J ... :]: :i:::::: :::: .. · ..... ·::: ........ :::::::::::::::::::L ..... ..... . ........................ ············=··· .................. : .. ........................ ......... =·· ···:······: .............. . .............. .... ················r· ............. ... . i .. ........... ............ ;••······i .... ..: .... ..: ···+· -1- .... . 0 .1 f -1 iso,µm 1.0 Figure 22: PSD versus/;so, A= 440.9 nm ... ... ... ............. . ·=···. _-_-_-_-_-_-_-_-_-_-· --- 1 Oct 9 7 _,.__ 5 Oct97 ....,...... 6 Oct 97 7-s~'.'.'.'.'.'.'.'-'-i'\' ...... :::::.·:::::.::::· ····· · ----------:":'i:':'·'·'·'·'-'·'·=k=.=.=.=.=.=.+·.-.-.-.-+.-.-.-.-i·.=.=.=L=.=.=.=.=.=.=.-.=.=.=.=.=.-.-.-.-.-.-.-.-.-.------------ ----.---.---.--- ····················------------------------