Juan, J.; Sanz, J.; Gonzalez-Casado, G.; Rovira-Garcia, Adrià. International Colloquium Scientific and Fundamental Aspects of the Galileo Programme p. 1-2 Data de presentació: 2017-10-25 Presentació treball a congrés
As direct consequence of Einstein’s Equivalence Principle, time runs (or clocks tick) more slowly near a massive body. This effect, named gravitational red-shift, can be detected when comparing time intervals measured by identical clocks placed at different positions in a gravitational field or when their tick rates, i.e. their requencies, are compared. Einstein’s prediction was tested to an uncertainty of 1.4·10-4 in the Gravity Probe A (GPA) experiment performed by Vessot et al. .
In August 2014, due to a technical problem, the Galileo 5 and 6 navigation satellites were injected into the wrong orbit. Because of the orbit eccentricity (~0.15), the on-board atomic clocks (passive H-maser) are now experiencing a gravitational red-shift effect with a peak-to-peak amplitude of 5·10-11. The clock stability, as low as 1·10-14, is now opening interesting perspectives to measure the gravitational time dilation effect to an uncertainty at the GPA level or even lower.
In the context of an ESA founded project gAGE/UPS is analysing the data delivered by Galileo 5 and 6 satellites to measure the Einstein’s gravitational red-shift effect. This challenge requires developing new and alternative techniques for improving the GNSS signal modelling and better handling the measurement systematics.
The guidelines of the study we are conducting are summarized next:
Raw data handling to diminish/mitigate some errors in the GNSS signals at receiver level
These errors (such as carrier phase multipath, thermal noise...) are typically below 1cm, but become more important when
forming carrier phase combinations. Diffractive scintillation, which is present in the observations gathered at low latitudes and affects the Ionosphere -Free (IF) combination, must be also included.
The solution is to model these effects, when possible, or to down-weight the data when the modelling of the effect is not possible. This last point requires to carry out a long term study, for each receiver in the network, in order to define a realistic covariance matrix.
Techniques for reducing the correlation between parameters
To reduce the correlation between parameters, it is necessary to impose confident constraints on the different observables. For
instance, a common solution for reducing correlations is the carrier phase ambiguity fixing. In this case, an un-differenced mode is applied: carrier ambiguities are fixed for all carrier phase data from different frequencies and different constellations used in the satellite clock estimations.
In addition, the models for the evaluation of effects, such as tropospheric delays, are being improved. This step is important to avoid incorrect modelling of these effects that, as mentioned above, would otherwise appear as a clock error.
A new technique for clock estimation
The clock solution is typically estimated from the IF combination of two carrier phases. However, this approach increases the noise by a factor 2.9 with the GPS L1/L2 frequencies and a factor 2.6 with the Galileo E1/E5a frequencies (in both cases, we assume the same noise for the 2 frequencies). With the new frequencies, it is possible to build up IF combinations involving more than two carriers and reducing the noise amplification. Still, these combinations present time varying IFBs should be carefully studied.
The proposed solution is based on a joint estimate of clock, IFBs, and ionospheric delays in the same filter, without building up IF combinations. In the case of 2 frequencies, and without mplementing any ionospheric model, the result is expected be equivalent to the classical solution using IF combinations. In this context, we intend to extend the method to handle more than 2 frequencies, at the same time introducing ionospheric models. A precision measurement of the gravitational redshift will require a correct modelling of IFBs, particularly of their dependence with temperature.
Juan, J.; Sanz, J.; Rovira-Garcia, Adrià.; Gonzalez-Casado, G.; Shao, Y.; Ibañez, D.; Alonso, M.; Segura, S.; Escudero, M. Multi-GNSS Asia Conference p. 1 Data de presentació: 2017-10-10 Presentació treball a congrés
navigation Satellite System (GNSS) provides different kinds of applications and accuracies, having an
increasing demand for precise navigation and positioning. Examples of applications requiring high accuracy
civil engineering, mapping and surveying, agricultural use
s, mining, marine navigation.
The Scintillation is one of the most challenging problems in GNSS navigation. This phenomenon appears
when the signal pass through ionospheric irregularities, producing rapid changes on refraction index and,
depending on the size of such irregularities, also diffractive effects affecting the signal amplitude and can
produce cycle slips. In this
we show that the scintillation effects on GNSS signal are quite different in
low and high latitudes.
For low latit
ude receivers, the main effects from the point of view of precise navigation, are the growing of
the carrier phase noise (sigma-phi) and a fading on the signal intensity (S4) that can produce cycle
the GNSS signal. The detection of these cycle-
slips is a challenging problem for precise navigation. Indeed, 1
cycle jump in the L1 carrier represents a jump of 48 cm in the ionosphere-
free combination (the
combination used in PPP), which, if it is not corrected, would derive in meters of position error.
In high latitude receivers the situation is not the same. In this region,
the size of the irregularities is typically
larger than the Fresnel scale, so the main effects are related with the fast change on the refractive index
associated to the fast movem
ent of the irregularities (which can reach up to several km/s). Consequently,
as we will show in the presentation, the main effect on the GNSS signal is a fast fluctuation of the carrier
phase (large sigma-
phi), but with a moderate fading in the intensity (moderate S4). Thus, on one hand, this
rapid fluctuation of carrier phases is mostly proportional to the inverse squared frequency of the signals,
being the effect quite limited (practically null) on the ionosphere
-free combination. On the other hand,
e fluctuations do not usually produce cycle
-slips. These two characteristics make feasible the use of the
ionospheric free combination for high accuracy navigation in high latitudes, also during high ionospheric
In this work
, we assess the high accuracy navigation in both scenarios, showing that in high latitude, dual
frequency users can navigate, also during high ionospheric activity, as the scintillation is mostly refractive
and does not produce frequent cycle
-slips. Nevertheless, due to the lar
temporal gradients, it is a
more challenging for single frequency users. Low latitude is a more difficult scenario where scintillation can
lead frequent cycle
-slips and loss of GNSS signals. Moreover, the carrier noise is increased, but high accuracy is still possible for dual-
frequency users, if the cycle
-slips were detected in a reliable way.
We address two main problems related to the receiver and satellite differential code biases (DCBs) determination. The first issue concerns the drifts and jumps experienced by the DCB determinations of the International GNSS Service (IGS) due to satellite constellation changes. A new alignment algorithm is introduced to remove these nonphysical effects, which is applicable in real time. The full-time series of 18 years of Global Positioning System (GPS) satellite DCBs, computed by IGS, are realigned using the proposed algorithm. The second problem concerns the assessment of the DCBs accuracy. The short- and long-term receiver and satellite DCB performances for the different Ionospheric Associate Analysis Centers (IAACs) are discussed. The results are compared with the determinations computed with the two-layer Fast Precise Point Positioning (Fast-PPP) ionospheric model, to assess how the geometric description of the ionosphere affects the DCB determination and to illustrate how the errors in the ionospheric model are transferred to the DCB estimates. Two different determinations of DCBs are considered: the values provided by the different IAACs and the values estimated using their pre-computed Global Ionospheric Maps (GIMs). The second determination provides a better characterization of DCBs accuracy, as it is confirmed when analyzing the DCB variations associated with the GPS Block-IIA satellites under eclipse conditions, observed mainly in the Fast-PPP DCB determinations. This study concludes that the accuracy of the IGS IAACs receiver DCBs is approximately 0.3–0.5 and 0.2 ns for the Fast-PPP. In the case of the satellite DCBs, these values are about 0.12–0.20 ns for IAACs and 0.07 ns for Fast-PPP.
Ionospheric scintillation produces strong disruptive effects on global navigation satellite system (GNSS) signals, ranging from degrading performances to rendering these signals useless for accurate navigation. The current paper presents a novel approach to detect scintillation on the GNSS signals based on its effect on the ionospheric-free combination of carrier phases, i.e. the standard combination of measurements used in precise point positioning (PPP). The method is implemented using actual data, thereby having both its feasibility and its usefulness assessed at the same time. The results identify the main effects of scintillation, which consist of an increased level of noise in the ionospheric-free combination of measurements and the introduction of cycle-slips into the signals. Also discussed is how mis-detected cycle-slips contaminate the rate of change of the total electron content index (ROTI) values, which is especially important for low-latitude receivers. By considering the effect of single jumps in the individual frequencies, the proposed method is able to isolate, over the combined signal, the frequency experiencing the cycle-slip. Moreover, because of the use of the ionospheric-free combination, the method captures the diffractive nature of the scintillation phenomena that, in the end, is the relevant effect on PPP. Finally, a new scintillation index is introduced that is associated with the degradation of the performance in navigation.
Sanz, J.; Juan, J.; Rovira-Garcia, Adrià.; Gonzalez-Casado, G.; Shao, Y.; Ibañez, D.; Romero-Sánchez, J.; Alonso, M.; Escudero, M. Multi-GNSS Asia Conference p. 66 Data de presentació: 2016-11-15 Presentació treball a congrés
Sanz, J.; Juan, J.; Rovira-Garcia, Adrià.; Gonzalez-Casado, G.; Shao, Y.; Ibañez, D.; Romero-Sánchez, J.; Alonso, M.; Escudero, M. Multi-GNSS Asia Conference p. 41 Data de presentació: 2016-11-14 Presentació treball a congrés
Juan, J.; Sanz, J.; Rovira-Garcia, Adrià.; Gonzalez-Casado, G.; Shao, Y.; Ibañez, D.; Alonso, M.; Segura, S.; Escudero, M. BELS Workshop: EGNSS Solutions for Sustainable Development p. 1-2 Data de presentació: 2016-10-10 Presentació treball a congrés
Global navigation Satellite System (GNSS) provides different kinds of applications and
accuracies, having an increasing demand for precise navigation and positioning.
Examples of applications requiring high accuracy are: civil engineering, mapping and
surveying, agricultural uses, mining, marine navigation...
The Scintillation is one of the most challenging problems in GNSS navigation. This
phenomenon appears when the signal pass through ionospheric irregularities,
producing rapid changes on refraction index and, depending on the size of such
irregularities, also diffractive effects affecting the signal amplitude and can produce
cycle slips. In this work we show that the scintillation effects on GNSS signal are quite
different in low and high latitudes.
For low latitude receivers, the main effects from the point of view of precise
navigation, are the growing of the carrier phase noise (sigma-phi) and a fading on the
signal intensity (S4) that can produce cycle-slips in the GNSS signal. The detection of
these cycle-slips is a challenging problem for precise navigation. Indeed, 1 cycle jump
in the L1 carrier represents a jump of 48 cm in the ionosphere-free combination (the
combination used in PPP), which, if it is not corrected, would derive in meters of
In high latitude receivers the situation is not the same. In this region the size of the
irregularities is typically larger than the Fresnel scale, so the main effects are related
with the fast change on the refractive index associated to the fast movement of the
irregularities (which can reach up to several km/s). Consequently, as we will show in
the presentation, the main effect on the GNSS signal is a fast fluctuation of the carrier
phase (large sigma-phi), but with a moderate fading in the intensity (moderate S4).
Thus, on one hand, this rapid fluctuation of carrier phases is mostly proportional to the
inverse squared frequency of the signals, being the effect quite limited (practically
null) on the ionosphere-free combination. On the other hand, these fluctuations do not
usually produce cycle-slips. These two characteristics make feasible the use of the
ionospheric free combination for high accuracy navigation in high latitudes, also
during high ionospheric activity.
In this work we assess the high accuracy navigation in both scenarios, showing that in
high latitude, dual-frequency users can navigate, also during high ionospheric activity,
as the scintillation is mostly refractive and does not produce frequent cycle-slips.
Nevertheless, due to the large spatial-temporal gradients, it is a more challenging for
single frequency users. Low latitude is a more difficult scenario where scintillation can
lead frequent cycle-slips and loss of GNSS signals. Moreover, the carrier noise is
increased, but high accuracy is still possible for dual-frequency users, if the cycle-slips
were detected in a reliable way.
Fast Precise Point Positioning (Fast-PPP) provides Global Navigation Satellite System corrections in real-time.
Satellite orbits and clock corrections are shown to be accurate to a few centimeters and a few tenths of a nanosecond
which, together with the determination of the fractional part of the ambiguities, enable global high-accuracy positioning with undifferenced Integer Ambiguity Resolution. The new global ionospheric model is shown to provide corrections accurate at the level of 1 Total Electron Content Unit over well-sounded areas and Differential Code Biases at the level of tenths of a nanosecond.
These corrections are assessed with permanent receivers, treated as rovers, located at 100 to 800 kilometers from the reference stations of the ionospheric model. Fast-PPP achieves decimeter-level of accuracy after few minutes, several times faster than single- and dual-frequency ionospheric-free solutions, using a month of Global Positioning System data close to the last Solar Maximum and including equatorial rovers.
Gonzalez-Casado, G.; Juan, J.; Sanz, J.; Shao, Y. International Conference on Localization and GNSS p. 1-5 DOI: 10.1109/ICL-GNSS.2016.7533857 Data de presentació: 2016-06-30 Presentació treball a congrés
Applying a methodology developed and tested in
previous studies, the contribution from the ionospheric and
plasmaspheric regions to the total electron content (measured by
ground receivers) is analyzed. The method is based in the
electron density profiles retrieved from radio occultations
observed with low Earth orbit satellites, combined with an
accurate empirical modeling of the topside-ionosphere electron
density. The results of a climatological study of the fractional
electron content from the ionospheric region are presented for a
year of low solar activity. It is shown that a simple parametric
model can be used to reproduce the electron content variations in
the ionosphere and the plasmasphere between sunrise and
midday, the period of the day showing the largest electron
Sanz, J.; Juan, J.; Rovira-Garcia, Adrià.; Gonzalez-Casado, G.; Ibañez, D.; Romero-Sánchez, J.; Alonso, M.; Shao, Y.; Escudero, M. BELS Short Workshops, Vientiame Data de presentació: 2016-04-24 Presentació treball a congrés
Sanz, J.; Juan, J.; Rovira-Garcia, Adrià.; Gonzalez-Casado, G.; Ibañez, D.; Romero-Sánchez, J.; Alonso, M.; Shao, Y.; Escudero, M. Worshops in Asia. GNSS solutions for sustainable development p. 1-2 Data de presentació: 2016-04-20 Presentació treball a congrés
Sanz, J.; Juan, J.; Rovira-Garcia, Adrià.; Gonzalez-Casado, G.; Ibañez, D.; Romero-Sánchez, J.; Alonso, M.; Shao, Y. Worshops in Asia. Multi-GNSS in Indonesia p. 1 Data de presentació: 2016-04-18 Presentació treball a congrés
Sanz, J.; Rovira-Garcia, Adrià.; Juan, J.; Gonzalez-Casado, G.; Ibañez, D.; Romero-Sánchez, J.; Alonso, M.; Shao, Y.; Escudero, M. Multi-GNSS Asia Conference p. 1 Data de presentació: 2015-12 Presentació treball a congrés
Two high-precision positioning techniques currently offer accur acy at the centimetre level: Real-Time Kinematics (RTK) and Precise Point Positioning (PPP). Both methods use carrier-phase measurements, 2 orders of magnitude more prec ise than pseudoranges. Classical single-baseline RTK (appeared in the 80’s) uses a nea rby reference station to compensate most of the delays (i.e., errors) affecting GNSS sig nals. RTK achieves centimetre-level of accuracy in seconds after the Double Differ ences of the carrier-phase ambiguities are fixed to integers. The drawbacks of RTK are: i) the bandwidth and continuity requirements to disseminate the measurements from th e reference receiver to the user and ii) the maximum distance to the reference station, which can range from 10-20 km (depending on the ionosph eric activity) to 50 km using Network-RTK. PPP (defined in the 90’s) overcomes the RTK limitations with du al-frequency measurements and orbit and clock products precise to a few cent imetres. PPP products require less bandwidth than RTK, with less continuity constrain s and allow world-wide coverage. However, PPP requires almost 1 hour to convergence th e un-differenced carrier-phase ambiguity estimation from the noisy pseudorange. This initialization is not acceptable in most professional kinematic applications (e.g. su rveying, farming) that usually rely on RTK. Recent improvements to PPP are: (i) the or bit and clock corrections are sent to users i n real-time, (ii) the user can f ix the carrier ambiguities in undifferenced mode, improving the accuracy, (iii) the multi-con stellation context. In this presentation we will review the main features of the Hi gh Accuracy Positioning techniques, from RTK to PPP. In particular we will address some the large convergence time of PPP and the lack of integrity in the user solution. Fin ally we will show how a World-Wide Ionospheric Model for Fast-PPP reduces the convergence time in PPP and, also, enables High-Accuracy navi gation with a single frequency receiver.
Rovira-Garcia, Adrià.; Juan, J.; Sanz, J.; Gonzalez-Casado, G.; Ibañez, D.; Romero-Sánchez, J. International Technical Meeting of the Satellite Division of the Institute of Navigation p. 3833-3840 Data de presentació: 2015-09-18 Presentació treball a congrés
The main objective of this work is to present a methodology to assess the accuracy of any ionospheric model used in Global Navigation Satellite System (GNSS) applications. A number of global and regional models (both in realtime and post-process) will be analyzed during the entire 2014, i.e. near to the last Solar Cycle Maximum, to identify seasonal characteristics. The new method uses as reference values the unambiguous and undifferenced geometry-free combination of carrier-phase measurements from a worldwide distribution of receivers. The differences between the Slant Total Electron Contents (STECs) of the model and the measurements are fit to constant hardware delays: a receiver plus a satellite Differential Code Bias (DCB). Once such DCBs are estimated, the post-fit residual of the adjustment to the reference values is computed. It is shown that this residual is a very suitable metric to represent the error of any ionospheric model tailored for GNSS-based navigation. Any miss-modeling present in the STECs predictions which cannot be represented by a constant parameter per station and a constant per satellite degrades the user positioning. The assessment includes the comparison of the 3D navigation error of some permanent stations, being processed in singlefrequency as kinematic rovers, using different ionospheric corrections and precise satellite orbits and clocks.
Gonzalez-Casado, G.; Juan, J.; Sanz, J.; Rovira-Garcia, Adrià.; Aragon-Angel, M.A. International Technical Meeting of the Satellite Division of the Institute of Navigation p. 3459-3468 Data de presentació: 2015-09-17 Presentació treball a congrés
Gonzalez-Casado, G.; Juan, J.; Sanz, J.; Rovira-Garcia, Adrià.; Aragon-Angel, M.A. Journal of geophysical research: space physics Vol. 120, num. 7, p. 5983-5997 DOI: 10.1002/2014JA020807 Data de publicació: 2015-07-04 Article en revista
We introduce a methodology to extract the separate contributions of the ionosphere and the plasmasphere to the vertical total electron content, without relying on a fixed altitude to perform that separation. The method combines two previously developed and tested techniques, namely, the retrieval of electron density profiles from radio occultations using an improved Abel inversion technique and a two-component model for the topside ionosphere plus protonosphere. Taking measurements of the total electron content from global ionospheric maps and radio occultations from the Constellation Observing System for Meteorology, Ionosphere, and Climate/FORMOSAT-3 constellation, the ionospheric and plasmaspheric electron contents are calculated for a sample of observations covering 2007, a period of low solar and geomagnetic activity. The results obtained are shown to be consistent with previous studies for the last solar minimum period and with model calculations, confirming the reversal of the winter anomaly, the hemispheric asymmetry of the semiannual anomaly, and the existence in the plasmasphere of an annual anomaly in the South American sector of longitudes. The analysis of the respective fractional contributions from the ionosphere and the plasmasphere to the total electron content shows quantitatively that during the night the plasmasphere makes the largest contribution, peaking just before sunrise and during winter. On the other hand, the fractional contribution from the ionosphere reaches a maximum value around noon, which is nearly independent of season and geomagnetic latitude.
Rovira-Garcia, Adrià.; Juan, J.; Sanz, J.; Gonzalez-Casado, G. IEEE transactions on geoscience and remote sensing Vol. 53, num. 8, p. 4596-4604 DOI: 10.1109/TGRS.2015.2402598 Data de publicació: 2015-03-03 Article en revista
Fast precise point positioning (Fast-PPP) is a satellite-based navigation technique using an accurate real-time ionospheric modeling to achieve high accuracy quickly. In this paper, an end-to-end performance assessment of Fast-PPP is presented in near-maximum Solar Cycle conditions; from the accuracy of the Central Processing Facility corrections, to the user positioning. A planetary distribution of permanent receivers including challenging conditions at equatorial latitudes, is navigated in pure kinematic mode, located from 100 to 1300 km away from the nearest reference station used to derive the ionospheric model.
It is shown that satellite orbits and clocks accurate to few centimeters
and few tenths of nanoseconds, used in conjunction with an ionosphere with an accuracy better than 1 Total Electron Content Unit (16 cm in L1) reduce the convergence time of dual-frequency Precise Point Positioning, to decimeter-level (3-D) solutions. Horizontal convergence times are shortened 40% to 90%, whereas the vertical components are reduced by 20% to 60%. A metric to evaluate the quality of any ionospheric model for Global Navigation Satellite System is also proposed. The ionospheric modeling accuracy is directly translated to mass-market single-frequency
users. The 95th percentile of horizontal and vertical accuracies is shown to be 40 and 60 cm for single-frequency users and 9 and 16 cm for dual-frequency users. The tradeoff between the formal and actual positioning errors has been carefully studied to set realistic confidence levels to the corrections.
Sanz, J.; Juan, J.; Gonzalez-Casado, G.; Prieto-Cerdeira, R.; Schlüter, S.; Orús, R. International Technical Meeting of the Satellite Division of the Institute of Navigation p. 1173-1182 Data de presentació: 2014 Presentació treball a congrés
This work introduces a novel ionospheric activity indicator useful for identifying disturbed periods affecting performance for GNSS users, at regional level. This indicator is based in the “Along Arc TEC Rate (AATR) and can be easily computed from GNSS data. The AATR indicator has been assessed over more than one Solar Cycle (2002-2013) involving 140 receivers distributed world-wide. Results show that it is well correlated with the ionospheric activity and, unlike other global indicators linked to the geomagnetic activity (i.e. DST, Ap), it is sensitive to regional behaviour the ionosphere and identifies specific effects on GNSS users. Moreover from a devoted analysis of EGNOS performances in different ionospheric conditions, it follows that the AATR indicator is able to predict SBAS user availability anomalies linked to the ionosphere. The AATR indicator has been chosen as the metric to characterise the ionosphere operational conditions in the frame of EGNOS activities. This indicator has been also proposed for joint analysis in the International SBAS-Ionosphere Working Group.
A simple model for the topside ionosphere region is introduced and applied to fit radio-occultation retrieved electron density profiles for altitudes above the F2-peak. The model considers two isothermal components representing the population of the O+ (ionosphere component) and the H+ (protonosphere component) ions. The purpose of the model is to achieve an accurate fit of the observed profiles in the topside ionosphere region while, at the same time, allowing a direct and simple derivation of two important ionospheric parameters, namely, the O+ vertical scale height and the upper transition height. Covering a time period of one year, the fits with the two-component model function are compared with those achieved with one-component functions commonly used in the literature and it is shown that the former provides significantly better fits than the later, with more than a factor of two improvement. The model predictions concerning: the correlation between the O+ vertical scale height and the upper transition height, the altitude dependence of the vertical scale height of the electron density, and the quantitative contribution of the protonosphere to the total electron content are examined and shown to be consistent with the observations and with previous studies. It is concluded that the model provides a realistic description of the vertical distribution of the two main ion constituents of the topside ionosphere.
Salvador, E.; Serra, S.; Manrique, A.; Gonzalez-Casado, G. Monthly notices of the Royal Astronomical Society Vol. 424, num. 4, p. 3129-3144 DOI: 10.1111/j.1365-2966.2012.21475.x Data de publicació: 2012-08-21 Article en revista
In a recent paper, Salvador-Solé et al. have derived the typical inner structure of dark matter haloes from that of peaks in the initial random Gaussian density field, determined by the power spectrum of density perturbations characterizing the hierarchical cosmology under consideration. In this paper, we extend this formalism to the typical kinematics and triaxial shape of haloes. Specifically, we establish the link between such halo properties and the power spectrum of density perturbations through the typical shape of peaks. The trends of the
predicted typical halo shape, pseudo-phase-space density and anisotropy profiles are in good agreement with the results of numerical simulations. Our model sheds light on the origin of the power-law-like pseudo-phase-space density profile for virialized haloes.
In the three preceding papers in the series, we presented a model dealing with the
global and small-scale structure and kinematics of hierarchically assembled, virialised,
collisionless systems, which correctly accounted for the typical properties of simulated
cold darkmatter (CDM) haloes. This model relied, however, on the spherical symmetry
assumption. Here we show that the foundations of the model hold equally well for
triaxial systems and extend it in a fully accurate way to objects that satisfy the latter
more general symmetry. The master equations in the new version take the same form
as in the version for spherically symmetric objects, but the profiles of all the physical
quantities are replaced by their respective spherical averages. All the consequences
of the model drawn under the spherical symmetry assumption continue to hold. In
addition, the new version allows one to infer the axial ratios of virialised ellipsoids from
those of the corresponding protoobjects. The present results generalise and validate
those obtained in Papers I, II and III for CDM haloes. In particular, they confirm that
all halo properties are the natural consequence of haloes evolving through accretion
and major mergers from triaxial peaks (secondary maxima) in the primordial density