Performance of GPS-based accelerometry: A simulation experiment

Recent studies have shown that with the availability of high-quality CHAMP and GRACE gravity field models, it is feasible to determine accurate non-gravitational accelerations for low Earth orbiting satellites indirectly from precise GPS satellite-to-satellite observations. Possible applications of this so-called GPS-based accelerometry approach consist of accelerometer calibration and atmospheric density and wind computations. With the growing number of high-quality space-borne GPS receivers, this method could be applied to a large range of satellites. In this paper an extensive simulation study has been carried out, based on real accelerometer data from the GRACE mission, in order to determine the optimal processing strategy and the resulting accuracy of the estimated non-gravitational accelerations. It is shown that the optimal processing strategy consists of a piecewise linear parameterization of the estimated empirical accelerations, together with short 6-h orbit arcs. The GPS-based accelerometry approach makes use of triple-differenced GPS observations and the impact of considering the correlated observation noise was found to be marginal in the presence of other error sources such as GPS ephemeris errors. Using a priori non-gravitational force models improves the recovery of low temporal resolution accelerations, except during huge geomagnetic storms. With this strategy, non-gravitational accelerations can be recovered during high solar activity with an accuracy of better than 10% of the total signal in along-track direction and around 25–40% in cross-track direction, at time resolutions of around 8–20 min. During solar minimum conditions, the relative recovery error will increase to approximately 50% in along-track direction and around 60–70% in cross-track direction, due to the reduced atmospheric drag signal. Unfortunately, GPS-based accelerometry is hardly sensitive in the radial direction.     

Measuring atmospheric density using GPS–LEO tracking data

We present a method to estimate the total neutral atmospheric density from precise orbit determination of Low Earth Orbit (LEO) satellites. We derive the total atmospheric density by determining the drag force acting on the LEOs through centimeter-level reduced-dynamic precise orbit determination (POD) using onboard Global Positioning System (GPS) tracking data. The precision of the estimated drag accelerations is assessed using various metrics, including differences between estimated along-track accelerations from consecutive 30-h POD solutions which overlap by 6 h, comparison of the resulting accelerations with accelerometer measurements, and comparison against an existing atmospheric density model, DTM-2000. We apply the method to GPS tracking data from CHAMP, GRACE, SAC-C, Jason-2, TerraSAR-X and COSMIC satellites, spanning 12 years (2001–2012) and covering orbital heights from 400 km to 1300 km. Errors in the estimates, including those introduced by deficiencies in other modeled forces (such as solar radiation pressure and Earth radiation pressure), are evaluated and the signal and noise levels for each satellite are analyzed. The estimated density data from CHAMP, GRACE, SAC-C and TerraSAR-X are identified as having high signal and low noise levels. These data all have high correlations with anominal atmospheric density model and show common features in relative residuals with respect to the nominal model in related parameter space. On the contrary, the estimated density data from COSMIC and Jason-2 show errors larger than the actual signal at corresponding altitudes thus having little practical value for this study. The results demonstrate that this method is applicable to data from a variety of missions and can provide useful total neutral density measurements for atmospheric study up to altitude as high as 715 km, with precision and resolution between those derived from traditional special orbital perturbation analysis and those obtained from onboard accelerometers.     

一种基于温度参数的热层密度修正方法

热层大气的阻力效应是影响低轨航天器大量空间操作的重要因素, 尤其是经验密度模式, 其固有的至少15%的内符合误差已严重制约航天器轨道计算精度的提高. 针对广泛应用的经验密度模式, 选择物理背景简明、关联参数较少的JACCHIA71模式, 以地磁平静条件下的全球散逸层顶温度最小值T_c及125 km高度拐点温度T_x为对象, 建立密度相对于上述温度参数的条件方程, 推导密度相对于温度参数的解析偏导数, 并给出其最小二乘解. 同时, 利用CHAMP卫星数据对模式进行修正, 模式平均误差从40%降低至3%左右. 通过TG01飞行器的轨道预报比较, 修正前后轨道预报位置精度从2 km提升至1 km左右. 经过CHAMP卫星和TG01飞行器的实测数据检验, 验证了修正算法的正确性和有效性.      

Improved kHz-SLR tracking techniques and orbit quality analysis for LEO missions

Satellite gravity field missions such as CHAMP, GRACE and GOCE are designed as low Earth orbiting spacecraft (LEO) with orbit heights of about 250–500 km. The challenging mission objectives require a very precise knowledge of the satellite orbit position in space. For these missions precise orbit information is typically provided by GPS satellite-to-satellite tracking (SST) observations supported by satellite laser ranging (SLR).     

High-fidelity geometry models for improving the consistency of CHAMP,GRACE, GOCE and Swarm thermospheric density data sets

During the last two decades, accelerometers on board of the CHAMP, GRACE, GOCE and Swarm satellites have provided high-resolution thermosphere density data to improve our knowledge on atmospheric dynamics and coupling processes in the thermosphere-ionosphere region. Most users of the data have focused on relative density variations. Scale differences between datasets and models have been largely neglected or removed using ad hoc scale factors. The origin of these scale differences arises from errors in the aerodynamic modelling, specifically in the modelling of the satellite outer surface geometry and of the gas-surface interactions. Therefore, the first step to remove the scale differences is to enhance the geometry modelling. This work forms the foundation for the future improvement of characterization of satellite aerodynamics and gas-surface interactions models at TU Delft, as well as for extending the use of sideways and angular accelerations in the aerodynamic analysis of accelerations and derivation of thermosphere datasets. Although work to improve geometry and aerodynamic force models by other authors has focused on CHAMP and GRACE, this paper includes the GOCE and Swarm satellites as well. In addition, it uses a density determination algorithm that is valid for arbitrary attitude orientations, enabling a validation making use of attitude manoeuvres. The results show an improvement in the consistency of density data between these four missions, and of data obtained before, during and after attitude manoeuvres of CHAMP and Swarm. The new models result in larger densities, compared to the previously used panel method. The largest average rescaling of density, by switching to the new geometry models is reached for Swarm at 32%, the smallest for GRACE at 5%. For CHAMP and GOCE, mean differences of 11% and 9% are obtained respectively. In this paper, an overview of the improvements and comparisons of data sets is provided together with an introduction to the next research phase on the gas-surface interactions.     

Onboard orbit determination using GPS observations based on the unscented Kalman filter

Spaceborne GPS receivers are used for real-time navigation by most low Earth orbit (LEO) satellites. In general, the position and velocity accuracy of GPS navigation solutions without a dynamic filter are 25 m (1σ) and 0.5 m/s (1σ), respectively. However, GPS navigation solutions, which consist of position, velocity, and GPS receiver clock bias, have many abnormal excursions from the normal error range for space operation. These excursions lessen the accuracy of attitude control and onboard time synchronization. In this research, a new onboard orbit determination algorithm designed with the unscented Kalman filter (UKF) was developed to improve the performance. Because the UKF is able to obtain the posterior mean and covariance accurately by using the second-order Taylor series expansion through the sampled sigma points that are propagated by using the true nonlinear system, its performance can be better than that of the extended Kalman filter (EKF), which uses the linearized state transition matrix to predict the covariance. The dynamic models for orbit propagation applied perturbations due to the 40 × 40 geo-potential, the gravity of the Sun and Moon, solar radiation pressure, and atmospheric drag. The 7(8)th-order Runge–Kutta numerical integration was applied for orbit propagation. Two types of observations, navigation solutions and C/A code pseudorange, can be used at the user’s discretion. The performances of the onboard orbit determination were verified using real GPS data of the CHAMP and KOMPSAT-2 satellites. The results of the orbit determination were compared with the precision orbit ephemeris (POE) of the CHAMP and KOMPSAT-2 satellites.     

NRLMSISE-00大气模型与GRACE和CHAMP卫星大气密度数据的对比分析

利用GRACE(Gravity Recovery And Climate Experiment)和CHAMP(Challenging Mini-Satellite Payload)卫星2002-2008年的大气密度数据与NRLMSISE-00大气模型密度结果进行比较,分析了模型密度误差及其特点.结果显示,NRLMSISE-00大气模型计算的密度值普遍偏大,其相对误差随经纬度变化,在高纬度相对较小;相对误差随地方时变化,在02:00LT和15:00LT左右较大,10:00LT和20:00LT左右较小.通过模型密度相对误差与太阳F_10.7指数的对比分析发现,在太阳活动低年模型相对误差最大,而在太阳活动高年相对误差较小;将模型结果分别与GRACEA/B双星和CHAMP卫星的密度数据进行比较,发现对于轨道高度更高的GRACE卫星轨道,模型相对误差更大;在地磁平静期,相对误差与地磁ap指数(当前3h)相关性不强,但是在大磁暴发生时,误差急剧增大.      

GRACE accelerometer calibration by high precision non-gravitational force modeling

The precise modeling and knowledge of non-gravitational forces acting on satellites is of big interest to many scientific tasks and missions. Since 2002, the twin GRACE satellites have measured these forces in a low Earth orbit with highly precise accelerometers, for about 15?years. Besides the significance for the GRACE mission, these measurement data allow the evaluation of modeling approaches and the improvement of force models. Unfortunately, before any scientific usage, the accelerometer measurements need to be calibrated, namely scale factor and bias have to be regularly estimated.In this study we demonstrate an accelerometer calibration approach, solely based on high precision non-gravitational force modeling without any use of empirically or stochastically estimated parameters, using our in-house developed satellite simulation tool XHPS. The aim of this work is twofold, first we use the accelerometer data and the residuals resulting from the calibration to quantitatively analyze and validate different non-gravitational force model approaches. In a second step, we compare the calibration results to three different calibration methods from different authors, based on gravity field recovery, GPS-based precise orbit determination, and based on modeled accelerations.We consider atmospheric drag forces and winds, as well as radiation forces due to solar radiation pressure, albedo, Earth infrared and thermal radiation (TRP) of the satellite itself. For TRP, we investigate different transient temperature calculation approaches for the satellite surfaces with absorbed power from the aforementioned radiation sources. A detailed finite element model of the satellite is utilized for every force, considering orientation, material properties and shadowing conditions for each element.For cross-track and radial direction, which are mainly affected by the radiative forces, our calibration residuals are quite small when drag is not super dominant (1–3?$\text{nm}/{\text{s}}^2$ for total accelerations around $\pm$50?$\text{nm}/{\text{s}}^2$). For these directions the calibration seems to perform better than the other compared methods, where some bigger differences were found. For the drag dominated along-track direction it is vice versa, here our method is not sensitive enough because the difference between modeled and measured drag is bigger (e.g. residuals around 10?$\text{nm}/{\text{s}}^2$ for total accelerations around $\pm$70?$\text{nm}/{\text{s}}^2$ for low solar activity). In along-track direction the orbit determination based methods are more sensitive and produce more reliable results. Results for the complete GRACE mission time span from 2003 to 2017 are shown, covering different seasonal environmental conditions.     

Global monitoring of tropospheric water vapor with GPS radio occultation aboard CHAMP

The global positioning system radio occultation (GPS RO) technique provides a powerful tool for atmospheric sounding which requires no calibration, is not affected by clouds, aerosols or precipitation, and provides an almost uniform global coverage. The paper deals with application of GPS RO measurements from CHAllenging Minisatellite Payload (CHAMP) for the retrieval of tropospheric water vapor profiles. CHAMP RO data are available since 2001 with up to 200 high resolution atmospheric profiles per day. We introduce a new direct method for water vapor retrieval from GPS RO data. Additionally, a 1Dvar algorithm is used for this purpose. The so derived CHAMP water vapor profiles are validated with radiosonde data on a global scale. Here, both methods come to statistically comparable results revealing a negative bias of less than 0.1 g/kg and a standard deviation of less than 1 g/kg specific humidity in the mid troposphere. Potentials of CHAMP RO retrievals for monitoring the mean tropospheric water vapor distribution on a global scale are presented.     

Use of two-line element data for thermosphere neutral density model calibration

Traditional empirical thermospheric density models are widely used in orbit determination and prediction of low-Earth satellites. Unfortunately, these models often exhibit large density errors of up to around 30% RMS. Density errors translate into orbit errors, adversely affecting applications such as re-entry operations, manoeuvre planning, collision avoidance and precise orbit determination for geodetic missions. The extensive database of two-line element (TLE) orbit data contains a wealth of information on satellite drag, at a sufficiently high spatial and temporal resolution to allow a calibration of existing neutral density models with a latency of one to two days. In our calibration software, new TLE data for selected objects is converted to satellite drag data on a daily basis. The resulting drag data is then used in a daily adjustment of density model calibration parameters, which modify the output of an existing empirical density model with the aim of increasing its accuracy. Two different calibration schemes have been tested using TLE data for about 50 objects during the year 2000. The schemes involve either height-dependent scale factors to the density or corrections to CIRA-72 model temperatures, which affect the density output based on a physical model. Both schemes have been applied with different spherical harmonic expansions of the parameters in latitude and local solar time. Five TLE objects, varying in perigee altitude between 280 and 530 km, were deliberately not used during calibration, in order to provide independent validation. Even with a single daily parameter, the RMS density model error along their tracks can already be reduced from the 30% to the 15% level. Adding additional parameters results in RMS errors lower than 12%.     

GPS single-frequency orbit determination for low Earth orbiting satellites

The determination of high-precision orbits for Low Earth Orbiting (LEO) satellites (e.g., CHAMP, GRACE, MetOp-A) is based on dual-frequency tracking data from on-board GPS (Global Positioning System) receivers. The two frequencies allow it to eliminate the first order ionosphere effects. Data screening and precise orbit determination (POD) procedures are optimized under the assumption of the availability of two frequencies.     

深空卫星重力测量计划研究综述

地球卫星重力测量计划CHAMP(CHAllenging Minisatellite Payload)、GRACE(Gravity Recovery and Climate Experiment)、GOCE(Gravity field and steady-state Ocean Circulation Explorer)和月球卫星重力测量计划(Gravity Recovery and Interior Laboratory,GRAIL)的成功实施,以及下一代地球重力卫星(GRACE Follow-On)的即将发射昭示着我们将迎来一个前所未有的高精度和高空间分辨的深空卫星重力探测时代。围绕深空卫星重力测量的研究背景、必要性、可行性、卫星重力反演软件平台构建、轨道摄动和未来研究方向开展了研究论证。研究表明:深空卫星重力测量作为新世纪重力探测技术,在精化量体重力场、提高惯性导航精度、天体动力学、天体物理学和军事技术的研究,以及促进国民经济发展和提高社会效益等方面具有广泛的应用前景。     

Performance improvement of real-time PPP ambiguity resolution using a regional integer clock

Obtaining reliable GNSS uncalibrated phase delay (UPD) or integer clock products is the key to achieving ambiguity-fixed solutions for real-time (RT) precise point positioning (PPP) users. However, due to the influence of RT orbit errors, the quality of UPD/integer clock products estimated with a globally distributed GNSS network is difficult to ensure, thereby affecting the ambiguity resolution (AR) performance of RT-PPP. In this contribution, by fully utilising the consistency of orbital errors in regional GNSS network coverage areas, a method is proposed for estimating regional integer clock products to compensate for RT orbit errors. Based on Centre National d’Études Spatiales (CNES) RT precise products, the regional GPS/BDS integer clock was estimated with a CORS network in the west of China. Results showed that the difference between the estimated regional and CNES global integer clock/bias products was generally less than 5 cm for GPS, whereas clock differences of greater than 10 cm were observed for BDS due to the large RT orbit error. Compared with PPP using global integer clock/bias products, the AR performance of PPP using the regional integer clock was obviously improved for four rover stations. For single GPS, the horizontal and vertical accuracies of ambiguity-fixed PPP solutions were improved by 56.2% and 45.3% on average, respectively, whereas improvements of 67.5% and 50.5% in the horizontal and vertical directions, respectively, were observed for the combined GPS/BDS situation. Based on a regional integer clock, the RMS error of a kinematic ambiguity-fixed PPP solution in the horizontal direction could reach 0.5 cm. In terms of initialisation time, the average time to first fix (TTFF) in combined GPS/BDS PPP was shortened from 18.2 min to 12.7 min. In view of the high AR performance realised with the use of regional integer clocks, this method can be applied to scenarios that require high positioning accuracy, such as deformation monitoring.     

Enhancing the kinematic precise orbit determination of low earth orbiters using GPS receiver clock modelling

Clock error estimation has been the focus of a great deal of research because of the extensive usage of clocks in GPS positioning applications. The receiver clock error in the spacecraft orbit determination is commonly estimated on an epoch-by-epoch basis, along with the spacecraft’s position. However, due to the high correlation between the spacecraft orbit altitude and the receiver clock parameters, estimates of the radial component are degraded in the kinematic approach. Using clocks with high stability, the predictable behaviour of the receiver oscillator can be exploited to improve the positioning accuracy, especially for the radial component. This paper introduces two GPS receiver clock models to describe the deterministic and stochastic property of the receiver clock, both of which can improve the accuracy of kinematic orbit determination for spacecraft in low earth orbit. In particular, the clock parameters are estimated as time offset and frequency offset in the two-state model. The frequency drift is also estimated as an unknown parameter in the three-state model. Additionally, residual non-deterministic random errors such as frequency white noise, frequency random walk noise and frequency random run noise are modelled. Test results indicate that the positioning accuracy could be improved significantly using one day of GRACE flight data. In particular, the error of the radial component was reduced by over 40.0% in the real-time scenario.     

Towards accurate atmospheric mass density determination Using precise positional information of space objects

This paper presents a new method of deriving atmospheric mass densities with a high temporal resolution from precise orbit data of low earth orbiting (LEO) space objects. This method is based on the drag perturbation equation of the semi-major axis of the orbit of LEO space objects which relates the change rate of the semi-major axis to the atmospheric mass density. The effectiveness of the new method is evaluated using the GFZ-ISDC GPS rapid science orbit (RSO) products of the CHAMP satellite over a time period of 3 months. The densities derived using this new method and obtained from accelerometer data are compared and good agreements are achieved. An example of using the derived density to generate good orbit prediction for CHAMP is presented.     

Comparison of the CHAMP radio occultation data with the Canadian advanced digital ionosonde in the Polar Regions

In this paper we present the results of the comparison of the retrieved electron density profiles of the Ionospheric Radio Occultation (IRO) experiment on board CHAMP (CHAllenging Minisatellite Payload), with the ground ionosonde profiles for the Polar Regions. IRO retrieved electron density profiles from CHAMP are compared with Canadian Advanced Digital Ionosonde (CADI) measurements at two vertical sounding stations well within the Polar Cap, Eureka (geog. 80°13′ N; 86°11′ W) and Resolute Bay (geog. 74°41′ N; 94°54′ W). We compared the ionospheric parameters such as the peak electron density of the F-layer (NmF₂) and the peak height of the F-layer (h_mF₂) for a 3-year period, 2004–2006. CHAMP derived NmF₂ shows reasonable agreement with the ionosonde retrieved NmF₂ for both the stations (0.76 and 0.71 correlation coefficient, for Eureka and Resolute Bay, respectively) whereas the h_mF₂ agreement is not that acceptable (0.25 and 0.37 correlation coefficient, respectively). The h_mF₂ from vertical sounding showed less spread than the CHAMP h_mF₂.     

Characteristics and accuracies of the GRACE inter-satellite pointing

For almost 10 years, the Gravity Recovery and Climate Experiment (GRACE) has provided information about the Earth gravity field with unprecedented accuracy. Efforts are ongoing to approach the GRACE baseline accuracy as there still remains an order of magnitude between the present error level of the gravity field solutions and the GRACE baseline. At the current level of accuracy, thorough investigation of sensor related effects is necessary as they are one of the potential contributors to the error budget. In the science mode operations, the twin satellites are kept precisely pointed with their KBR antennas towards each other. It is the task of the onboard attitude and orbit control system (AOCS) to keep the satellites in the required formation. We analyzed long time series of the inter-satellite pointing variations as they reflect the AOCS performance and characteristics. We present significant systematic effects in the inter-satellite pointing and discuss their possible sources. Prominent features are especially related to the magnetic torquer characteristics, star cameras’ performance and KBR antenna calibration parameters. The relation between the magnetic torquer attitude control and the Earth magnetic field, impact of the different performance of the two star camera heads on the attitude control and the features due to uncertainties in the calibration parameters relating the star camera frame to K-frame are discussed in detail. Proper understanding of these effects will help to reduce their impact on the science data and subsequently increase the accuracy of the gravity field solutions. Moreover, understanding the complexity of the onboard system is essential not only for increasing the accuracy of the GRACE data but also for the development of the future gravity field satellite missions.     

An alternative computation of a gravity field model from GOCE

GOCE is the first satellite with a gravitational gradiometer (SGG). This allows to determine a gravity field model with high spatial resolution and high accuracy. Four of the six independent components of the gravitational gradient tensors (GGT) are measured with high accuracy in the so-called measurement band (MB) from 5 to 100 mHz by the GOCE gradiometer. Based on more than 1 year of GOCE measurements, two gravity field models have been derived. Here, we introduce a strategy for spherical harmonic analysis (SHA) from GOCE measurements, with a bandpass filter applied to the SGG data, combined with orbit analysis based on the integral equation approach, and additional constraints (or stabilization) in the polar areas where no observation is available due to the orbit geometry. In addition, we combined the GOCE SGG part with a set of GRACE normal equations. This improves the accuracy of the gravity field in the long-wavelength parts, due to the complementarity of GOCE and GRACE. Comparison with other models and with external data shows that our results are rather close to the GPS-levelling data in well-selected test regions, with an uncertainty of 4–7 cm, for truncation at degree 200.     

Storm-time atmospheric density modeling using neural networks and its application in orbit propagation

Upper atmospheric densities during geomagnetic storms are usually poorly estimated due to a lack of clear understanding of coupling mechanisms between the thermosphere and magnetosphere. Consequently, the orbit determination and propagation for low-Earth-orbit objects during geomagnetic storms have large uncertainties. Artificial neural networks are often used to identify nonlinear systems in the absence of rigorous theory. In the present study, an attempt has been made to model the storm-time atmospheric density using neural networks. Considering the debate over the representative of geomagnetic storm effect, i.e. the geomagnetic indices ap and Dst, three neural network models (NNM) are developed with ap, Dst and a combination of ap and Dst respectively. The density data used for training the NNMs are derived from the measurements of the satellites CHAMP and GRACE. The NNMs are evaluated by looking at: (a) the mean residuals and the standard deviations with respect to the density data that are not used in training process, and (b) the accuracy of reconstructing the orbits of selected objects during storms employing each model. This empirical modeling technique and the comparisons with the models NRLMSIS-00 and Jacchia-Bowman 2008 reveal (1) the capability of neural networks to model the relationship between solar and geomagnetic activities, and density variations; and (2) the merits and demerits of ap and Dst when it comes to characterizing density variations during storms.     

Quantifying barystatic sea-level change from satellite altimetry,GRACE and Argo observations over 2005–2016

Time-varying spherical harmonic coefficients determined from the Gravity Recovery and Climate Experiment (GRACE) data provide a valuable source of information about the water mass exchange that is the main contributor to the Earth’s gravity field changes within a period of less than several hundred years. Moreover, by measuring seawater temperature and salinity at different layers of ocean depth, Argo floats help to measure the steric component of global mean sea level (GMSL). In this study, we quantify the rate of barystatic sea-level change using both GRACE RL05 and RL06 monthly gravity field models and compare the results with estimates achieved from a GMSL budget closure approach. Our satellite altimetry-based results show a trend of 3.90 ± 0.14 mm yr⁻¹ for the GMSL rise. About 35% or 1.29 ± 0.07 mm yr⁻¹ of this rate is caused by the thermosteric contribution, while the remainder is mainly due to the barystatic contribution. Our results confirm that the choice of decorrelation filters does not play a significant role in quantifying the global barystatic sea-level change, and spatial filtering may not be needed. GRACE RL05 and RL06 solutions result in the barystatic sea-level change trends of 2.19 ± 0.13 mm yr⁻¹ and 2.25 ± 0.16 mm yr⁻¹, respectively. Accordingly, the residual trend, defined as the difference between the altimetry-derived GMSL and sum of the steric and barystatic components, amounts to 0.51 ± 0.51 and 0.45 ± 0.44 mm yr⁻¹ for RL05 and RL06-based barystatic sea-level changes, respectively, over January 2005 to December 2016. The exclusion of the halosteric component results in a lower residual trend of about 0.36 ± 0.46 mm yr⁻¹ over the same period, which suggests a sea-level budget closed within the uncertainty. This could be a confirmation on a high level of salinity bias particularly after about 2015. Moreover, considering the assumption that the GRACE-based barystatic component includes all mass change signals, the rather large residual trend could be attributed to an additional contribution from the deep ocean, where salinity and temperature cannot be monitored by the current observing systems. The errors from various sources, including the model-based Glacial Isostatic Adjustment signal, independent estimation of geocenter motion that are not quantified in the GRACE solutions, as well as the uncertainty of the second degree of zonal spherical harmonic coefficients, are other possible contributors to the residual trend.