** 3 Regional gravity field modeling with the point mass method**

**3.5 Point mass method with free positions**

**3.5.2 Two-step point mass method and relevant model factors**

Defined model factors

Choose the initial position of the first point mass radial basis function (RBF) by placing it below the observation point with maximum absolute value at a proper depth

Refine the model parameters of the first point mass RBF by the chosen iteration algorithm

if (i< K) then
*i = i+1*
RBF 1

RBF*i*
(i≥ 2)

**First step (search process)**

Residual gravity data

*K*point mass RBFs with known magnitudes and positions that
can be used to predict other gravity field quantities

Readjust the magnitudes of K searched point mass RBFs by solving a linear equation system w i t h k n o w n p o s i t i o n s obtained from the first step

For the case of the ill-conditioned linear problem, Tikhonov regularization is applied and the regularization parameter is chosen by one of the three approaches: (1) Empirical RMS minimization, (2) GCV, and (3) VCE

Choose the i-th RBF by placing it below the point with maximum absolute residual value at a proper depth

Refine the model parameters of the new point mass RBF by the chosen iteration algorithm

Select nearest point mass RBFs around the new point mass RBF from already searched point mass RBFs

Refine the model parameters of the new point mass RBF as well as its nearest RBFs by using the chosen iteration algorithm

if (i= K)
*N*_{ε}*N*_{ε}

**Second step (readjustment)**

**Figure 3.8:** Computation procedure of the two-step method. The mentioned “maximum absolute residual
value” is described as max

√**P*** _{e}*[l−

*F*(m

^{0})]

with **m**^{0} the model parameters for all previously
selected point mass RBFs and**P***e*the diagonal weight matrix for the input data.

• The initial depths and depth limits for the point mass RBFs: The depths of the point mass RBFs are directly related to their spatial bandwidths, playing a crucial role on the performance of the point mass RBFs in regional gravity field modeling. On the one hand, the point mass RBFs should not be located too deep (i.e., too large spatial bandwidths), as the input data for regional gravity field modeling are residuals with the removal of the long-wavelength components derived from a global geopotential model. The deeper the point mass RBFs are, the less orthogonal the basis functions become (e.g., Claessens et al., 2001). In addition, these deep point mass RBFs will also cause serious numerical instabilities, and represent only smooth signals. On the other hand, since the very-short-wavelength components computed by a DTM are also subtracted from the original input data, the point mass RBFs should not be too shallow either (i.e., too small spatial bandwidths). Otherwise, the fit of the data to the model can be achieved in terms of small RMS errors, but the fit of independent control values may become worse. Accordingly, it is necessary to limit the chosen RBFs in a layer with upper and lower bounds to guarantee the quality of the approximation. This means that we should introduce appropriate depth

constraints to the nonlinear problems to be solved for finding the point mass RBFs in the first step. In the context of this thesis, we employ simple bound constraints on the depths, and the L-BFGS-B algorithm is consequently a suitable choice. However, the other three iteration algorithms (i.e., LM, NLCG, and L-BFGS) introduced in Section 3.3.6 are also implemented in the first step. A numerical comparison between the four algorithms will be carried out in Chapter 4. Before choosing the depth limits, the initial depth for each newly found point mass RBF should be fixed first. It may be chosen based on the data spacing or the correlation length of the empirical covariance function derived from the observations. When the initial depth is fixed, the upper depth limit should be shallower than it, while the lower depth limit should be deeper. More numerical comparisons about the depth limits can be found in Chapter 4.

• The total number *K* of point mass RBFs: The required number of point mass RBFs for a good
representation of regional gravity field depends on the data variability, data distribution, and
targeted level for data misfit. More point mass RBFs are needed in an area with rough gravity
field features while fewer RBFs are sufficient for representing smooth features. The addition of
each new point mass RBF may decrease the data misfit, but it also reduces the redundancy. In
the two-step point mass method, the total number of point mass RBFs is determined by one of
the following three stop criteria for the search process: (1) a predefined data misfit; (2) the trend
of the reduction of the data misfit in terms of the number of RBFs; and (3) a predefined number
of point mass RBFs. Often, we prefer the first one, and then the second one follows. The target
data misfit is usually determined based on the accuracy of the input data. If the accuracy of
the input data is unknown, the target data misfit should be chosen carefully. If it is too small,
we may also fit the errors in the data. Sometimes, the decreasing of the data misfit becomes
very slow when the number of searched point mass RBFs is larger than a specific number *N*. In
this case, the search process can stop at reaching *N* point mass RBFs according to the second
criterion. Compared to the first two criteria, the last one is more empirical. No direct relation
between the number of observations and the number of RBFs can be found. Klees et al. (2008)
gave the empirical criterion that the number of RBFs is usually less than 25% of the number of
data points. However, this ratio may change in different research areas.

• The spectral bandwidths of the point mass RBFs (i.e., the choice of*N*min): In the framework of
the RCR technique, when using the full point mass RBFs (i.e., *N*_{min}= 0), the errors caused by
different spectral bandwidths of the input residuals and the RBFs are significant in the solutions
of the gravity field quantities that are in the long-wavelength domain (e.g., height anomalies,
geoid heights, disturbing potentials). In this case, the use of reduced point mass RBFs (i.e.,
*N*_{min} *>*0) will reduce the errors and provide better solutions. Related numerical comparisons
will be carried out in Chapter 4. However, for the solutions of the gravity field quantities that
are in the short-wavelength domain (e.g., gravity anomalies, gravity disturbances), the spectral
bandwidths of the RBFs have minor effects on them. Related numerical investigations will also
be given in Chapter 4. It has to be noted that, too much computation time is needed when
using the reduced point mass RBFs in the first step of PM-FRE without obtaining significantly
improved results, the full RBFs are employed for finding the point mass RBFs in the search
process. The choice of the spectral bandwidths of the point mass RBFs is only considered in
the second step for the readjustment of the magnitudes of all selected RBFs. In this case, the
use of reduced point mass RBFs can be regarded as a modification of the spatial bandwidths of
the RBFs obtained from the first step. In addition, the solution scheme of using the full RBFs
and considering the constraints in the least-squares adjustment in the second step of PM-FRE
(i.e., constrained solution, see also Section 3.4.1) will also be investigated.

• The optimization direction: Besides the magnitude of each new point mass RBF, the position

can be optimized in all directions or only in a radial direction (i.e., depth). Regarding the optimization in all directions in the test case using real gravity data, usually some point mass RBFs tend to move into the gaps to minimize the data misfit, and if the upper depth limit is too shallow, the estimated magnitudes for these RBFs may become too large and lead to unreasonable small-scale local features. In addition, the all-direction optimization may relocate the point mass RBFs near the edge of the data area far away from the initial locations, resulting in serious numerical instabilities. Hence, additional horizontal constraints must be added. Such circumstances can be avoided by using the radial-direction optimization, which is more simple and stable. Related numerical tests will be given in Chapter 4.

• The iteration limit*N** _{it}*for each new point mass RBF: The magnitude and position for each new
point mass RBF are estimated iteratively in the first step. The required number of iterations
for finding the local minimum for each RBF is different. For the sake of simplicity, an iteration
limit

*N*

*is usually defined for all searched point mass RBFs. Sometimes, we divide the point mass RBFs to be sought into several groups. For each group, there is an iteration limit. Often, the larger the*

_{it}*N*

*is, the longer the computation time is. A numerical test by using different*

_{it}*N*

*will be carried out in Chapter 4.*

_{it}• The number *N** _{ε}* of the nearest point mass RBFs: Ideally, for each new point mass RBF, the
magnitudes and positions of this and all other searched point mass RBFs should be recomputed.

However, this is a very time-consuming process. Barthelmes (1986) showed that it was sufficient
to recompute only a certain number*N** _{ε}* of the nearest point mass RBFs together with the new
one when the RBFs are nearly orthogonal. The value of

*N*

*ε*was increased when the RBFs were not sufficiently orthogonal. For the sake of simplicity, the same

*N*

*is often used for each new point mass RBF. A related numerical comparison will also be given in Chapter 4.*

_{ε}