The ideal network can be defined to have a homogeneous and isotropic variance-covariance matrix, which is called the criterion matrix (Grafarend, 1972). “Homogeneity” means that it is invariant with respect to a translation, and “Isotropy” refers to its rotational symmetry. Thus, the homogeneous and isotropic ideal network has a uniform quality throughout the network, and the local error ellipses will be circular and equally sized. The criterion matrix can be described by the Taylor-Karman structure, which is defined by the variance-covariance matrix between two points in the network. The general form of the 3 × 3 criterion matrix of the Taylor-Karman structure for two points is given by Schaffrin (1985, p. 586) as

where *r*_{
i
}, *r*_{
j
} represent the coordinates of *P*_{
i
} and *P*_{
j
}, respectively, and *s* indicates the distance between these two points related to the criterion matrix. The directional covariance functions, and , correspond to the longitudinal and cross directions, respectively, and the unitless variance component, , is multiplied by the criterion matrix. The desired variance of the estimated coordinates in the network conforms to the covariance function of zero distance; that is, the case of two identical points. The analytical formula of covariance functions are given by Grafarend and Schaffrin (1979) as

where *K*_{0} and *K*_{1} are the modified Bessel function of the second kind, i.e., zero and first order, respectively. The characteristic distance, *d*, of the network can be chosen so that the maximum distance of the network is the upper limit of 10*d* according to Wimmer (1982). Another type of characteristic distance can be found in Schmitt (1980).

From Eq. (1), it is clear that the criterion matrix is not dependent on the measurement types or linear models with specific rank-deficiency. The second order Design (SOD) was successfully implemented for the network design of ground tracking stations for the Low Earth Orbiter (LEO) orbit determination (Bae, 2005). This approach optimizes the cofactor matrix by determining the measurement weights between candidate and network stations. Other approaches, such as zero, first, and third order design (ZOD/FOD/TOD) can be referred to Schaffrin (1985) and/or Kuang (1996).

The basic idea of the SOD is to minimize the difference between the criterion and cofactor matrix of the estimated point coordinates in the network by estimating the weight for each measurement between two points. The ideal cofactor matrix (“criterion” matrix) can be computed from the Taylor-Karman structure, and the cofactor matrix of the estimated point coordinates are obtained from the design matrix of the assumed observation model (distance measurements in this study). Let us assume the linearized Gauss Markov Model (GMM) of

with the diagonal weight matrix *P*, where *y* and *A* are the *n* × 1 observation vector and the *n* × *m* design matrix, respectively. The term ξ represents the unknown vector for the point coordinates, and the *n* × 1 random error *e* is assumed to have a zero-mean and same variance component as in Eq. (1). Then, the condition equation for SOD can be represented by

where is the cofactor matrix of the estimated points coordinates f given by

Without any loss of generality, Eq. (5) can be rewritten as

The final solution for the weight components is given by

where * defines the *Hadamard product* of matrices with equal size. For the detailed derivation of the above equation, see Bae (2005).