شنبه ۱۴ آذر ۱۳۹۴

GeoGIS

سایتی در زمینه علوم ژئوماتیک

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ارسال توسط راستگو در تاریخ ۲۴ - آبان - ۱۳۹۰
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Traverse Computations

  • Used to perform geometric checks of observed distances and angles
  • With today’s computational abilities, can often be performed in field.
  • Provides methods for distributing random errors throughout observations
  • Two common methods used
    • (Bowditch) Compass-rule adjustment (We will focus on this method in this class.)
      • Adjust angles for geometric closure
      • Compute geometric closures, and proportionately distribute errors based on length of course versus overall perimeter of traverse
      • Easily performed on hand-held calculator, but not mathematically rigorous. In fact, it is an arbitrary method!
    • Least-squares adjustment
      • Based on probability theory
      • Distributes errors based on their estimate errors
      • Provides the most probable values for the observations and station coordinates.
      • Forces geometric closures.
      • Provides only one solution for any given set of observations.
      • Easily performed by computers.

 

Balancing Angles

  • There are two procedures to balance angles
    1. Make corrections to angles in a proportionate method.
      • Angles receive equal corrections. This is the preferred method, but does not match reality.
    2. Make larger corrections to angles where poor observing conditions exist.
      • This method is dependent on judgment, and thus is arbitrary

EXAMPLE

 

Given the following traverse observations, compute the adjusted angles and azimuths for each course given the azimuth of  AB is 31°۰۵’۱۳”.

Course

Length (ft)

Station

Angle

AB

۴۰۵٫۷۰

D-A-B

۱۳۰°۳۱’۴۱”

BC

۴۱۸٫۶۷

A-B-C

۸۶°۴۴’۴۳”

CD

۸۱۴٫۴۱

B-C-D

۹۹°۰۵’۲۶”

DA

۶۵۸٫۹۲

C-D-A

۴۳°۳۸’۰۵”

S angles = (4 – 2) 180° = ۳۶۰°

Method 1

Station

Angle

Multiples of correction

Rounded Correction

Successive Differences

Adjusted Angles

A

۱۳۰°۳۱’۴۱”

B

۸۶°۴۴’۴۳”

C

۹۹°۰۵’۲۶”

D

۴۳°۳۸’۰۵”

Compute Preliminary Azimuths

Course

Azimuth

Course

Azimuth

AB

۳۱°۰۵’۱۳”

CD

BA

DC

BC

DA

CB

AD

CD

AB

Computation of Latitudes and Departures

  • Departure – the change in X, or easting (DX)
  • Latitude – the change in Y, or northing (DY)
  • From sketch it can be seen that departure (Dep) and latitude (Lat) can be computed as

Dep = L sin Az

Lat = L cos Az

  • This is obvious for first quadrant. It is also true for remaining quadrants since
    • Sine of an angle between 0° and 180° is “+” and the sine of an angle between 180° and 360° is “-“
    • Cosine of an angle between 0° to 90° and between 270° to 360° is “+” and the cosine of an angle between 90° and 270° is “-“

Example: Compute the unadjusted latitude and departure for the traverse example given above.

Station

Distance (ft)

Azimuth

Latitude

Departure

A

۴۰۵٫۷۰

B

۴۱۸٫۶۷

C

۸۱۴٫۴۱

D

۶۵۸٫۹۲

A

Totals

 ۲۲۹۷٫۷۰

Traverse Closure

  • This is a measure of precision of the work, although not a good one. It will not account for compensating errors.
  • Still heavily used in industry.
  • Linear misclosure: The linear distance from the final computed coordinates to the known coordinates for the point.
  • Relative precision: The ratio of the linear misclosure to the total traverse length expressed as a ratio of 1:xxxxx.

Example: Compute the linear misclosure and relative precision of the example traverse in this lesson.

Linear misclosure: ______________________

Relative precision: ______________________

 

 

Traverse Adjustment

  • Compass (Bowditch) Rule
    • Prorate misclosures based on ratio of course length to traverse length (see Equations 10-5 and 10-6).

and

    • Carry computations to one more decimal place than distance accuracy to minimize rounding errors.

Example: Compute the adjusted departures and latitudes for the example in this lesson.

Course

Length (ft)

Dep

Lat

Adj. Dep.

Adj. Lat.

AB

۴۰۵٫۷۰

BC

۴۱۸٫۶۷

CD

۸۱۴٫۴۱

DA

۶۵۸٫۹۲

Totals

۲۲۹۷٫۷۰

  • Least-squares – see Chapter 15 of text book for an introduction to least-squares.
    • Provide a single solution to a set of observations.
    • Solution is the most-probable value for the set of observations.
    • Can easily be done with software like WolfPack.

 

Rectangular Coordinates

  • Advantageous for computing elements not measured.
  • Can correct for curvature of earth by using map projection surface such as state plane coordinates (see Chapter 20.) This is covered in SUR 262.
  • X (easting) = Xprevious station + Depcourse
  • Y (northing) = Yprevious station + Latcourse

Example: Compute the coordinates for each station in the example for this lesson.

Course

Length (ft)

Adj. Dep.

Adj. Lat.

X (easting)

Y (northing)

A

۱۰,۰۰۰٫۰۰

۵۰۰۰٫۰۰

۴۰۵٫۷۰

B

۴۱۸٫۶۷

C

۸۱۴٫۴۱

D

۶۵۸٫۹۲

A

Computation of Lengths and Directions

Adjusted course lengths and azimuths can be computed from adjusted departures and latitudes, or coordinates as:

  • Azimuths

 

where C is dependent on the quadrant of the coordinates.

Values for C

Quadrant

Sign of Dep

Sign of Lat

C

I

+

+

۰°

II

+

۱۸۰°

III

۱۸۰°

IV

+

۳۶۰°

  • Lengths

 

Example: Compute the course course lengths and directions from the adjusted latitudes and departures.

Course

Adj. Dep

Adj. Lat

Length (ft)

Azimuth

AB

BC

CD

DA

Personal Practice Problem

Given the following coordinates, compute the course length and azimuth on scrap paper and compare with your response with the correct solution.

Station

X (m)

Y (m)

A

۴۹۳,۲۹۰٫۸۱۰

۸۹,۶۰۹٫۴۲۰

B

۴۹۵,۵۸۲٫۰۶۰

۸۹,۹۵۳٫۹۱۰

C

۴۹۶,۱۲۰٫۶۶۰

۸۷,۳۶۳٫۴۱۰

D

۴۹۸,۰۵۸٫۷۰۰

۸۷,۷۹۱٫۷۸۰

E

۴۹۷,۸۰۱٫۵۰۰

۹۰,۲۹۹٫۲۲۰

Course

Length (m)

Azimuth

 

 

COMPUTATIONS FOR A LINK TRAVERSE

  • Link traverses connect two stations of known coordinates
  • The angular closure is determined by use of a starting and ending azimuth.
  • This form of traverse is commonly used to densify control in a region.

EXAMPLE

Using the following observed values for the link traverse shown above, compute the adjusted coordinates. Use the Bowditch Compass Rule adjustment procedure.

Observations

Station

Angle

Distance (ft)

Azimuth

X

Y

۱

۱۸°۵۲’۴۲”

A

۴۸°۱۸’۴۲”

۱۳,۶۶۴٫۲۳

۴۶,۱۴۵٫۶۰

۱۲,۲۴۴٫۷۴

B

۲۴۹°۲۲’۱۶”

۱۶,۹۷۸٫۶۸

C

۱۷۶°۵۰’۰۳”

۱۳,۵۰۴٫۲۱

D

۱۳۳°۳۱’۴۴”

۱۶,۳۸۳٫۰۷

E

۱۰۱°۵۳’۱۸”

۶۲,۷۹۶٫۱۴

۳۰,۱۶۴٫۰۸

۸°۴۸’۵۲”

۲

Computation of Preliminary Azimuths

Course

Azimuth

Correction

Course

Azimuth

Correction

A1

۱۸°۵۲’۴۲”

CD

۱۳۳°۲۳″۴۳″

۱۳۳°۲۳’۴۷”

+۴۸°۱۸’۴۲”

+۱٫۴″

+۱۸۰°

AB

۶۷°۱۱’۲۴”

۶۷°۱۱″۲۵″

DC

۳۱۳°۲۳’۴۳”

+۱۸۰°

+۱۳۳°۳۱’۴۴”

+۵٫۶″

BA

۲۴۷°۱۱’۲۴”

DE

۸۶°۵۵’۲۷”

+۲۴۹°۲۲’۱۶”

+۲٫۸″

+۱۸۰°

BC

۱۳۶°۳۳’۴۰”

ED

۲۶۶°۵۵’۲۷”

۲۶۶°۵۵’۳۳”

+۱۸۰°

+۱۰۱°۵۳’۱۸”

+۷٫۰″

CB

۳۱۶°۳۳’۴۰”

E2

۸°۴۸″۴۵″

۸°۴۸’۵۲” (check)

+۱۷۶°۵۰’۰۳”

۴٫۲″

Given Az

-۸°۴۸″۵۲″

CD

۱۳۳°۲۳’۴۳”

۱۳۳°۲۳’۴۷”

Misclosure

-۷″

  • -۷″ misclosure for 5 angles yields +7″/5 = +1.4″ per angle. For azimuths, angular correction is cumulative. Thus, the corrections for each azimuth are the properly rounded values of 1.4″, 2.8″, 4.2″, 5.6″, and 7.0″. These corrections and the adjusted preliminary azimuths are shown in the above table under the corrections column.
  • The remainder the traverse computations are similar to a polygon traverse. See Section 10-9 of your textbook for an example and discussion on these computations.

Locating Blunders

  • Blunders in observations are inevitable.
  • A distance blunder will create a computation shift in the traverse. The bearing of the misclosure line will closely match the line the created the blunder. Small discrepancies in the bearings can be expected because of other random errors.
  • An angle blunder will rotate the computational path of the traverse. The perpendicular bisector of the misclosure line will point to the station with the blunder. You can expect the bisector to miss the precise center of the offending angle due to other random errors in the traverse.

 

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