Scaled TDOA 3D positioning

Abstract

Systems and methods are disclosed herein that relate to positioning in a cellular communications system. In some embodiments, a method for determining a three dimensional location of a wireless device in a cellular communications network comprises obtaining a plurality of measurements for a wireless device, where the plurality of measurements are Time Difference of Arrival (TDOA) related measurements. The method further comprises computing a three dimensional position of the wireless device using the plurality of measurements and a vertical surface model, wherein the vertical surface model is a translated and scaled version of an initial vertical surface model. The new vertical surface model provides translation and scaling of the initial vertical surface model to a suitable range before it is used to determine the three dimensional position of the wireless device such that accuracy is improved, e.g., in large rural cells.

Claims

1. A method for determining a three dimensional location of a wireless device in a cellular communications network, the method comprising: obtaining a plurality of measurements for a wireless device, the plurality of measurements being Time Difference of Arrival, TDOA, related measurements; and computing a three dimensional position of the wireless device using the plurality of measurements and a vertical surface model, wherein the vertical surface model is a translated and scaled version of an initial vertical surface model.

2. The method of claim 1 wherein both the vertical surface model and the initial vertical surface model are models of, for each cell j of one or more cells in the cellular communications network, altitude, z, at points within an interior of a polygon that represents the cell j.

3. The method of claim 2 wherein both the vertical surface model and the initial vertical surface model are based on polygon corners of the polygon in the horizontal plane augmented with vertical information for each corner of the polygon.

4. The method of claim 3 wherein the vertical surface model is further based on a translation vector and a scaling matrix.

5. The method of claim 4 wherein the vertical surface model is further based on virtual points on three dimensional line segments between the augmented corners of the polygon.

6. The method of claim 2 wherein the vertical surface model is defined as:
{tilde over (z)}=f.sup.T({tilde over (x)},{tilde over (y)}){tilde over ({circumflex over (d)})}.sub.j,({tilde over (x)}{tilde over (y)}0).sup.T∈j where: {tilde over (x)}, {tilde over (y)}, and {tilde over (z)} are a Cartesian x-coordinate, a Cartesian y-coordinate, and an altitude value, respectively, in the translated and scaled version of the initial surface model; f.sup.T({tilde over (x)}, {tilde over (y)}) is defined as:
f({tilde over (x)},{tilde over (y)})=(1 . . . {tilde over (y)}.sup.Q(j){tilde over (x)}{tilde over (x)}{tilde over (y)} . . . {tilde over (x)}{tilde over (y)}.sup.Q(j) . . . {tilde over (x)}.sup.P(j){tilde over (y)}.sup.Q(j)).sup.T where P(j) and Q(j) are each a bi-nominal degree; {tilde over ({circumflex over (d)})}.sub.j is defined as: $$\begin{gathered} {\hat{\tilde{d}}}_j={\tilde{R}}_j^{-1}{\tilde{r}}_j, \\ {\tilde{R}}_j=\underset{i=1}{\overset{M\left(j\right)}{.Math.}}f\left({\tilde{x}}_{i,j},{\tilde{y}}_{i,j}\right)f^T\left({\tilde{x}}_{i,j},{\tilde{y}}_{i,j}\right), \\ {\tilde{r}}_j=\underset{i=1}{\overset{M\left(j\right)}{.Math.}}{\tilde{z}}_{i,j}f\left({\tilde{x}}_{i,j},{\tilde{y}}_{i,j}\right) \end{gathered}$$ where {tilde over (x)}.sub.i,j, {tilde over (y)}.sub.i,j, and {tilde over (z)}.sub.i,j are altitude value, respectively, for each i-th scaled polygon corner of the polygon that represents the cell j in the vertical surface model, where: $\left(\begin{array}{c}{\tilde{x}}_{i,j}\\ {\tilde{y}}_{i,j}\\ {\tilde{z}}_{i,j}\end{array}\right)=\left(\begin{array}{ccc}{\sigma }_{x,j}^{-1}& 0& 0\\ 0& {\sigma }_{y,j}^{-1}& 0\\ 0& 0& {\sigma }_{z,j}^{-1}\end{array}\right)\left(\begin{array}{c}{x}_{i,j}-.Math.x_j.Math.\\ y_{i,j}-.Math.y_j.Math.\\ z_{i,j}-.Math.z_j.Math.\end{array}\right)$ and: {tilde over (x)}.sub.i,j is a Cartesian x-coordinate of the i-th scaled and translated polygon corner of the polygon that represents the cell j in the vertical surface model; {tilde over (y)}.sub.i,j is a Cartesian y-coordinate of the i-th scaled and translated polygon corner of the polygon that represents the cell j in the vertical surface model; {tilde over (z)}.sub.i,j is an altitude value of the i-th scaled and translated polygon corner of the polygon that represents the cell j in the vertical surface model; x.sub.i,j is a Cartesian x-coordinate of the i-th polygon corner of the polygon that represents the cell j in the initial vertical surface model; y.sub.i,j is a Cartesian y-coordinate of the i-th polygon corner of the polygon that represents the cell j in the initial vertical surface model; z.sub.i,j is an altitude value of the i-th polygon corner of the polygon that represents the cell j in the initial vertical surface model; x.sub.j is a translation value applied to x.sub.i,j; y.sub.j is a translation value applied to y.sub.i,j; z.sub.j is a translation value applied to z.sub.i,j; σ.sub.x,j.sup.−1 is a scaling factor applied to x.sub.i,j; σ.sub.y,j.sup.−1 is a scaling factor applied to y.sub.i,j; and σ.sub.z,j.sup.−1 is a scaling factor applied to z.sub.i,j.

7. The method of claim 6 wherein the translation values and scaling factors are defined as: $$\begin{gathered} \left(\begin{array}{c}.Math.x_j.Math.\\ .Math.y_j.Math.\\ .Math.z_j.Math.\end{array}\right)=\frac{1}{M\left(j\right)}\underset{i=1}{\overset{M\left(j\right)}{.Math.}}\left(\begin{array}{c}{x}_{i,j}\\ y_{i,j}\\ z_{i,j}\end{array}\right), \\ {\sigma }_{x,j}=c_x\underset{i_1,i_2}{\max }.Math.x_{i_1,j}-x_{i_2,j}.Math.,{\sigma }_{y,j}=c_y{\max }_{i_1,i_2}.Math.y_{i_1,j}-y_{i_2,j}.Math.,\mathrm{and}{\sigma }_{z,j}=c_z{\max }_{i_1,i_2}.Math.z_{i_1,j}-z_{i_2,j}.Math., \end{gathered}$$ where M(j) is a number of polygon corners for the cell j and c.sub.x, c.sub.y, c.sub.z are scaling constants.

8. The method of claim 1 wherein the method is performed by a positioning node.

9. A positioning node in a cellular communications network, the positioning node comprising: a network interface; and processing circuitry configured to cause the positioning node to: obtain a plurality of measurements for a wireless device, the plurality of measurements being Time Difference of Arrival, TDOA, related measurements; and compute a three dimensional position of the wireless device using the plurality of measurements and a vertical surface model, wherein the vertical surface model is a translated and scaled version of an initial vertical surface model.

10. A method for determining a translated and scaled vertical surface model for cells in a cellular communications network, comprising: obtaining an initial vertical surface model for cells in a cellular communications network; and translating and scaling the initial vertical surface model to provide a new vertical surface model, wherein both the new vertical surface model and the initial vertical surface model are models of, for each cell j, altitude, z, at points within an interior of a polygon that represents the cell j, wherein the new vertical surface model is defined as:
{tilde over (z)}=f.sup.T({tilde over (x)},{tilde over (y)}){tilde over ({circumflex over (d)})}.sub.j,({tilde over (x)}{tilde over (y)}0).sup.T∈j where: {tilde over (x)}, {tilde over (y)}, and {tilde over (z)} are a Cartesian x-coordinate, a Cartesian y-coordinate, and an altitude value, respectively, in the new vertical surface model; f.sup.T({tilde over (x)},{tilde over (y)}) is defined as:
f({tilde over (x)},{tilde over (y)})=(1 . . . {tilde over (y)}.sup.Q(j){tilde over (x)}{tilde over (x)}{tilde over (y)} . . . {tilde over (x)}{tilde over (y)}.sup.Q(j) . . . {tilde over (x)}.sup.P(j){tilde over (y)}.sup.Q(j)).sup.T where P(j) and Q(j) are each a bi-nominal degree; and {tilde over ({circumflex over (d)})}.sub.j is defined as: $$\begin{gathered} {\hat{\tilde{d}}}_j={\tilde{R}}_j^{-1}{\tilde{r}}_j, \\ {\tilde{R}}_j=\underset{i=1}{\overset{M\left(j\right)}{.Math.}}f\left({\tilde{x}}_{i,j},{\tilde{y}}_{i,j}\right)f^T\left({\tilde{x}}_{i,j},{\tilde{y}}_{i,j}\right), \\ {\tilde{r}}_j=\underset{i=1}{\overset{M\left(j\right)}{.Math.}}{\tilde{z}}_{i,j}f\left({\tilde{x}}_{i,j},{\tilde{y}}_{i,j}\right) \end{gathered}$$ where x.sub.i,j, {tilde over (y)}.sub.i,j, and {tilde over (z)}.sub.i,j are a Cartesian x-coordinate, a Cartesian y-coordinate, and an altitude value, respectively, for each i-th scaled polygon corner of the polygon that represents the cell j in the new vertical surface model, where: $\left(\begin{array}{c}{\tilde{x}}_{i,j}\\ {\tilde{y}}_{i,j}\\ {\tilde{z}}_{i,j}\end{array}\right)=\left(\begin{array}{ccc}{\sigma }_{x,j}^{-1}& 0& 0\\ 0& {\sigma }_{y,j}^{-1}& 0\\ 0& 0& {\sigma }_{z,j}^{-1}\end{array}\right)\left(\begin{array}{c}{x}_{i,j}-.Math.x_j.Math.\\ y_{i,j}-.Math.y_j.Math.\\ z_{i,j}-.Math.z_j.Math.\end{array}\right)$ and: {tilde over (x)}.sub.i,j is a Cartesian x-coordinate of the i-th scaled and translated polygon corner of the polygon that represents the cell j in the new vertical surface model; {tilde over (y)}.sub.i,j is a Cartesian y-coordinate of the i-th scaled and translated polygon corner of the polygon that represents the cell j in the new vertical surface model; {tilde over (z)}.sub.i,j is an altitude value of the i-th scaled and translated polygon corner of the polygon that represents the cell j in the new vertical surface model; x.sub.i,j is a Cartesian x-coordinate of the i-th polygon corner of the polygon that represents the cell j in the initial vertical surface model; y.sub.i,j is a Cartesian y-coordinate of the i-th polygon corner of the polygon that represents the cell j in the initial vertical surface model; z.sub.i,j is an altitude value of the i-th polygon corner of the polygon that represents the cell j in the initial vertical surface model; x.sub.j is a translation value applied to x.sub.i,j; y.sub.j is a translation value applied to y.sub.i,j; z.sub.j is a translation value applied to z.sub.i,j; σ.sub.x,j.sup.−1 is a scaling factor applied to x.sub.i,j; σ.sub.y,j.sup.−1 is a scaling factor applied to y.sub.i,j; and σ.sub.z,j.sup.−1 is a scaling factor applied to z.sub.i,j.

11. The method of claim 10 wherein the translation values and scaling factors are defined as: $$\begin{gathered} \left(\begin{array}{c}.Math.x_j.Math.\\ .Math.y_j.Math.\\ .Math.z_j.Math.\end{array}\right)=\frac{1}{M\left(j\right)}\underset{i=1}{\overset{M\left(j\right)}{.Math.}}\left(\begin{array}{c}{x}_{i,j}\\ y_{i,j}\\ z_{i,j}\end{array}\right), \\ {\sigma }_{x,j}=c_x\underset{i_1,i_2}{\max }.Math.x_{i_1,j}-x_{i_2,j}.Math.,{\sigma }_{y,j}=c_y{\max }_{i_1,i_2}.Math.y_{i_1,j}-y_{i_2,j}.Math.,\mathrm{and}{\sigma }_{z,j}=c_z{\max }_{i_1,i_2}.Math.z_{i_1,j}-z_{i_2,j}.Math., \end{gathered}$$ where M(j) is a number of polygon corners for the cell j and c.sub.x, c.sub.y, c.sub.z are scaling constants.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The accompanying drawing figures incorporated in and forming a part of this specification illustrate several aspects of the disclosure, and together with the description serve to explain the principles of the disclosure.

(2) FIG. 1 illustrates the Long Term Evolution (LTE) positioning architecture for the control plane;

(3) FIG. 2 illustrates the LTE user plane positioning architecture;

(4) FIG. 3 illustrates an example part of the time-frequency grid of the LTE downlink that depicts the locations of Positioning Reference Signals (PRSs);

(5) FIG. 4 illustrates an example cell polygon format that gives a geographical extension of a cell;

(6) FIG. 5 illustrates an example data structure for representing a cell polygon;

(7) FIG. 6 illustrates Observed Time Difference of Arrival (OTDOA);

(8) FIG. 7 illustrates one example of a cellular communications network in which embodiments of the present disclosure may be implemented;

(9) FIGS. 8 through 11 provide an illustration of the performance of the new scaled and translated surface model as disclosed herein;

(10) FIG. 12 illustrates a method in accordance with at least some of the embodiments described herein;

(11) FIG. 13 is an illustration of the performance of one example implementation of the process of FIG. 12;

(12) FIG. 14 illustrates new architecture aspects of the present disclosure;

(13) FIG. 15 illustrates a method in accordance with at least some of the embodiments described herein;

(14) FIGS. 16 through 18 illustrate example embodiments of a radio access node; and

(15) FIGS. 19 and 20 illustrate example embodiments of a User Equipment device (UE).

DETAILED DESCRIPTION

(16) The embodiments set forth below represent information to enable those skilled in the art to practice the embodiments and illustrate the best mode of practicing the embodiments. Upon reading the following description in light of the accompanying drawing figures, those skilled in the art will understand the concepts of the disclosure and will recognize applications of these concepts not particularly addressed herein. It should be understood that these concepts and applications fall within the scope of the disclosure.

(17) Generally, all terms used herein are to be interpreted according to their ordinary meaning in the relevant technical field, unless a different meaning is clearly given and/or is implied from the context in which it is used. All references to a/an/the element, apparatus, component, means, step, etc. are to be interpreted openly as referring to at least one instance of the element, apparatus, component, means, step, etc., unless explicitly stated otherwise. The steps of any methods disclosed herein do not have to be performed in the exact order disclosed, unless a step is explicitly described as following or preceding another step and/or where it is implicit that a step must follow or precede another step. Any feature of any of the embodiments disclosed herein may be applied to any other embodiment, wherever appropriate. Likewise, any advantage of any of the embodiments may apply to any other embodiments, and vice versa. Other objectives, features, and advantages of the enclosed embodiments will be apparent from the following description.

(18) Some of the embodiments contemplated herein will now be described more fully with reference to the accompanying drawings. Other embodiments, however, are contained within the scope of the subject matter disclosed herein, the disclosed subject matter should not be construed as limited to only the embodiments set forth herein; rather, these embodiments are provided by way of example to convey the scope of the subject matter to those skilled in the art.

(19) Radio Node: As used herein, a “radio node” is either a radio access node or a wireless device.

(20) Radio Access Node: As used herein, a “radio access node” or “radio network node” is any node in a Radio Access Network (RAN) of a cellular communications network. Some examples of a radio access node include, but are not limited to, a base station (e.g., a New Radio (NR) base station (gNB) in a Third Generation Partnership Project (3GPP) Fifth Generation (5G) NR network or an enhanced or evolved Node B (eNB) in a 3GPP Long Term Evolution (LTE) network), a high-power or macro base station, a low-power base station (e.g., a micro base station, a pico base station, a home eNB, or the like), and a relay node.

(21) Core Network Node: As used herein, a “core network node” is any type of node in a core network. Some examples of a core network node include, e.g., a Mobility Management Entity (MME), a Packet Data Network Gateway (P-GW), a Service Capability Exposure Function (SCEF), or the like.

(22) Wireless Device: As used herein, a “wireless device” is any type of device that has access to (i.e., is served by) a cellular communications network by wirelessly transmitting and/or receiving signals to a radio access node(s). Some examples of a wireless device include, but are not limited to, a User Equipment device (UE) in a 3GPP network and a Machine Type Communication (MTC) device.

(23) Network Node: As used herein, a “network node” is any node that is either part of the RAN or the core network of a cellular communications network/system.

(24) Note that the description given herein focuses on a 3GPP cellular communications system and, as such, 3GPP terminology or terminology similar to 3GPP terminology is oftentimes used. However, the concepts disclosed herein are not limited to a 3GPP system.

(25) Note that, in the description herein, reference may be made to the term “cell;” however, particularly with respect to 5G NR concepts, beams may be used instead of cells and, as such, it is important to note that the concepts described herein are equally applicable to both cells and beams.

(26) There currently exist certain challenge(s) with respect to positioning in a cellular communications system. In particular, there is no known Time Difference of Arrival (TDOA) positioning method that can provide three dimensional (3D) accuracies meeting E-911 requirements in large rural cells covering hilly terrain, where the sites are coarsely distributed. This also impacts the lawful intercept possibilities since Assisted Global Navigation Satellite System (A-GNSS) can be turned off in mobiles, which is well known to criminals. The hybrid TDOA method proposed in the present disclosure discloses a positioning method with the potential to solve this problem.

(27) In addition, the hybrid positioning method using Geographical Information System (GIS) information of U.S. Pat. No. 7,203,499 does not account for scaling or translation of the lateral variables, the reason for this being listed below. Nor does any other such known method. The method of U.S. Pat. No. 7,203,499 can therefore not be used in a straightforward manner, together with the disclosed surface modeling technique of the present disclosure.

(28) In addition, the surface modeling solution of U.S. Pat. No. 7,676,232 is based on a bi-polynomial model of the surface, in the lateral x and y coordinates. The coefficients of this model determine the vertical model inside of a polygon. However, the solution of the Least Squares (LS) problem that determine the parameters become ill-conditioned, close to singular and produces very poor solutions whenever the variation of x, y, and z is far from 0. In addition, if the mean value of the polygon corner is large, this problem is amplified, rendering severe accuracy problems in practical systems, where said variation is far from zero. This problem is not mentioned or solved in prior solutions.

(29) Still further, the surface modeling solution of U.S. Pat. No. 7,676,232 is obtaining the altitude information of the corners of the polygon from maps, i.e. a GIS system. In case no GIS system is available, the prior art surface modeling method can therefore not be used.

(30) As shown by the present disclosure, a solution to the problem is to translate and scale the position variables to a suitable range before the problem is solved. To further improve, additional points on the boundary of the polygon may be introduced. In order to signal all needed information to nodes that, e.g., compute UE positions in the RAN, core network, and the end user, the information about the scaling, translation, and additional points needs to be signaled as well. However, no standardized reporting format or proposed augmented reporting format support this need.

(31) In 5G, it is likely that the position calculation may be moved to the cloud and distributed within the cloud. However, there is no signaling defined in prior solutions that allow for signaling of the polygon information, augmented with altitude, together with scaling, translation, and additional point parameters.

(32) Certain aspects of the present disclosure and their embodiments may provide solutions to the aforementioned or other challenges. The present disclosure discloses methods that solve the above problems. There are two problems, namely, (i) alack of hybrid TDOA methods with needed functionality and (ii) lack of support for preferred signaling. The first problem is addressed by the present disclosure, the second problem by U.S. Provisional Patent No. 62/633,251 which is the priority document for PCT Application No. [P74279 WO1], entitled SIGNALING OF SCALED 3D POSITION INFORMATION. Note that the solutions to the first and second problems addressed by the present disclosure and the parallel disclosure may be used separately (i.e., each in a standalone manner) or used in combination.

(33) In detail the present disclosure discloses: 1. A new surface modeling method for modeling of the vertical coordinate of the interior of a polygon, the method being based on polygon corners in the horizontal plane augmented with vertical information for each polygon corner, a translation vector and a scaling matrix, and virtual points on the 3D line segment between the augmented corner points. 2. A new TDOA hybrid position calculation method based on the surface model of item 1 above. 3. A new positioning method architecture, where the polygon with corners augmented with vertical information is obtained by a preceding Adaptive Enhanced Cell Identity (AECID) 3D polygon computation step.
Each of these aspects are described in detail below.

(34) FIG. 7 illustrates one example of a cellular communications network 700 in which embodiments of the present disclosure may be implemented. In the embodiments described herein, the cellular communications network 700 is a LTE or 5G NR network, but the present disclosure is not limited thereto. In this example, the cellular communications network 700 includes base stations 702-1 and 702-2, which in LTE are referred to as eNBs and in 5G NR are referred to as gNBs, controlling corresponding macro cells 704-1 and 704-2. The base stations 702-1 and 702-2 are generally referred to herein collectively as base stations 702 and individually as base station 702. Likewise, the macro cells 704-1 and 704-2 are generally referred to herein collectively as macro cells 704 and individually as macro cell 704. The cellular communications network 700 may also include a number of low power nodes 706-1 through 706-4 controlling corresponding small cells 708-1 through 708-4. The low power nodes 706-1 through 706-4 can be small base stations (such as pico or femto base stations) or Remote Radio Heads (RRHs), or the like. Notably, while not illustrated, one or more of the small cells 708-1 through 708-4 may alternatively be provided by the base stations 702. The low power nodes 706-1 through 706-4 are generally referred to herein collectively as low power nodes 706 and individually as low power node 706. Likewise, the small cells 708-1 through 708-4 are generally referred to herein collectively as small cells 708 and individually as small cell 708. The base stations 702 (and optionally the low power nodes 706) are connected to a core network 710.

(35) The base stations 702 and the low power nodes 706 provide service to wireless devices 712-1 through 712-5 in the corresponding cells 704 and 708. The wireless devices 712-1 through 712-5 are generally referred to herein collectively as wireless devices 712 and individually as wireless device 712. The wireless devices 712 are also sometimes referred to herein as UEs.

(36) While not illustrated, the cellular communications network 700 includes positioning nodes (e.g., Evolved Serving Mobile Location Center (E-SMLC) in LTE or similar positioning node in 5G) and, in some embodiments, other nodes such as, e.g., nodes hosting a positioning database in the radio access network or core network. Further, in some embodiments, the cellular communications network 700 may be connected to a Public Safety Answering Point (PSAP) or other emergency center, a cloud-based positioning database, where these nodes may or may not be considered part of the cellular communications network 700.

(37) As described below, a system and methods related to positioning of a UE (e.g., a wireless device 712) are disclosed herein. Notably, some aspects of the present disclosure may be performed by or implemented in a network node, which may be referred to herein as a positioning node. While 5G positioning standardization is not close to finalization at the time of writing of the present disclosure, the positioning architecture in 5G may, for example, be similar to that used in 3GPP LTE.

I. New Scaled and Translated Surface Model

(38) A. Description of the New Scaled and Translated Surface Model

(39) In some embodiments, a method of generating a new (scaled and translated) surface model is provided. The description of the method first defines the unscaled surface modeling algorithm, after which translation, scaling, and additional points are introduced. The additional points are also referred to herein as virtual corners of the polygon, or virtual polygon corners.

(40) The following unscaled bi-polynomial vertical surface model is first selected,

(41) $z=\underset{l=0}{\overset{P\left(j\right)}{.Math.}}\underset{m=0}{\overset{Q\left(j\right)}{.Math.}}{d}_{l,m,j}{x}^l{y}^m,\left(xy0\right)\in \mathrm{cell}j$
where z is altitude, P(j) is a bi-nominal degree, Q(j) is another bi-nomial degree, d.sub.l,m,j are the unknowns in the surface model, and x and y are the two dimensional (2) position variables. Note that the model is here assumed to be Cartesian, i.e. transformed from the latitude-longitude system to a local Cartesian coordinate system, e.g. with the x-axis pointing east, the y-axis pointing north, and the z-axis pointing up, and with the origin somewhere on the global ellipsoidal Earth model used to define latitude and longitude. The above model is linear in the unknowns d.sub.l,m,j that determine the surface. These unknowns will be determined from the points that define the corners of the polygon. Towards that end, note that the surface model above can be written as
z=f.sup.T(x,y)d.sub.j,
f(x,y)=(1 . . . y.sup.Q(j)xxy . . . xy.sup.Q(j) . . . x.sup.P(j)y.sup.Q(j)).sup.T,
d.sub.j=(d.sub.0,0,j . . . d.sub.0,Q(j),jd.sub.1,0,j . . . d.sub.1,Q(j),j . . . d.sub.P(j),Q(j),j).sup.T
These equations are in linear regression form and LS modeling is therefore applicable. Assume that the number of points of the polygon fulfils M(j)≥(P(j)+1)(Q(j)+1), where M(j) is the number of corners of the polygon. Then, there are at least as many data points as there are unknowns and the LS problem is normally non-singular. To formulate the LS solution in detail, define the LS modeling error as

(42) 0$\stackrel{\_}{E}\left(d_j\right)=\underset{i=1}{\overset{M\left(j\right)}{.Math.}}{\left(z_{i,j}-f^T\left(x_{i,j},y_{i,j}\right)d_j\right)}^2.$
Differentiation of the error with respect to d and equating the result to 0 results in the following LS estimate

(43) $$\begin{gathered} {\hat{d}}_j=R_j^{-1}{r}_j,R_j=\underset{i=1}{\overset{M\left(j\right)}{.Math.}}f\left(x_{i,j},y_{i,j}\right)f^T\left(x_{i,j},y_{i,j}\right), \\ {r}_j=\underset{i=1}{\overset{M\left(j\right)}{.Math.}}{z}_{i,j}f\left(x_{i,j},y_{i,j}\right) \end{gathered}$$
The altitudes in interior points of the cell are obtained from
z=f.sup.T(x,y){circumflex over (d)}.sub.j,(xy0).sup.T∈cell j.

(44) As stated above, the LS algorithm does work well close to the origin of the Cartesian coordinate system, but it does not work well when the numerical values of x, y, and z become large.

(45) One solution, disclosed here, is to apply a change of coordinates for each cell j, thereby translating the surface modeling problem back to a situation where the numerical properties are good. To do so, a transformation could, e.g., rescale from m to km, or use the maximum distance between two polygon corners in each of the dimensions. As a first step, a subtraction of the 3D mean value of the polygon corners is needed to translate the cell polygon to be centered around zero. The resulting scaling method can then be defined in terms of the scaled corner variables
{tilde over (P)}.sub.i,j.sup.C=({tilde over (x)}.sub.i,j{tilde over (y)}.sub.i,j{tilde over (z)}.sub.i,j).sup.T.
The scaling transformation then becomes

(46) $\left(\begin{array}{c}{\tilde{x}}_{i,j}\\ {\tilde{y}}_{i,j}\\ {\tilde{z}}_{i,j}\end{array}\right)=\left(\begin{array}{ccc}{\sigma }_{x,j}^{-1}& 0& 0\\ 0& {\sigma }_{y,j}^{-1}& 0\\ 0& 0& {\sigma }_{z,j}^{-1}\end{array}\right)\left(\begin{array}{c}{x}_{i,j}-.Math.x_j.Math.\\ y_{i,j}-.Math.y_j.Math.\\ z_{i,j}-.Math.z_j.Math.\end{array}\right),$
where: {tilde over (x)}.sub.i,j is a Cartesian x-coordinate of the i-th scaled and translated polygon corner of the polygon that represents cell j in the vertical surface model; {tilde over (y)}.sub.i,j is a Cartesian y-coordinate of the i-th scaled and translated polygon corner of the polygon that represents cell j in the vertical surface model; {tilde over (z)}.sub.i,j is an altitude value of the i-th scaled and translated polygon corner of the polygon that represents cell j in the vertical surface model; x.sub.i,j is a Cartesian x-coordinate of the i-th polygon corner of the polygon that represents cell j in the initial vertical surface model; y.sub.i,j is a Cartesian y-coordinate of the i-th polygon corner of the polygon that represents cell j in the initial vertical surface model; z.sub.i,j is an altitude value of the i-th polygon corner of the polygon that represents cell j in the initial vertical surface model; x.sub.j is a translation value applied to x.sub.i,j; y.sub.j is a translation value applied to y.sub.i,j; z.sub.j is a translation value applied to z.sub.i,j; σ.sub.x,j.sup.−1 is a scaling factor applied to x.sub.i,j; σ.sub.y,j.sup.−1 is a scaling factor applied to y.sub.i,j; and σ.sub.z,j.sup.−1 is a scaling factor applied to z.sub.i,j.
Note that, in some embodiments, the scaled polygon corners represented by {tilde over (x)}.sub.i,j, {tilde over (y)}.sub.i,j, and {tilde over (z)}.sub.i,j include one or more virtual points on one or more 3D line segments between the polygon corners of the vertical surface model. Further, the translation values and scaling factors are defined as:

(47) $$\begin{gathered} \left(\begin{array}{c}.Math.x_j.Math.\\ .Math.y_j.Math.\\ .Math.z_j.Math.\end{array}\right)=\frac{1}{M\left(j\right)}\underset{i=1}{\overset{M\left(j\right)}{.Math.}}\left(\begin{array}{c}{x}_{i,j}\\ y_{i,j}\\ z_{i,j}\end{array}\right), \\ {\sigma }_{x,j}=c_x\underset{i_1,i_2}{\max }.Math.x_{i_1,j}-x_{i_2,j}.Math.,{\sigma }_{y,j}=c_y\underset{i_1,i_2}{\max }.Math.y_{i_1,j}-y_{i_2,j}.Math.,{\sigma }_{z,j}=c_z\underset{i_1,i_2}{\max }.Math.z_{i_1,j}-z_{i_2,j}.Math.. \end{gathered}$$
Here, M(j) is a number of corners for cell j, c.sub.x, c.sub.y, c.sub.z denote scaling constants that may be used for tuning, max.sub.i.sub.1.sub.,i.sub.2|x.sub.i.sub.1.sub.,j−x.sub.i.sub.2.sub.,j| denotes a search for a maximum distance between any two corner points in the Cartesian x-coordinate plane, max.sub.i.sub.1.sub.,i.sub.2|y.sub.i.sub.1.sub.,j−y.sub.i.sub.2.sub.,j| denotes a search for a maximum distance between any two corner points in the Cartesian y-coordinate plane, and max.sub.i.sub.1.sub.i.sub.2|z.sub.i.sub.1.sub.,j−z.sub.i.sub.2.sub.,j| denotes a search for a maximum distance between any two corner points in the Cartesian z-coordinate plane. The LS problem defined above is then solved for the scaled and translated corner points. This results in the following scaled and translated surface model
{tilde over (z)}=f.sup.T({tilde over (x)},{tilde over (y)}){tilde over ({circumflex over (d)})}.sub.j,({tilde over (x)}{tilde over (y)}0).sup.T∈j
where (as naturally follows from the LS problem defined above): {tilde over (x)}, {tilde over (y)}, and {tilde over (z)} are Cartesian x-coordinate, Cartesian y-coordinate, and altitude value, respectively, in the scaled and translated surface model; f({tilde over (x)}, {tilde over (y)}) is defined as:
f({tilde over (x)},{tilde over (y)})=(1 . . . {tilde over (y)}.sup.Q(j){tilde over (x)}{tilde over (x)}{tilde over (y)} . . . {tilde over (x)}{tilde over (y)}.sup.Q(j) . . . {tilde over (x)}.sup.P(j){tilde over (y)}.sup.Q(j)).sup.T {tilde over ({circumflex over (d)})}.sub.j is defined as:

(48) $$\begin{gathered} {\hat{\tilde{d}}}_j={\tilde{R}}_j^{-1}{\tilde{r}}_j, \\ {\tilde{R}}_j=\underset{i=1}{\overset{M\left(j\right)}{.Math.}}f\left({\tilde{x}}_{i,j},{\tilde{y}}_{i,j}\right)f^T\left({\tilde{x}}_{i,j},{\tilde{y}}_{i,j}\right), \\ {\tilde{r}}_j=\underset{i=1}{\overset{M\left(j\right)}{.Math.}}{\tilde{z}}_{i,j}f\left({\tilde{x}}_{i,j},{\tilde{y}}_{i,j}\right). \end{gathered}$$

(49) In order to use the result in original coordinates, a back transformation results in the following interior surface model of the original augmented polygon

(50) $z=.Math.z_j.Math.+{\sigma }_{z,j}{f}^T\left(\frac{x-.Math.x_j.Math.}{{\sigma }_{x,j}},\frac{y-.Math.y_j.Math.}{{\sigma }_{y,j}}\right){\hat{\tilde{d}}}_j,{\left(xy0\right)}^T\in \mathrm{cell}j.$

(51) This end result is useful to solve practical problems in a cellular network. In case P(j)=Q(j)=3, 16 corners are needed to avoid a singular LS problem, leading to a need to add points on the polygon boundary. Numerical experiments also show that an augmentation of the problem with additional polygon boundary points on the 3D line segments in between polygon corners can provide significantly improved interior surface models. It is therefore recommended to introduce one or several equidistant additional points for every 3D line segment of the augmented polygon. Note that these points are uniquely defined by the information encoded by the augmented polygon format and no additional signaling is therefore required.

(52) It needs to be kept in mind that also other schemes for addition of points on the boundary can be used. This is treated in the parallel disclosure.

(53) B. Performance Illustration

(54) In order to obtain an indication of the accuracy of the proposed method given, a rural 10 cell geometry was set up on the artificial topography of FIG. 8. The topography is intended to model a coastal region, with zero altitude representing the sea level. This represents an additional modeling difficulty, since some of the cells cover both the sea and the hilly terrain. The cells are depicted with overlaid 3D lines. The LS algorithm was then used to compute the 3D surfaces of each cell, applying scaling according to the above description. A simplified approach was used, with horizontal down scaling a factor of 1000 and vertical down scaling a factor of 100. Additional boundary points, centered on the connecting line segments between corners, were used for regularization and the bi-polynomial degrees were selected as P(j)=Q(j)=3, j=1, . . . , 10. The promising result is depicted in FIG. 9. To further visualize the performance, vertical errors were computed on a grid covering the topography of FIG. 8. The vertical errors can be obtained directly by a comparison between the true altitude and that obtained by the surface model. The results are depicted in FIG. 10. The statistical result depicted by the Cumulative Distribution Function (CDF) of FIG. 11 was computed only for points in the interior of each polygon. As can be seen, the proposed modeling method is remarkably accurate and reduces the maximal vertical errors by about 80%. For the treated topography the vertical Minimum Mean Square Error (MMSE) is less than 3 meters in almost 50% of the coverage of the cells. This indicates that the Federal Communications Commission (FCC) E-911 positioning requirements can be met by the new method, also in quite difficult scenarios.

II. New TDOA Algorithm

(55) A. The Algorithm

(56) This section describes how the computed 3D surface information can be used to enhance the accuracy of TDOA positioning methods, of which the Uplink TDOA (U-TDOA) method is considered here. TDOA methods are subject to at least two problems. First, additional base stations exceeding the three required for 2D localization and the four required for 3D localization need to be detected to discriminate outliers, e.g., due to Non-Line-Of-Sight (N-LOS) radio propagation. Secondly, the poor Vertical Dilution Of Precision (VDOP) often magnifies vertical measurement errors. The 3D surface model is used to mitigate both problems and/or enabling 3D localization with only three involved sites by formulation of the following optimization problem:

(57) $\left(\begin{array}{c}{\hat{x}}_{\mathrm{UE}}\\ {\hat{y}}_{\mathrm{UE}}\end{array}\right)=\arg \underset{x_{\mathrm{UE}},y_{\mathrm{UE}}}{\min }V\left(x_{\mathrm{UE}},y_{\mathrm{UE}}\right),V\left(x_{\mathrm{UE}},y_{\mathrm{UE}}\right)=\underset{k=1,k\ne r}{\overset{n}{.Math.}}({\mathrm{OTD}}_{k,r}-{\mathrm{RTD}}_{k,r}-\frac{1}{c}\left({\left(x_k-x_{\mathrm{UE}}\right)}^2+{\left(y_k-y_{\mathrm{UE}}\right)}^2+{{\left(z_k-.Math.z_j.Math.-{\sigma }_{z,j}{f}^T\left(\frac{\left(x_{\mathrm{UE}}-.Math.x_j.Math.\right)}{{\sigma }_{x,j}},\frac{\left(y_{\mathrm{UE}}-.Math.y_j.Math.\right)}{{\sigma }_{y,j}}\right){\hat{\tilde{d}}}_j\right)}^2)}^{1/2}\right)$
Here, OTD.sub.k,r, k=1, . . . , n, denotes the observed time differences in the mobile device for base station k=1, . . . , n; RTD.sub.k,r, k=1, . . . , n, denotes the real time (transmission) differences from each base station; r.sub.k=(x.sub.k y.sub.k z.sub.k).sup.T, k=1, . . . , n denotes the position vectors of the base stations; r.sub.UE=(x.sub.UE y.sub.UE).sup.T denotes the unknown position vector of the mobile device; c denotes the speed of light; and the subscript r denotes the selected reference base station. The above equation thus represents an original criterion for solution of a 3D OTDOA problem, but where the 3D interior surface model is inserted in the criterion to reduce it to an altitude compensated 2D OTDOA problem. With the correct 2D position of the mobile device computed, the altitude follows which results in a complete hybrid 3D OTDOA positioning method. Finally, note that the real time differences are continuously estimated by measurements in Location Measurement Units (LMUs) at surveyed locations as discussed above.

(58) FIG. 12 illustrates a method in accordance with at least some of the embodiments described herein. This method may be performed by a positioning node (e.g., E-SMLC in LTE or corresponding positioning node in 5G). As illustrated, measurements are obtained (step 1200). These measurements include observed time differences (OTD.sub.k,r, k=1, . . . , n) in a wireless device (e.g., a mobile device) for a base station k=1, . . . , n and real time (transmission) differences (RTD.sub.k,r, k=1, . . . , n) for each base station k=1, . . . , n.

(59) Next, a 3D position of the wireless device is computed using the new TDOA algorithm, as described above (step 1202). In other words, the 3D position of the wireless device is computed using a TDOA scheme that uses the new (scaled and translated) vertical surface model described herein. As discussed above, the new vertical surface model can be represented as:

(60) $\tilde{z}=f^T\left(\tilde{x},\tilde{y}\right){\hat{\tilde{d}}}_j,{\left(\tilde{x}\tilde{y}0\right)}^T\in jz=.Math.z_j.Math.+{\sigma }_{z,j}{f}^T\left(\frac{x-.Math.x_j.Math.}{{\sigma }_{x,j}},\frac{y-.Math.y_j.Math.}{{\sigma }_{y,j}}\right){\hat{\tilde{d}}}_j,{\left(xy0\right)}^T\in \mathrm{cell}j.$
Optionally, the 3D position of the wireless device is provided to another node (e.g., step 1204).

(61) B. Performance Illustration

(62) Vertical errors lead to additional horizontal errors by the coupling introduced by the TDOA equations. To address these errors, the performance of the proposed new scaled TDOA positioning method was evaluated over the same typography as used for the illustration of the surface modeling performance. To assess the performance bound related to the surface modeling, no detections were discarded and measurement error free Line-Of-Sight (LOS) radio propagation was assumed in all cases. The results are plotted in terms of the CDF of FIG. 13, which used a 100 megahertz (MHz) bandwidth and a time resolution of 4e-8 to account for the difference formation of measured TOAs in the UE and of the real time differences.

III. New Architecture Aspects

(63) The last characterizing feature of the present disclosure is related to the generation of polygons, where the corners are augmented with altitude information. In prior solutions, GIS systems, i.e. maps, have been assumed to provide this information. The present sections present an architecture that avoids the need for a GIS system by application of the original version of the AECID fingerprinting algorithm. The architecture is depicted in FIG. 14.

(64) The first step is to produce 3GPP polygons with the AECID method. The extension of that method listed above is then used to augment altitude information to each corner of the horizontal polygons, thereby producing the input needed for the surface modeling method of the present disclosure. The augmented polygons are stored and continuously updated in the AECID database. The surface modelling algorithm of the present disclosure may then be run continuously creating surface models that can be stored in a database associated with the AECID database. During the optimization of the scaled TDOA method, the needed surface model parameters including scaling parameters are fetched from the surface model data base and used in the running computation of the UE position estimate.

(65) Note that all this functionality can be performed by the positioning node. In case it shall be distributed, standardization of the signaling over the relevant 3GPP interfaces needs to be in place.

IV. Example Process

(66) FIG. 15 illustrates a method in accordance with at least some of the embodiments described herein. This method may be performed by a positioning node or distributed across two or more network nodes. As illustrated, an initial surface model (e.g., a conventional surface model such as the unscaled surface model described above) is obtained (step 1500). Next, the initial surface model is translated and scaled to provide a new surface model (e.g., the new surface model described above) (step 1502). For example, a change of coordinates for each cell is applied to thereby translate the initial surface model to a situation where the numerical properties are good (e.g., x, y, and z are small), as described above. As described above, in some embodiments, the cells are defined by polygons, and this translation is performed by a subtraction of the 3D mean value of the polygon corners. As described above, this results in the new 3D model, and can be represented as

(67) $\tilde{z}=f^T\left(\tilde{x},\tilde{y}\right){\hat{\tilde{d}}}_i,{\left(\tilde{x}\tilde{y}0\right)}^T\in jz=.Math.z_j.Math.+{\sigma }_{z,j}{f}^T\left(\frac{x-.Math.x_j.Math.}{{\sigma }_{x,j}},\frac{y-.Math.y_j.Math.}{{\sigma }_{y,j}}\right){\hat{\tilde{d}}}_i,{\left(xy0\right)}^T\in \mathrm{cell}j.$

(68) Optionally, the new surface model is used, e.g., to compute a position of a wireless device using the new TDOA scheme described herein and/or information related to the new surface model is provided to another node (e.g., another network node or the wireless device) (step 1504).

V. Additional Implementation Aspects

(69) FIG. 16 is a schematic block diagram of a radio access node 1600 according to some embodiments of the present disclosure. The radio access node 1600 may be, for example, a base station 702 or 706. As illustrated, the radio access node 1600 includes a control system 1602 that includes one or more processors 1604 (e.g., Central Processing Units (CPUs), Application Specific Integrated Circuits (ASICs), Field Programmable Gate Arrays (FPGAs), and/or the like), memory 1606, and a network interface 1608. In addition, the radio access node 1600 includes one or more radio units 1610 that each includes one or more transmitters 1612 and one or more receivers 1614 coupled to one or more antennas 1616. In some embodiments, the radio unit(s) 1610 is external to the control system 1602 and connected to the control system 1602 via, e.g., a wired connection (e.g., an optical cable). However, in some other embodiments, the radio unit(s) 1610 and potentially the antenna(s) 1616 are integrated together with the control system 1602. The one or more processors 1604 operate to provide one or more functions of a radio access node 1600 as described herein. In some embodiments, the function(s) are implemented in software that is stored, e.g., in the memory 1606 and executed by the one or more processors 1604.

(70) FIG. 17 is a schematic block diagram that illustrates a virtualized embodiment of the radio access node 1600 according to some embodiments of the present disclosure. This discussion is equally applicable to other types of network nodes. Further, other types of network nodes may have similar virtualized architectures.

(71) As used herein, a “virtualized” radio access node is an implementation of the radio access node 1600 in which at least a portion of the functionality of the radio access node 1600 is implemented as a virtual component(s) (e.g., via a virtual machine(s) executing on a physical processing node(s) in a network(s)). As illustrated, in this example, the radio access node 1600 includes the control system 1602 that includes the one or more processors 1604 (e.g., CPUs, ASICs, FPGAs, and/or the like), the memory 1606, and the network interface 1608 and the one or more radio units 1610 that each includes the one or more transmitters 1612 and the one or more receivers 1614 coupled to the one or more antennas 1616, as described above. The control system 1602 is connected to the radio unit(s) 1610 via, for example, an optical cable or the like. The control system 1602 is connected to one or more processing nodes 1700 coupled to or included as part of a network(s) 1702 via the network interface 1608. Each processing node 1700 includes one or more processors 1704 (e.g., CPUs, ASICs, FPGAs, and/or the like), memory 1706, and a network interface 1708.

(72) In this example, functions 1710 of the radio access node 1600 described herein are implemented at the one or more processing nodes 1700 or distributed across the control system 1602 and the one or more processing nodes 1700 in any desired manner. In some particular embodiments, some or all of the functions 1710 of the radio access node 1600 described herein are implemented as virtual components executed by one or more virtual machines implemented in a virtual environment(s) hosted by the processing node(s) 1700. As will be appreciated by one of ordinary skill in the art, additional signaling or communication between the processing node(s) 1700 and the control system 1602 is used in order to carry out at least some of the desired functions 1710. Notably, in some embodiments, the control system 1602 may not be included, in which case the radio unit(s) 1610 communicate directly with the processing node(s) 1700 via an appropriate network interface(s).

(73) In some embodiments, a computer program including instructions which, when executed by at least one processor, causes the at least one processor to carry out the functionality of radio access node 1600 or a node (e.g., a processing node 1700) implementing one or more of the functions 1710 of the radio access node 1600 in a virtual environment according to any of the embodiments described herein is provided. In some embodiments, a carrier comprising the aforementioned computer program product is provided. The carrier is one of an electronic signal, an optical signal, a radio signal, or a computer readable storage medium (e.g., a non-transitory computer readable medium such as memory).

(74) FIG. 18 is a schematic block diagram of the radio access node 1600 according to some other embodiments of the present disclosure. The radio access node 1600 includes one or more modules 1800, each of which is implemented in software. The module(s) 1800 provide the functionality of the radio access node 1600 described herein. This discussion is equally applicable to the processing node 1700 of FIG. 17 where the modules 1800 may be implemented at one of the processing nodes 1700 or distributed across multiple processing nodes 1700 and/or distributed across the processing node(s) 1700 and the control system 1602.

(75) FIG. 19 is a schematic block diagram of a UE 1900 according to some embodiments of the present disclosure. As illustrated, the UE 1900 includes one or more processors 1902 (e.g., CPUs, ASICs, FPGAs, and/or the like), memory 1904, and one or more transceivers 1906 each including one or more transmitters 1908 and one or more receivers 1910 coupled to one or more antennas 1912. In some embodiments, the functionality of the UE 1900 described above may be fully or partially implemented in software that is, e.g., stored in the memory 1904 and executed by the processor(s) 1902.

(76) In some embodiments, a computer program including instructions which, when executed by at least one processor, causes the at least one processor to carry out the functionality of the UE 1900 according to any of the embodiments described herein is provided. In some embodiments, a carrier comprising the aforementioned computer program product is provided. The carrier is one of an electronic signal, an optical signal, a radio signal, or a computer readable storage medium (e.g., a non-transitory computer readable medium such as memory).

(77) FIG. 20 is a schematic block diagram of the UE 1900 according to some other embodiments of the present disclosure. The UE 1900 includes one or more modules 2000, each of which is implemented in software. The module(s) 2000 provide the functionality of the UE 1900 described herein.

(78) While FIGS. 16 through 20 focus on the radio access node 1600 and the UE 1900, other types of network nodes (e.g., positioning nodes, node implementing a positioning database, etc.) may have a similar architecture. For instance, in general, a network node includes processing circuitry (e.g., similar to the processor(s) of the radio access node 1600 and the UE 1900) configured to cause the network node to perform the functionality of that network node as described herein, e.g., via execution of respective software instructions that are stored in memory. In addition, the network node includes one or more communication interfaces (e.g., a network interface(s) and/or a radio interface(s)). For example, a positioning node (e.g., such as the one that performs the method of FIG. 12, FIG. 14, or FIG. 15) includes processing circuitry that causes the positioning node to perform the disclosed method. The same is true of the other network nodes described herein.

(79) Any appropriate steps, methods, features, functions, or benefits disclosed herein may be performed through one or more functional units or modules of one or more virtual apparatuses. Each virtual apparatus may comprise a number of these functional units. These functional units may be implemented via processing circuitry, which may include one or more microprocessor or microcontrollers, as well as other digital hardware, which may include Digital Signal Processor (DSPs), special-purpose digital logic, and the like. The processing circuitry may be configured to execute program code stored in memory, which may include one or several types of memory such as Read Only Memory (ROM), Random Access Memory (RAM), cache memory, flash memory devices, optical storage devices, etc. Program code stored in memory includes program instructions for executing one or more telecommunications and/or data communications protocols as well as instructions for carrying out one or more of the techniques described herein. In some implementations, the processing circuitry may be used to cause the respective functional unit to perform corresponding functions according one or more embodiments of the present disclosure.

(80) While processes in the figures may show a particular order of operations performed by certain embodiments of the present disclosure, it should be understood that such order is exemplary (e.g., alternative embodiments may perform the operations in a different order, combine certain operations, overlap certain operations, etc.).

VI. Example Embodiments

(81) Some example embodiments are as follows.

(82) Embodiment 1: A method for determining a three dimensional location of a wireless device in a cellular communications network, the method comprising: obtaining a plurality of measurements for a wireless device, the plurality of measurements being Time Difference of Arrival, TDOA, related measurements; and computing a three dimensional position of the wireless device using the plurality of measurements and a vertical surface model, wherein the vertical surface model is a translated and scaled version of an initial vertical surface model.

(83) Embodiment 2: The method of embodiment 1 wherein the both the vertical surface model and the initial vertical surface model are models of, for each cell j, altitude, z, at points within an interior of a polygon that represents the cell j.

(84) Embodiment 3: The method of embodiment 2 wherein both the vertical surface model and the initial vertical surface model are based on polygon corners of the polygon in the horizontal plane augmented with vertical information for each corner of the polygon.

(85) Embodiment 4: The method of embodiment 3 wherein the vertical surface model is further based on a translation vector and a scaling matrix.

(86) Embodiment 5: The method of embodiment 4 wherein the vertical surface model is further based on virtual points on three dimensional line segments between the augmented corners of the polygon.

(87) Embodiment 6: The method of any one of embodiments 2 to 5 wherein the vertical surface model is defined as:

(88) $\tilde{z}=f^T\left(\tilde{x},\tilde{y}\right){\hat{\tilde{d}}}_j,{\left(\tilde{x}\tilde{y}0\right)}^T\in jz=.Math.z_j.Math.+{\sigma }_{z,j}{f}^T\left(\frac{x-.Math.x_j.Math.}{{\sigma }_{x,j}},\frac{y-.Math.y_j.Math.}{{\sigma }_{y,j}}\right){\hat{\tilde{d}}}_i,{\left(xy0\right)}^T\in \mathrm{cell}j.$

(89) Embodiment 7: The method of any one of embodiments 1 to 6 wherein the method is performed by a positioning node.

(90) Embodiment 8: A positioning node in a cellular communications network, the positioning node adapted to perform the method of any one of embodiments 1 to 6.

(91) Embodiment 9: A method for determining a translated and scaled vertical surface model for cells in a cellular communications network, comprising: obtaining an initial vertical surface model for cells in a cellular communications network; and translating and scaling the initial vertical surface model to provide a new vertical surface model.

(92) Embodiment 10: The method of embodiment 9 further comprising using the new vertical surface model.

(93) Embodiment 11: The method of embodiment 10 wherein using the new vertical surface model comprises using the new vertical surface model to compute a three dimensional position of a wireless device.

(94) Embodiment 12: The method of any one of embodiments 9 to 11 wherein the both the new vertical surface model and the initial vertical surface model are models of, for each cell j, altitude, z, at points within an interior of a polygon that represents the cell j.

(95) Embodiment 13: The method of embodiment 12 wherein both the vertical surface model and the initial vertical surface model are based on polygon corners of the polygon in the horizontal plane augmented with vertical information for each corner of the polygon.

(96) Embodiment 14: The method of embodiment 13 wherein translating and scaling the initial vertical surface model to provide the new vertical surface model comprises translating and scaling the initial vertical surface model based on a translation vector and a scaling matrix.

(97) Embodiment 15: The method of embodiment 14 wherein the new vertical surface model is further based on virtual points on three dimensional line segments between the augmented corners of the polygon.

(98) Embodiment 16: The method of any one of embodiments 12 to 15 wherein the new vertical surface model is defined as:

(99) 0$\tilde{z}=f^T\left(\tilde{x},\tilde{y}\right){\hat{\tilde{d}}}_j,{\left(\tilde{x}\tilde{y}0\right)}^T\in jz=.Math.z_j.Math.+{\sigma }_{z,j}{f}^T\left(\frac{x-.Math.x_j.Math.}{{\sigma }_{x,j}},\frac{y-.Math.y_j.Math.}{{\sigma }_{y,j}}\right){\hat{\tilde{d}}}_i,{\left(xy0\right)}^T\in \mathrm{cell}j.$

(100) Embodiment 17: The method of any one of embodiments 9 to 16 wherein the method is performed by a positioning node.

(101) Embodiment 18: A positioning node in a cellular communications network, the positioning node adapted to perform the method of any one of embodiments 9 to 16.

(102) At least some of the following abbreviations may be used in this disclosure. If there is an inconsistency between abbreviations, preference should be given to how it is used above. If listed multiple times below, the first listing should be preferred over any subsequent listing(s). 2D Two Dimensional 3D Three Dimensional 3GPP Third Generation Partnership Project 4G Fourth Generation 5G Fifth Generation AECID Adaptive Enhanced Cell Identity A-GNSS Assisted Global Navigation Satellite System A-GPS Assisted Global Positioning System AoA Angle of Arrival AP Application Protocol ASIC Application Specific Integrated Circuit CDF Cumulative Distribution Function CID Cell Identity CPU Central Processing Unit DSP Digital Signal Processor E-CID Enhanced Cell Identity (positioning method) eNB Enhanced or Evolved Node B E-SMLC Evolved Serving Mobile Location Center FCC Federal Communications Commission FPGA Field Programmable Gate Array GAD Geographical Area Description GIS Geographical Information System GLONASS Global Navigation Satellite System GMLC Gateway Mobile Location Center gNB New Radio Base Station GNSS Global Navigation Satellite System GPS Global Positioning System ID Identifier IP Internet Protocol kHz Kilohertz LBS Location Based Services LCS Location Service LMU Location Measurement Unit LOS Line-Of-Sight LPP Long Term Evolution Positioning Protocol LPPa Long Term Evolution Positioning Protocol Annex LS Least Squares LTE Long Term Evolution MBS Metropolitan Beacon System MHz Megahertz MME Mobility Management Entity MMSE Minimum Mean Square Error mmw Millimeter Wave ms Millisecond MSC Mobile Switching Center MTC Machine Type Communication N-LOS Non-Line-Of-Sight NR New Radio O&M Operation and Maintenance OFDM Orthogonal Frequency Division Multiplexing OFDMA Orthogonal Frequency Division Multiple Access OTDOA Observed Time Difference of Arrival P-GW Packet Data Network Gateway PRS Positioning Reference Signal PSAP Public Safety Answering Point RAM Random Access Memory RAN Radio Access Network ROM Read Only Memory RRC Radio Resource Control RRH Remote Radio Head SCEF Service Capability Exposure Function SET Secure User Plane Location Enabled Terminal SLC Secure User Plane Location Location Center SLP Secure User Plane Location Location Platform SON Self-Organizing Network SPC Secure User Plane Location Positioning Center SRS Sounding Reference Signal SUPL Secure User Plane Location TA Timing Advance TBS Terrestrial Beacon System TCP Transmission Control Protocol TDOA Time Difference of Arrival TOA Time of Arrival UE User Equipment U-TDOA Uplink Time Difference of Arrival VDOP Vertical Dilution Of Precision WGS World Geodetic System

(103) Those skilled in the art will recognize improvements and modifications to the embodiments of the present disclosure. All such improvements and modifications are considered within the scope of the concepts disclosed herein.