Relative Positional
Precision
ALTA/ACSM Land Title Surveys
3.E.i. - Relative Positional Precision
We know that every point located on a
survey has an uncertainty in its location that
can be represented by an error ellipse.
ALTA/ACSM Land Title Surveys
Relative Positional Precision
Target
Measured angle and distance
Occupied
Station
ALTA/ACSM Land Title Surveys
Relative Positional Precision
Target
Occupied
Station
Uncertainty in the angle measurement
(exaggerated)
ALTA/ACSM Land Title Surveys
Relative Positional Precision
Target
Occupied
Station
Uncertainty in the distance measurement
(exaggerated)
ALTA/ACSM Land Title Surveys
Relative Positional Precision
ALTA/ACSM Land Title Surveys
Relative Positional Precision
ALTA/ACSM Land Title Surveys
Relative Positional Precision
ALTA/ACSM Land Title Surveys
3.E.i. - Relative Positional Precisio
The size (length of the semi-major and
semi-minor axes), and orientation of the
error ellipse is a function of:
(1) the uncertainties in the measurements
that were made and,
(2) the confidence level we are using (95%).
ALTA/ACSM Land Title Surveys
3.E.i. - Relative Positional Precisio
Think about the “Bell Curve” –
If we want a high probability (confidence)
in our measurements, our uncertainty will
be larger
ALTA/ACSM Land Title Surveys
3.E.i. - Relative Positional Precisio
95% Probability, but large
uncertainty
ALTA/ACSM Land Title Surveys
3.E.i. - Relative Positional Precisio
Think about the “Bell Curve” –
If we want a lower amount of uncertainty,
we can accomplish that, but with a lower
probability (i.e. a lower percentage of the
time)
ALTA/ACSM Land Title Surveys
3.E.i. - Relative Positional Precisio
Smaller Uncertainty,
but only with
5% probability
ALTA/ACSM Land Title Surveys
3.E.i.- R.P.P. defined
“the length of the semi-major axis,
expressed in feet or meters, of the error
ellipse representing the uncertainty due to
random errors in measurements in the
location of the monument, or witness,
marking any corner of the surveyed
property relative to the monument, or
witness, marking any other corner of the
surveyed property at the 95 percent
confidence level (two standard deviations).
ALTA/ACSM Land Title Surveys
Relative Positional Precision
The length of the semi-major axis in this error ellipse
ALTA/ACSM Land Title Surveys
3.E.v.- Relative Positional Precisio
The maximum allowable Relative Positional
Precision [the maximum length of the semimajor axis] for an ALTA/ACSM Land Title
Survey is 2 cm (0.07 feet) plus 50 parts per
million (based on the direct distance
between the two corners being tested)
ALTA/ACSM Land Title Surveys
Relative Positional Precision
Note that R.P.P. is not expressed at a
point, but rather it is related to the
relationship between the two points
being tested…
The size of the error ellipse is a function
of the error ellipses at each of the two
end points of that line.
ALTA/ACSM Land Title Surveys
Relative Positional Precision
ALTA/ACSM Land Title Surveys
Relative Positional Precision
The RPP for the line is compared to the
allowable RPP (0.07 feet (2 cm) plus 50
ppm).
The 50 ppm is based on the length of the
line connecting the two points being tested.
ALTA/ACSM Land Title Surveys
Relative Positional Precision
Example:
If the line being tested is 1,000 feet long,
what is the allowable RPP for that line?
0.07 feet+[(50)/1,000,000]*1000= 0.12
feet
Can we achieve this?
ALTA/ACSM Land Title Surveys
3.E.iv.- Measurement Procedure
To achieve a low level of random error
uncertainty, the surveyor must:
(1) use appropriately trained personnel
(2)compensate for systematic errors,
including those associated with
instrument calibration
ALTA/ACSM Land Title Surveys
3.E.iv.- Measurement Procedure
The surveyor shall:
(3) use appropriate error propagation and
measurement design theory (selecting the
proper instruments, geometric layouts, and
field and computational procedures) … to
control random errors such that the
maximum allowable Relative Positional
Precision is not exceeded.
ALTA/ACSM Land Title Surveys
Relative Positional Precision
If the RPA for the line being tested exceeds
the allowable 0.07 feet and 50 ppm, we can…
• make redundant measurements
• use higher precision equipment
• develop a different survey design
… to decrease the size of the error ellipses
at the endpoints of the line
ALTA/ACSM Land Title Surveys
3.E.v.- Relative Positional Precisio
It is recognized that in certain
circumstances, the size or configuration
of the surveyed property, or the relief,
vegetation or improvements on the
surveyed property will result in survey
measurements for which the maximum
allowable Relative Positional Precision may
be exceeded. This must be explained (see
Section 6.B.ix)
ALTA/ACSM Land Title Surveys
Relative Positional Precision
A simple 3-step process to address R.P.P.
on your survey:
1. Understand: What are the primary
sources of random errors in your
measurements ?
ALTA/ACSM Land Title Surveys
Relative Positional Precision
2. Develop reliable estimates of the
magnitude of those random errors from
each of those sources
ALTA/ACSM Land Title Surveys
Relative Positional Precision
3. Use a correctly weighted Least Squares
adjustment to process your data
ALTA/ACSM Land Title Surveys
Estimating R.P.P.
1. “What are the primary sources of
random errors in your measurements ?”
ALTA/ACSM Land Title Surveys
Estimating R.P.P.
“What are the primary sources of random
errors in your measurements ?”
In an
•
•
•
•
angle (with electronic total station):
Pointing
Centering the instrument
Centering the target
Reading (straight out of the manual)
ALTA/ACSM Land Title Surveys
Estimating R.P.P.
“What are the primary sources of random
errors in your measurements ?”
In a distance (with EDM):
• Centering the instrument
• Centering the target
• Reading (straight out of the manual)
ALTA/ACSM Land Title Surveys
Estimating R.P.P.
“What are the primary sources of random
errors in your measurements ?”
With
•
•
•
GPS:
Centering the instrument
Centering the target
“Reading” (this is a function of the
quality of your observation (PDOP,
SNR, length of observation, etc.)
and the length of your baseline)
ALTA/ACSM Land Title Surveys
Estimating R.P.P.
2. “Develop reliable estimates of the
magnitude of the random errors from
each of those sources.”
ALTA/ACSM Land Title Surveys
Estimating R.P.P.
Develop reliable estimates of the magnitude
of the random errors from each of those
sources
• These cannot be “WAGs”
• They need be statistically based
(Note that random reading errors for angles and
distances come directly from the equipment manual;
and random ‘reading’ errors for GPS will be
generated by the processing software)
ALTA/ACSM Land Title Surveys
Estimating R.P.P.
3. These uncertainties are used in the least
squares program for a weighted (best fit)
solution.
ALTA/ACSM Land Title Surveys
Uncertainty in Conventional
Measurements
Following is an example of the determination
of the magnitude of significant random
errors in a conventional measurement…
ALTA/ACSM Land Title Surveys
Uncertainty in Conventional
Measurements
Analyze the angular
measurement
at station 2 (BS
station 1 and FS
station 3).
Analyze the distance
measurement
from 2 to 3
ALTA/ACSM Land Title Surveys
Uncertainty in Conventional
Measurements
• Significant sources of Uncertainty
(random error) in Angle
Measurements
• Significant sources of Uncertainty
(random error) in Distance
Measurements
ALTA/ACSM Land Title Surveys
Significant Sources of Uncertainty in
Angular Measurements
•
•
•
•
Pointing*
Centering the instrument
Centering the target
Reading *
* These two are a function of how many pointings &
readings we make. Let’s assume we are turning 1 direct &
1 reverse (or turning the angle and closing the horizon) with
an electronic total station (where the standard deviation is
2” DIN from the manual). This is 4 pointings & 4 readings.
ALTA/ACSM Land Title Surveys
Develop reliable estimates of the
magnitude of the random errors in
Angular Measurements
Pointing
Ed Mikhail and Gordon Gracie in their book Analysis
and Adjustments of Survey Measurements, estimate
the standard deviation of a single pointing (p) to be 1
to 4 seconds. We will assume p = 2 seconds. But we
need the standard deviation for 4 pointings (n = 4).
 for 4 pointings  pn  ”
ALTA/ACSM Land Title Surveys
Develop reliable estimates of the
magnitude of the random errors in
Angular Measurements
Centering the instrument
Mikhail and Gracie estimate the standard deviation of the
instrument centering (c) to be 0.5 to 3 mm. We will
assume c = 1 mm (0.0033 feet). But we need to know how
this impacts the angle. (Where D1= length of backsight. D2=
length of foresight. D3= Distance between backsight and
foresight)
  cD3)DD

Cont’d
  cD3)DD

  X  rads  ”
ALTA/ACSM Land Title Surveys
Develop reliable estimates of the
magnitude of the random errors in
Angular Measurements
Centering the target
What is the standard deviation of your target centering (T)?
If the target is on a tripod, it should be the same as the
standard deviation of your instrument centering.
But what if your target is a prism on 8 feet of range pole on
a windy day? What is T then?!
We will assume the target is on a tripod (T= 1 mm).
Again, we need to know how this impacts the angle.
ALTA/ACSM Land Title Surveys
Develop reliable estimates of the
magnitude of the random errors in
Angular Measurements
Centering the target
(where D1= length of backsight. D2= length of foresight and
T is 1mm (0.0033 feet)(target is on tripod)
  T(D

2
+ D 2)1/2D D 
 

1
2
  X  rads  ”
ALTA/ACSM Land Title Surveys
Develop reliable estimates of the
magnitude of the random errors in
Angular Measurements
Centering the target
But what if our target was on 8 feet of range pole on a windy
day?
Estimate T is 2 cm (0.066 feet) [Is this even a
reliable estimate??]
(Where D1= length of backsight. D2= length of foresight and
T is 2 cm (0.066 feet)
  T(D12 + D22)1/2DD
  
 X  rads  ”
ALTA/ACSM Land Title Surveys
Develop reliable estimates of the
magnitude of the random errors in
Angular Measurements
Reading
From manufacturer the Standard Deviation is 2” for
one D&R (2” DIN). Remember, we turned a direct and
reverse (or turned the angle and closed the horizon) –
either way, we had 4 readings, so the DIN out of the
manual is the number we need.
The equation:
 Rn
(where n = number of readings)
Cont’d
 Rn
 
ALTA/ACSM Land Title Surveys
Develop reliable estimates of the
magnitude of the random errors in
Angular Measurements
Pointing = 1.0”
Centering the Instrument = 1.2”
Centering the Target (on a tripod) = 1.0”
Reading = 1.0”
ALTA/ACSM Land Title Surveys
Develop reliable estimates of the
magnitude of the random errors in
Angular Measurements
 = ( p2 +  i2 +  t2 +  r2)1/2
Cont’d
 = (1.02 + 1.22 + 1.02 + 1.02)1/2
= 2.1 seconds standard
deviation in our angle
measurement
ALTA/ACSM Land Title Surveys
Develop reliable estimates of the
magnitude of the random errors in
Angular Measurements
But what if we used 8 feet of range pole?
Pointing = 1.0”
Centering the Instrument = 1.2”
Centering the Target = ?
Reading = 2.0”
ALTA/ACSM Land Title Surveys
Develop reliable estimates of the
magnitude of the random errors in
Angular Measurements
 = (1.02 + 1.22 + 20.02 + 1.02)1/2
= ??? standard deviation
in our angle
measurement
ALTA/ACSM Land Title Surveys
Significant Sources of Uncertainty in
Distance Measurements
• Centering the instrument
• Centering the target
• Instrument error
ALTA/ACSM Land Title Surveys
Develop reliable estimates of the
magnitude of the random errors in
Distance Measurements
Centering the instrument
If our instrument is on a tripod, remember we
already estimated the standard deviation as 1
mm (0.0033 feet).
There is no equation to use; the impact of the
standard deviation of the instrument centering
on the distance is simply that number (1 mm
or 0.0033 feet)
ALTA/ACSM Land Title Surveys
Develop reliable estimates of the
magnitude of the random errors in
Distance Measurements
Centering the target
If our target is on a tripod, remember we
already estimated the standard deviation as 1
mm (0.0033 feet).
There is no equation to use; the impact of the
standard deviation of the instrument centering
on the distance is simply that number (1 mm
or 0.0033 feet)
ALTA/ACSM Land Title Surveys
Develop reliable estimates of the
magnitude of the random errors in
Distance Measurements
Centering the target
But what if our target was on 8 feet of range
pole on a windy day? We estimated (guessed)
the standard deviation as 2 cm (0.066 feet).
Again, there is no equation to use; the impact
of the standard deviation of the instrument
centering on the distance is simply that
number (2 cm or 0.066 feet)
ALTA/ACSM Land Title Surveys
Develop reliable estimates of the
magnitude of the random errors in
Distance Measurements
Instrument Error
From the manufacturer, Standard Deviation is
2 mm and 2 ppm
ALTA/ACSM Land Title Surveys
Develop reliable estimates of the
magnitude of the random errors in
Distance Measurements
Centering the Instrument = 1 mm
Centering the Target (on a tripod) = 1 mm
Instrument Error = 2 mm + 2 ppm
ALTA/ACSM Land Title Surveys
Develop reliable estimates of the
magnitude of the random errors in
Distance Measurements
What is total standard
deviation?
 = (0.00332 + 0.00332 + 0.00662 +
(1250.00*2) 2/1,000,0002)1/2
 = 0.008 feet
standard deviation in our distance
measurement
ALTA/ACSM Land Title Surveys
Develop reliable estimates of the
magnitude of the random errors in
Distance Measurements
But what if we used 8 feet of range pole?
Centering the Instrument = 1 mm
Centering the Target = 2 cm
Instrument Error = 2 mm + 2 ppm
ALTA/ACSM Land Title Surveys
Develop reliable estimates of the
magnitude of the random errors in
Distance Measurements
What is total standard
deviation?
 = (0.00332 + 0.0662 + 0.00332 +
(1250.00*2) 2/1,000,0002)1/2
 = 0.066 feet
standard deviation in our distance
measurement
ALTA/ACSM Land Title Surveys
Total Uncertainty in this
Conventional Measurement 1
Using the semi-major axis as the defining
factor for Relative Positional Precision, we
need to check to see which is greater, the
angular error or the distance error (target
on tripod)…
Angular:
1250.00’ X sin 2.7” = 0.016 feet
Distance:
0.008 feet
Therefore, angular error is greater
ALTA/ACSM Land Title Surveys
Total Uncertainty in this
Conventional Measurement 1
But this is for  What about 95% 
 = 0.016 feet
95% C.I.  = 0.032 feet for course 2-3
(target on tripod)
ALTA/ACSM Land Title Surveys
Total Uncertainty in this
Conventional Measurement 1
Using the semi-major axis as the defining
factor for Relative Positional Precision, we
check to see which is greater, the angular
error or the distance error (target on 8 feet
of range pole)…
Angular: 1250.00’ X sin 20.2” = feet
Distance: 0.066 feet
Therefore, angular error is greater
ALTA/ACSM Land Title Surveys
Total Uncertainty in this
Conventional Measurement 1
But this is for  What about 95% or 
 = 0.016 feet
95% C.I.  = 0.032 feet for course 2-3
(target on 8 feet of range pole)
ALTA/ACSM Land Title Surveys
Allowable Relative Positional
Precision in this Conventional
Measurement
What is the allowable RPP?
 = 0.07’ + 50 ppm (from the Standards)