Station Coordinates Research Articles

GPS horizontal deformation model in the southern region of the Iberian Peninsula and northern Africa (SPINA)

Global Navigation Satellite System (GNSS), and in particular Global Positioning System (GPS) technology, provides a powerful tool for studying geodynamic processes. As a consequence of GPS studies, it is now possible to analyze the interaction between tectonic plates in order to evaluate and establish the characteristics of their boundaries. In this study, our main interest is to focus on the time series analysis obtained from observations of GNSS-GPS satellites. Each GPS observation session provides topocentric geodetic coordinates (east, north, elevation) of the permanent stations that constitute the geodetic network established for this purpose. This paper shows a detailed topocentric coordinate time-series study for sites belonging to what we call the SPINA network, which stands for south of the Iberian Peninsula, north of Africa region. The series under study are processed by techniques of relative positioning with respect to the IGS (International GNSS Service) reference station located in Villafranca. These times series have been analyzed using filter processes, harmonic adjustments, and wavelets. A surface velocity field is derived from the time series of daily solutions for each station, whose observations span 8 years or longer. This allows us to obtain a horizontal displacement model to show the regional geodynamic main characteristics.

Geocenter variations derived from a combined processing of LEO- and ground-based GPS observations

GNSS observations provided by the global tracking network of the International GNSS Service (IGS, Dow et al. in J Geod 83(3):191–198, 2009) play an important role in the realization of a unique terrestrial reference frame that is accurate enough to allow a detailed monitoring of the Earth’s system. Combining these ground-based data with GPS observations tracked by high-quality dual-frequency receivers on-board low earth orbiters (LEOs) is a promising way to further improve the realization of the terrestrial reference frame and the estimation of geocenter coordinates, GPS satellite orbits and Earth rotation parameters. To assess the scope of the improvement on the geocenter coordinates, we processed a network of 53 globally distributed and stable IGS stations together with four LEOs (GRACE-A, GRACE-B, OSTM/Jason-2 and GOCE) over a time interval of 3 years (2010–2012). To ensure fully consistent solutions, the zero-difference phase observations of the ground stations and LEOs were processed in a common least-squares adjustment, estimating all the relevant parameters such as GPS and LEO orbits, station coordinates, Earth rotation parameters and geocenter motion. We present the significant impact of the individual LEO and a combination of all four LEOs on the geocenter coordinates. The formal errors are reduced by around 20% due to the inclusion of one LEO into the ground-only solution, while in a solution with four LEOs LEO-specific characteristics are significantly reduced. We compare the derived geocenter coordinates w.r.t. LAGEOS results and external solutions based on GPS and SLR data. We found good agreement in the amplitudes of all components; however, the phases in x- and z-direction do not agree well.

Adding source positions to the IVS combination—First results

The consistent estimation of terrestrial reference frames (TRF), celestial reference frames (CRF) and Earth orientation parameters (EOP) is still an open subject and offers a large field of investigations. Until now, source positions resulting from Very Long Baseline Interferometry (VLBI) observations are not routinely combined on the level of normal equations in the same way as it is a common process for station coordinates and EOPs. The combination of source positions based on VLBI observations is now integrated in the IVS combination process. We present the studies carried out to evaluate the benefit of the combination compared to individual solutions. On the level of source time series, improved statistics regarding weighted root mean square have been found for the combination in comparison with the individual contributions. In total, 67 stations and 907 sources (including 291 ICRF2 defining sources) are included in the consistently generated CRF and TRF covering 30 years of VLBI contributions. The rotation angles \(A_1\), \(A_2\) and \(A_3\) relative to ICRF2 are −12.7, 51.7 and 1.8 \({\upmu }\) as, the drifts \(D_\alpha \) and \(D_\delta \) are −67.2 and 19.1 \(\upmu \) as/rad and the bias \(B_\delta \) is 26.1 \(\upmu \) as. The comparison of the TRF solution with the IVS routinely combined quarterly TRF solution shows no significant impact on the TRF, when the CRF is estimated consistently with the TRF. The root mean square value of the post-fit station coordinate residuals is 0.9 cm.

A Study on In Situ Calibration of an Off-The-Shelf Digital Camera Integrated to a Lidar System

Photogrammetric procedures using the integration of imagery and light detection and ranging (Lidar) datasets are becoming common in mapping companies nowadays. When photogrammetric and Lidar surveys are performed simultaneously while having the camera connected to the Lidar system, the image's exterior orientation parameters computed from the Lidar system can be used to perform photogrammetric applications. To implement such approach, camera calibration procedures should be applied to obtain accurate interior orientation parameters. This paper shows the performed study and experimental results from an in situ self-calibration using simultaneously collected imagery and Lidar datasets. The experimental study is conducted to devise a methodology for the in situ self-calibration that uses a set of vertical control points, extracted from Lidar point cloud, as well as the three-dimensional coordinates of the camera exposure stations, computed from the Lidar trajectory. For the small photogrammetric block used in this study, the results from the performed experiments demonstrated that the in situ self-calibration is required to obtain horizontal accuracies of approximately one image pixel on the ground in bundle adjustment assisted by direct measured of the coordinates of the camera exposure stations.

A coordinate-invariant model for deforming geodetic networks: understanding rank deficiencies, non-estimability of parameters, and the effect of the choice of minimal constraints

By considering a deformable geodetic network, deforming in a linear-in-time mode, according to a coordinate-invariant model, it becomes possible to get an insight into the rank deficiency of the stacking procedure, which is the standard method for estimating initial station coordinates and constant velocities, from coordinate time series. Comparing any two out of the infinitely many least squares estimates of stacking unknowns (initial station coordinates, velocity components and transformation parameters for the reference system in each data epoch), it is proven that the two solutions differ only by a linear-in-time trend in the transformation parameters. These pass over to the initial coordinates (the constant term) and to the velocity estimates (the time coefficient part). While the difference in initial coordinates is equivalent to a change of the reference system at the initial epoch, the differences in velocity components do not comply with those predicted by the same change of reference system for all epochs. Consequently, the different velocity component estimates, obtained by introducing different sets of minimal constraints, correspond to physically different station velocities, which are therefore non-estimable quantities. The theoretical findings are numerically verified for a global, a regional and a local network, by obtaining solutions based on four different types of minimal constraints, three usual algebraic ones (inner or partial inner) and the lately introduced kinematic constraints. Finally, by resorting to the basic ideas of Felix Tisserand, it is explained why the station velocities are non-estimable quantities in a very natural way. The problem of the optimal choice of minimal constraints and, hence, of the corresponding spatio-temporal reference system is shortly discussed.

The AUSTRAL VLBI observing program

The AUSTRAL observing program was started in 2011, performing geodetic and astrometric very long baseline interferometry (VLBI) sessions using the new Australian AuScope VLBI antennas at Hobart, Katherine, and Yarragadee, with contribution from the Warkworth (New Zealand) 12 m and Hartebeesthoek (South Africa) 15 m antennas to make a southern hemisphere array of telescopes with similar design and capability. Designed in the style of the next-generation VLBI system, these small and fast antennas allow for a new way of observing, comprising higher data rates and more observations than the standard observing sessions coordinated by the International VLBI Service for Geodesy and Astrometry (IVS). In this contribution, the continuous development of the AUSTRAL sessions is described, leading to an improvement of the results in terms of baseline length repeatabilities by a factor of two since the start of this program. The focus is on the scheduling strategy and increased number of observations, aspects of automated operation, and data logistics, as well as results of the 151 AUSTRAL sessions performed so far. The high number of the AUSTRAL sessions makes them an important contributor to VLBI end-products, such as the terrestrial and celestial reference frames and Earth orientation parameters. We compare AUSTRAL results with other IVS sessions and discuss their suitability for the determination of baselines, station coordinates, source coordinates, and Earth orientation parameters.

Impact of Baseline Distance and Interstation Height Difference On the Accuracy of GPS-Derived Station Coordinates

This paper investigates the accuracy comparison of scientific and commercial GPS processing software for varying length of baselines and interstation height differences. European Reference (EUREF) Permanent Network Stations were used for the processing. GAMIT/GLOBK research GPS processing software (version 10.6) and Leica Geo Office (version 8.4) commercial GPS processing software were chosen for the experiment. TUBO EUREF GPS station was taken as a known station whose three-dimensional (3D) published coordinates are precisely known and the other stations were taken as unknown points whose 3D coordinates are calculated after the processing. In this way, minimally constrained adjustment was performed for each unknown station within the EUREF network and their coordinates were calculated w.r.t. TUBO station with using each software. Only independent (non-trivial) baselines are processed between known station and unknown stations. Processing was conducted for 10 different days with 24 hours rinex data of the stations. When choosing the GPS stations, we consider the interstation height difference and baseline distance between the chosen GPS stations. Baseline distances varying from 131 km to 495 km. Interstation height differences varying from 4 m to 1620 m. After the processing, differences between the calculated 3D Cartesian coordinates of the stations whose coordinates were assumed unknown and the published true coordinates of these stations were transformed to topocentric coordinates (north, east, up) to observe the error component in 3D space. Results show that vertical accuracy obtained from commercial software is significantly low and not consistent comparing with the scientific software. There is no significant difference observed for horizontal accuracy between the software but the horizontal accuracy obtained from the scientific software is slightly better than the commercial one. There is no strong correlation found between the baseline distance and the accuracy for each software but this is not the issue for interstation height differences. Regarding the interstation height differences, it is observed that height difference between station pairs effect the accuracy in up component. As height difference between baseline points grow vertical, error becomes larger, especially for commercial software.

Is the Jason-2 DORIS oscillator also affected by the South Atlantic Anomaly?

We analyzed time series of daily DORIS and GPS station coordinate estimates derived from Precise Point Positioning (PPP). The DORIS coordinates were estimated using Jason-2 precise orbits based on GPS data only, implying that the station positions from the two techniques are expressed in the same GPS-based terrestrial reference frame. Comparisons of 3-D vectors of such co-located stations show systematic biases in position around South America when compared to local geodetic ties. We conclude that these results could be explained by a sensitivity of the Jason-2/DORIS oscillator to radiation when the satellite passes over the South Atlantic Anomaly (SAA). The effect for Jason-2 manifests mainly as an offset in station coordinates, though there is also evidence of a drift at the start of the mission thatdiminishes in time. This contrasts with the experience on Jason-1, wherein large, persistent drifts were observed for stations in this same (SAA) region. The spurious drift is much (∼90%) smaller for Jason-2, which may be attributable to the steps taken prior to launch to harden the oscillator. Analysis of DORIS Doppler residuals may indicate some small degradation after 2009 for these stations.

The impact of the antenna phase center models on the coordinates in the EUREF Permanent Network

Precise positioning using the signals of the Global Position System requires correcting the distance between the points of reception of the signal carrier phase and the antenna reference point. Sophisticated models to account for these phase errors are available and widely in use. These models are usually based on calibrations of several antennas of the same type to derive a mean model. There is also the possibility of applying individual correction models that are derived for one single antenna, since the individual antennas may still vary by a few millimeters. We prepared two sets of station position time series to understand the impact of using type mean or individual antenna correction models. At first, we estimated the station positions using the type mean model for the antenna corrections as it is provided by the IGS. Second, the same processing strategy was used, but this time with individual antenna correction models provided they were available. Then, based on these two solutions, we evaluated how different antenna corrections impact the network alignment and subsequently station coordinates. We showed that mixing two sources of antenna phase center corrections may influence the reference frame realization. However, this impact is very small and can be reduced by an appropriate selection of reference stations. Finally, we demonstrated that for some antennas, the application of type mean or individual antenna calibration models may cause a discrepancy in the position of over 10 mm for the horizontal and vertical component. However, for most of the antennas these offsets are below 2 mm for horizontal components and below 4 mm for vertical component. Thereby, for common antennas we confirmed the results obtained by others.

Activity of the CNES/CLS Analysis Center for the IDS contribution to ITRF2014

Within the frame of the International DORIS Service, the CNES/CLS Analysis Center, previously known as LCA and renamed GRG, contributes to the geodetic and geophysical research activity through DORIS data analysis. The main work carried out in the past two years concerns the processing of the measurements collected by the DORIS-equipped satellites over 22years, in order to provide a homogeneous series of station coordinates and Earth pole parameters for the IDS contribution to the ITRF2014. First, we brought several upgrades to the processing and the modeling. Some of them are corrective actions to issues raised during or shortly after the production of our contribution to the ITRF2008 (ground station frequency offsets, attitude laws and macromodels). Recent models have been assessed with the aim to update our analysis configuration. Among others, we adopted the time variable gravity (TVG) model EIGEN-6S2 and applied tropospheric gradients. Then we processed almost all the DORIS data collected between January 1993 and December 2014. The series of weekly SINEX solutions derived from this processing is labeled grgwd40. This new series performs better than the series produced for ITRF2008. Especially, the results discussed in this paper show a decrease of 2% of the DORIS orbit residuals as well as a strong reduction of the annual terms of the TRF scale and Tz translation which can be explained by the application of the TVG model.

Change in air temperature over Sudan and South Sudan with time and station coordinates and elevation during the last century

Annual mean air temperature for Sudan and South Sudan for the three periods 1900-1940, 1961- 1990 and 1981-2010 for 12 stations was analyzed with objectives of studying changes in air temperature over the area during the last century and also to study the linkages between mean, maximum and minimum air temperature on the one hand and latitudes, longitudes and elevations on the other hand. The data was obtained as normal data for each period from Sudan Meteorological Authority and other sources. The results showed an increase of about 0.60 degrees Celsius in the mean temperature of the 12 stations over the last century. The maximum temperature dropped in the second period by about 0.40 degrees Celsius and increased in the third period by about 0.35 degrees making almost no net change over the century. The minimum temperatures increased on the average by about 1.2 degrees since the turn of the last century and increased by about 0.32 degrees between the last two periods. The net increase in air temperature for the 12 stations over the last century was mainly a result of an increase in the minimum air temperature. The results also showed strong linkages between mean and minimum temperatures on one hand and longitudes on the other hand. Keywords: Air temperature, Variation, Coordinates, Elevation, Sudan

Horizontal and vertical velocities derived from the IDS contribution to ITRF2014, and comparisons with geophysical models

In the context of the 2014 realization of the International Terrestrial Reference Frame (ITRF2014), the International DORIS Service (IDS) has delivered to the IERS a set of 1140 weekly SINEX files including station coordinates and Earth orientation parameters, covering the time period from 1993.0 to 2015.0. From this set of weekly SINEX files, the IDS Combination Center estimated a cumulative DORIS position and velocity solution to obtain mean horizontal and vertical motion of 160 stations at 71 DORIS sites. The main objective of this study is to validate the velocities of the DORIS sites by comparison with external models or time series. Horizontal velocities are compared with two recent global plate models (GEODVEL 2010 and NNR-MORVEL56). Prior to the comparisons, DORIS horizontal velocities were corrected for Global Isostatic Adjustment (GIA) from the ICE-6G (VM5a) model. For more than half of the sites, the DORIS horizontal velocities differ from the global plate models by less than 2-3 mm/yr. For five of the sites (Arequipa, Dionysos/Gavdos, Manila, Santiago) with horizontal velocity differences wrt these models larger than 10 mm/yr, comparisons with GNSS estimates show the veracity of the DORIS motions. Vertical motions from the DORIS cumulative solution are compared with the vertical velocities derived from the latest GPS cumulative solution over the time span 1995.0-2014.0 from the University of La Rochelle (ULR6) solution at 31 co-located DORIS-GPS sites. These two sets of vertical velocities show a correlation coefficient of 0.83. Vertical differences are larger than 2 mm/yr at 23 percent of the sites. At Thule the disagreement is explained by fine-tuned DORIS discontinuities in line with the mass variations of outlet glaciers. Furthermore, the time evolution of the vertical time series from the DORIS station in Thule show similar trends to the GRACE equivalent water height.

Assessment of the accuracy of global geodetic satellite laser ranging observations and estimated impact on ITRF scale: estimation of systematic errors in LAGEOS observations 1993–2014

Satellite laser ranging (SLR) to the geodetic satellites LAGEOS and LAGEOS-2 uniquely determines the origin of the terrestrial reference frame and, jointly with very long baseline interferometry, its scale. Given such a fundamental role in satellite geodesy, it is crucial that any systematic errors in either technique are at an absolute minimum as efforts continue to realise the reference frame at millimetre levels of accuracy to meet the present and future science requirements. Here, we examine the intrinsic accuracy of SLR measurements made by tracking stations of the International Laser Ranging Service using normal point observations of the two LAGEOS satellites in the period 1993 to 2014. The approach we investigate in this paper is to compute weekly reference frame solutions solving for satellite initial state vectors, station coordinates and daily Earth orientation parameters, estimating along with these weekly average range errors for each and every one of the observing stations. Potential issues in any of the large number of SLR stations assumed to have been free of error in previous realisations of the ITRF may have been absorbed in the reference frame, primarily in station height. Likewise, systematic range errors estimated against a fixed frame that may itself suffer from accuracy issues will absorb network-wide problems into station-specific results. Our results suggest that in the past two decades, the scale of the ITRF derived from the SLR technique has been close to 0.7 ppb too small, due to systematic errors either or both in the range measurements and their treatment. We discuss these results in the context of preparations for ITRF2014 and additionally consider the impact of this work on the currently adopted value of the geocentric gravitational constant, GM.

Improved Method to Detect and Repair Cycle Slip for GNSS Medium–Long Baseline in Real-Time Marine Surveys

ABSTRACTDistance-related errors complicate the resolution of real-time ambiguity in medium–long baseline marine surveys. Therefore, detection and recovery of cycle slips in real time is required to ensure high accuracy of global navigation satellite system positioning and navigation in marine surveys. To resolve this, an improved method was presented, where linear combinations of the triple-differenced (TD) between carriers L1 and L2 were formed for a wide lane and free ionosphere. To overcome severe ill-conditioned problems of the normal equation, the Tikhonov regularization method was used. The construction of a regularized matrix by combining a priori information of known coordinates of reference stations, followed by the determination of the corresponding regularized parameter are suggested. A float solution was calculated for the TD ambiguity. The search cycle slip (TD integer ambiguity) was obtained using the least-squares ambiguity decorrelation adjustment (LAMBDA) method. Using our method, cycle slips of several reference station baselines with lengths of a few hundred to one thousand kilometers were detected in real time. The results were consistent with professional software, with a success rate of 100%.

Testing impact of the strategy of VLBI data analysis on the estimation of Earth Orientation Parameters and station coordinates

Abstract Very Long Baseline Interferometry (VLBI) is the only space geodetic technique capable to realise the Celestial Reference Frame and tie it with the Terrestrial Reference Frame. It is also the only technique, which measures all the Earth Orientation Parameters (EOP) on a regular basis, thus the role of VLBI in determination of the universal time, nutation and polar motion and station coordinates is invaluable. Although geodetic VLBI has been providing observations for more than 30 years, there are no clear guidelines how to deal with the stations or baselines having significantly bigger post-fit residuals than the other ones. In our work we compare the common weighting strategy, using squared formal errors, with strategies involving exclusion or down-weighting of stations or baselines. For that purpose we apply the Vienna VLBI Software VieVS with necessary additional procedures. In our analysis we focus on statistical indicators that might be the criterion of excluding or down-weighting the inferior stations or baselines, as well as on the influence of adopted strategy on the EOP and station coordinates estimation. Our analysis shows that in about 99% of 24-hour VLBI sessions there is no need to exclude any data as the down-weighting procedure is sufficiently efficient. Although results presented here do not clearly indicate the best algorithm, they show strengths and weaknesses of the applied methods and point some limitations of automatic analysis of VLBI data. Moreover, it is also shown that the influence of the adopted weighting strategy is not always clearly reflected in the results of analysis.

Towards the new Croatian terrestrial reference frame based on CROPOS - preliminary results

Since the establishment of CROPOS, almost seven years have passed. Initial and also the last adjustment CROPOS network was carried out in 2008 using only seven 24h session data in GPS Week 1503 (ETRF2000 R05 ; e2008, 83). In the recent period there have been significant changes - Croatian State Geodetic Administration (CSGA) signed an agreement on the exchange of data of national positioning systems neighboring countries: Slovenia (SIGNAL), Hungary (GNSSnet.hu), Montenegro (MontePOS), Federation of Bosnia and Herzegovina (FBiHPOS) and Republika Srpska (SRPOS). This paper presents the results of long-term stability of coordinates of CROPOS stations through absolute comparison of ETRF2000 coordinates from initial adjustment with the combined networked solution of CROPOS and nearby positioning systems, calculated on the basis of 1463 daily solutions for the time span of four years.

Integration of soft computational simulator and strapdown inertial navigation system for aerial surveying project planning

Aerial photogrammetry is an engineering surveying operation, which is a traditional methodology that can be used to map the earth. The rapid development in instrumentation and computer dependency led to the need to develop photogrammetric software. Inertial navigation system (INS) can provide light weight positioning methodology in aerial imagery. This research is a trial to integrate INS with a pre-developed soft computational simulator of 3A3P in order to obtain a dynamic virtual map. The developed system supports inertial measurement unit (IMU) and link the theoretical exposure station coordinates and the inertial measurements. The developed system enables photogrammetrists to visualize the aerial imaging process and it can be used to monitor the imaging process. It makes all the calculations that are required to obtain the theoretical coordinates and produces a virtual aerial map. The system converts the measurements of the IMU into space coordinates that can be useful to monitor the imaging process.

IVS contribution to ITRF2014

Every few years the International Terrestrial Reference System (ITRS) Center of the International Earth Rotation and Reference Systems Service (IERS) decides to generate a new version of the International Terrestrial Reference Frame (ITRF). For the upcoming ITRF2014 the official contribution of the International VLBI Service for Geodesy and Astrometry (IVS) comprises 5796 combined sessions in SINEX file format from 1979.6 to 2015.0 containing 158 stations, overall. Nine AC contributions were included in the combination process, using five different software packages. Station coordinate time series of the combined solution show an overall repeatability of 3.3 mm for the north, 4.3 mm for the east and 7.5 mm for the height component over all stations. The minimum repeatabilities are 1.5 mm for north, 2.1 mm for east and 2.9 mm for height. One of the important differences between the IVS contribution to the ITRF2014 and the routine IVS combination is the omission of the correction for non-tidal atmospheric pressure loading (NTAL). Comparisons between the amplitudes of the annual signals derived by the VLBI observations and the annual signals from an NTAL model show that for some stations, NTAL has a high impact on station height variation. For other stations, the effect of NTAL is low. Occasionally other loading effects have a higher influence (e.g. continental water storage loading). External comparisons of the scale parameter between the VTRF2014 (a TRF based on combined VLBI solutions), DTRF2008 (DGFI-TUM realization of ITRS) and ITRF2008 revealed a significant difference in the scale. A scale difference of 0.11 ppb (i.e. 0.7 mm on the Earth’s surface) has been detected between the VTRF2014 and the DTRF2008, and a scale difference of 0.44 ppb (i.e. 2.8 mm on the Earth’s surface) between the VTRF2014 and ITRF2008. Internal comparisons between the EOP of the combined solution and the individual solutions from the AC contributions show a WRMS in X- and Y-Pole between 40 and 100 $$\upmu $$ as and for dUT1 between 5 and 15 $$\upmu $$ s. External comparisons with respect to the IERS-08-C04 series show a WRMS of 132 and 143 $$\upmu $$ as for X- and Y-Pole, respectively, and 13 $$\upmu $$ s for dUT.

The Study of Seasonal Changes of Permanent Stations Coordinates based on Weekly EPN Solutions

Abstract Daily and weekly coordinates solutions of GNSS permanent stations operating within EPN network allows to track long-term changes of coordinates caused e.g. by the local and global movements of tectonic plates. They are therefore an excellent tool for testing stability and repeatability of stations position. The article presents an analysis of coordinates changes of selected reference stations based on weekly EPN solutions. In addition the author proposes parameters of approximating function by assuming an existence of periodic, annually repeatable trend. The author performed also an independent fitting function for two different periods of two ITRF frames of routine time analysis and reprocessing.

A Circular Interference Model for Heterogeneous Cellular Networks

Scaling up the number of base stations per unit area and augmenting the heterogeneity of networks are two major trends in mobile cellular systems of the fourth (4G)- and fifth generation (5G), making it increasingly difficult to characterize aggregate interference statistics with system models that are characterized by a low number of parameters. This paper proposes a new circular interference model that aggregates given interferer topologies to power profiles along circles. We introduce a mapping procedure that preserves the aggregate interference statistics at arbitrary user locations within the associated cell with a certain desired accuracy. At the same time, our method identifies the amount of nodes in a given interferer topology that principally determine the shape of the interference distribution. To further refine the analysis of given power profiles, a new finite sum representation for the sum of Gamma random variables with integer-valued shape parameter is introduced. The approach allows decomposing the distribution of the aggregate interference into the contributions of the individual interferers. Such knowledge is particularly expedient for the application of base station coordination- and cooperation schemes. Moreover, the proposed method enables accurate prediction of the corresponding signal-to-interference-ratio- and rate statistics.