Tropospheric Delay Estimation Research Articles

On the real-time tropospheric delay estimates using low-cost GNSS receivers and antennas

On the real-time tropospheric delay estimates using low-cost GNSS receivers and antennas

BDS/GPS Zenith Tropospheric Delay Estimation and Its Effect on Precise Point Positioning

BDS/GPS Zenith Tropospheric Delay Estimation and Its Effect on Precise Point Positioning

A global grid model for the estimation of zenith tropospheric delay considering the variations at different altitudes

Abstract. The accuracy of tropospheric delay correction heavily depends on the quality of the tropospheric model, and the zenith tropospheric delay (ZTD) is an important factor affecting the tropospheric delay. Therefore, it is essential to establish a precise ZTD empirical model. The existing ZTD models are constrained by a single fitting function, lack consideration for daily cycle variations, and rely solely on data with one resolution for modeling. To address these limitations, we proposed a global piecewise ZTD empirical grid (GGZTD-P) model. This model considers the daily cycle variation and latitude factor of ZTD, using the sliding window algorithm based on fifth-generation European Centre for Medium-Range Weather Forecasts atmospheric reanalysis data (ERA5). The ZTD data from 545 radiosonde stations and the second Modern-Era Retrospective analysis for Research and Applications (MERRA-2) atmospheric reanalysis data are used to validate the accuracy of the GGZTD-P model. The results indicate that the GGZTD-P model outperforms the global pressure and temperature 3 (GPT3) model, exhibiting 26 % and 53 % lower bias and rms, respectively, when using radiosonde stations as reference values. Furthermore, when evaluated using MERRA-2 atmospheric reanalysis data, the GGZTD-P model consistently exhibits superior performance across various latitude regions. It is expected that the application of this new model will provide improved services for high-precision global navigation satellite system (GNSS) positioning and GNSS meteorology.

Real-time GNSS tropospheric delay estimation with a novel global random walk processing noise model (GRM)

Real-time GNSS tropospheric delay estimation with a novel global random walk processing noise model (GRM)

Estimation and Evaluation of Zenith Tropospheric Delay from Single and Multiple GNSS Observations

Multi-Global Navigation Satellite Systems (multi-GNSS) (including GPS, BDS, Galileo, and GLONASS) provide a significant opportunity for high-quality zenith tropospheric delay estimation and its applications in meteorology. However, the performance of zenith total delay (ZTD) retrieval from single- or multi-GNSS observations is not clear, particularly from the new, fully operating BDS-3. In this paper, zenith tropospheric delay is estimated using the single-, dual-, triple-, or four-GNSS Precise Point Positioning (PPP) technique from 55 Multi-GNSS Experiment (MGEX) stations over one year. The performance of GNSS ZTD estimation is evaluated using the International GNSS Service (IGS) standard tropospheric products, radiosonde, and the fifth-generation European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis (ERA5). The results show that the GPS-derived ZTD time series is more consistent and reliable than those derived from BDS-only, Galileo-only, and GLONASS-only solutions. The performance of the single-GNSS ZTD solution can be enhanced with better accuracy and stability by combining multi-GNSS observations. The accuracy of the ZTD from multi-GNSS observations is improved by 13.8%, 43.8%, 27.6%, and 22.9% with respect to IGS products for the single-system solution (GPS, BDS, Galileo, and GLONASS), respectively. The ZTD from multi-GNSS observations presents higher accuracy and a significant improvement with respect to radiosonde and ERA5 data when compared to the single-system solution.

Estimation of tropospheric parameters with GNSS smartphones in a differential approach

With the introduction of the operating system Android 7 Nougat in the year 2016, it became possible to access Global Navigation Satellite System (GNSS) code and carrier phase observations. These observations can be processed with the state-of-the-art GNSS processing software packages, which allows an in-depth evaluation of the smartphone’s GNSS performance. The availability of carrier phase observations enables sub-decimeter-level positioning. A few years ago, smartphones wearing dual-frequency GNSS chipsets hit the mass market. In this study, we investigate the capability of such a device for the estimation of tropospheric delays. Static measurements carried out over the period of two weeks are performed using a Google Pixel 4 XL smartphone. The measurements are processed using relative positioning methods with a baseline length of about 33 kilometers, where a continuously operating reference station (CORS) acts as a base. The estimated differential zenith tropospheric wet delay (dZWD), obtained for the smartphone are then combined with absolute values computed at the reference station, in order to obtain time series of Zenith Total Delay (ZTD). Using this method, we demonstrate that high-precision ZTDs can be successfully determined from smartphone GNSS observations. When comparing the estimated tropospheric delays with those determined at a nearby geodetic receiver to assess the accuracy of the acquired time series of ZTD, differences in the range of few millimeters to a centimeter are visible. We examine the impact of various error sources, such as antenna phase center variations and residual effects of the ionosphere. Given that the obtained accuracies are at the level of a centimeter and below, the suggested method shows the potential to resolve small-scale tropospheric structures in near real-time, and thus, could be an interesting data source for numerical weather prediction models or related GNSS crowdsourcing projects.

Multi-temporal InSAR tropospheric delay modelling using Tikhonov regularization for Sentinel-1 C-band data

The increased availability of satellite SAR data coupled with improved InSAR processing algorithms has led to higher accuracy of InSAR-derived displacements. However, obtaining millimeter-level accuracy still remains a challenge due to the inability to accurately model the tropospheric delay, which is sometimes larger than the actual deformation. We propose to estimate the relative daily SAR tropospheric delay by double differencing interferograms that share a common image by leveraging Sentinel-1’s short revisit time and the interchanging role of the common image. The resulting double-differenced interferograms consist of tropospheric delay and processing noise. This combination leads to an inverse problem as a result of the differential nature of the interferograms. The unknown SAR tropopsheric delay is solved by Tikhonov regularization and the processing errors are accounted for by covariance modelling. In cases of rapid localized displacement, the deforming region is masked and the tropospheric delay for the deforming region is determined through kriging. Validation is performed using simulated SAR data and GNSS tropospheric delays. We find a good correlation between estimated SAR tropospheric delays and GNSS tropospheric delays, with an average root mean square error (RMSE) of 1.93 radians across six GNSS locations. In addition, we examine the performance of the correction on interferograms through variogram modelling, which indicates improved correction both for short-wavelength and long-wavelength tropospheric noise. SAR tropospheric delay estimates are integrated into a multitemporal InSAR workflow through either interferometric subtraction or stochastic modelling. Validation using GNSS measurements indicates that SAR tropospheric delay estimates significantly improve the accuracy of the InSAR-derived time-series displacement.

Quality control of SIRGAS ZTD products

AbstractThe SIRGAS-CON network currently has more than 450 continuous GNSS stations, and it is used for geodetic purposes. In atmospheric studies, it is used for ionospheric monitoring and for the estimation of zenith tropospheric delays (ZTDs). From the Neutral Atmosphere Analysis Center of SIRGAS, Centro de Ingeniería Mendoza Argentina, the final tropospheric products of this network are generated after several stages of quality controls and filtering, in order to be published on a daily basis in the official website of SIRGAS, since 2014 (https://sirgas.ipgh.org/en/products/tropospheric-delays). These products arise from adjusting the solutions estimated by different SIRGAS analysis centers. Prior to the combination, a quality control of the individual solutions is carried out, based on the precision estimator of each parameter and an internal control of each solution with respect to the combined value. In this work, we show the quality control process of the inputs, the selected tolerance and its justification. The internal consistency analysis of tropospheric parameters for a period of 7 years was carried out. We also exposed the improvements in the estimation of tropospheric parameters implemented during 2021 and its impact in the generation of the final ZTD products (in 99% of the stations the mean standard deviation of ZTD is less than 1 mm).

Comparative evaluation and analysis of different tropospheric delay models in Ghana

Tropospheric delay prediction models have become increasingly important in Global Navigation Satellite System (GNSS) as they play a critical role in GNSS positioning applications. Due to the different atmospheric conditions over the earth regions, tropospheric effect on GNSS signals also differs, influencing the performance of these prediction models. Thus, the choice of a particular prediction model can significantly degrade the positioning accuracy especially when the model does not suit the user’s environs. Therefore, a performance assessment of existing prediction models in various regions for a suitable one is very imperative. This paper evaluates and analyses seven commonly used tropospheric delay models in Ghana in terms of performances in Zenith Tropospheric Delay (ZTD) estimation and baseline positional accuracies using data from six selected Continuously Operating Reference Stations (CORS). The 1˚x1˚ gridded Vienna Mapping Functions 3 (VMF3) ZTD product and coordinates solutions from the CSRS-PPP positioning service were respectively used as references. The results show that the Black model performed better in estimating the ZTD, followed by Askne and Nordius model. The Saastamoinen, Marini and Murray, Niell, Goads and Goodman and Hopfield models respectively performed poorly. However, the result of the baseline solutions did not show much variation in the coordinate difference provided by the use of the prediction models, nonetheless, the Black and Askne and Nordius models continue to dominate the other models. Of all the models evaluated, either Black or Askne and Nordius model is recommended for use to mitigate the ZTD in the study area, however, the choice of the Black model will be more desirable.

Troposphere Delay Remote Sensing Using Single GPS Receiver

The most prominent spatially correlated errors in GNSS observations are well known to be atmospheric effects. The ionosphere and troposphere are the two main layers of the Atmosphere that cause delays in GNSS observations. A linear combination of the dual-frequency data can be used to reduce ionospheric delay. Unlike the ionospheric delay, the tropospheric delay cannot be eliminated using the same methods. The troposphere is primarily associated with GPS. The delay it causes in GPS signals is regarded as one of the primary sources of errors that must be eliminated to determine accurate positions. This paper's main purpose is to develop a new source code that can estimate the effect of tropospheric delay over any GPS station. The tropospheric delay in this proposed code is estimated utilizing sequential least-squares adjustment using a model depending on Niell Mapping Function (NMF). This model, known as the Tropospheric Delay Estimation program, was created in the MATLAB® environment (TDE). This research presents the results of tropospheric delay during DOY 2, 2020 of actual data from ten ground-based IGS stations distributed over Antarctica, China, Canada, Fiji, Russia, Greenland, and Portugal IGS stations worldwide. For validation of the proposed code results, they were compared with troposphere delay results of the International GNSS Service (IGS). Good agreement and high correlation were found between both results. In comparison to IGS, the proposed code's standard deviations range from 0.0000525 m to 0.008154 m, indicating how accurate this study is in terms of agreement of solutions provided by IGS. Finally, the MATLAB software can accurately estimate troposphere delay with an adaptable temporal resolution for GPS users.

Modeling and performance assessment of precise point positioning with multi-frequency GNSS signals

The development of global navigation satellite system (GNSS), especially BeiDou-3 Navigation Satellite System (BDS-3) and Galileo system, promotes the progress of the precise point positioning (PPP) technology. Considering deep integration and full utilization of GNSS signals, this study dedicates to developing three triple-frequency PPP models apply to multi-constellation including BDS-3. Besides, some biases such as differential code bias (DCB), inter-frequency bias (IFB) and inter-system bias (ISB) are also addressed. Through the ten-day datasets collected from 160 stations, the multi-frequency PPP models for single- and multi-constellation solutions are analyzed and compared from static positioning performance, Zenith Tropospheric Delay (ZTD) estimates, ISB estimates, and IFB estimates. The results indicate that the triple-frequency PPP models, especially two dual-frequency ionosphere-free (IF) model namely TIF with triple-constellation which can provide the positioning accuracy of 4.89, 7.31, 12.1 mm after the convergence time of 10.75 min in east, north and up directions, perform better than the dual-frequency model, apart from the a single IF model with GPS. Interestingly, the positioning performance is not effectively improved with the addition of GLONASS constellation. But the performance of GAL/BDS combined constellation is comparable to GPS single constellation in TIF model.

Accuracy Evaluation and Analysis of GNSS Tropospheric Delay Inversion from Meteorological Reanalysis Data

Accurate estimation of tropospheric delay is significant for global navigation satellite system’s (GNSS) high-precision navigation and positioning. However, due to the random and contingent changes in weather conditions and water vapor factors, the classical tropospheric delay model cannot accurately reflect changes in tropospheric delay. In recent years, with the development of meteorological observation/detection and numerical weather prediction (NWP) technology, the accuracy and resolution of meteorological reanalysis data have been effectively improved, providing a new solution for the inversion and modeling of regional or global tropospheric delays. Here, we evaluate the consistency and accuracy of three different types of reanalysis data (i.e., ERA5, MERRA2, and CRA40) used to invert the zenith tropospheric delay (ZTD) from 436 international GNSS service (IGS) stations in 2020, based on the integral method. The results show that the ZTD inversion of the three types of reanalysis data was consistent with the IGS ZTD, even in heavy rain conditions. Furthermore, the average precision of the ZTD inversion of the ERA5 reanalysis data was higher, where the mean deviation (bias), mean absolute error (MAE), and root mean square (RMS) were –3.39, 9.69, and 12.55 mm, respectively. The ZTD average precisions of the MERRA2 and CRA40 inversions were comparable, showing slightly worse performance than the ERA5. In addition, we further analyzed the global distribution characteristics of the ZTD errors inverted from the reanalysis data. The results show that ZTD errors inverted from the reanalysis data were highly correlated with station latitude and climate type, and they were mainly concentrated in the tropical climate zone at low latitudes. Compared to dividing error areas by latitude, dividing error areas by climatic category could better reflect the global distribution of errors and would also provide a data reference for the establishment of tropospheric delay models considering climate type.

Küresel BeiDou Sistemi (BDS-3) için hassas nokta konumlama performansının değerlendirilmesi

The main objective of this study is to evaluate the positioning performance of the global BeiDou system (BDS-3) in terms of Precise Point Positioning (PPP). The PPP performance of BDS-3 was compared with both GPS and the regional BeiDou system (BDS-2). In addition, the contribution of BDS-3 to the PPP performance was analyzed through multi-GNSS combinations of these three systems. For this purpose, seven processing scenarios in total were composed, and the observation dataset collected at 15 International GNSS Service (IGS) stations over the period of November 21 and 30, 2021 were processed under these scenarios. The results showed that positioning accuracies of 0.0152, 0.0605, and 0.0203 m could be acquired from the GPS, BDS-2, and BDS-3 PPP solutions, respectively. Also, average convergence times obtained from these solutions were respectively computed as 60.94, 261.71, and 63.06 minutes. While results of the BDS-2 PPP solution remained substantially behind the GPS results, comparable results with the GPS PPP were acquired from the BDS-3 PPP solution. Besides, the obtained results revealed that the positioning performance could be improved significantly when double and triple combinations of these navigation systems were employed. The positioning accuracy and convergence time obtained from the PPP solution where GPS, BDS-2, and BDS-3 systems were used in common were 0.0136 m and 26.45 minutes. In this study, tropospheric delay estimations were also evaluated to analyze the PPP performance comprehensively. The results demonstrated that the BDS-3 solution could provide close results to the GPS PPP solution in terms of tropospheric delay estimation and the best troposphere estimation performance could similarly be acquired from the multi-GNSS solution where three systems were employed in common.

PERFORMANCE OF NUMERICAL WEATHER PRODUCTS FOR INSAR TROPOSPHERIC CORRECTION: A CASE STUDY OF A TROPICAL REGION

Abstract. Tropospheric delay variability remains a significant source of error in the InSAR-derived measurements. Numerical weather models have been proposed as an alternative technique to mitigate tropospheric delays in InSAR and have become a standard procedure for some multi-temporal InSAR processing workflows. This study evaluates the viability of three numerical weather models for mitigating tropospheric delay in InSAR for a tropical region. We assess their performance in correcting tropospheric delay in Sentinel-1 interferograms at different spatial wavelengths using variograms. Their performance is validated using GNSS tropospheric delay and our proposed SAR-derived tropospheric delay estimates. The results indicate that numerical weather model estimates do not mitigate short-wavelength turbulent delays, but can mitigate long-wavelength stratified delays to some extent, which may also introduce additional errors in interferograms. At a spatial wavelength of 40 km, 36% of the interferograms showed increased spatial autocorrelation after correction with GACOS, 55% with ERA-5, and 51% with MERRA-2. In contrast the InSAR-derived tropospheric delays resulted in a significant reduction in variance at all wavelengths indicating the ability to mitigate both turbulent and stratified delays. Our study demonstrates the limited potential of numerical weather model estimates to satisfactorily mitigate trophospheric noise in InSAR and the capability of InSAR-derived trophospheric delay to significantly correct tropospheric noise in InSAR.

An improved weighting strategy for tropospheric delay estimation with real-time single-frequency precise positioning

An improved weighting strategy for tropospheric delay estimation with real-time single-frequency precise positioning

Adaptive Fusion of Multi-Source Tropospheric Delay Estimates for InSAR Deformation Measurements

Atmospheric propagation delay correction is the key to improving the accuracy of deformation measurement of satellite interferometric synthetic aperture radar (InSAR). The empirical phase-elevation models and external data-based models present uneven performances of atmospheric delay correction for InSAR deformation monitoring. In this study, based on our previous fusion of delays predicted by multiple weather models (FDWM), we propose a new approach of adaptive fusion of multi-source tropospheric delay (AFMTD) estimates derived from multiple models over wide areas, i.e., ERA5, GACOS, WRF, MERRA2, NARR, MODIS, Linear model, and Powerlaw model. The spatially varying scaling algorithm is employed to refine the tropospheric delays predicted by the weather models. Meanwhile, we adopt a multiple-window strategy to cope with the spatially lateral variation of tropospheric delays. The AFMTD not only improves the spatial heterogeneity of tropospheric delay, but also adaptively combines multiple models to achieve a more reliable delay estimation. This AFMTD method is incorporated into the StaMPS-SBAS procedure. We compared the AFMTD with other single models using ENVISAT ASAR and Sentinel-1 datasets over Los Angeles of Southern California. The result of ASAR first demonstrates the effectiveness and reliability of the AFMTD method by referring to the assumed ground truth of simultaneous MERIS observations. The results of Sentinel-1 data show that over 95% of unwrapped interferograms have the minimum root-mean-square values after AFMTD correction for both descending and ascending tracks. The validation against GPS observation presents that the RMSEs of InSAR displacement time series after AFMTD correction decreases at more than 90% of 125 GPS stations. The average reductions of RMSE are 35.79% and 36.28% for descending and ascending data, respectively, and the maximum improvement is more than 70%. Overall, the proposed AFMTD method outperforms any single model for InSAR tropospheric delay correction and provides an open framework to fuse multi-source tropospheric delay estimates.

Forecasting GNSS Zenith Troposphere Delay by Improving GPT3 Model with Machine Learning in Antarctica

Antarctica has a significant impact on global climate change. However, to draw climate change scenarios, there is a need for meteorological data, such as water vapor content, which is scarce in Antarctica. Global navigation satellite system (GNSS) networks can play a major role in overcoming this problem as the tropospheric delay that can be derived from GNSS measurements is an important data source for monitoring the variation of water vapor content. This work intends to be a contribution for improving the estimation of the zenith tropospheric delay (ZTD) obtained with the latest global pressure–temperature (GPT3) model for Antarctica through the use of long short-term-memory (LSTM) and radial basis function (RBF) neural networks for modifying GPT3_ZTD. The forecasting ZTD model is established based on the GNSS_ZTD observations at 71 GNSS stations from 1 January 2018 to 23 October 2021. According to the autocorrelation of the bias series between GNSS_ZTD and GPT3_ZTD, we predict the LSTM_ZTD for each GNSS station for period from October 2020 to October 2021 using the LSTM day by day. Based on the bias between LSTM_ZTD and GPT3_ZTD of the training stations, the RBF is adopted to estimate the LSTM_RBF_ZTD of the verified station, where the LSTM_ZTD represents the temporal forecasting ZTD at a single station, and the LSTM_RBF_ZTD represents the predicted ZTD obtained from space. Both the daily and yearly RMSE are calculated against the reference (GNSS_ZTD), and the improvement of predicted ZTD is compared with GPT3_ZTD. The results show that the single-station LSTM_ZTD series has a good agreement with the GNSS_ZTD, and most daily RMSE values are within 20 mm. The yearly RMSE of the 65 stations ranges from 6.4 mm to 32.8 mm, with an average of 10.9 mm. The overall accuracy of the LSTM_RBF_ZTD is significantly better than that of the GPT3_ZTD, with the daily RMSE of LSTM_RBF_ZTD significantly less than 30 mm, and the yearly RMSE ranging from 5.6 mm to 50.1 mm for the 65 stations. The average yearly RMSE is 15.7 mm, which is 10.2 mm less than that of the GPT3_ZTD. The LSTM_RBF_ZTD of 62 stations is more accurate than GPT3_ZTD, with the maximum improvement reaching 76.3%. The accuracy of LSTM_RBF_ZTD is slightly inferior to GPT3_ZTD at three stations located in East Antarctica with few GNSS stations. The average improvement across the 65 stations is 39.6%.

Assessment of tropospheric modeling on PPP-AR performances under brazilian atmospheric condition

In the Global Navigation Satellite System (GNSS), ambiguity resolution (AR) can shorten observation time and increase the positioning quality. The correct tropospheric modeling is intrinsically related to the ability to perform AR. Here, we assessed the influence of different tropospheric correction alternatives on AR for static Precise Point Positioning (PPP) in Brazilian territory. Our goal was to provide directions to users when choosing a suitable tropospheric model for application in PPP-AR under Brazilian atmospheric conditions. Thus, this study was carried out using well-known models such as the Saastamoinen model and the Zenith Tropospheric Delay (ZTD) Estimation and Numerical Weather Prediction (NWP) model from CPTEC/INPE. Six GNSS stations from the Brazilian Network for Continuous Monitoring (RBMC) (BRAZ, UFPR, RNNA, POVE, SMAR, and SAGA) were selected. Different GNSS processing setups were considered for GNSS data registered at selected stations during summer and winter. The assessment was based on a statistical analysis of positioning accuracy during one-hour sessions. The results indicated that such as the ZTD Estimation, the NWP model provides an accuracy of a few centimeters. On the other hand, the Saastamoinen model provided decimeter level accuracy, thus it is not the recommended choice for PPP-AR in the experimental conditions.

Differential Tropospheric Delay Estimation by Simultaneous Multi-Angle Repeat-Pass InSAR

Tropospheric delays are one of the main contributors to the interferometric phase in synthetic aperture radar (SAR) interferometry. When the phase contributions from surface deformation, topography, and ionospheric delays are negligible or known, the interferogram can be used to estimate the differential tropospheric delay (DTD), which can help to improve tropospheric delay predictions from weather models and in situ measurements. In conventional repeat-pass interferometric SAR (InSAR), however, the estimation of the DTD can still be significantly hindered by baseline errors. In addition, a single interferogram provides only relative DTDs, as the delays can be retrieved up to an unknown offset. To address such issues, this article presents a method for the estimation of DTDs on large scales by using repeat-pass simultaneous multi-angle SAR systems. Complementary simultaneous observations of the correlated troposphere from multiple angles are used to retrieve estimates of the absolute DTD and, at the same time, to mitigate the effect of baseline knowledge errors. Finally, a performance evaluation is presented for the Harmony Earth Explorer 10 candidate mission. A centimeter-level absolute accuracy and a submillimeter-level relative accuracy of the DTD estimation are achieved under the multistatic Harmony case when at least one companion satellite has an inter-satellite distance longer than 300 km to provide enough sensitivity.

Wet tropospheric delay estimation of satellite altimeters based on machine learning

Wet path delay (WPD) is an important source of satellite altimeters’ transmission error, which must be fully corrected to ensure data quality. After decades of study, WPD can now be accurately negated by measuring the brightness temperature with a microwave radiometer (MWR). However, due to hardware and financial limitations, many satellites are not equipped with MWRs, and WPD can only be corrected by mathematical models. Currently available empirical models, which are mostly established by traditional fitting methods, are unable to reach the accuracy required by altimeters, so it is necessary to further improve the correction accuracy of the models. Based on three machine learning algorithms, we established three high-precision wet tropospheric correction (WTC) models suitable for satellite altimeters. We analyzed the global navigation satellite system dataset and three MWR datasets, i.e., Jason-3, Sentinel-3A, and Sentinel-3B, for data quality and consistency and combined these datasets for modeling. In addition, several input features were extracted from existing knowledge on tropospheric delay. Through a comparative analysis of different feature combinations, temperature (T), water vapor pressure (e), and total content of water vapor were selected as the input features. Then we established three WTC models based on multilayer perceptron, random forest, and the adaptive network-based fuzzy inference system (ANFIS), respectively. With a dataset from Jason-2 that was not used during modeling as the verification set, we verified the generalizability of these WTC models from the global, latitudinal, and seasonal perspectives. Moreover, three traditional WTC models were compared with the ones developed here. The results show that the WTC models established based on machine learning algorithms outperform traditional WTC models. The WTC model based on ANFIS boasts the best calculation accuracy in all aspects, over 50% higher than that of traditional WTC models.