Global Mapping Function Research Articles

Symmetric Neutral-Atmosphere Mapping Functions: A Review of the State-Of-The-Art

ABSTRACT The aim of this paper is to review of six recent symmetric mapping functions. The mapping function can be largely used for GPS meteorological measurements, InSAR atmospheric corrections and precise measurements of very long baseline interferometry (VLBI). These spacebased techniques use radio signal that propagate through the Earth's atmosphere. The electrically-neutral region, predominantly the troposphere, affects the speed and direction of travel of radio waves leading to existence of excess path. The mapping function models the elevation angle dependence of the delay. Within the past decade, significant improvements have been achieved in order to use of Numerical Weather Models (NWM) for geodetic positioning. Ray-tracing algorithms have been performed through refractivity shells retrieved from NWMs in order to relate zenith delays to slant delays. Therefore, there seems to be a real need for deep review of recent developments in the mapping function domain. This paper proposes a comprehensive review of the symmetric mapping functions state of the art, their spatio-temporal variations and used NWM and generic models. Niell Mapping Function (NMF), Vienna Mapping Function (VMF1), University of New Brunswick-VMF1 (UNB-VMF1) mapping functions, Global Mapping Function (GMF) and Global Pressure and Temperature (GPT2)/GMF are reviewed in this paper.

One-to-Many Registration of Landsat Imagery

Abstract. This paper presents an innovative methodology for the registration of multiple satellite scenes by using a global Least Squares adjustment. The main advantage in relation to traditional approaches concerns the combined use of all the images of the dataset to obtain a more robust and comprehensive registration along with a statistical evaluation of the solution accuracy. This technique avoids standard matching solutions (implemented in several commercial software packages) where pairs of images are independently co-registered on the basis of an "one-to-one" approach, starting from points extracted via independent "image-to-master" matching. Once a set of corresponding multi-image features is extracted from the whole dataset, the implemented algorithm provides a global mapping function for an adjustment process that simultaneously includes all the available data. The multi-image matching process (coined "one-to-many" approach) is performed by exploiting multi-temporal image combinations to obtain features visible in as many scenes as possible. The features are then clustered to generate a regular structure of image coordinates for the following Least Squares adjustment phase. The method is able to manage features visible in as many images as possible and is therefore a powerful alternative for registering satellite data which do not directly share common features with the master. Some examples will be illustrated to report the feasibility of the registration algorithm and to prove its sub-pixel accuracy for the specific case of Landsat images.

Assessment of troposphere mapping functions using three-dimensional ray-tracing

The troposphere delay is an important source of error for precise GNSS positioning due to its high correlation with the station height parameter. It has been demonstrated that errors in mapping functions can cause sub-annual biases as well as affect the repeatability of GNSS solutions, which is a particular concern for geophysical studies. Three-dimensional ray-tracing through numerical weather models (NWM) is an excellent approach for capturing the directional and daily variation of the tropospheric delay. Due to computational complexity, its use for positioning purposes is limited, but it is an excellent tool for evaluating current state-of-the-art mapping functions used for geodetic positioning. Many mapping functions have been recommended in the past such as the Niell Mapping Function (NMF), Vienna Mapping Function 1 (VMF1), and the Global Mapping Function (GMF), which have been adopted by most IGS analysis centers. A new Global Pressure Temperature model (GPT2) has also been developed, which has been shown to improve upon the original atmospheric model used for the GMF. Although the mapping functions mentioned above use the same functional formulation, they vary in terms of their atmospheric source and calibration approach. A homogeneous data set of three-dimensional ray-traced delays is used to evaluate all components of the mapping functions, including their underlying functional formulation, calibration, and compression method. Additionally, an alternative representation of the VMF1 is generated using the same atmospheric source as the truth data set to evaluate the differences in ray-tracing methods and their effect on the end mapping function. The results of this investigation continue to support the use of the VMF1 as the mapping function of choice when geodetic parameters are of interest. Further support for the GPT2 and GMF as reliable back-ups when the VMF1 is not available was found due to their high consistency with the NWM-derived mapping function. Additionally, a small latitude-dependent bias in station height was found in the current mapping functions. This bias was identified to be due to the assumption of a constant radius of the earth and was largest at the poles and at the equator. Finally, an alternative version of the VMF1 is introduced, namely the UNB-VMF1 which provides users with an independent NWM-derived mapping function to support geodetic positioning.

GPT2: Empirical slant delay model for radio space geodetic techniques.

Up to now, state-of-the-art empirical slant delay modeling for processing observations from radio space geodetic techniques has been provided by a combination of two empirical models. These are GPT (Global Pressure and Temperature) and GMF (Global Mapping Function), both operating on the basis of long-term averages of surface values from numerical weather models. Weaknesses in GPT/GMF, specifically their limited spatial and temporal variability, are largely eradicated by a new, combined model GPT2, which provides pressure, temperature, lapse rate, water vapor pressure, and mapping function coefficients at any site, resting upon a global 5° grid of mean values, annual, and semi-annual variations in all parameters. Built on ERA-Interim data, GPT2 brings forth improved empirical slant delays for geophysical studies. Compared to GPT/GMF, GPT2 yields a 40% reduction of annual and semi-annual amplitude differences in station heights with respect to a solution based on instantaneous local pressure values and the Vienna mapping functions 1, as shown with a series of global VLBI (Very Long Baseline Interferometry) solutions.

Tropospheric Modeling and Fixed Stations Constraints in Precise GPS Computations: Case Study

Accurate GPS positioning requires correct modeling of tropospheric refraction, the phenomenon responsible for tropospheric delay (TD). The Niell mapping function, which is used in almost all processing packages to model TD, is being replaced by improved models. This study analyzes the influence of TD models and absolute antenna phase center variation (PCV) models. Solutions computed with the Niell mapping function and global mapping function TD models were compared at several European International Global Navigation Satellite Systems (GNSS) Service (IGS) stations. GPS sessions of 24 hours, observed on five different days in 2008, were used for this investigation. Two independent network-station constraint approaches are considered: a GPS network fixing a single station was implemented and was followed by a second network solution in which several IGS stations were constrained. The elevation-cutoff angle was set to 0 to obtain the maximum influence of the TD model at altitudes close to the horizon. Results of the final solutions depend more on the constrained stations within the study area than on the type of tropospheric model or antenna type. DOI:10.1061/(ASCE)SU.1943-5428.0000039.

Assessment of ECMWF-derived tropospheric delay models within the EUREF Permanent Network

The Global Positioning System (GPS) observations from the EUREF Permanent Network (EPN) are routinely analyzed by the EPN analysis centers using a tropospheric delay modeling based on standard pressure values, the Niell Mapping Functions (NMF), a cutoff angle of 3° and down-weighting of low elevation observations. We investigate the impact on EPN station heights and Zenith Total Delay (ZTD) estimates when changing to improved models recommended in the updated 2003 International Earth Rotation and Reference Systems Service (IERS) Conventions, which are the Vienna Mapping Functions 1 (VMF1) and zenith hydrostatic delays derived from numerical weather models, or the empirical Global Mapping Functions (GMF) and the empirical Global Pressure and Temperature (GPT) model. A 1-year Global Positioning System (GPS) data set of 50 regionally distributed EPN/IGS (International GNSS Service) stations is processed. The GPS analysis with cutoff elevation angles of 3, 5, and 10° revealed that changing to the new recommended models introduces biases in station heights in the northern part of Europe by 2---3 mm if the cutoff is lower than 5°. However, since large weather changes at synoptic time scales are not accounted for in the empirical models, repeatability of height and ZTD time series are improved with the use of a priori Zenith Hydrostatic Delays (ZHDs) derived from numerical weather models and VMF1. With a cutoff angle of 3°, the repeatability of station heights in the northern part of Europe is improved by 3---4 mm.

Numerical simulation of troposphere-induced errors in GPS-derived geodetic time series over Japan

Troposphere-induced errors in GPS-derived geodetic time series, namely, height and zenith total delays (ZTDs), over Japan are quantitatively evaluated through the analyses of simulated GPS data using realistic cumulative tropospheric delays and observed GPS data. The numerical simulations show that the use of a priori zenith hydrostatic delays (ZHDs) derived from the European Centre for Medium-Range Weather Forecasts (ECMWF) numerical weather model data and gridded Vienna mapping function 1 (gridded VMF1) results in smaller spurious annual height errors and height repeatabilities (0.45 and 2.55 mm on average, respectively) as compared to those derived from the global pressure and temperature (GPT) model and global mapping function (GMF) (1.08 and 3.22 mm on average, respectively). On the other hand, the use of a priori ZHDs derived from the GPT and GMF would be sufficient for applications involving ZTDs, given the current discrepancies between GPS-derived ZTDs and those derived from numerical weather models. The numerical simulations reveal that the use of mapping functions constructed with fine-scale numerical weather models will potentially improve height repeatabilities as compared to the gridded VMF1 (2.09 mm against 2.55 mm on average). However, they do not presently outperform the gridded VMF1 with the observed GPS data (6.52 mm against 6.50 mm on average). Finally, the commonly observed colored components in GPS-derived height time series are not primarily the result of troposphere-induced errors, since they become white in numerical simulations with the proper choice of a priori ZHDs and mapping functions.

Effects of Tropospheric Mapping Functions on GPS Data Processing

In processing space geodetic data, mapping functions are used to convert the tropospheric signal delay along the zenith direction to the line of sight direction. In this study, we compared three mapping functions by evaluating their effects on the tropospheric signal delay and position estimates in GPS data processing. The three mapping functions tested are Niell Mapping Function (NMF), Vienna Mapping Function 1 (VMF1), and Global Mapping Function (GMF). The tropospheric delay and height estimates from VMF1 and GMF are compared with the ones obtained with NMF. The differences among mapping functions show annual signals with the maximum occurring in February or August. To quantitatively estimate the discrepancies among mapping functions, we calculated the maximum difference and the amplitude using a curve fitting technique. Both the maximum difference and amplitude have high correlations with the latitude of the site. Also, the smallest difference was found around <TEX>$30^{\circ}N$</TEX> and the amplitudes increase toward higher latitudes. In the height estimates, the choice of mapping function did not significantly affect the vertical velocity estimate, and the precision of height estimates was improved at most of the sites when VMF1 or GMF was used instead of NMF.

A new hydrostatic mapping function for tropospheric delay estimation

A new hydrostatic mapping function is proposed by modifying the analytical solution of the Chapman grazing incident (Chi) function. Applicability of this mapping function is evaluated over the Indian region by comparing it with the true value of mapping function estimated through ray-tracing the refractivity profile of neutral atmosphere as well as by comparing it with the Niell and Global Mapping Functions. This comparison shows that the new mapping function, agrees very well with the ray-traces values and it is in par with if not better than the other global mapping functions for very high zenith angles up to ∼88°.

Comparison of GMF/GPT with VMF1/ECMWF and implications for atmospheric loading

This paper compares estimates of station coordinates from global GPS solutions obtained by applying different troposphere models: the Global Mapping Function (GMF) and the Vienna Mapping Function 1 (VMF1) as well as a priori hydrostatic zenith delays derived from the Global Pressure and Temperature (GPT) model and from the European Centre for Medium-Range Weather Forecasts (ECMWF) numerical weather model data. The station height differences between terrestrial reference frames computed with GMF/GPT and with VMF1/ECMWF are in general below 1 mm, and the horizontal differences are even smaller. The differences of annual amplitudes in the station height can also reach up to 1 mm. Modeling hydrostatic zenith delays with mean (or slowly varying empirical) pressure values instead of the true pressure values results in a partial compensation of atmospheric loading. Therefore, station height time series based on the simple GPT model have a better repeatability than those based on more realistic ECMWF troposphere a priori delays if atmospheric loading corrections are not included. On the other hand, a priori delays from numerical weather models are essential to reveal the full atmospheric loading signal.

A theoretical analysis of unsteady incompressible flows for time-dependent constitutive equations based on domain transformations

In this paper, we propose an analysis that allows calculation of kinematic histories in unsteady problems of continuum mechanics, in relation to the use of memory-integral constitutive equations. Such cases particularly concern flow conditions of processing rheology, requiring evaluation of strain or deformation rate tensors, for viscoelastic incompressible fluids as polymers. In two- and three-dimensional cases, we apply concepts of the stream-tube method (STM) initially given for stationary conditions, where unknown local or global mapping functions are defined instead of classic velocity–pressure formulations, leading to consider the flow parameters in domains where the streamlines and trajectories are parallel straight lines. The approach enables us to provide accurate formulae for evaluating the kinematics histories that can be used later for computing the stresses for a given memory-integral model.

Testing of global pressure/temperature (GPT) model and global mapping function (GMF) in GPS analyses

Several sources of a priori meteorological data have been compared for their effects on geodetic results from GPS precise point positioning (PPP). The new global pressure and temperature model (GPT), available at the IERS Conventions web site, provides pressure values that have been used to compute a priori hydrostatic (dry) zenith path delay z h estimates. Both the GPT-derived and a simple height-dependent a priori constant z h performed well for low- and mid-latitude stations. However, due to the actual variations not accounted for by the seasonal GPT model pressure values or the a priori constant z h, GPS height solution errors can sometimes exceed 10 mm, particularly in Polar Regions or with elevation cutoff angles less than 10 degrees. Such height errors are nearly perfectly correlated with local pressure variations so that for most stations they partly (and for solutions with 5-degree elevation angle cutoff almost fully) compensate for the atmospheric loading displacements. Consequently, unlike PPP solutions utilizing a numerical weather model (NWM) or locally measured pressure data for a priori z h, the GPT-based PPP height repeatabilities are better for most stations before rather than after correcting for atmospheric loading. At 5 of the 11 studied stations, for which measured local meteorological data were available, the PPP height errors caused by a priori z h interpolated from gridded Vienna Mapping Function-1 (VMF1) data (from a NWM) were less than 0.5 mm. Height errors due to the global mapping function (GMF) are even larger than those caused by the GPT a priori pressure errors. The GMF height errors are mainly due to the hydrostatic mapping and for the solutions with 10-degree elevation cutoff they are about 50% larger than the GPT a priori errors.

Effect of different tropospheric mapping functions on the TRF, CRF and position time-series estimated from VLBI

This paper compares estimated terrestrial reference frames (TRF) and celestial reference frames (CRF) as well as position time-series in terms of systematic differences, scale, annual signals and station position repeatabilities using four different tropospheric mapping functions (MF): The NMF (Niell Mapping Function) and the recently developed GMF (Global Mapping Function) consist of easy-to-handle stand-alone formulae, whereas the IMF (Isobaric Mapping Function) and the VMF1 (Vienna Mapping Function 1) are determined from numerical weather models. All computations were performed at the Deutsches Geodatisches Forschungsinstitut (DGFI) using the OCCAM 6.1 and DOGS-CS software packages for Very Long Baseline Interferometry (VLBI) data from 1984 until 2005. While it turned out that CRF estimates only slightly depend on the MF used, showing small systematic effects up to 0.025 mas, some station heights of the computed TRF change by up to 13 mm. The best agreement was achieved for the VMF1 and GMF results concerning the TRFs, and for the VMF1 and IMF results concerning scale variations and position time-series. The amplitudes of the annual periodical signals in the time-series of estimated heights differ by up to 5 mm. The best precision in terms of station height repeatability is found for the VMF1, which is 5–7% better than for the other MFs.

Global Mapping Function (GMF): A new empirical mapping function based on numerical weather model data

Troposphere mapping functions are used in the analyses of Global Positioning System and Very Long Baseline Interferometry observations to map a priori zenith hydrostatic and wet delays to any elevation angle. Most analysts use the Niell Mapping Function (NMF) whose coefficients are determined from site coordinates and the day of year. Here we present the Global Mapping Function (GMF), based on data from the global ECMWF numerical weather model. The coefficients of the GMF were obtained from an expansion of the Vienna Mapping Function (VMF1) parameters into spherical harmonics on a global grid. Similar to NMF, the values of the coefficients require only the station coordinates and the day of year as input parameters. Compared to the 6‐hourly values of the VMF1 a slight degradation in short‐term precision occurs using the empirical GMF. However, the regional height biases and annual errors of NMF are significantly reduced with GMF.

Global mapping functions for the atmosphere delay at radio wavelengths

I have developed expressions for calculating the ratios (mapping functions) of the “line of sight” hydrostatic and wet atmospheric path delays to their corresponding zenith delays at radio wavelengths for elevation angles down to 3°. The coefficients of the continued fraction representation of the hydrostatic mapping function depend on the latitude and height above sea level of the observing site and on the day of the year; the dependence of the wet mapping function is only on the site latitude. By comparing with mapping functions calculated from radiosonde profiles for sites at latitudes between 43°S and 75°N, the hydrostatic mapping function is seen to be more accurate than, and of comparable precision to, mapping functions currently in use, which are parameterized in terms of local surface meteorology. When the new mapping functions are used in the analysis of geodetic very long baseline interferometry (VLBI) data, the estimated lengths of baselines up to 10,400 km long change by less than 5 mm as the minimum elevation of included data is reduced from 12° to 3°. The independence of the new mapping functions from surface meteorology, while having comparable accuracy and precision to those that require such input, makes them particularly valuable for those situations where surface meteorology data are not available.

Correlation of image intensity on the Turin Shroud with the 3-D structure of a human body shape

The frontal image on the Shroud of Turin is shown to be consistent with a body shape covered with a naturally draping cloth in the sense that image shading can be derived from a single global mapping function of distance between these two surfaces. The visible image on the Shroud does not appear to be the work of an artist in an eye/brain/hand coordination sense nor does it appear to be the result of direct contact only, diffusion, radiation from a body shape or engraving, dabbing powder on a bas-relief, or electrostatic imaging. The visible image on the Shroud is probably not the result of a hot bas-relief impressed into cloth, but such a mechanism seems capable of accounting for the Shroud image’s distance correlation, resolution, and similar chemical structure. It does not simultaneously account for (1) the 3-D image residing on one side of the Shroud, (2) observed lateral image distortions (consistent with a draping cloth over a body shape), or (3) expected thermal perturbations associated with physically thick superimposed blood images. A complex mechanism involving more than one process may account for some of the Shroud image’s characteristics, but potential inconsistencies in shading continuity, cloth contact, lateral distortions, and pressure independence may exist.