International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 7-4-3 W6, Valladolid, Spain, 3-4 June, 1999

84

errors). Height measurement fails for various reasons all over

the DEM (e.g. temporal (radiometric or geometric) changes or

sensor noise), therefore all resulting random errors will be

summarized in a statistic error. The definition of blunders is

adopted from the common multi-measurement case (e.g. wrong

determination of ground control points). Note that blunders and

systematic errors have (statistically) the same effect, as they

cause an offset in the expectation values of the observations.

3. CORRELATION COEFFICIENT

The correlation coefficient R is defined as

[x\xi]

a/ £ 1

[xix\]

\e\

[X2X2)

(-1

Several authors give surveys of InSAR and stereo-optical DEM

errors. Grim (1986), who refers to the Baarda error model, gives

a detailed error description for photogrammetric sensor

modelling. Zebker et al. (1994) gave an evaluation of ERS

topographic map accuracy, dividing errors according to their

effects in statistic and systematic errors.

Each error type will occur in both InSAR and stereo-optical

DEMs. In contrast to systematic errors and blunders, statistic

errors of phase and parallax measurements occur locally, but

may turn affected regions unusable. Typical reasons for statistic

errors in spacebome DEM generation are temporal changes

between both passes. As the InSAR procedure is very sensitive

to a change of backscatter geometry, the errors in InSAR DEMs

increase with the repeat pass time interval. Additional problems

with the handling of mountainous terrain lead, in comparison to

optical sensors, to a generally less robust performance.

The presentation of erroneous heights is different in both

DEMs. The stereo-optical DEM is affected by spikes, which

make the DEM appear rugged. These spikes originate from the

mismatching of conjugate points in the stereo pair. The

resulting parallax causes a wrong height estimation, i.e. the

point lies above or below the terrain (Fig. 4). The localization of

these errors seems to be much easier than the error detection in

the InSAR DEM. Although residues in the interferogram appear

as phase jumps higher than 7t, these values are either not used or

replaced by surrounding phase estimates, depending on the

phase unwrapping algorithm. For this study, the least squares

(LS) unwrapping algorithm (Ghiglia and Romero, 1994) has

been applied, which estimates heights by integrating the phase

vector gradient field under a smooth solution constraint, thus

suppressing residuals and yielding a global solution. Although

this algorithm performs robustly in comparison to the tree

algorithm (Goldstein et al., 1988), errors are still localized in

areas of low correlation in the LS DEM (Zebker and Lu, 1998).

As stated above, interferometric and stereo DEM generation

requires the correct identification of the same point in both

images, in order to measure the phase difference or the parallax.

Unlike systematic errors, which can be reduced by suitable

choice of parameters, the reduction of statistic errors is the

challenging act for the fusion process. The next section shows

the meaning of the correlation coefficient for both techniques. It

will be used as an indicator of the presence and extent of noise

and consequently of the statistic error of each measurement.

where E{Xi} denotes the expectation value of the statistic

variable Xi, hence E{(Xi) 2 } is the variance and E{X1X2} the

covariance of XI and X2. Note that coherence denotes the

absolute value of p in interferometry.

The correlation coefficient between two images is used in both

techniques, but has for each technique a different meaning. In

the stereo-optical case, points are matched by maximising the

correlation coefficient within the correlation window.

Therefore, the correlation coefficient of each DEM point shows

the maximum correlation value occurring during the matching

process. Therefore, low correlation indicates not necessarily a

point error, but points, where matching problems may have

occurred.

Low correlation of grey values results from radiometric

differences between the images, due to multitemporal and

across track data acquisition (case of SPOT stereo viewing),

sensor noise or atmospheric effects due to scattering and

absorption, etc. Therefore, a low correlation coefficient will

occur randomly all over a DEM, wherever differences between

the stereo images exist (Fig. 3). Image pre-processing, like

filtering and introduction of geometric and radiometric

constraints are highly needed in the multitemporal case in order

to reduce the effects of radiometric differences on the matching

algorithm (Baltsavias and Stallmann, 1993).

In the InSAR case, a high coherence indicates those areas,

where phase estimation for DEM generation is reasonable. Even

more, the cross-correlation would give an estimate of the height

error, if it was known precisely, as

a = K

(2)

with

P

1-/7

(3)

a: height error

K: constant term

E: signal to noise ratio

Note that only a correlation estimate is computed within an

estimator window from the flattened interferogram (Small et al.,

1995). Therefore, the estimated height error is dependent on the

size of the estimator window (Prati et al., 1994). Still, the

correlation coefficient can be used as a confidence map of the

InSAR DEM.