sample size N. This simplifying assumption perhaps needs justification since the station network configuration in reality is somewhat fixed. Our justification for it here is that, for instance, the locations of the earthquake epicentres are rather unpredictable and are not strongly correlated with the geodetic site locations. Thus in the epicentral coordinate system the locations of the geodetic stations may instead be regarded as randomly chosen. For the surface water loading case, this assumption may be rationalized through our expectation that the locations of the geodetic sites are not strongly correlated with the locations of hydrologically active regions. The validation of this expectation is ultimately provided by our results reported in Chapter5.
Under this assumption, TCN, DCN, and RCN are themselves random variables. Their probability distributions are called, in this context, the sampling distributions of the sample means. It follows that
〈TCN〉=TCF (3.16a)
〈DCN〉=DCF (3.16b)
〈RCN〉=RCF (3.16c)
where definition of the expectation value in Equation 3.13has been generalised to several observation points,
〈X〉= 1 (4π)N Z ∂⊕ . . . Z ∂⊕ X(r1, . . . ,rN)dΩ1. . .dΩN (3.17)
3.3
Expected bias as standard deviation
We are however primarily interested in the network effect, that is, the deviation of a CN parameter from its corresponding CF parameter. Fortunately, a fundamental theorem of probability theory lets us estimate this deviation.
Central Limit Theorem. Let {X1, . . . ,XN} be a set of independent and identically distributed random variables, that is, a random sample, drawn from a distribution with population meanµand standard deviationσ. Then asN increases, the sampling
distribution of the sample mean XN = 1 N N X k=1 Xk
approaches the normal distribution with meanµand standard deviationσ/pN. In our case, the random variable X is any of the components of the fields u,s, orw. The population meanµis the corresponding CF parameter, and the sample meanXN is the respective CN parameter,
TCN=uN (3.18a)
DCN=sN (3.18b)
RCN=wN (3.18c)
Thus, the network effect in the notation of the theorem is XN−µ. As expected for an error term, its population mean is zero by Equation3.16.
The standard deviation of the network effect therefore provides a natural mea- sure of the expected magnitude of the network effect. Since
std(XN −µ) =std(XN) (3.19)
this measure is the same as the standard deviation of the sample mean itself and, therefore, we define the expected biases in the Helmert parameters to be,
∆T =std(TCN) (3.20a)
∆D=std(DCN) (3.20b)
∆R=std(RCN) (3.20c)
wherestd(X)denotes the standard deviation of a random variable X. The Central Limit Theorem (CLT) guarantees that the expected bias scales as1/pN, that is, for largeN, ∆T ≈stdp(u) N (3.21a) ∆D≈stdp(s) N (3.21b) ∆R≈stdp(w) N (3.21c)
3.3. Expected bias as standard deviation It is instructive here to view the CF frame parameters to be the sample means when the network is infinite and uniformly distributed so that there is no error associated with network bias.
This scaling may also be understood from an alternative viewpoint. Here, the value of a field at a single point is regarded as a measurement of the corresponding CF parameter. Then the deviation of that measurement from the true value of the CF parameter may be regarded as the error in that measurement. ForN such points that are randomly chosen, the sum of these uncorrelated errors may be interpreted as a random walk ofN steps. Probability theory than predicts that this sum scales aspN and, therefore, the average of these N errors scales as1/pN.
This measure of the NE can in principle be applied to any theoretical model for crustal deformation. Besides the geodynamical processes investigated in this thesis, it could also be applied to postseismic rebound, atmospheric loading, or tectonic plate motion, for instance, to assess the agreement between the prediction of the GC motion by their theoretical models and the space geodetic observations. Such validation of geophysical models against space geodetic observations have been increasingly useful to researchers [see, for example, Xu and Chao,2015].
The expected bias may be interpreted as the statistical uncertainty, or the expected error, in the determination of model-predicted GC motion due to the finite size and unevenness of the geodetic networks. Moreover, since it ignores the details of the configuration of the observing network except for the size, we may also regard it as a formal error associated with the crustal deformation model itself due to discrete sampling.
Furthermore, it is often desirable for the stability of the RF to realise it using sites that have relatively higher reliability and consistency, longer timespan of operation, and weaker non-linear signals in their motion. However, as our criteria for selection of sites get stricter, the choice of sites becomes increasingly more restricted. A possible application of the expected bias is to decide on the optimal size of the network that ensures that the NE does not overwhelm the advantages of a strict selection criteria.
In addition, geodetic observations are often compatible with ranges of values for the input parameters of a model such as, for example, the fault slip model of an earthquake. In that case the deterministic calculation of the NE might not be useful.
However, our stochastic approach to estimate the NE could still be applied here to obtain a measure of the expected error associated with it, and thereby, to partially validate its prediction for the GC motion. Likewise, this approach may also be useful when the spatio-temporal domain of applicability of a model is incomplete.