GPS pseudorange and carrier phase observations may be modelled as

1
GPS pseudo-range and carrier phase observations
may be modelled as

• Obsji ji bj bi bji vji
• where
• jiis geometric range from stn.j to sat.i,biare
  the satellite dependent biases,bjare the station
  dependent biases,bjiare the observation dependent
  biases, andvjiis the measurement noise

2

• sample data file record for one epoch (date
  26/7/92, time 65230 UT) to the five satellites
  PRN2, 11, 18, 19, 28 (one record of five
  observation data types to each satellite) will
  help illustrate the impact of some of the
  measurement biases and errors

3
C/A pseudo-range(m)L1 phase(L1 cycles)L2
phase(L2 cycles)L1 pseudo-range(m)L2
pseudo-range(m)

• PRN2 --22333042.96600 -12176065.17700
  -9487835.16600 22333025.72200 22333027.39100

4
comments can be made concerning the various
quantities

• All five observation types are biased by the same
  amount (equivalent range) by the receiver and
  satellite clock errors, and the tropospheric
  delay.
• The phase observations have negligible noise. The
  P code pseudo-ranges have observation noise of a
  few decimetres, while the C/A code pseudo-ranges
  are the "noisiest". The multipath error (if
  present) is greatest for the C/A pseudo-ranges,
  and least for the phase measurements.
• The ionosphere accounts for most of the
  difference in the pseudo-range measurements on L1
  and L2. This is equivalent to the difference in
  the L1 and L2 phase observations when they are
  converted to range (metric) equivalent values.
• The ionospheric delay on the C/A pseudo-range is
  equal to that on the L1 pseudo-range, and equal
  in value, though not in sign, to that of the L1
  phase (when expressed in metric units).
• The ionospheric delay on pseudo-range
  measurements means that they measure range that
  is longer than "true", but that the phase
  observations are shorter than "true".
• The (unknown) ambiguity on the L1 phase is
  different to that of the L2 phase, and is
  different for each satellite.

5
Options for handling GPS measurement biases and
error

• End of 6.3.1

6
SETTING UP THE SOLUTION

• Defining the apriori groundmark coordinates,
  including that for the "datum" station to be held
  fixed (for example, from a pseudo-range point
  position solution, or a triple-difference phase
  solution, or a previous solution when "chaining"
  baselines). This involves correctly setting up
  the station file (usually from the recorded field
  data), with information on antenna height, etc.
• Identifying the ephemeris file to be used (may be
  a Navigation Message file, or the Precise
  Ephemerides).
• Any satellites to be excluded from solution (for
  example, because of known health problems).
• Identifying the baseline to be processed, by
  selecting the data files to be used (generally
  from a database of GPS files).
• Inputting the standard deviation of the
  differenced observations.
• If option is available for taking correlations
  into account, this may be exercised.
• Minimum elevation cutoff angle for data culling
  to low satellites.
• Data selection for solution (all data or some
  sample rate, for example every 5th data epoch).
• Tropospheric refraction model for bias may be
  activated, based on input met data or "standard
  atmosphere" values.
• Dual-frequency processing options to be
  exercised.
• Whether to attempt ambiguity resolution or not
  (in the case of double-differenced solution), and
  the test/validation parameters associated with
  the algorithm.

7
"hard-wired" processing options for
double-differenced data solutions

• internally defined by the software
• Differencing strategy (between-satellites) to be
  used.
• Ambiguity parameter model, usually single- or
  double-differenced model.
• Criteria for judging success of ambiguity
  resolution on a parameter-by-parameter basis.
• Solution convergence criteria.
• Internal modelling of the satellite orbit.
• Ordering of the satellites (influences the
  between-satellite differencing process

8
PROCESSING OF DIFFERENCED DATA

•  
•  

9
GPS manufacturers provide guidelines for the
selection of the "optimum" solution, on the basis
of the length of the baseline

• recommends that the "optimum" solution for
  baselines is
• (a) if shorter than 15km it is the
  double-difference ambiguity-fixed solution
  (assuming ambiguity resolution was possible),
• (b) if baseline lengths greater than 15km and
  less than 50km it is the double-difference
  ambiguity-free solution unless session length is
  greater than 30mins, and
• (c) for baselines greater than 50km it is the
  triple-difference solution. (Note
  recommendations vary from manufacturer to
  manufacturer and "age" of the hardware/software.)

10
(No Transcript)
11
The triple-differences solution algorithm

• Difference epoch data between-satellites, form
  double-differences.
• Difference double-differences between epochs at
  some sample rate (for example, every 5th
  observation epoch), form triple-differences.
• Assume all triple-difference observations are
  independent when forming Weight Matrix (no
  correlations taken into account), define P
  matrix.
• Form Observation Equations, construct the A
  matrix.
• Accumulate Normal Equations, scaled by the
  Weight Matrix ATPA.
• At end of data set, invert Normal Matrix and
  obtain geodetic parameter solution,
  (ATPA)-1.ATP .
• Update parameters.
• Optionally scan triple-difference residuals for
  cycle slips in double-difference observables.

12
Triple diff. aspects

• Triple-difference solutions are "robust", being
  relatively immune to the effect of cycle slips in
  the data, which have the characteristics of
  "spikes" in the data (see Table in section 6.3.7
  and Figure in section 7.3.5).
• This low susceptibility to data that is not free
  from cycle slips is due to the correlations in
  the differenced data not being taken into account
  (assume a diagonal Weight Matrix P).
• The algorithm used to construct the
  triple-differenced observables is ideally suited
  for detecting and repairing cycle slips in the
  double-differenced data. Hence these solutions
  are generally carried out as part of the overall
  data cleaning (pre-)process (section 7.3.1). An
  automatic procedure would be based on scanning
  the residuals of the triple-difference solution
  for those close to an integer value of one or
  more cycles.
• Relatively simple algorithm that can easily
  handle a changing satellite constellation.
• The triple-difference solution provides good
  apriori values for the baseline components.
• Under extreme circumstances the triple-difference
  solution may be the only one that is reliable.

13

• Triple-difference solutions are "robust", being
  relatively immune to the effect of cycle slips in
  the data, which have the characteristics of
  "spikes" in the data (see Table in section 6.3.7
  and Figure in section 7.3.5).
• This low susceptibility to data that is not free
  from cycle slips is due to the correlations in
  the differenced data not being taken into account
  (assume a diagonal Weight Matrix P).
• The algorithm used to construct the
  triple-differenced observables is ideally suited
  for detecting and repairing cycle slips in the
  double-differenced data. Hence these solutions
  are generally carried out as part of the overall
  data cleaning (pre-)process (section 7.3.1). An
  automatic procedure would be based on scanning
  the residuals of the triple-difference solution
  for those close to an integer value of one or
  more cycles.
• Relatively simple algorithm that can easily
  handle a changing satellite constellation.
• The triple-difference solution provides good
  apriori values for the baseline components.
• Under extreme circumstances the triple-difference
  solution may be the only one that is reliable.

14
The triple-differences solution algorithm

• Difference epoch data between-satellites, form
  double-differences.
• Difference double-differences between epochs at
  some sample rate (for example, every 5th
  observation epoch), form triple-differences.
• Assume all triple-difference observations are
  independent when forming Weight Matrix (no
  correlations taken into account), define P
  matrix.
• Form Observation Equations, construct the A
  matrix.
• Accumulate Normal Equations, scaled by the
  Weight Matrix ATPA.
• At end of data set, invert Normal Matrix and
  obtain geodetic parameter solution,
  (ATPA)-1.ATP .
• Update parameters.
• Optionally scan triple-difference residuals for
  cycle slips in double-difference observables

15
Double-Differenced Phase Solution (Ambiguity-Free

• The following are some characteristics of
  double-differenced phase (ambiguity-free)
  solutions
• The functional model for the solution is eqn
  (7.2-4), containing both coordinate parameters
  and ambiguity parameters (the exact form
  depending upon the ambiguity parameter model used
  -- section 7.2.1).
• Double-difference solutions are vulnerable to
  cycle slips in the (double-differenced) data.
• The solution can be quite sensitive to a number
  of internal software factors such as
• between-satellite differencing strategy (see
  below),
• data rejection criteria,
• whether correlations are taken into account
  during differencing (see below),
• whether the observation time-tags are in the GPS
  Time system (section 6.3.8 and section 7.3.3).
• The solution is also sensitive to such external
  factors as
• length of observation session,
• receiver-satellite geometry (including the number
  of simultaneously tracked satellites),
• residual biases in the double-differenced data
  due to such things as atmospheric disturbances,
  multipath, etc.,
• the length of the baseline.
• Only the independent epoch double-differences are
  constructed (S-1) double-differences per
  baseline per epoch, where S is the number of
  satellites tracked.
• The algorithm used to construct the independent
  double-differenced observables must take into
  account the situations such as the rising and
  setting of a satellite during an observation
  session (and the appropriate definition of the
  ambiguity parameters in such a case).

16
The double-difference solution algorithm

• Difference epoch data between-satellites, form
  double-differences.
• Apply data reductions, such as a troposphere
  bias model.
• Construct Weight Matrix (depending on whether
  correlations are to be taken into account),
  define the P matrix.
• Form Observation Equations -- construct the A
  matrix.
• Accumulate Normal Equations, scaled by the
  Weight Matrix ATPA.
• At end of session, invert Normal Matrix and
  obtain geodetic and ambiguity parameter solution,
  (ATPA)-1.ATP .
• Update parameters.
• Decide (a) iterate solution, or (b) iterate
  solution only after ambiguity resolution attempted

17
Double differences

The quantities on the first line are known. The
only unknown quantities are the coordinates of
receiver 2. If the measurement noise is at the
few millimetre level (no multipath and residual
atmospheric biases are assumed to be present in
the double-differences), then it would be
possible to determine the three coordinate
components of receiver 2 to centimetre level
accuracy with just three independent
double-differences (from the simultaneous
tracking of four GPS satellites)!
18

• Unambiguous carrier phase (or "carrier-range")
  positioning has all the advantages of
  pseudo-range positioning, such as instantaneous
  single epoch results, but with unprecedented
  precision.
• The Figure below illustrates a series of repeated
  single epoch results (the "carrier-range" data
  was processed in the "kinematic" mode) for the
  length of a static baseline. Note that the
  baseline length variability is of the order of a
  centimetre (similarly for the other components).
  The following comments can be made

19

• The signature in figure above is largely due to
  the impact of multipath on the phase observations
  -- the multipath effect on pseudo-range would be
  up to several orders of magnitude greater.
• A change in the constellation of satellites from
  one epoch to the next will cause a "jump" in the
  solution.
• As the baseline was static, the redundant
  measurements are not contributing to the solution
  (solutions are carried out on a single epoch
  basis) -- redundant observations increase
  precision according to the "averaging law" ( ).
• The precision is influenced by the instantaneous
  satellite geometry as represented by navigation
  DOPs such as PDOP -- this varies smoothly over
  time except when satellites set below the
  tracking horizon or new satellites rise above it.
  
• For each epoch the repeated baseline estimates
  are derived from double-differenced precise
  "carrier-range" observations (eqn (8.1-1) as long
  as the ambiguities used in the previous epoch
  remain valid -- hence GPS hardware must continue
  to track the satellites (new satellites require
  new ambiguities to be estimated) and cycle slips
  must be avoided.

20
Double-Differenced Phase Solution
(Ambiguity-Fixed)

• The following are some characteristics of
  double-differenced phase (ambiguity-fixed)
  solutions
• The functional model for the solution is eqn
  (7.2-4), containing coordinate parameters and any
  unresolved ambiguity parameters (or none if all
  ambiguities have been resolved to their integer
  values). As ambiguities are resolved the
  (integer) value of the ambiguity becomes part of
  the apriori known information, and this has the
  effect of converting ambiguous phase observations
  into unambiguous range observations.
• Such a double-difference solution is
  comparatively strong (there are less parameters
  to estimate!), but is reliable only if the
  correct integer values of the ambiguities have
  been identified.
• The solution can be quite sensitive to the
  strategy used to resolve the ambiguities, for
  example
• whether all ambiguities are to be resolved as a
  set, or only a subset,
• the resolution criteria used for decision making,
  
• the search strategy used for integer values.
• The ambiguity resolution process is also
  sensitive to such external factors as
• length of observation session,
• receiver-satellite geometry,
• residual biases in the double-differences due to
  such things as atmospheric disturbances,
  multipath, etc.,
• whether satellites rise or set during the
  session,
• the length of the baseline

21
The ambiguity-fixed solution algorithm

• Difference epoch data between-satellites, form
  double-differences as before but without
  ambiguities as solve-for parameters.
• Apply data reductions, such as a troposphere
  bias model.
• Construct Weight Matrix (depending on whether
  correlations are to be taken into account),
  define the P matrix.
• Form Observation Equations, construct the A
  matrix.
• Accumulate Normal Equations, scaled by the
  Weight Matrix ATPA.
• At end of session, invert Normal Matrix and
  obtain geodetic parameter solution,
  (ATPA)-1.ATP .
• Update parameters.
• This process can be iterated to resolve other
  ambiguities until (a) all have been resolved (and
  "fixed" to integers), or (b) no more can be
  reliably resolved.

22

• where t is the time-tag of the observable
  constructed from four one-way phase observations
  which have been made within 30 millisecond of
  each other (section 6.3).
• Note that there are now only two classes of
  parameters coordinates and ambiguities.
• There are a number of alternative ambiguity
  modelling options (see discussion later in this
  section). In the event that the ambiguities are
  resolved, eqn (7.2-4) can be rewritten as a
  double-differenced range equation in which the
  ambiguity terms are eliminated from the parameter
  set.
• A solution using this "reduced" phase observable
  is known as an "ambiguity-fixed" solution.

23
Some of the issues that must be addressed in
order to make "carrier-range" positioning
reliable are

• How to prevent cycle slips?
• Requires new ambiguity resolution to be carried
  out, but how to do this quickly, reliably and
  without too great a nuisance to the surveyor?
•  
• How to improve the precision of the solution and
  increase its "robustness"?
• Extra satellites, Kalman filter algorithms,
  dual-frequency observations, precise pseudo-range
  data, multipath resistant antennas and receiver
  electronics, etc.
•  
• Is real-time "carrier-range" positioning
  possible, or desirable?
• It is possible for pseudo-range positioning.
  "Real-time kinematic" (RTK) is very attractive
  for many users, it would also help indicate to
  field staff if something goes wrong (loos of
  lock, etc.).

24
What is ambiguity resolution?

• The mathematical process of converting ambiguous
  ranges (integrated carrier phase) to unambiguous
  ranges of millimetre measurement precision ...
• For conventional GPS surveying, corresponds to
  converting real-valued ambiguity parameter values
  to the likeliest integer values

25

• In an ambiguity-free solution, no advantage is
  taken of the integer nature of n as it is
  indistinguishable from other, non-integer, biases
  such as orbit uncertainties, multipath and
  atmospheric refraction (ionosphere and
  troposphere), and is in fact "contaminated" by
  them.
• Thus, "ambiguity resolution" as it is generally
  known, is only possible after all biases are
  eliminated or otherwise accounted for to better
  than one cycle (20cm wavelength on L1).
• Constraint of an ambiguity value to its correct
  integer will improve the estimation of the
  remaining geodetic (station coordinate)
  parameters, as an inspection of figure below
  indicates.

26
The Figure below illustrates what happens in a
sequential transition from an 100 ambiguity-free
solution to an 100 ambiguity-fixed solution.

• For the first ten epochs the solution is an
  ambiguity-free one, but after 10-15 epochs, when
  the ambiguities have been resolved, the precision
  of the remaining estimable coordinate parameters
  improves significantly. It should be noted that
  the precisions (as well as the numeric values of
  the parameters) have virtually converged to their
  final values immediately after all ambiguities
  have been resolved. As a corollary therefore,
  phase data collected beyond the minimum necessary
  to ensure an ambiguity-fixed solution is obtained
  has almost no influence on the final results.
• Considerable R D effort has been invested in
  so-called "rapid ambiguity resolution"
  algorithms, which are the basis of the "rapid
  static" GPS surveying techniques (section 5.5.2
  and section 8.3.1).

The change in quality of baseline components in
an ambiguity-free compared to an ambiguity-fixed
solution.
27

• Clearly, an ambiguity-fixed solution is very
  desirable, and all efforts should be made to
  obtain one. There are several steps involved in
  obtaining an ambiguity-fixed solution
• Define the apriori values of the ambiguity
  parameters.
• Use a search algorithm to identify likely integer
  values.
• Employ a decision-making algorithm to select the
  "best" set of integer values.
• Apply ambiguities to the new (ambiguity-fixed)
  solution.
• Ambiguity resolution is discussed in detail in
  section 8.2.1 and section 8.3.1. Although
  ambiguity resolution can be considered largely an
  optional solution strategy for conventional
  (static) GPS surveying, it is a vital operation
  for some of the modern high productivity GPS
  surveying techniques.

28
Maximising the Chances of Successful Ambiguity
Resolution

• There are some well-known strategies that can be
  employed for maximising the chances of resolving
  ambiguities (though it cannot be guaranteed) for
  conventional static GPS surveying, including
• Single frequency observations for short baselines
  (
• Dual-frequency observations for longer baselines.
  
• Adequate length observation session ( 1/2 hour
  -- 2 hours ).
• Minimise interference (for example, no multipath,
  low unmodelled ionosphere, etc.), by good
  selection of sites, observing at night, etc.
• Observe as many satellites as possible to ensure
  good receiver-satellite geometry.
• Use precise pseudo-range data if available (and
  the ambiguity resolution algorithm can make use
  of this data -- see sections 8.2, 8.3 and 8.4).

29
Factors making ambiguity resolution difficult
include (section 8.2.4)

• The degree to which the geodetic parameters are
  reliably separated from the ambiguity parameters.
  
• The magnitude of any unmodelled biases present in
  the double-differenced phase data.
• The length of the baseline.
• The quality of the receiver-satellite geometry,
  and how much it has changed during the
  observation session.
• The data quality.
• Sub-optimal algorithms

30
RELIABLE ambiguity resolution is essential

• ...
• Incorrect resolution of some ambiguities will
  lead to a poor solution (worse than
  ambiguity-free or triple-difference solution).
• Hint has the solution changed by more than 10cm?
• How does one know when Ambiguity Resolution is
  possible?
• IT IS NOT POSSIBLE to be certain!! but with 1hour
  sessions to 4 satellites baselines rapidly changing pdop it should be possible to
  resolve ambiguities.

31
no standard form for the baseline solution
output.

• With regards to the output of a baseline
  solution, the following information is usually
  provided in some form or another
• Type of solution whether triple-, ambiguity-free
  or ambiguity-fixed.
• Input and output coordinates (solve-for and
  fixed), in various systems, for example
  Cartesian, geodetic, baseline components.
• Echo of receiver (serial numbers, etc.) and
  station information (site I.D., antenna height,
  etc.).
• Standard deviation of estimated coordinate
  components.
• The correlation matrix or variance-covariance
  (VCV) matrix for all coordinate parameters (and
  perhaps for the ambiguities as well).
• Optional estimate of quality of satellite
  geometry, for example the RDOP value (derived
  from the VCV matrix).
• Tracking data acquired, logging times at
  individual sites, tracking channels used,
  satellites tracked, signal quality flags, etc.
• Number of observations used in solution, as well
  as those rejected, the sampling rate used, and
  the data edit criteria (usually some factor times
  "RMS tracking").
• Summary of ephemeris information, health warning
  flags in the Navigation Message, etc.
• Any data pre-processing performed (for example,
  tropospheric delay model).
• Indicator of fit of observations to final
  solution (that is, the residuals),usually in the
  form of an "RMS tracking" value (though the label
  may vary for different processing software). The
  magnitude is an indicator of the quality and
  reliability of solution, and whether ambiguity
  resolution is possible (or resolution was
  successful).
• Results of statistical tests of the residuals may
  also be displayed.
• If an ambiguity-fixed solution was attempted,
  then a summary of the number of ambiguity
  parameters resolved.
• Possibly a summary recommendation on the quality
  of the solution.

32
in order to be able to compare one baseline
solution to another, some "rule-of-thumb" remarks
may be kept in mind

• In conventional static GPS surveying successful
  ambiguity resolution is basically a function of
  baseline length. For baselines over 15km in
  length it can be a problem. If ambiguity
  resolution was not possible for baselines check the continuity of tr acking, length of
  observation session, data "noise", and take steps
  such as reduce the resolution decision making
  criteria, and perhaps rerun the solution.
•  
• Having sessions of the same length (each day and
  throughout the day) is not sufficient to ensure
  similar quality results. Nor are sessions
  observed at the same time each day a guarantee of
  consistent quality results, even though the
  receiver-satellite geometry are the same.
  Residual biases due to orbit error, atmospheric
  effects, etc. are dynamic in nature and will
  influence the results as much as (and often more
  than) the receiver-satellite geometry during a
  session.
•  
• The "RMS of tracking" quantity tends to increase
  with increasing baseline length. (This quantity
  may have a different label in some software.)
•  
• The accuracy of the baseline components,
  expressed in metres, will degrade with increasing
  baseline length. Expressed as "parts per million"
  it should be a constant.
•  
• A "total" quality indicator, generally on the
  basis of a number of internal "tests" such as
  whether the ambiguities were resolved, the "RMS
  tracking", etc. may be provided, but is
  invariably software dependent -- therefore READ
  THE MANUAL.
•  
• The coordinate standard deviations are lower for
  an ambiguity-fixed solution than for an
  ambiguity-free solution, which are in turn lower
  than for a triple-difference solution. (Though
  that does not mean that the ambiguity-fixed
  solution is correct!)

33
Solution quality

• Solution statistics based on the VCV information
  are often optimistic. Beware of software packages
  that may artificially inflate the uncertainties
  of the parameters in order for them to appear
  more "realistic".
•  
• There is no measure for "reliability", the output
  VCV information will not reflect the influence of
  systematic biases.
•  
• It is good practice to first carry out
  preliminary baseline reductions with a minimum of
  options. This would permit the output to be
  assessed for bad data, residual cycle slips,
  etc., and the final adjustment may be carried out
  using the best dataset.
•  
• If there is any doubt about the quality of the
  ambiguity-fixed solution, it is preferable to
  accept the ambiguity-free solution in its place.
  If the ambiguity-free solution indicates high a
  "RMS tracking" value (for example because the
  baseline 50km), the triple-difference solution
  may be the preferred solution. However, check
  that the recommended standards practices for a
  certain class of GPS survey will accept such a
  solution (see section 10.3.1).
•  
• Improvement in the modelling of biases (for
  example, through the use of dual-frequency
  observations), or increase in the sophistication
  of the solution ("rigorous" adjustment, inclusion
  of additional parameters, etc.) leads to better
  and more reliable results for the same length
  baseline and observation session compared with
  the basic single baseline processing strategy.
•  
• Additional statistical testing may be carried out
  (for example, using "chi squared" and "variance
  factor" tests). In addition, external tests may
  be based on loop closure statistics, or the
  results of a network adjustment. This is dealt
  with further in Chapter 9.

34
There a re a number of "Quality Indicators" that
may be monitored, including

• RMS of observation residuals.
• Number of rejected observations.
• Statistical tests on residuals or parameters.
• Aposteriori variance factor.
• VCV matrix of solution.
• The type of "optimal" solution obtained.
• "Trustworthiness" of solution.
• Measures of reliability of selected ambiguity
  parameter set.

35
comments may be made with respect to the "RMS of
residuals" and "rejected observations"

• A "low" RMS value and a "low" number of rejected
  observations often indicates that both the data
  and solution quality are OK.
• Manufacturers often give recommended maximum
  values of RMS. Generally a function of baseline
  length, observation type (L1, L2, L3), etc.
• In general, an RMS value below 0.1 cycles is
  considered acceptable.
• Data editing is often carried out during solution
  iterations. Generally based on some factor (say
  3) x RMS.
• Possible reasons for high RMS and data rejection
  rates are the presence of multipath and
  uncorrected cycle slips. Residual plots are good
  tools to verify this.
• Some phase data processing software permits the
  residuals to be plotted. Residuals should be
  examined.

36
The following comments may be made with respect
to the "statistical tests" and "VCV information"

• In general, little statistical testing is carried
  out on parameters or residuals.
• If the aposteriori variance factor is unity then
  it is likely that the VCV matrix has been
  adaptively scaled to ensure this happens.
• In general however, the output VCV matrix is too
  optimistic , suggesting higher precisions for the
  parameters than is warranted. Does not take into
  account unmodelled systematic biases (atmospheric
  refraction, satellite orbit and fixed station
  errors, etc.).
• The standard deviations of baseline components
  vary considerably as a function of the type of
  phase solution (triple-difference,
  double-difference ambiguity-free,
  double-difference ambiguity-fixed).

37
There are other several Quality Indicators
related to "solution characteristics", including

• What is the "optimal" solution? Was an
  ambiguity-fixed solution obtained? Was it
  expected?
• If an ambiguity-fixed solution was obtained,
  check the absolute and relative RMS of the
  residuals. Are resolved ambiguities reliable?
• If an ambiguity-fixed solution was obtained,
  check baseline components. Did baseline solution
  change by more than 10cm compared to the
  ambiguity-free solution?
• The formal accuracy estimates for the vertical
  component is usually twice the size of the
  horizontal components.
• Verify solution characteristics such as
• satellites used -- any health problems?
• sample data rate
• common tracking period -- as planned?
• apriori station coordinates -- were the correct
  WGS84 values used?
• antenna heights
• elevation mask angle
• troposphere reduction applied?
• satellite geometry indicator -- PDOP, RDOP, etc.
  

38
Advantage can also be taken of "external
evidence", including

• Hierarchy of solutions -- Comparison of
  triple-difference and double-difference
  solutions.
• "decimetre" triple-difference solution precisions
  
• "centimetre" double-difference ambiguity-free
  solution precisions
• "millimetre" double-difference ambiguity-fixed
  solution precisions
• Dual-frequency solutions --Comparison of L1, L2
  and L3 solutions.
• Single baseline vs multi-baseline solutions
  --Different analysts? Process using different
  software packages?
• Repeat baseline solutions from different
  sessions.
• Solutions involving tracking to 4 or more
  satellites, over a period of 30-60 minutes, for
  baselines less than about 15km, should be high
  quality ambiguity-fixed solutions.
• Compare with ground control --Usually just
  distance.
• Check BBS or Integrity Monitoring Service

39
How does one really estimate the quality of
individual solutions? 

• Combine the baselines into a network adjustment

40
What is ambiguity resolution?

• A means of improving the accuracy of GPS
  surveying ...
• The mathematical process of converting ambiguous
  ranges (integrated carrier phase) to unambiguous
  ranges of millimetre measurement precision ...
• For conventional GPS surveying, corresponds to
  converting real-valued ambiguity parameter values
  to the likeliest integer values ...
• For modern GPS surveying, corresponds to
  discriminating the likeliest set of integer
  values from many alternative sets ...

41
Two scenarios for ambiguity resolution can be
distinguished

• In the case of conventional static GPS surveying,
  as developed since the early 1980's, the lengths
  of the observation sessions are sufficient to
  ensure a reliable ambiguity-free solution, and
  the successful resolution of the ambiguities to
  their integer values is a useful "bonus".
• For modern GPS surveying techniques (section
  5.5.1), ambiguity resolution is a critical
  operation, and if not successfully carried out,
  the integrity and reliability of the baseline
  solution can be degraded significantly.

42
GENERAL OVERVIEW OF AMBIGUITY RESOLUTION

• Several steps in the ambiguity resolution
  process can be identified
• Define the apriori values of the ambiguity
  parameters.
• Use a search algorithm to identify likely integer
  values.
• Employ a decision-making algorithm to select the
  "best" set of integer values.
• Apply ambiguities to the new (ambiguity-fixed)
  solution.

43
The apr iori values of the ambiguities are
generally provided by a double-difference
solution in the form of real-valued quantities
plus the variance-covariance information.

• The likeliest values of the ambiguities therefore
  are the nearest, "round-off" integer values. In
  some cases the estimate may be very near an
  integer, but in other cases the real-valued
  estimate is not obviously near an integer. The
  reliability of these estimates is a function of
  the baseline length, satellite-receiver geometry
  and length of observation session, and they are
  affected by multipath, residual biases and cycle
  slips.
• There are several approaches which can be used,
  some of which are particularly useful for modern
  GPS surveying techniques. These include
• Estimate ambiguities with the aid of pseudo-range
  data
• (8.2-1)
•  
• Use other geometric information, such as the
  known length of the baseline
• (8.2-2)
•  
• Make use of dual-frequency relationships that
  permit the L1 (or L2) ambiguities to be estimated
  from the wide-lane or ionosphere-free "lumped"
  ambiguity terms (section 8.4.2).

44

• All techniques rely on some "search" technique
  that tests a range of neighbouring values around
  the initial ambiguity values (see Figure below).
  For example, in the case of six tracked
  satellites there are five (double-differenced)
  ambiguities to be resolved. If the search window
  is three integers wide (one on either side of the
  round-off value), then there are 35 ambiguity set
  to be tested

45
The st andard criteria for successful ambiguity
resolution is if the identified ambiguity set (
n ) clearly fits the double-differenced phase
data better than any other ambiguity set
(8.2-3)

• The testing criteria is generally the lowest
  weighted root-sum-of-squares (RSS) of the
  double-differenced data residuals
• (8.2-4)
• where v is the vector of residuals, P is the
  observation weight matrix.
• This procedure requires
• The computation of residuals for each
  ambiguity-fixed solution being tested -- estimate
  a new baseline for each ambiguity set.
• Assumes unbiased (for example, no residual
  atmospheric refraction), clean (that is, no cycle
  slips), low noise (for example, no multipath)
  data.
• Observation data series long enough for reliable
  residual testing.

46
There are several factors that make ambiguity
resolution difficult, including

• The degree to which the geodetic parameters are
  reliably separated from the ambiguity parameters.
  
• The magnitude of any unmodelled biases still
  present in the double-differenced phase data.
• The length of the baseline.
• The quality of the receiver-satellite geometry,
  and how much it has changed during the
  observation session.
• The data quality.
• Sub-optimal algorithms.
• When the baseline is comparatively long (20 km),
  or there are some unmodelled biases present (high
  ionospheric activity, etc.), some ambiguities may
  be far from an integer value but still have small
  standard deviations. Or, alternatively, the
  ambiguity values may be close to integers but the
  standard deviations may be too large (arising
  from poor receiver-satellite geometry, data
  outages, etc.).

47
AMBIGUITY RESOLUTION AND WAVEFRONT GEOMETRY

• Imagine t he carrier phase wavefronts from
  satellites 1 and 2, as illustrated in Figure 1
  below in a 2-D representation. The grid has a
  mesh which is wide ( 19cm wavelength on L1).
  (In reality these wavefronts can be considered to
  be the result of between-receiver differencing of
  data to each of the satellites in turn.)
• Figure 1. Wavefront grid formed from two
  satellites.
• The two sets of parallel lines (in 3-D they are
  surfaces) can be combined into lines of
  double-differenced ambiguities (each line is the
  intersection of two wavefronts, and represents a
  constant double-differenced integer ambiguity
  value), as illustrated in Figure 2 below.
• Figure 2. Constant double-differenced
  "lines-of-ambiguities" involving two satellites.
• In the case of (m1) satellites, the geometric
  lattice is formed by the intersection of m sets
  of "lines-of-ambiguities". In next the figure,
  pairs of candidate ambiguities n12 and n23 are
  located at the intersection of the resulting
  lattice formed from two sets of
  "lines-of-ambiguities".

48
(No Transcript)
49

• The two sets of parallel lines (in 3-D they are
  surfaces) can be combined into lines of
  double-differenced ambiguities (each line is the
  intersection of two wavefronts, and represents a
  constant double-differenced integer ambiguity
  value), as illustrated in Figure 2 below

50

• However, there is no redundant information to
  allow for the unambiguous selection of the
  correct pair of ambiguities (corresponding to one
  intersection point of the "lines-of-ambiguities").
  The observations from a fourth satellite would
  permit another set of parallel "lines-of-ambiguiti
  es" to be overlain on Figure below. There may be
  one intersection that satisfies all geometric
  conditions, or more likely the case, several
  which are "close" intersections. As data is
  accumulated and the satellite geometry changes
  (due to the motion of the satellites), each set
  of "lines-of-ambiguities" (involving a pair of
  satellites) rotates by a different amount. Hence
  the total lattice pattern changes in a manner
  similar to interference fringe lines, and the one
  correct ambiguity set may become steadily more
  obvious (it is the only intersection point about
  which all the grids rotate).
• Figure 3. Two sets of constant double-differenced
  "lines-of-ambiguities" involving three
  satellites.

51
Hence what is required is either

• a significant change in satellite-receiver
  geometry over an observation session so that the
  intersection point representing the correct
  resolved integer ambiguity values becomes
  obvious. This is generally the situation for
  conventional static GPS surveying with long
  observation sessions.
• good geometry at a single epoch, or over a very
  short time period (a matter of minutes), when
  there is close to orthogonal intersection of the
  "lines-of-ambiguities" and sufficient redundancy
  so that there is only one candidate intersection
  point within the grid. This is the requirement
  for modern GPS surveying techniques.