esa-sceva/satcom-chunk-collection huggingface dataset:
# Topp-Leone Modified Weibull Model: Theory and Applications to Medical and Engineering Data
## 4 Three Different Estimation Methods
### Maximum Product of Spacing Method  
In this subsection, the MPS will be used, a substitute for the MLE method that is used to estimate the parameters of continuous uni-variate models. This method was developedby [40; 41; 42; 43; 44; 45; 46]. The uniform spacing of the TLMW distribution can be characterised as follows, \(z_{1}<...<z_{n}\) assuming a random sample of size \(n\):  
\[D_{i}(\varphi)=F(z_{(i)},\varphi)-F(z_{(i-1)},\varphi);\ i=1,...,n+1, \tag{26}\]  
where \(D_{i}(\varphi)\) denotes to the uniform spacing, \(F(z_{(0)},\varphi)=0\), \(F(z_{(n+1)},\varphi)=1\) and \(\sum_{i=1}^{n+1}D_{i}(\varphi)=1\).  
\[G(\varphi)=\frac{1}{n+1}\sum_{i=1}^{n+1}\log\biggl{\{}\left[1-e^{-2(az_{i}+ \beta z_{i}^{\theta})}\right]^{\lambda}-\left[1-e^{-2(az_{i-1}+\beta z_{i-1}^{ \theta})}\right]^{\lambda}\biggr{\}}. \tag{27}\]  
The MPS estimators (MPSE) of the TLMW distribution parameters may be constructed by maximizing Equation (27) with regards to \(\alpha\) and \(\delta\).  
\[\frac{\partial G(\varphi)}{\partial\alpha}=\frac{1}{n+1}\sum_{i=1}^{n+1}2 \lambda\frac{\left[1-e^{-2(az_{i}+\beta z_{i}^{\theta})}\right]^{\lambda-1}e ^{-2(az_{i}+\beta z_{i}^{\theta})}z_{i}-\left[1-e^{-2(az_{i-1}+\beta z_{i-1}^{ \theta})}\right]^{\lambda-1}e^{-2(az_{i-1}+\beta z_{i-1}^{\theta})}z_{i-1}}{ \left[1-e^{-2(az_{i}+\beta z_{i}^{\theta})}\right]^{\lambda}-\left[1-e^{-2( az_{i-1}+\beta z_{i-1}^{\theta})}\right]^{\lambda}}, \tag{28}\]  
\[\frac{\partial G(\varphi)}{\partial\beta}=\frac{1}{n+1}\sum_{i=1}^{n+1}2 \lambda\frac{\left[1-e^{-2(az_{i}+\beta z_{i}^{\theta})}\right]^{\lambda-1}e ^{-2(az_{i}+\beta z_{i}^{\theta})}z_{i}^{\theta}\left[1-e^{-2(az_{i-1}+\beta z _{i-1}^{\theta})}\right]^{\lambda-1}e^{-2(az_{i-1}+\beta z_{i-1}^{\theta})}z_ {i-1}^{\theta}}{\left[1-e^{-2(az_{i}+\beta z_{i}^{\theta})}\right]^{\lambda}- \left[1-e^{-2(az_{i-1}+\beta z_{i-1}^{\theta})}\right]^{\lambda}}, \tag{29}\]  
\[\frac{\partial G(\varphi)}{\partial\theta}=\frac{1}{n+1}\sum_{i=1}^{n+1}2 \beta\lambda\frac{\left[1-e^{-2(az_{i}+\beta z_{i}^{\theta})}\right]^{\lambda -1}e^{-2(az_{i}+\beta z_{i}^{\theta})}z_{i}^{\theta}\ln(z_{i})-\left[1-e^{-2( az_{i-1}+\beta z_{i-1}^{\theta})}\right]^{\lambda-1}e^{-2(az_{i-1}+\beta z_{i-1}^{ \theta})}z_{i-1}^{\theta}\ln(z_{i-1})}{\left[1-e^{-2(az_{i}+\beta z_{i}^{ \theta})}\right]^{\lambda}-\left[1-e^{-2(az_{i-1}+\beta z_{i-1}^{\theta})} \right]^{\lambda}}, \tag{30}\]  
and  
\[\frac{\partial G(\varphi)}{\partial\lambda}=\frac{1}{n+1}\sum_{i=1}^{n+1} \frac{\left[1-e^{-2(az_{i}+\beta z_{i}^{\theta})}\right]^{\lambda}\ln\left[1- e^{-2(az_{i}+\beta z_{i}^{\theta})}\right]^{\lambda}\ln\left[1-e^{-2(az_{i}+ \beta z_{i}^{\theta})}\right]-\left[1-e^{-2(az_{i-1}+\beta z_{i-1}^{\theta})} \right]^{\lambda}\ln\left[1-e^{-2(az_{i-1}+\beta z_{i-1}^{\theta})}\right]}{ \left[1-e^{-2(az_{i}+\beta z_{i}^{\theta})}\right]^{\lambda}-\left[1-e^{-2(az_{i- 1}+\beta z_{i-1}^{\theta})}\right]^{\lambda}}. \tag{31}\]  
Additionally, using the R application to solve Equations (28)-(31) to determine the MPSE of the parameters.
## 2 Materials and Methods
### Nearest-Neighbor-Based Randomized Algorithm  
The original nearest neighbor algorithm is perhaps the simplest heuristic for the traveling salesman problem (TSP) [14], which is a classical optimization problem closely related to the VRP. The key to this algorithm is to always visit the nearest unvisited city (customer). It is well known that the nearest neighbor algorithm, a popular choice for tour construction heuristics, works at an acceptable level for the Euclidean TSP, but it can produce very poor results for the general TSP [15; 16]. We now introduce our nearest-neighbor-based randomized algorithm to tackle the HVRPSPD, as shown in Algorithm 1. In our approach, at each iteration of the main loop of the algorithm (line 2), a feasible solution to the HVRPSPD (i.e., a set of feasible routes, each of them associated with a vehicle type, to cover all customers) is constructed from scratch. We keep track of the best solution obtained during iterations, but it is not used to define a neighborhood structure or to guide the search for better solutions and, in this respect, it can be seen as a blind search. The construction of a feasible solution, however, is probabilistically guided by two alternative greedy criteria to select the next customer to be visited on the current route: the nearest neighbor or the first one from a given list of unvisited customers. In the sequel, we explain the construction of a feasible solution in more detail. ```
parameters:Computation time limit \(\tau\), probability \(\rho\) of using the nearest neighbour strategy. input:Set of customers and depot, set of vehicle types. output:Set of vehicle routes (bestRoutes). \(bestRoutes\gets nil\)// list of best routes found so far while\(\tau\)not exceededdo \(currentRoutes\gets nil\)// current list of routes under construction \(unvisitedCustomers\gets permute(customers)\)// random permutation of all customers while\(unvisitedCustomers\)notemptydo \(route\gets nil\)// list of customers on the route under construction \(vehicle\gets random(types)\)// random selection of the vehicle to serve the route \(route.add(depot)\)// the route starts from depot \(endCustomer\gets route.last()\)// last customer (depot) from route \(first\leftarrow\)true// there is no customer added to route yet \(feasibleCustomers\leftarrow\)true// there are customers that can be possibly added to route for each customer in unvisitedCustomersdo \(customer.infeasible\leftarrow\)false// customer is not proven infeasible to be part of route while\(feasibleCustomers\)do \(trial\gets nil\)// customer under consideration to be added to route ifnot\(first\)and\(random(0,1)\leq\rho\)then \(trial\gets nearestNeighbour(unvisitedCustomers,endCustomer)\)// trial is chosen among customers from unvisitedCustomers that has not yet been proven infeasible to be part of route by means of nearest neighbour greedy strategy else \(trial\gets firstPossiblyFeasible(unvisitedCustomers)\)// trial is set to be the first customer from unvisitedCustomers that is not proven infeasible to be part of route if\(checkRouteFeasibility(route,trial,vehicle)\)then // in case route remains feasible (w.r.t.
## 5 Collision Avoidance Trajectory Tracking Control
### Tracking Controller Design  
The position tracking error is  
\[\left\{\begin{array}{l}x_{e}=x-x_{d}\\ y_{e}=y-y_{d}.\end{array}\right. \tag{26}\]  
The derivation of time \(t\) on both sides of (26) can be obtained  
\[\left[\begin{array}{l}\dot{x}_{e}\\ \dot{y}_{e}\end{array}\right]{=}R(\psi)\left[\begin{array}{l}u\\ v\end{array}\right]-\left[\begin{array}{l}\dot{x}_{d}\\ \dot{y}_{d}\end{array}\right], \tag{27}\]  
with  
\[R(\psi){=}\left[\begin{array}{cc}\cos\psi&-\sin\psi\\ \sin\psi&\cos\psi\end{array}\right].\]  
For \(u\) and \(v\), the virtual control laws \(\alpha_{u}\) and \(\alpha_{v}\) with \(\Lambda=\sqrt{x_{e}^{2}+y_{e}^{2}+\
abla}\) are  
\[\left[\begin{array}{l}\alpha_{u}\\ \alpha_{v}\end{array}\right]{=}R(\psi)^{-1}\left[\begin{array}{l}\dot{x}_{d }-\kappa x_{e}/\Lambda\\ \dot{y}_{d}-\kappa y_{e}/\Lambda\end{array}\right], \tag{28}\]  
Figure 7: The actual and desired trajectories for collision avoidance.  
where \(\kappa>0\) and \(\
abla>0\). When \(u_{eg}=u-\alpha_{u}v_{eg}=v-\alpha_{v}\), then  
\[\left[\begin{array}{c}\dot{x}_{e}\\ \dot{y}_{e}\end{array}\right]= R(\psi)\left[\begin{array}{c}u_{eg}\\ v_{eg}\end{array}\right]-\left[\begin{array}{c}xx_{e}/\Lambda\\ xy_{e}/\Lambda\end{array}\right]. \tag{29}\]  
When \(u_{eg}\to 0\), \(v_{eg}\to 0\), (29) is updated to  
\[\left[\begin{array}{c}\dot{x}_{e}\\ \dot{y}_{e}\end{array}\right]= -\left[\begin{array}{c}\kappa x_{e}/\Lambda\\ xy_{e}/\Lambda\end{array}\right]. \tag{30}\]  
Consider the following candidate Lyapunov function as  
\[V_{0}= \frac{1}{2}x_{e}^{2}+\frac{1}{2}y_{e}^{2}. \tag{31}\]  
Deriving from (31) we can obtain  
\[\dot{V}_{0}= x_{e}\dot{x}_{e}+y_{e}\dot{y}_{e}=-\kappa\left(x_{e}^{2}+y_{e}^{2} \right)/\Lambda. \tag{32}\]  
According to (32) and (30), when \(\lim\limits_{t\to\infty}\dot{V}_{0}\leq 0\), the position error converges to zero gradually.  
The PID controller is introduced as the tracking controller, and its control law is as follows  
\[\mathbf{u}=K_{p}\Gamma(t)+K_{i}\int_{0}^{t}\Gamma(t)dt+K_{d}\dot{\Gamma}(t), \tag{33}\]  
where \(\Gamma=\left[u_{eg},v_{eg}\right]^{T}\), and the proportional coefficient, integral coefficient and differential coefficient of the PID controller are \(K_{p}=diag(K_{pu},K_{pv})\), \(K_{i}=diag(K_{iu},K_{iv})\) and \(K_{d}=diag(K_{du},K_{dv})\), respectively.
## 3 Materials and Methods
### Experimental Setup
#### 3.2.4 Evaluation Metrics  
The experiments were evaluated using the following metrics commonly used in the segmentation task [9,18,38]. These metrics use the True Positives (TP), True Negatives (TN), False Positives (FP), and False Negatives (FN) obtained from the confusion matrix computed from the label predictions of the model and the ground truth map.  
* Intersection over Union (IoU) evaluates the overlap between two areas, such as the predicted and ground truth masks. It is calculated as the intersection divided by the union of two sets: \[IoU=\frac{TP}{TP+FP+FN}.\]  
Figure 3: F1-Scores resulting from changing key hyperparameters \(\lambda_{FL}=\) FL, \(\lambda_{GDL}=\) GDL. The combination leading to the best results is highlighted in orange. (**a**) Experiments performed on FLAIR sequences of ISLES 2015. (**b**) Experiments performed on DWI modality of ISLES 2022.  
* F1-Score represents the harmonic mean between precision and sensitivity. It measures the model's performance in balancing false positives and false negatives: \[F1\text{-}Score=\frac{2 TP}{2 TP+FP+FN}.\] (8)
* The Hausdorff Distance (HD) measures the maximum distance between the nearest points in two sets. It evaluates the similarity between shapes or contours: \[HD(Y,G)=\max\biggl{(}\max_{y\in Y}\min_{g\in G}d(y,g),\max_{g\in G}\min_{y\in Y} d(g,y)\biggr{)},\] (9) where \(Y\) is the set of labeled pixels in the foreground class predicted by the model, \(G\) corresponds to the set of real foreground pixels given in the ground truth map, and \(d(\bullet,\bullet)\) is the Euclidean distance between two pixels.
* Accuracy assesses the overall performance of a classification model, describing the percentage of correct predictions out of the total predictions: \[Accuracy=\frac{TP+TN}{TP+FP+TN+FN}.\] (10)
* Precision gives the accuracy of the positive predictions: \[Precision=\frac{TP}{TP+FP}.\] (11)
* Sensitivity, also known as Recall, evaluates the model's ability to identify positive cases correctly: \[Sensitivity=\frac{TP}{TP+FN}.\] (12)
* Specificity measures the model's ability to identify the negative cases correctly: \[Specificity=\frac{TN}{FP+TN}.\] (13)
# Passive Radar-Based Parameter Estimation of Low Earth Orbit Debris Targets
## 5 Simulations
### Moving Target over a Forward Scatter Receiver Array
#### 5.2.1 Single Receiver Geometry  
Figure 6 displays the geometry involved for a single receiver-transmitter pair and an approaching target. This scenario is considered to be in a single bistatic plane. Therefore, the perpendicular distance to the target from the baseline is equal to the horizontal distance along the ground. The transmitter and receiver look-angles and bistatic angles can then be derived and are presented in Equations (16)-(18).  
\[\theta=\tan^{-1}\left(\frac{h}{D}\right) \tag{16}\]  
\[\gamma=\tan^{-1}\left(\frac{D}{L-h}\right) \tag{17}\]  
\[\beta=180^{\circ}-\left(\gamma+\left(90^{\circ}-\theta\right)\right) \tag{18}\]  
In the above equations, \(h\) is the target altitude, \(D\) is the horizontal distance along the ground, \(\theta\) is the receiver look-angle, and \(\gamma\) is the transmitter look-angle.
# Diversity of Physical Processes: Challenges and Opportunities for Space Electric Propulsion
###### Abstract  
The renewed interest in space exploration has led to the growth in research efforts pertaining to advanced space propulsion systems, including highly efficient electric propulsion systems. Although already tested in space many decades ago and being currently employed on various space platforms and thousands of satellites, these systems are yet to reach their full potential for applications on orbit and in deep space. One specific feature of space electric propulsion is the large diversity of physical processes used in this technology, which is not typical for many other types of propulsion systems used in transport, such as those used by airplanes or automobiles. Various physical processes and mechanisms underpin different electric propulsion technologies and should be integrated to drive the future science and technology of space electric propulsion systems. This opinion article briefly highlights this feature of space electric propulsion and outlines some challenges and opportunities that follow from this diversity.  
electric propulsion; thrusters; CubeSats +
Footnote †: journal: Applied Sciences
## 3 Results
### Canopy Coverage
#### 3.3.1 Canopy Zone Level Coverage  
There was no significant difference in spray coverage among canopy zones regardless of the SSCDS treatments (Table 5). Moreover, no significant interaction was reported between canopy zone and SSCDS treatment. The bottom zone coverage of modified irrigation micro-emitter treatment i.e., T1 (34.6 \(\pm\) 5.3%) was highest followed by bottom zone coverage of hollow cone nozzle treatment i.e., T2 (26.0 \(\pm\) 5.7%) (Figure 7a). Moreover, top zone coverage of T1 (15.5 \(\pm\) 3.9%) was significantly lower than bottom zone coverage. The least canopy coverage was reported for mid zone canopy coverage of treatment T2  
Figure 6: Mean spray deposition evaluated at (a) different canopy zones and (b) leaf surfaces for modified irrigation micro-emitter (T1) and hollow cone nozzle (T2) treatment. Different lowercase letters above individual bar plots indicate significance of mean differences in transformed data at 5% level and associated error bars indicate standard error; non-transformed deposition (ng cm\({}^{-2}\)) values are presented.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variables** & **df** & **MS** & **F** & \(p\) \\ \hline Main plot & & & & \\ Block & 2 & 0.04 & & \\ Treatment & 1 & 2.00 & 1.20 & 0.27 \\ Error(a) & 1 & 9.85 & & \\ Canopy zone & 2 & 0.60 & 0.36 & 0.70 \\ Leaf surface & 1 & 1.00 & 8.91 & 0.02 \\ Treatment \(\times\) Canopy zone & 2 & 3.30 & 1.99 & 0.14 \\ Treatment \(\times\) Leaf surface & 1 & 2.24 & 1.35 & 0.24 \\ Canopy zone \(\times\) Leaf surface & 2 & 0.42 & 0.25 & 0.78 \\ Treatment \(\times\) Canopy zone \(\times\) Leaf surface & 2 & 0.68 & 0.41 & 0.66 \\ Error(b) & 192 & 1.66 & & \\ \hline \hline \end{tabular}
\end{table}
Table 5: ANOVA result of cube root transformed canopy coverage data.  
(15.3 \(\pm\) 3.9%). Furthermore, unlike T1, the differences in bottom and top zone coverage (15.8 \(\pm\) 4.0%) for T2 were statistically non-significant.
## 2 Electromagnetic Wave Propagation Behavior and the Principle of the Reflected Voltage
### Reflected Voltage Mode of Electromagnetic Waves  
During the propagation of the electromagnetic waves of the ground penetrating radar in the medium, the reflected voltage power of the object can be used as an indicator to determine whether there is a sufficient reflection signal, and its value depends on the characteristic impedance and the resistance of the object and medium. When the electromagnetic wave propagates in the medium interface \(A\), the reflected coefficient between the air and the medium interface is the ratio of the reflected voltage to the incident voltage, as shown in Equation (7):  
\[R_{I}=\frac{r_{I}(t)}{s(t)} \tag{7}\]  
where \(r_{I}(t)\) is the reflected voltage from the air to the concrete medium, and \(s(t)\) is the incident voltage at interface \(I\). Therefore, the reflected voltage of the interface \(I\) between the air and the medium can be calculated by Equation (8):  
\[r_{I}(t)=R_{I}\cdot s(t) \tag{8}\]  
The reflection behavior of the interface between the air and the concrete is shown in Figure 4.  
In a double-layered medium, the reflected wave \(A\) in medium \(I\) will generate another reflected wave, as shown in Figure 5.  
Figure 4: Electromagnetic Wave Reflection Behavior in Medium.  
Figure 5: Electromagnetic Wave Reflection Behavior in Double-layered Medium.  
The reflection coefficient of the interface can be calculated by Equation (9):  
\[R_{II}=\frac{r_{II}(t)}{s(t)\cdot w_{i}} \tag{9}\]  
The reflected voltage of the interface _II_ can be calculated by Equation (10):  
\[r_{II}(t)=R_{II}\cdot s(t)\cdot w_{i} \tag{10}\]  
where \(r_{II}(t)\) is the reflected voltage of the interface between medium \(A\) and medium \(B\), _s(t)_ is the incident voltage at interface _II_, _w\({}_{i}\)_ is the reflected power, calculated by (_w\({}_{i}\)_ = 1 - _R\({}_{II}\)_2), and _R\({}_{II}\)_2 is the reflected power of the interface \(I\).  
The total reflected voltage of the electromagnetic wave of the ground penetrating radar can be expressed by Equation (11):  
\[r(t)=r_{I}(t)+r_{II}(t) \tag{11}\]  
Then, Equation (12) can be derived from Equation (11):  
\[\sum r(t)=R_{I}s(t-t_{A})+R_{II}\Big{(}1-{R_{I}}^{2}\Big{)}s(t-t_{A}-t_{B}) \tag{12}\]  
This equation is the function of the incident signal voltage, the reflection coefficient of the medium \(I\), and the reflection coefficient of the interface _II_ in the double-layer medium consisting of medium \(A\) and medium \(B\).
## 4 Monitoring Scheme  
The MEWMA control chart is used for the simultaneous monitoring of several variables [32]. In this research, the means of latent node positions for all nodes and node connection rules are monitored. We consider \(R=2\) dimensions for latent positions and connection rules. Therefore, vector \(L_{t}=(\overline{u}_{1t},\overline{u}_{2t},v_{1t},v_{2t})\) is monitored over time. To obtain MEWMA statistics, vector \(Z_{t}\) is defined as the following relation:  
\[Z_{i}=\Lambda L_{t}+(1-\Lambda)Z_{t-1} \tag{6}\]  
In which, \(\Lambda\) is the vector of smoothing parameters, which is a diagonal matrix of \(\lambda_{1},\ \lambda_{2},\ldots,\lambda_{p}\), where \(p\) is the number of variables. If all the variables are equally important, then \(\lambda_{1}=\lambda_{2}=\ldots=\lambda_{p}\)[33]. The statistic to be calculated for the MEWMA control chart is given as,  
\[T_{t}^{2}=Z_{t}Q^{-1}Z_{t}{}^{\prime} \tag{7}\]  
where, \(Q\) is the variance-covariance matrix of the variables. \(T^{2}\) statistic is monitored using a control chart with an upper control limit defined by \(\frac{(n-1)p}{(n-p)}F_{(\alpha,p,n-p)}^{-1}\) where \(F_{(\alpha,p,n-p)}^{-1}\) denotes the upper \(100(1-\alpha)\) percentile of the \(F\) distribution with \(p\) and \(n\)-\(p\) degrees of freedom, respectively [32].  
In order to design the MEWMA control chart in phase I, a number of in-control networks are collected over the course of time. The parameters of these networks are estimated with the MCMC algorithm and accordingly the \(T^{2}\) statistics of the networks are calculated. Then, based on the \(T^{2}\) statistics, an upper control limit (UCL) is found so that a type I error \((\alpha)\) is met. For phase II, new snapshots of networks are collected and their \(T^{2}\) statistics are compared to the UCL. An out of control signal is triggered if the \(T^{2}\) statistic of any network exceeds the UCL.
# Improving Short Term Clock Prediction for BDS-2 Real-Time Precise Point Positioning
###### Abstract  
Although there are already several real-time precise positioning service providers, unfortunately, not all users can use the correction information due to either cost of the service and limitation of their equipment or out of the service coverage. An alternative way is to enhance the accuracy of the predicted satellite clocks for precise real-time positioning. Based on the study of existing prediction models, an improved model combing the spectrum analysis (SA) and the generalized regression neural network (GRNN) model is proposed especially for BeiDou satellite navigation system (BDS)-2 satellites. The periodic terms and GRNN-related parameters including length and interval of sample data, as well as a smooth factor, are optimized satellite by satellite to consider satellite-specific characteristics for all the fourteen BDS-2 satellites. The improved model is validated by comparing the predicted clocks of existing models and the improved model with precisely estimated ones. The bias of the predicted clock is within \(\pm\)0.5 ns over three hours and better than that of the other models and can be used for several real-time precise applications. The clock prediction is further evaluated by applying clock corrections to precise point positioning (PPP) in both static and kinematic mode for eight IGS (International GNSS Service) MGEX (Multi-GNSS Experiment) stations in the Asia-Pacific region. In the static PPP, the improved model is validated to be effective, and position accuracies of some IGS MGEX stations achieve more than 30.0% improvements on average for each component, which enables us to obtain sub-decimeter positioning. In the kinematic PPP, the improved model performs much better than the others in terms of both the convergence time and the position accuracy. The convergence time can be shortened from 1-2 h to 0.5-1 h, while the position accuracy is enhanced by 15.4%, 21.6% and 19.3% on average in east, north and up component, respectively.  
BeiDou satellite navigation system (BDS); satellite clock prediction; generalized regression neural network; precise point positioning +
Footnote †: journal: Sensors
## 1 Introduction
* Q2: Is the impact of the existence of a bottleneck on the efficacy of PCM different depending on the level of variability, processing time, as well as the level of work-in-process (defined by the number of PCM control cards in the process)?
# Estimating Position from Millisecond Samples of GPS Signals (the \"FastFix\" Algorithm)
## 3 The FastFix Positioning Algorithm  
The FastFix algorithm uses both the Doppler-shift and the codephase measurements to solve the inverse problem of estimating the position, \(\mathbf{r}_{0}\), and time-of-fix, \(t_{0}\) from stored GPS \(L_{1}\)-band data. The forward problem assumes that the position \(\mathbf{r}\), local time \(t_{0}\) and local clock offset \(t_{u}\) are known, and satellite ephemerides are available. This is sufficient to determine from Equation (5) the relative codephase, \(\delta_{\phi}^{ij}\), and the Doppler-shift of the signal from the \(i\)th satellite,  
\[\Delta f^{i}(t_{0},\mathbf{r})=-\frac{1}{c}\frac{d\Delta_{r}^{i}}{dt}f_{0}, \tag{6}\]  
where \(f_{0}\sim 1575.42\) MHz, \(c\) is the speed of light, and \(\Delta_{r}^{i}\) is the relative distance between the \(i\)th satellite and the receiver position \(\mathbf{r}\). If \(\Delta_{r}^{i}\) is decreasing, then satellite is moving towards the receiver, and the Doppler-shift is positive.  
This method is rather like the method of exact fractions from interferometry (see, for example, Baird [23]), the measurements comprise only the fractional part of the codephase for the signal from each satellite and its Doppler shift. At any given moment, there are only a small number of positions near the earth's surface that will have any specific set of relative-codephases.