The Search for Azimuth

An Evolution of Methodology with

Technology

Azimuth definition

noun

1.The Horizontal Angular distance from a Reference Direction,usually the Northern* Point of the horizon, to a Target Point.The measurement is made clockwise through from 0° to 360°.

2.The Horizontal Angle of the Observer's Bearing in Surveying,measured Clockwise from a Reference Direction, as from the North*.

* Astronomers use the South Reference Direction

Why do weneed Azimuth?

Azimuth on Cadastral Plans

The Need for AccurateAzimuth on Cadastral Plans

How do weget Azimuth?

dx

dy

q

= q tan-1 [dx/dy]dy ≠ 0

From Control:

How elsedo we getAzimuth?

Solar Obs:

Altitude SolarThe Astronomical Triangle

What datado we need?

AltitudeSolar

Location: f,lHoriz. Reading Sun & R.O.Vertical Reading SunApproximate Time (± 1minute)Ephemeris: Sun’s Declination

Getting Location

Before Now

- GPS- Google Maps- L&S Tiles: UTM Inverse E,N → f,l (Sub-second Accuracy)

- Ward Sheets- Topo Maps(1’ Accuracy)

Getting Sun’s Declination

Primary Source:Star AlmanacforLand Surveyors

1989 Editionissued to N. Abdulby L&S Division Libraryon 14 Sept. 1988

Getting Sun’s Declination

Star AlmanacforLand Surveyors

2016 EditionNow includesa CD-ROMwith tablesin PDF/TXT Formats

Getting Sun’s Declination

First Attempts toAutomate the Process (1991)

Fourier Coefficients

a1 = 0.3631751a2 = 4.0944611a3 = -22.8147677a4 = -0.224888a5 = -0.4492882a6 = 0.1045454a7 = -0.2316979a8 = 0.0251665a9 = -0.0027951a10 = 0.0019216a11 = 0.0019295

d =

a1 + a2 sin x + a3 cos x +a4 sin 2x + a5 cos 2x +a6 sin 3x + a7 cos 3x +a8 sin 4x + a9 cos 4x +a10 sin 5x + a11 cos 5x

The Fourier Equation

x = 0.017 *[ days since 1990.1231 +(hr + 4 + min/60)/24 ]

Available Technology

The HP41CX hada function DDAYSto subtract datesto get the number ofdays.

The HP42s did not!

The Fourier Coefficientswere only valid for6 months at a time.

Obtaining Sun’s Declinationwith Fourier Series Example

Date: 01 Jan 1991Local Time 07:30

Result: -23° 01’ 22”Error* = 0”

http://ephemeris.com/

Date/Time: 1991.01.0111:30:00 UTCDeclinationSun -23°01'22"

Over 6 monthsThe max. error was ±3”

Getting Sun’s DeclinationGetting Long-Term Programmable Equations from the WWW

Obtaining Sun’s Declinationwith Long-Term Equations

Example

Date: 01 Jan 1991Local Time 07:30

Result: -23° 01’ 18”Error = -6”

http://ephemeris.com/

Date/Time: 1991.01.0111:30:00 UTCDeclinationSun -23°01'22"

SDEC \<< Wait -42 SFDATE IP 'day' STODATE FP 100 * DUPIP 'mnt' STO FP10000 * 'year' STOTIME 2 RND 'time'STO"SUN'S DECLINATION"{ { "DAY : ""ENTER DAY" 0 } {"MONTH : ""ENTER MONTH" 0 } {"YEAR : ""ENTER YEAR" 0 } {"TIME : ""ENTER LOCAL TIME"0 } } { } DUP daymnt year time 4\->LIST { day mntyear time } PURGE IF INFORM NOT THEN CLEARKILL END PROCLIST\-> DROP SWAP'year' STO 'LT'STOK 4 + HMS\-> 24 /ROT ROT STD \->STR IF DUP SIZE 2 < THEN "0" SWAP

+ END DUP "/"year + + "/" SWAP +'dmy' STO SWAP \->STRDUP dmy + 'dmy' STO"." + SWAP + year +STR\-> 31.121999 SWAPDDAYS 6 FIX + 1.5 -'d' STOK .98560028* 357.529 + 'g' STO280.459 .98564736 d* + g SIN 1.915 * +g 2 * SIN .02 * +SIN 23.439.00000036 d * - SIN* ASIN \->HMS 4 RND'Dec' STO "Date : "dmy + 2 FIX"Time : " LT 2 RND\->STR + IF LT 12 > THEN 1 SF"PM OBS: FLAG 1 SET" ELSE "" END 4 FIX"Sun's Declination :"Dec DMS """SUN [ENTER] / Quit[+]"{ d g q year dmy LT} PURGE 7 dispSTKSUN \>>

Major limitations with computingThe Sun’s Declination on an HPCalculator were:

1. Memory (RAM)2. The Interface (9 Lines of Text)3. The CPU (very slow)4. The Programming Language5. No File System6. Limited I/O Capability7. No Storage Capability

In 2013Android Devices

caused aParadigm Shift

The ComputationMethod for

Altitude Solarsbecame moresophisticated

- Built-in GPS Chips*- Better User Interface- Powerful Multi-Core CPUs- Powerful Java Language- Better Declination Algorithm**- GBs of RAM- MS Filing System- USB/Wi-Fi/Bluetooth I/O

** From G.G. Bennett (Australian Surveyor 1980)* GPS/GLONASS on Some Models

The ComputationMethod for

Altitude Solarsbecame more

sophisticated…

… but the fieldobs. procedurewas identical

Revisiting theAstronomical Triangle

Instead ofMeasuring Altitude (h)

we could insteadMeasure Time

to get theLocal Hour Angle

(LHA)

Since the Earth rotatesthrough 360° every 24 Hours

(more or less)

then

we can compute that itrotates 15” (arc seconds)every 1 second of Time

(more or less)

In theory we canmeasure Time

more accurately thanwe can measure theAltitude of the Sun

Basis for Hour Angle Method

Altitude Obs. Uncertainty

“…the refraction determinationis to some degree an educated guess.”

- Jerry L. Wahl (American Land Surveyor)

GPS Time Internet Time

Android App “GPS Test” “Time.is” Website

AndroidTimeCertainty

GPS Time is basedon Atomic Clocksand is never adjustedfor Terrestrial Phenomenalike the slowing of theEarth’s rate of rotation.

Instead an offset to UTis broadcast in the GPS Signal.

From the GPS Time &broadcast correction to UT,we get Accurate Universal Time.

The Android OS makes thecorrection automatically.

Values of delta-T(seconds)

2014 1 1 67.2810 2014 2 1 67.3136 2014 3 1 67.3457 2014 4 1 67.3890 2014 5 1 67.4318 2014 6 1 67.4666 2014 7 1 67.4858 2014 8 1 67.4989 2014 9 1 67.5111 2014 10 1 67.5353 2014 11 1 67.5711 2014 12 1 67.6070 2015 1 1 67.6439 2015 2 1 67.6765 2015 3 1 67.7117 2015 4 1 67.7591 2015 5 1 67.8012 2015 6 1 67.8402 2015 7 1 67.8607

UTUniversal Time.Defined by the Earth's rotation, formerly determinedby astronomical observations but today GPS satellites are usedinstead. This time scale is slightly irregular.

TTTerrestial Time.Was defined in 1991 to beconsistent with the SI second andthe General Theory of Relativity.

Terrestial Time is used to Computethe Astronomical Ephemerides.We need to add delta-T to UT toget TT.

TT = UT + delta-T

What datado we need?

Hour AngleSolar

Location: f,lHoriz. Reading Sun & R.O.Vertical Reading SunAccurate Time (± 1second)Ephemeris: Sun’s Declination,GHAAires , Right Ascension &Sun’s Semi-Diameter

GHASun = GHAAires – RASun

LHA = GHASun – Longitudew

To compute the required Ephemerides(Right Ascension, Declination & Semi-Diameter of the Sun) to maximum accuracy, the effects of Precession & Nutation

on the Earth’s position have to taken into account

Putting all theRequired Codeto process Location,Time and Ephemeridesin an Android App

SolarHA

Using realtime GPSLocation & Time

We can Predict theAzimuth andGrid Bearingof the Suneliminating the need forobservations thencomputations to orienta Total Station

Finally using GPSLocation &Time Correction

We can compute theAzimuth andGrid Bearingof the Sunat a given Time Instanceto orient a Total Station.

Thank You!