NASA Goddard Space Flight Center Research Articles

Multidecadal Arctic sea ice thickness and volume derived from ice age

Abstract. Sea ice is a key component of the Arctic climate system, and has impacts on global climate. Ice concentration, thickness, and volume are among the most important Arctic sea ice parameters. This study presents a new record of Arctic sea ice thickness and volume from 1984 to 2018 based on an existing satellite-derived ice age product. The relationship between ice age and ice thickness is first established for every month based on collocated ice age and ice thickness from submarine sonar data (1984–2000) and ICESat (2003–2008) and an empirical ice growth model. Based on this relationship, ice thickness is derived for the entire time period from the weekly ice age product, and the Arctic monthly sea ice volume is then calculated. The ice-age-based thickness and volume show good agreement in terms of bias and root-mean-square error with submarine, ICESat, and CryoSat-2 ice thickness, as well as ICESat and CryoSat-2 ice volume, in February–March and October–November. More detailed comparisons with independent data from Envisat for 2003 to 2010 and CryoSat-2 from CPOM, AWI, and NASA GSFC (Goddard Space Flight Center) for 2011 to 2018 show low bias in ice-age-based thickness. The ratios of the ice volume uncertainties to the mean range from 21 % to 29 %. Analysis of the derived data shows that the ice-age-based sea ice volume exhibits a decreasing trend of −411 km3 yr−1 from 1984 to 2018, stronger than the trends from other datasets. Of the factors affecting the sea ice volume trends, changes in sea ice thickness contribute more than changes in sea ice area, with a contribution of at least 80 % from changes in sea ice thickness from November to May and nearly 50 % in August and September, while less than 30 % is from changes in sea ice area in all months.

Development of a Daily GRACE Mascon Solution for Terrestrial Water Storage

AbstractThe Gravity Recovery and Climate Experiment (GRACE) and GRACE‐Follow On missions have provided a global history of terrestrial water storage changes since 2002. Traditional GRACE products resolve monthly time‐variable gravity at spatial resolutions of 300–500 km, with many recent efforts focusing on regularized mass concentration (mascon) solutions to better resolve signals spatially. However, monthly resolution inhibits the applicability of GRACE to investigations of sub‐monthly signals. This study presents a new daily mass change solution estimated as deviations from the NASA Goddard Space Flight Center (GSFC) converged monthly mascon product and quantifies the fundamental trade‐off between temporal and spatial resolution in GRACE‐only solutions. As an iteration of the GSFC product, this daily solution represents a mixing of monthly information at higher relative spatial resolution with daily information at lower relative spatial resolution, resolving each temporal scale at the best resolution achievable without requiring hydrological model dependencies in the estimation. The resolution of the daily mascons is 300–400 km at high latitudes and 600–1,000 km lower, depending on each daily orbital track and the proximity of each mascon to constraint region boundaries (e.g., coastlines). Through simulations and model comparisons, we demonstrate that daily signals over areas larger than 400,000 km are recoverable, with basins larger than 800,000 km exhibiting strong signal recovery relative to leakage. This analysis establishes baselines for daily signal recovery from GRACE in the context of longer time scale solutions, characterizes leakage inherent in daily GRACE information, and creates new opportunities for applying GRACE to investigations of sub‐monthly signals.

From Physics to Climate Science

AbstractI began my career in climate science in 1974 when I went on Sabbatical from a tenured position in a physics department to National Center for Atmospheric Research (NCAR). I began working on simple climate models that year, and it led a few years later to a senior scientist position at NASA Goddard Space Flight Center (1978–1986). During this time, colleagues and I continued working on simple models expanding them to two horizontal dimensions and with many applications in paleoclimatology. When the models were forced by white noise the response was remarkably close to those of general circulation models. My group did numerous studies on the sampling errors in estimating empirical orthogonal functions and in the identification, evaluation, and estimation of errors for satellite retrieval of rain. I was the first study scientist for the Tropical Rainfall Measuring Mission. I moved to Texas A&M University in 1986 and served as department head from 1995–2003. At A&M I continued working on statistical problems related to Tropical Rain Measuring Mission (TRMM) and other applications such as detection of climate signals. I served in many service roles such as on boards, award committees, and as editor‐in‐chief for a journal and an encyclopedia. I also wrote, coauthored, or edited books on paleoclimatology, thermodynamics, climate models, and memoirs. I became Distinguished Professor Emeritus in 2014 and Research Professor until 2020, when I left the payroll.

A Frequency Diversity Algorithm for Extending the Radar Doppler Velocity Nyquist Interval

Compact millimeter wavelength radars have been widely used for applications such as remote sensing of clouds, guidance avionics, and recently, automotive navigation. However, the short wavelength of these radars limits their maximum unambiguous Doppler velocity. A common solution to this problem is to subsequently unfold the Doppler velocity estimate with the staggered pulse repetition time (PRT) algorithm, which requires two different PRTs to be employed in sequence. This article investigates a potentially more rapid method to extend the Doppler velocity Nyquist interval. We estimate Doppler velocity using a pair of frequency diverse pulses separated by a time lag that is significantly shorter than the PRT. During the first PRT, two pulses with center frequencies f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , followed by f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , separated by a lag τ are transmitted. During the next PRT, the pulses transmitted are in the order f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> followed by f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> . Doppler velocity is then estimated using the sum of the Doppler phases derived from f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> /f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> and f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> /f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> sequences. The focus of this article is the demonstration of this algorithm in a beam-filled scenario and the error canceling algorithm design. Based on Monte-Carlo simulations and data collected with the NASA Goddard Space Flight center's Cloud Radar System, the algorithm is demonstrated on nearly static surface echoes. Projected aircraft speeds and traditional pulse-pair estimates are employed as references. The algorithm was found to perform adequately on surface echoes for a low spectrum width of the order of 0.5 m/s and SNR comparable to 20 dB. Doppler velocity retrievals in a cirrus cloud layer with low velocity turbulence retained some qualitative structure, albeit with significantly degraded precision due to the lost coherence.

Method and apparatus for measuring the stability of large lateral transfer retroreflectors through a single optical port.

Achieving precise alignment between spatially separated optical axes is a challenging problem in optical applications such as boresighting, rangefinding, laser tracking, and optical beam delivery. The Advanced Topographic Laser Altimeter System (ATLAS) developed for NASA's Ice, Cloud and Land Elevation Satellite-II (ICESat-2) mission launched in 2018 was one such application where the alignment of the transmitter laser beam to the receiver telescope is critical for the operation of the instrument. This paper discusses a novel approach for verifying the stability of high-precision beam pointing in the conditions of the Thermal Vacuum (TVAC) test, developed for ATLAS testing at NASA Goddard Space Flight Center. It includes a method and an optical assembly that enables measuring tiny variations in the pointing of the beams of a large lateral transfer retroreflector through a single optical access port such as a TVAC chamber window. The method can be applied for verifying the alignment of optical beams separated by extended distances of tens of meters in vacuum.

Comparison of bidirectional transmittance distribution function (BTDF) measurements on fused silica and sintered polytetrafluoroethylene diffusers

Accurate determination of the bidirectional transmittance distribution function (BTDF) of transmissive diffusers is critical for the on-orbit spectral radiance calibration of several satellite-based, Earth remote sensing instruments. This study presents the results of the comparison of BTDF measurements by NASA Goddard Space Flight Center’s Diffuser Calibration Laboratory and the National Institute of Standards and Technology’s Spectral Tri-function Automated Reference Reflectometer facility on two transmissive diffusers: HOD-500, a synthetic fused silica sample manufactured by Hereaus Quarzglas, and Spectralon-250, a sintered polytetrafluoroethylene sample manufactured by Labsphere, Incorporated. BTDF measurements were acquired at seven wavelengths from 290 nm to 740 nm, at incident elevation angles of 0° and 30°, and at scatter elevation angles from 1° to 15°. Comparison of the measurements made by the two facilities revealed excellent agreement within their combined standard uncertainties. NASA chose the parameters for the BTDF measurements to be identical to those NASA used when measuring the BTDF of the flight diffusers to be flown onboard the Tropospheric Emissions: Monitoring of Pollution (TEMPO) and the Geostationary Environment Monitoring Spectrometer (GEMS) satellite instruments. Successful agreement between NASA and NIST of BTDF results, therefore, effectively validates the BTDF measurements NASA made for these satellite flight programs.

Future space drilling and sample acquisition: A collaborative industry-government workshop

A workshop on Planetary Drilling and Sample Acquisition was held at NASA-Goddard Space Flight Center, its purpose was to bring together the past decade of research and development results to define a framework for consistently and completely describing sample acquisition tasks for future Solar System missions, to survey the current state of knowledge of sample acquisition methods, and for each high priority application of sample acquisition technology, determine what had been demonstrated thus far vs. unaddressed areas. Combining inputs from government, academic and industrial participants, the workshop concluded that drilling missions that go 1-2 m deep (on the Moon, Mars, Europa or small bodies) are now feasible to propose in the 2020s, with potentially-high scientific payoff. Further, in developing complex sample acquisition systems for Solar System targets, the workshop agreed that both chamber tests and field tests at terrestrial analog sites are required. The workshop proposed adding cross-contamination and planetary protection requirements explicitly to instrument and surface mission flight calls, and also to add these as part of the technology readiness evaluation for NASA mission proposals.

HgCdTe avalanche photodiode array detectors with single photon sensitivity and integrated detector cooler assemblies for space lidar applications

A HgCdTe avalanche photodiode (APD) focal plane array assembly with linear mode photon-counting capability has been developed for space lidar applications. An integrated detector cooler assembly (IDCA) has been built using a miniature Stirling cooler. A microlens array has been included to improve the fill factor. The HgCdTe APD has a spectral response from 0.9- to 4.3-μm wavelengths, a photon detection efficiency as high as 70%, and a dark count rate of <250 kHz at 110 K. The mass of the IDCA is 0.8 kg and the total electrical power consumption is about 7 W. The HgCdTe APD arrays have been characterized at NASA Goddard Space Flight Center. A series of environmental tests have been conducted for the IDCAs, including vibration, thermal cycling, and thermal vacuum tests. A description of the device and the test results at NASA are given in this paper.

The role of the polar vortex strength during winter in Arctic ozone depletion from late winter to spring

The polar ozone destruction occurs as a result of the chemical processes and catalytic cycles with the participation of polar stratospheric clouds (PSCs) in the presence of weak solar radiation inside the polar vortex from late winter to spring. However, in some years, ozone depletion was not observed under strong polar vortex conditions (in the presence of PSCs). Based on the ERA-Interim reanalysis data and the NASA Goddard Space Flight Center (GSFC) satellite data, we show that the polar vortex strength in early winter plays an important role in polar ozone depletion. If the short-term weakening or splitting of the polar vortex, occurring with PSC melting, happened during the winter, then, in most cases, ozone destruction during the subsequent period from late winter to spring was not observed even under conditions of the strong polar vortex containing PSCs.

Potential improvements in global carbon flux estimates from a network of laser heterodyne radiometer measurements of column carbon dioxide

Abstract. We present observing system simulation experiments (OSSEs) to evaluate the impact of a proposed network of ground-based miniaturized laser heterodyne radiometer (mini-LHR) instruments that measure atmospheric column-averaged carbon dioxide (XCO2) with a 1 ppm precision. A particular strength of this passive measurement approach is its insensitivity to clouds and aerosols due to its direct sun pointing and narrow field of view (0.2∘). Developed at the NASA Goddard Space Flight Center (GSFC), these portable, low-cost mini-LHR instruments were designed to operate in tandem with the sun photometers used by the AErosol RObotic NETwork (AERONET). This partnership allows us to leverage the existing framework of AERONET's global ground network of more than 500 sites as well as providing simultaneous measurements of aerosols that are known to be a major source of error in retrievals of XCO2 from passive nadir-viewing satellite observations. We show, using the global 3-D GEOS-Chem chemistry transport model, that a deployment of 50 mini-LHRs at strategic (but not optimized) AERONET sites significantly improves our knowledge of global and regional land-based CO2 fluxes. This improvement varies seasonally and ranges 58 %–81 % over southern lands, 47 %–76 % over tropical lands, 71 %–92 % over northern lands, and 64 %–91 % globally. We also show significant added value from combining mini-LHR instruments with the existing ground-based NOAA flask network. Collectively, these data result in improved a posteriori CO2 flux estimates on spatial scales of ∼10 km2, especially over North America and Europe, where the ground-based networks are densest. Our studies suggest that the mini-LHR network could also play a substantive role in reducing carbon flux uncertainty in Arctic and tropical systems by filling in geographical gaps in measurements left by ground-based networks and space-based observations. A realized network would also provide necessary data for the quinquennial global stocktakes that form part of the Paris Agreement.

Regularization and error characterization of GRACE mascons.

We present a new global time-variable gravity mascon solution derived from Gravity Recovery and Climate Experiment (GRACE) Level 1B data. The new product from the NASA Goddard Space Flight Center (GSFC) results from a novel approach that combines an iterative solution strategy with geographical binning of inter-satellite range-acceleration residuals in the construction of time-dependent regularization matrices applied in the inversion of mascon parameters. This estimation strategy is intentionally conservative as it seeks to maximize the role of the GRACE measurements on the final solution while minimizing the influence of the regularization design process. We fully reprocess the Level 1B data in the presence of the final mascon solution to generate true post-fit inter-satellite residuals, which are utilized to confirm solution convergence and to validate the mascon noise uncertainties. We also present the mathematical case that regularized mascon solutions are biased, and that this bias, or leakage, must be combined with the estimated noise variance to accurately assess total mascon uncertainties. The estimated leakage errors are determined from the monthly resolution operators. We present a simple approach to compute the total uncertainty for both individual mascon and regional analysis of the GSFC mascon product, and validate the results with comparisons to independent mascon solutions and calibrated Stokes uncertainties. Lastly, we present the new solution and uncertainties with global analyses of the mass trends and annual amplitudes, and compute updated trends for the global ocean, and the respective contributions of the Greenland Ice Sheet, Antarctic Ice Sheet, Gulf of Alaska, and terrestrial water storage. This analysis highlights the successful closure of the global mean sea level budget; i.e. the sum of global ocean mass from the GSFC mascons and the steric component from Argo floats agrees well with the total determined from sea surface altimetry.

Welcome to the New L&amp;O: Methods Associate Editors!

Welcome to the New <i>L&amp;O</i>: <i>Methods</i> Associate Editors!

SOFIA-HIRMES: Looking Forward to the HIgh-Resolution Mid-infrarEd Spectrometer

The HIgh-Resolution Mid-infrarEd Spectrometer (HIRMES) is the 3rd Generation Instrument for the Stratospheric Observatory For Infrared Astronomy (SOFIA), currently in development at the NASA Goddard Space Flight Center (GSFC), and due for commissioning in 2019. By combining direct-detection Transition Edge Sensor (TES) bolometer arrays, grating-dispersive spectroscopy, and a host of Fabry-Perot tunable filters, HIRMES will provide the ability for high resolution ([Formula: see text]), mid-resolution ([Formula: see text]), and low-resolution ([Formula: see text]) slit-spectroscopy, and 2D Spectral Imaging ([Formula: see text] at selected wavelengths) over the 25–122[Formula: see text][Formula: see text]m mid to far infrared waveband. The driving science application is the evolution of proto-planetary systems via measurements of water-vapor, water-ice, deuterated hydrogen (HD), and neutral oxygen lines. However, HIRMES has been designed to be as flexible as possible to cover a wide range of science cases that fall within its phase-space, all whilst reaching sensitivities and observing powers not yet seen thus far on SOFIA, providing unique observing capabilities which will remain unmatched for decades.

Multiband Simulations of Multistream Polarimetric Microwave Radiances Over Aspherical Hydrometeors

AbstractA numerical precision assessment and simulation results of the Unified Microwave Radiative Transfer (UMRT) model incorporating aspherical frozen hydrometeors based on the NASA Goddard Space Flight Center (NASA/GSFC) OpenSSP database are presented. It is shown that UMRT maintains unconditional numerical stability and computational efficiency for absorbing and scattering clouds. UMRT requires symmetry of the transition matrix for the discretized planar‐stratified radiative transfer equation to realize numerically stable and accurate matrix operations as required by the discrete ordinate eigenanalysis method. UMRT has heretofore been restricted to spherical polydispersive hydrometeors. In this study, the necessary block‐diagonal structure of the full Stokes matrix for randomly oriented OpenSSP aspherical hydrometeors is shown to be maintained, albeit with small asymmetric deviations, which introduce small asymmetric components into the transition matrix that are negligible for most remote sensing applications. An upper bound of the brightness temperature error calculated by neglecting the asymmetric components of the transition matrix under even extreme atmospheric conditions is shown to be small. Hence, the OpenSSP hydrometeor database can be reliably used within the UMRT model. Block‐diagonal Stokes matrix elements along with other single‐scattering hydrometeor parameters were subsequently used in radiative simulations of multistream dual‐polarization radiances for a simulated hurricane event to demonstrate the inherent numerical stability and utility of the enhanced UMRT model. An intercomparison of computed upwelling radiances for a multiphase distribution of aspherical OpenSSP hydrometeors versus a mass‐equivalent Mie hydrometeor polydispersion for key sensing frequencies from 10 to 874 GHz shows the considerable impact of complex (vs. simple spherical) hydrometeors on predicted microwave radiances.

The SPECTRAL Ice Chamber: Application to Titan’s Stratospheric Ice Clouds

Abstract An ice chamber for obtaining thin-film transmission spectra of Titan-relevant organic ices between 50 and 11,700 cm−1 (200–0.85 μm) is described in detail in this work. The ice chamber, called the SPECtroscopy of Titan-Related ice AnaLogs chamber, is located in the Spectroscopy for Planetary ICes Environments laboratory at NASA Goddard Space Flight Center. Organic vapors are typically deposited between 30 and 150 K via a variable leak valve onto a chemical vapor deposition diamond substrate, where condensation directly into the solid state occurs. Quantitative thin-film infrared transmission spectra of the given ice or ice mixture are then measured, from which optical constants (complex indices of refraction) are computed. These optical constants in turn provide the necessary input for interpreting radiative transfer analyses of Cassini’s Composite InfraRed Spectrometer observations of Titan’s organic stratospheric ice clouds to determine their chemical compositions, abundances, and vertical distributions as functions of Titan latitude and season.

Advanced illumination modeling for data analysis and calibration. Application to the Moon

We present a new illumination modeling tool, called IllumNG, developed at NASA Goddard Space Flight Center (GSFC). We describe its capabilities to enhance the analysis and calibration of science data collected by planetary missions. We highlight these with examples making use of lunar data, particularly the topographic and radiometric measurements collected by the Lunar Orbiter Laser Altimeter (LOLA) instrument, with applications to radiometric measurements from other LRO instruments as well. The unique features of IllumNG are its accuracy and flexibility to handle multiple types of observers and light sources, and its ability to accurately model both singly- and doubly-scattered radiation to an observer.

The implementation of NEMS GFS Aerosol Component (NGAC) Version 2.0 for global multispecies forecasting at NOAA/NCEP – Part 1: Model descriptions

Abstract. The NEMS GFS Aerosol Component Version 2.0 (NGACv2) for global multispecies aerosol forecast has been developed at the National Centers of Environment Prediction (NCEP) in collaboration with the NESDIS Center for Satellite Applications and Research (STAR), the NASA Goddard Space Flight Center (GSFC), and the University at Albany, State University of New York (SUNYA). This paper describes the continuous development of the NGAC system at NCEP after the initial global dust-only forecast implementation (NGAC version 1.0, NGACv1). With NGACv2, additional sea salt, sulfate, organic carbon, and black carbon aerosol species were included. The smoke emissions are from the NESDIS STAR's Global Biomass Burning Product (GBBEPx), blended from the global biomass burning emission product from a constellation of geostationary satellites (GBBEP-Geo) and GSFC's Quick Fire Emission Data Version 2 from a polar-orbiting sensor (QFED2). This implementation advanced the global aerosol forecast capability and made a step forward toward developing a global aerosol data assimilation system. The aerosol products from this system have been used by many applications such as for regional air quality model lateral boundary conditions, satellite sea surface temperature (SST) physical retrievals, and the global solar insolation estimation. Positive impacts have been seen in these applications.

Performance Analysis of Airborne Photon- Counting Lidar Data in Preparation for the ICESat-2 Mission

Two airborne photon-counting laser altimeters have been deployed in direct support of National Aeronautics and Space Adminsitration (NASA’s) upcoming Ice, Cloud, and land Elevation Satellite-2 (ICESat-2) mission. Multiple Altimeter Beam Experimental Lidar (MABEL) was developed specifically for ICESat-2 testing and development. MABEL data are used to simulate key aspects of the ICESat-2 measurement strategy and are critical to the development of the algorithms for geophysical data-product generation. Slope Imaging Multi-polarization Photon-counting Lidar (SIMPL) is a NASA Goddard Space Flight Center instrument that has also been deployed in support of ICESat-2 performance discovery. Both instruments are photon-counting, small footprint laser altimeters that sample in both the 532- and 1064-nm wavelengths. And both instruments serve as a proxy for ICESat-2 operational performance and error assessment and a basis for the development of potential validation strategies. This paper provides an overview of how data from MABEL and SIMPL overflights have specifically provided the foundation for understanding the quality of ICESat-2 data and how we can plan to evaluate ICESat-2 products through comparison with other modalities of lidar data and/or ground-truth locations using ground fiducials.

EcoSAR

This article contains information on EcoSAR. EcoSAR is the advanced airborne polar metric and single pass interferometric P-band SAR instruments under development at the NASA Goddard Space Flight Center, through NASA's instrument Incubator Program (IIP). EcoSar will provide two and three-dimensional fine scale measurements of terrestrial ecosystem structure and biomass onboard the NASA P3 aircraft. These measurements directly support science requirements for the study of the carbon cycle and its relationship to climate change. To evaluate plant growth on earth from an airborne platform, both construction of the radar and subsequent testing of the radar using advanced radar test equipment to help develop and improve the radar system was necessary. This article describes the development, testing, and deployment of the EcoSAR system.

Measurement-based perturbation theory and differential equation parameter estimation with applications to satellite gravimetry

The numerical integration method has been routinely used by major institutions worldwide, for example, NASA Goddard Space Flight Center and German Research Center for Geosciences (GFZ), to produce global gravitational models from satellite tracking measurements of CHAMP and/or GRACE types. Such Earth’s gravitational products have found widest possible multidisciplinary applications in Earth Sciences. The method is essentially implemented by solving the differential equations of the partial derivatives of the orbit of a satellite with respect to the unknown harmonic coefficients under the conditions of zero initial values. From the mathematical and statistical point of view, satellite gravimetry from satellite tracking is essentially the problem of estimating unknown parameters in the Newton’s nonlinear differential equations from satellite tracking measurements. We prove that zero initial values for the partial derivatives are incorrect mathematically and not permitted physically. The numerical integration method, as currently implemented and used in mathematics and statistics, chemistry and physics, and satellite gravimetry, is groundless, mathematically and physically. Given the Newton’s nonlinear governing differential equations of satellite motion with unknown equation parameters and unknown initial conditions, we develop three methods to derive new local solutions around a nominal reference orbit, which are linked to measurements to estimate the unknown corrections to approximate values of the unknown parameters and the unknown initial conditions. Bearing in mind that satellite orbits can now be tracked almost continuously at unprecedented accuracy, we propose the measurement-based perturbation theory and derive global uniformly convergent solutions to the Newton’s nonlinear governing differential equations of satellite motion for the next generation of global gravitational models. Since the solutions are global uniformly convergent, theoretically speaking, they are able to extract smallest possible gravitational signals from modern and future satellite tracking measurements, leading to the production of global high-precision, high-resolution gravitational models. By directly turning the nonlinear differential equations of satellite motion into the nonlinear integral equations, and recognizing the fact that satellite orbits are measured with random errors, we further reformulate the links between satellite tracking measurements and the global uniformly convergent solutions to the Newton’s governing differential equations as a condition adjustment model with unknown parameters, or equivalently, the weighted least squares estimation of unknown differential equation parameters with equality constraints, for the reconstruction of global high-precision, high-resolution gravitational models from modern (and future) satellite tracking measurements.