DIFFFLH is used to express the results of the significance analysis by supplying coordinates and displacement vectors with their error and reliability ellipsoids. The error ellipsoid defines the precision of each displacement vector, whereas the reliability ellipsoid represents the maximum error on this vector due to a hidden (undiscovered) error in the baselines of both the compared surveys [Teunissen, 1985].

A Test Network

The software package was tested by surveying twice a 4-station network located in the Modena region of northern Italy (Figure 2). The 4 stations are numbered and named respectively 1-CAST (Castelfranco Emilia), 3-PESC (Pescale), 22-NAVI (Naviglio), and 10-SLITTA (Osservatorio Geofisico di Modena). The antenna at 10-SLITTA is mounted on a device which allows displacement along three orthogonal axes (aligned in this case north, east, and up) with a precision of 0.1 mm [Balestri et al., 1994] (Figure 3).

Two GPS antenna positions in 10-SLITTA were considered, one taken as zero position and the second defined with respect the first by setting a displacement of 15 mm along each axis. Two observing sessions of 6 hours each were performed for each of the 2 antenna locations (Table 1).

Table 1. Observing Sessions.

Session	CAST	PESC	NAVI		SLITTA
	1	3	22			10
					DN	DE	DH
					(mm)	(mm)	(mm)
1	X	X	X		0	0	0
2	X	X	X		15	-15	15
3	X	X	X		0	0	0
4	X	X	X		15	-15	15

Data Analysis

The GPS phase measurements were processed by GPSurvey [Trimble Navigation, 1995] following the suggestions given in the user manual. In particular, the ionospheric-free observable was adopted to process baselines longer than 10 km, initial phase ambiguities were solved when possible, and the troposphere delay was modeled according to the Saastamoinen model with a minimum elevation angle of 20E.

NET	EQ	UNK	CON	RED	a priori 02	estim 02	theor 2	exper 2	SMAX (m)	SM (m)
0 15	18 18	12 12	3 3	9 9	1.00 1.00	1.03 1.01	16.52 16.52	9.57 9.17	0.013 0.005	0.005 0.001

EQ=equations, UNK= unknowns, CON= constraints, RED= redundancy, ₀²= variance of unit weight, SMAX= maximum value of error ellipsoid semiaxis, SM= mean value of error ellipsoid semiaxis

The significance analysis of the observed coordinate differences between the two repeated surveys performed by DENETGPS starts from the evaluation of the significance of the coordinate differences with respect to the associated error ellipsoids. This is done by an iterative testing procedure based on the Fisher variate which allows a separation of the points into two groups: those whose coordinates are significantly changed and those whose coordinates are not changed. The test is based on the assumption that both the estimated coordinates and their differences x₂ = X₂-X₁ are normally distributed. Starting from the null hypothesis

H₀ : X₂ = X₁, it is possible to define a statistic F_OT that, if H_O is true, follows a Fisher distribution [Koch, 1984]
_X^T Q^-1 _X/m_o² = F_O ~ F_m, _{(r1 + r2)} = F_t where Q is the cofactor matrix of the coordinate differences, (r₁, r₂ ) the redundancies of the two adjustments, m the number of the unknown parameters to be tested, and _o² = (r₁ ² ₀₁ + r₂²₀₂)/(r₁ + r₂) with (²₀₁, ²₀₂) the variances of the unit weight in the two compared surveys. If the experimental F value exceeds the theoretical F value, the coordinates are significantly changed.

The significance analysis related to the discussed example was performed without considering any a priori information on the stations' behavior; in other words, the goal was to identify which of the sites suffered displacement starting from a noninformative condition. The two sets of adjusted coordinates (0 and 15) with the corresponding covariance matrices were submitted to DENETGPS. As expected, the program detected significant coordinate differences only for the site of SLITTA, which is clearly shown by the comparison of the experimental F values with respect to the theoretical value at the chosen significance level (5%) (Table 3).

Table 3. Fisher Test (F_t = 3.16).

In the end, displacement vector and error ellipse on the local tangent plane were computed by DIFFFLH. Table 4 shows a simplified view of DIFFFLH output, where WGS84 coordinates (, , h) related to the solutions 0 and 15, their differences (15-0), the corresponding RMS, three-dimensional and two-dimensional vectors with their azimuth, ellipsis semiaxes (with major semiaxis azimuth), and confidence interval on heights (95% confidence level) are reported. Figure 4 shows the two network adjustments together with the error ellipse and the unique significant displacement vector detected in the whole network, the step imposed on SLITTA.

Table 4. Simplified Diagram of DIFFFLH Output.

S I T E	0 15 (dmS) =15-0 (m)	RMS0 RMS15 (m) RMSD (m)	0 15 (dpS) =15-0 (m)	RMS0 RMS15 (m) RMSD (m)	h0 h15 (m) h (m)	RMS0 RMS15 (m) RMSD (m)	SMAX0 SMAX15 (m) VT 3D (m)	SMIN0 SMIN15 (m) VT 2D (m)	AZIM0 AZIM15 (dms) AZ VT2D (dms)	CI h0 CI h15 (m)
1     3     2 2   10	44 36 06.91490 44 36 06.91489 0.000 44 29 35.06664 44 29 35.06670 0.002 44 39 59.01375 44 39 59.01370 - 0.002 44 37 54.24366 44 37 54.24408 0.013	0.003 0.001 0.003 0.004 0.001 0.004 0.002 0.001 0.002 0.002 0.001 0.002	11 03 21.12020 11 03 21.12018 - 0.001 10 43 05.31313 10 43 05.31339 0.001 10 56 41.28676 10 56 41.28672 - 0.001 10 56 43.26372 10 56 43.26299 - 0.016	0.002 0.001 0.003 0.003 0.001 0.003 0.002 0.001 0.002 0.002 0.001 0.002	76.447 76.452 0.005 225.359 225.351 -0.008 65.599 65.602 0.003 93.862 93.875 0.012	0.007 0.003 0.007 0.009 0.003 0.009 0.005 0.002 0.006 0.005 0.002 0.005	0.007 0.003 0.005 0.009 0.003 0.008 0.005 0.002 0.004 0.005 0.002 0.024	0.006 0.002 0.001 0.008 0.003 0.002 0.004 0.002 0.002 0.004 0.002 0.021	1 21 08 6 22 42 -104 44 24 1 5 39 7 20 21 41 32 45 1 6 43 6 57 50 -148 01 49 1 12 53 6 59 26 -50 49 6	0.013 0.005   0.017 0.006   0.010 0.004   0.010 0.004

Currently available is a version of the program executables prepared by the Lahey compiler and linker for PC-DOS systems where DOS extender is installed. A VAX-OpenVMS 6.2 version will be provided on request from Mattia Crespi (crespi@dits.ing.uniroma1.it) or Federica Riguzzi (riguzzi@ing750.ingrm.it). Crespi can also provide more detailed theoretical explanations about the software package, and Riguzzi can supply more information about data files managing.