ARTIFICIAL SATELLITES, Vol. 41, No DOI: /v

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1 ARTIFICIAL SATELLITES, Vol. 41, No DOI: /v On InSAR AMBIGUITY RESOLUTION FOR DEFORMATION MONITORING P.J.G. Teunissen Department of Earth Observation and Space Sstems (DEOS) Delft Universit of Technolog Kluverweg 1, Delft, The Netherlands Fax: ; ABSTRACT Integer carrier phase ambiguit resolution is the ke to fast and highprecision satellite positioning and navigation. It applies to a great variet of current and future models of GPS, modernized GPS and Galileo. It also applies to stacked radar interferometr for deformation monitoring, see e.g. [Hanssen, et al, 001]. In this contribution we appl the integer least-squares principle to the rank defect model of stacked InSAR carrier phase data. We discuss two was of dealing with the rank defect for ambiguit resolution. One is based on the use of a priori data, the other is based on the use of an interval constraint on the deformation rate. 1 INTEGER LEAST-SQUARES (ILS) Consider the sstem of observation equations = Ax + Bz + e (1) where R m is the vector of observations, x R p and z Z n are the vectors of unknown parameters and e R m is the noise vector. Matrix (A, B) is given and assumed to be of full column rank. The structure of the sstem (1) is tpical for carrier phase GPS applications. In that case consists of the carrier phase and pseudo range data, z consists of the integer double difference ambiguities and x consists of the baseline components and possibl other real-valued parameters (e.g. atmospheric dela parameters). The model is however also applicable to radar interferometric phase ambiguit resolution problems, see e.g. [Bamler and Hartl, 1998], [Hanssen, 001], [Hanssen et al, 001], [Kampes and Hanssen, 004], [Kampes, 005]. In order to solve (1) in a least-squares sense, we follow the approach of [Teunissen, 1993]. Let Q be the variance matrix of and let. into a sum of three quadratic terms: e =(.) T (.). We first decompose e = Ax Bz = ê + ẑ z ẑ + ˆx(z) x ˆx z ()

2 18 in which ê = Aˆx Bẑ, ˆx and ẑ are the least-squares solutions for x and z, respectivel, assuming that z is real-valued instead of integer-valued, and ˆx(z) =(A T A) 1 A T ( Bz) (3) is the least-squares solution of x based on the assumption that z is known. In GPS terminolog, the solutions ˆx and ẑ are also referred to as the float solutions. Solution ˆx(z) is a conditional least-squares solution. Matrices Qẑ and Qˆx z are the variance matrices of ẑ and ˆx(z), respectivel. Note that the float solutions ˆx and ẑ would not be unique in case matrix (A, B) is rank defect. From () follows that min Ax + x R p,z Z Bz = ê n +min ẑ z +min ˆx(z) z Z n ẑ x R x p ˆx z This shows that the least-squares solutions for x R p and z Z n are given as (4) ž =argmin ẑ z Z z n ẑ and ˇx =ˆx(ž) (5) In [Teunissen, 1995] it is shown how ž can be computed in an efficient manner b means of the LAMBDA method. In [Teunissen, 1999] it is shown that the integer least-squares estimator ž is optimal in the sense that it maximizes the probabilit of correct integer estimation. For a brief review of statistical carrier phase ambiguit resolution, see e.g. [Teunissen, 001]. The probabilit of correct integer estimation (the so-called success-rate) is driven b the variance matrix of ẑ. The ADOP (Ambiguit Dilution of Precision), defined as ADOP = Qẑ 1 n (ccle) (6) is a simple scalar measure for the precision of ẑ. It is the geometric mean of the conditional standard deviations of ẑ. It has the advantage of being invariant for the class of admissible integer transformations. The ADOP can be used to get a first quick impression on the model s abilit to achieve successful ambiguit resolution. Fo this to be the case, the ADOP needs to be small enough (e.g. well below the ccle level). InSAR ILS WITH DEFORMATION DATA One of the simplest models of phase ambiguit resolution for stacked radar interferometr takes the form (see e.g. [Hanssen et al., 001]), i = a i x + z i + e i,i=1,...,n (7) with i the observed phase difference between two permanent scatterers (expressed in ccles), x the unknown deformation rate, z i the unknown integer ambiguit, e i the noise term, and a i = λ Δt i,whereλ denotes the wavelength (e.g. 5.8cm) andδt i is a time interval (expressed in ears) between the current time and a reference time. The model can be written in vector form as = ax + z + e (8)

3 This model appears to be a special case of (1), since A a, B I, p =1andm = n. Note however that the condition of having a coefficient matrix of full column rank is not met anmore, since matrix (a, I n ) has a rank defect of one. Hence, the sstem contains insufficient information to compute a unique float solution for x and z. This situation can be remedied b including data x o on the deformation rate x. This could be actual data (from earlier studies) or pseudo-data. As a result we obtain the full rank model [ ] = x o [ a 1 ] x + [ I 0 ] [ ] e z + e x with variance matrix [ Q 0 0 σ x in which σx is used to express the a priori uncertaint of the deformation rate. Now that the model is of full rank, the method of the previous section can be applied again. This is the approach which has been followed in [Hanssen et al, 001]. To determine the ADOP, we need the variance matrix of ẑ. It is given as Qẑ = Q + σx aat, from which the ADOP follows as ADOP = Qẑ 1 n = Q 1 n (1 + σ x a T a) 1 n (10) With Q = σφi n, a i = Δt λ i and small σ φ,weget ADOP σ φ ( λ σ x σ φ Δt i ) 1 n ] (9) (ccle) (11) This simple result can be used to get a quick impression of whether the model has enough strength for successful ambiguit resolution. 3 InSAR ILS WITH DEFORMATION CONSTRAINT Instead of working with a priori data on the deformation rate x, one ma also consider using a constraint in the form of an interval α x β, withα and β given. In practice one usuall has a fair idea over which range one can expect the deformation to occur. In that case it is not difficult to choose α and β. Hence, the ILS-problem now becomes min ax x R,z Z z subject to α x β (1) n Note that we could have added the constraint also to the model based on the a priori deformation data. In the following, however, we will work with the model (1). The procedure for solving it can also be used for the model based on the a priori deformation data. Unconstrained minimization Let us first consider minimizing the least-squares objective function without the constraint. With the use of the orthogonal projectors P a = a(a T a) 1 a T and Pa = I n P a,wehave min x R,z Z n ax z = min x R,z Z n Pa ( z) + P a ( z) ax = min z Z n Pa ( z) +min x R P a ( z) ax = min z Z n Pa ( z) +min x R ˆx(z) x a T a (13) 19

4 0 with ˆx(z) =(a T a) 1 a T ( z). This shows that the solution is given as ž =argmin P z Z n a ( z) Q 1 and ˇx =ˆx(ž) (14) This result seems to impl that in order to find ž, we need to search over the complete space of integers Z n. Clearl, this is not practicall feasible. Fortunatel, we can confine the search to a subset of Z n. The relevant integer vectors Let S : R n Z n be the integer map of componentwise integer rounding. Then all R n which are mapped b S to z Z n form the set S z = R n S() =z (15) This set is the n-dimensional unit-cube centred at z. Note,sinceQ is assumed to be a diagonal matrix, that S has the propert min z Z n z = S().Wema therefore decompose the least-squares objective function also as min ax x R,z Z z n =min min ax x R z Z z =min ax S( n x R ax) (16) This shows that the relevant integer vectors are those which are the centres of the unitcubes through which the line ax passes. The form the set Thus instead of (14) we ma now write Ω z = z Z n S( ax) =z, x R (17) ž =argmin Pa ( z) z Ω z and ˇx =ˆx(ž) (18) However, since the range of x has not et been bounded, the set Ω z still contains an infinite number of integer vectors. The finite set Ω z We will now make use of the constraint α x β. This allows us to reduce the infinite set Ω z to the finite set Ω z = z Zn S( ax) =z, α x β (19) Hence, we are now onl considering a line segment of length a T a(β α) of the line ax. Note that the length of the line segment is an approximate upper bound for the number of integer vectors contained in Ω z. Finding the integer vectors z Ω z To find all integer vectors that satisf S( ax) =z for α x β, we first make the problem a bit more tractable. Let x = x α, z = S( aα), z = z z,and = aα z.thenz = S( ax) =S( ax )+z and thus z = S( ax ) with 0 x γ =(β α) and S 0 (0) Hence, x is nonnegative and all components of lie within the interval [ 1, + 1 ]. Note that z = S( ax )isequivalenttothen intervals z i 1 i a i x z i + 1, i =1,...,n.

5 To find the elements of Ω z, one can now proceed as follows. Consider for each i the graph of the line ɛ i (x) = i a ix as function of x. The line starts at the point (0, i ), where for each i, i [ 1, + 1 ]. The line ma be located in either the first or the fourth quadrant, depending on whether a i 0ora i 0. Starting with x = 0, the first candidate integer vector is z = 0. The next candidate integer vector follows from which of the n lines ɛ i (x) first crosses one of the levels ± 1 modulus 1. For instance, if it is the line ɛ j (x) which first crosses the level ± 1, then the second candidate integer vector is given as z =(...,0, 1, 0,...) T,withthe1inslotj. In this wa one can proceed collecting all candidate integer vectors z for the range 0 x γ. From them the elements of Ω z are obtained as z = z + S( aα). Computing the final solution Now that the finite set Ω z is given, we compute 1 ž =argmin P z Ω a ( z) z and ˇx =ˆx(ž) (1) On this result one last check has to be performed, namel whether the constraint α ˇx =ˆx(ž) β is satisfied or not. If it is, then (1) is the solution sought. If it is not, then the solution is given b one of the two endpoints of the line segment, x = α or x = β. The solution is x = α if aα S( aα) < aβ S( aβ). Note that in practice one ma question the choice of the constraint if one of the two endpoints turns out to be the solution. The purpose of the constraint in our present application is namel to be able to construct a finite integer search space Ω z, but it is not chosen so as to influence the final solution. Thus if one of the two endpoints turns out to be the solution, it is likel that the constraint has been chosen too strict. 4 A POSTERIORI PROBABILITY OF THE INTEGER SOLUTION Once the integer vector ž has been computed, one would like to know whether one can have an trust in the solution. Intuitivel, this boils down to a comparison of the values taken b Pa ( z) for z Ω z. One would then have more trust in the solution if Pa ( ž) is significantl the smallest value, than when P a ( ž) is comparable to one or more of the other values. As shown below this is what is measured b the conditional distribution of z. The following assumptions are made: x, z N(ax + z, Q ), with conditional probabilit densit function (pdf) f( x, z). Thus f( x, z) exp 1 ax z (where stands for proportional to). x is uniforml distributed over the interval [α, β]. Its pdf g(x) is therefore proportional to the indicator function δ α,β (x) of the interval, g(x) δ α,β (x). For the distribution of z, we assume a noninformative prior. x and z are assumed to be independent and the distribution of z is denoted as h(z). We therefore have for the joint distribution f(x,, z) = f( x, z)g(x)h(z), from which the a posteriori distribution of z follows as h(z ) = f( x, z)g(x)dx z Z n f( x, z)g(x)dx = exp 1 P a ( z) k(z) z Z n exp 1 P a ( z) k(z) ()

6 where k(z) = 1 β σ exp 1 ( ) x ˆx(z) dx with σ 1 = π α σ a T a (3) For sufficientl small σ, k(z) will be close to one for ˆx(z) [α, β] and close to zero outside this range. Under this assumption () ma be approximated as h(z ) exp 1 P a ( z) z Ω z exp 1 P a ( z) (4) Note that ž is the maximizer of the right hand side. The a posteriori probabilit of the integer solution (i.e. the probabilit conditioned on the data) is given as h(ž ). The closer this value is to one, the more trust one will have in the computed solution. 5 REFERENCES Bamler, R. and Hartl, P. (1998): Snthetic aperture radar interferometr. Problems, 14, R1-R54. Inverse Hanssen, R.F. (001): Radar Interferometr: Data Interpretation and Error Analsis. Kluwer Academic Publishers, Dordrecht. Hanssen, R.F., P.J.G. Teunissen and P. Joosten (001): Phase ambiguit resolution for stacked radar interferometric data. In: Proc. KIS001, International Smposium on Kinematic Sstems in Geodes, Geomatics and Navigation, Banff, Canada, pp Kampes, B.M. and R.F. Hanssen (004): Ambiguit resolution for permanent scatterer interferometr. IEEE Transactions on Geoscience and Remote Sensing, 4(11), Kampes, B.M. (005): Displacement Parameter Estimation using Permanent Scatterer Interferometr. Netherlands Geodetic Commission. Publications on Geodes, 168 p. Teunissen, P.J.G. (1993): Least-squares estimation of the integer GPS ambiguities. Invited Lecture, Section IV Theor and Methodolog, IAG General Meeting, Beijing, China, August Also in: LGR Series, No. 6, Delft Geodetic Computing Centre. Teunissen, P.J.G. (1995): The least-squares ambiguit decorrelation adjustment: a method for fast GPS integer ambiguit estimation. Journal of Geodes, 70: Teunissen, P.J.G. (1999): An optimalit propert of the integer least- squares estimator. Journal of Geodes, 73: Teunissen, P.J.G. (001): Statistical GNSS carrier phase ambiguit resolution: a review. IEEE, /01, pp Received: , Reviewed: , b W. Pachelski, Accepted: