POTW #13-11 Photographic Dating

Transcription

1 POTW #13-11 Photographic Dating Finding When A Photograph On Google Earth Was Taken John Snyder, FSA October 26, :30 EST Problem Locate West Point on Google Maps. Determine the time, day, month, and year that the satellite photograph was taken. Give several pieces of evidence to support your conclusion. Solution Summary Of Results I estimate that the photograph was take on October 7, 2011 at 11:45 AM EDT. Photographic Analysis We begin by loading the raw photograph of the Battle Monument as copied from Google Earth.

2 2 PhotoDate1311.nb In[1]:= raw Import NotebookDirectory "Battle Monument.png" Out[1]= We want to measure the sun s azimuth and the length of the shadow using this photograph. We first simplify things by detecting edges in the photograph and then drawing a north-south line and the vector from the base of the monument to the tip of the shadow cast on the ground.

3 PhotoDate1311.nb 3 In[2]:= img EdgeDetect raw, 8 ; Show img, Graphics Thickness 0.005, Orange, Line 214.5, 0, 214.5, 512, Arrow 214.5, 222, 125.5, Out[3]= In[4]:= The angle between the two orange lines can now be found. This subtracted from 180 gives the sun s azimuth measured north through east. 180 VectorAngle 125.5, , 222, 214.5, , 222 Degree Out[4]= In[5]:= The height to the top of the statue of Fame is listed in Internet sources as being 75 feet. The length of the shadow cast on the ground as measured from the above picture using the Google Earth ruler is 86 feet. This makes the sun s altitude in degrees: ArcTan Degree Out[5]= Astronomical Calculations At this point we need to undertake some astronomical calculations. The following function will compute the Julian Day Number for any date. The Julian calendar is used prior to October 15, 1582 and the Gregorian calendar on or after that date. See Meeus p60-62.

4 4 PhotoDate1311.nb In[6]:= In[7]:= JD m_?numericq, d_?numericq, y_?numericq : Module month, year, a, b, If m 2, year y 1, year y ; If m 2, month m 12, month m ; a IntegerPart year 100 ; b 2 a IntegerPart a 4 ; If y m 100 d `12, b 0 ; IntegerPart `12 year 4716 IntegerPart `12 month 1 d b `12 ; To calculate the Sun s position we ll use the French VSOP87 analytic theory of the motion of the Earth. We load a package which I wrote for another project that computes these values and make its contents available to the parallel kernels. Needs "VSOP87`" ; ParallelNeeds "VSOP87`" ; To calculate the Sun s altitude and azimuth we ll use the following function based on formulas given in Meeus. Since the precision with which we can measure angles and distances from the photograph is severely limited we ll not introduce any significant refinements. We ignore the Sun s latitude, parallax, light travel time, nutation, variation in obliquity during the year, aberration, and the T time differential; all these factors being immaterial for the present purpose. The only refinement we introduce is a correction for atmospheric refraction in the Sun s apparent altitude if the Sun is above the horizon. Note that the latitude and longitude of the monument as determined from Google Earth is defined in this function. The function also includes an allowance for the fact that West Point is 5 hours behind Greenwich in time. For this reason all local times used in this function are in terms of Eastern Standard Time so any allowance for daylight savings time must be made manually. In[9]:= sun JDay_?NumericQ : Block jd, t, tz 5 24, L , Φ , Θ, Λ, Ε 23.44, Α,, azimuth, altitude, add time zone correction to put JD on Greenwich basis jd JDay tz; jd t ; compute local mean sidereal time in radians Θ Mod jd t 2 t L, 360 Degree; call VSOP87 to get Earth's ecliptic longitude, assume latitude is zero Λ Mod Π Earth t 10 1, 2 Π ; calculate right ascension and declination in radians Α ArcTan Cos Λ, Sin Λ Cos Ε ; ArcSin Sin Ε Sin Λ ; Print Α 15,,Θ 15 Degree ; compute azimuth N E S W and altitude in degrees azimuth Mod Π ArcTan Cos Θ Α Sin Φ Tan Cos Φ, Sin Θ Α, 2 Π Degree; altitude ArcSin Sin Φ Sin Cos Φ Cos Cos Θ Α Degree; increase altitude for refraction if Sun above horizon 50' If altitude , altitude altitude ; 10.3 Tan altitude Degree altitude 5.11 return azimuth and altitude in decimal degrees azimuth, altitude ; In fitting the Sun s position to the photographic data we ll minimize the sum of the squares of the errors in azimuth and altitude measured in degrees. The next cell computes this error. In[10]:= error az_?numericq, al_?numericq : az al 2 ; We ll run the function for all the days during In the next two cells we compute the starting and ending Julian day numbers for these calculations.

5 PhotoDate1311.nb 5 In[11]:= NumberForm JD 1, 1, 2011, 10, ExponentFunction If 10 10, Null, & Out[11]//NumberForm= In[12]:= NumberForm JD 12, 31, 2011, 10, ExponentFunction If 10 10, Null, & Out[12]//NumberForm= In[13]:= In[14]:= The next cell carries out the calculations using parallel kernels. For each day of the year we find the time of day that best fits the photographic data for the Sun s altitude and azimuth. The variable Τ is the fraction of a day measured from the preceding midnight. We assume that Τ must range over the interval of 6:00 am to 6:00 PM EST. data ParallelTable FindMinimum error sun jd Τ, Τ, 0.25, 0.75, jd, , ; Because the Sun passes through the same declination twice in any year there must be two possible solutions for the time the photograph was taken. These solutions will be the values having the smallest errors. We find the days of the year that correspond to these two minima. Position data All, 1, RankedMin data All, 1, & 1, 2 Flatten Out[14]= In[15]:= Out[15]= 65, 280 And the corresponding possible dates on which the photograph was taken are then as follows: DatePlus 2010, 12, 31, & 2011, 3, 6, 2011, 10, 7 In[16]:= The time of day at which the photograph could have been taken can now be determined. We first find the Julian day numbers at the time of the preceding midnight. NumberForm Table jd, jd, , , 280, 10, ExponentFunction If 10 10, Null, & Out[16]//NumberForm= , In[17]:= Out[17]= In[18]:= Out[18]= For the date March 6, 2011 the photograph could have been taken at the following EST. DMSList Τ. Rest FindMinimum error sun Τ, Τ, 0.25, Round 11, 8, 36 For the date October 7, 2011 the photograph could have been taken at the following EST. DMSList Τ. Rest FindMinimum error sun Τ, Τ, 0.25, Round 10, 45, 13 Now from the Google Earth photograph it appears that the picture must have been taken in October instead of early March. In evidence of this we note that there appear to be mature leaves on the trees. In addition, looking at the West Point football stadium it appears to be ready for play. In light of this, the photograph seems to have been taken on October 7. In October West Point would be on Eastern Daylight Time or EDT. From the calculations in the prior cell we would estimate that the photograph was taken at 11:45 AM EDT. It makes sense that Google would take all photographs of this type near noon in order to minimize the length of shadows which can other obscure details in the picture. This leaves us with only the task of finding the year in which the photograph was taken. There may have been construction or other changes at West Point which would be unfamiliar to me. In my experience, however, Google updates photos of important areas frequently. It seems too soon for most 2012 photographs to have published, so we ll guesstimate that the photograph was taken in I estimate that the photograph was take on October 7, 2011 at 11:45 EDT.

6 6 PhotoDate1311.nb References Meeus, Jean, Astronomical Algorithms, Second Edition, Willimann-Bell, 1998