Fig. 10.3 Stereo aerial photography of a forest scene viewed in a digital photogrammetric environment using color anaglyph, upper Tanana valley of interior Alaska, USA. (Red-blue glasses are necessary to view this scene in stereo). Significant radial displacement: apparent shift of an object having height in relation to its base in an image with a central projection of trees (layover) is evident in the top left corner of the scene

crown base (tree height) or a (planimetrically-correct) polygon delineating a distinct forest condition class, such as its size class2 or species class3 or density class.4

10.2.1.2 Ground Control and Direct Georeferencing

As the collinearity conditions indicate above, it is essential to know the coordinates of each camera station and the elements of the rotation matrix (M) of each camera before aerial photographs can be used to acquire three-dimensional measurements of forest features (Xp , Yp , Zp ). Typically, this information is obtained through the use of ground control points, which are features on the ground, with known (Xp , Yp , Zp ) coordinates, that are also visible in the overlap area of the stereo pair of aerial photographs (technically, three vertical control points for leveling the model and two horizontal control points for scaling the model is the minimum requirement for controlling a single stereo pair [56]). If additional ground control points are available, a least-squares solution can be obtained with an estimate of uncertainty. When there are a large number of overlapping stereo models covering an area, the requirement of three control points per stereo model is relaxed, and the exterior orientation parameters for all photos within the block can be obtained through a procedure known as a bundle block adjustment [56]. As stated

2Size class refers to the predominant size (diameter, or DBH) of the trees within a stand; e.g. regeneration (DBH < 12.7 cm), poletimber (12.7 cm < DBH < 30.48 cm), sawtimber (DBH > 30.48 cm).

3Species class refers to the predominant species (or species mixture) of the stand (e.g. black spruce, mixed spruce-hemlock, etc.).

4Density class refers to the number of tree stems in a given area (i.e. trees per hectare).

in [56], a bundle adjustment is a procedure for simultaneously “adjusting all photogrammetric measurements to ground control values in a single solution.” However, when using aerial photographs in a sampling mode, where plots are widely spaced, the need for sufficient ground control can introduce a significant and potentially prohibitive additional cost to the aerial photo-based inventory.

If smaller scale (i.e. lower resolution), controlled aerial photography is available that covers the same area as the uncontrolled large scale photography, ground control points can be measured photogrammetrically in the small scale photography and subsequently used to control the large scale photography (a method known as bridging control) [46]. However, in very remote areas, such as interior Alaska, it is unlikely that even recent small-scale, controlled imagery will be available. Fortunately, in recent years, technology has become available that allows for precise, and accurate, measurement of the position and orientation of the camera in the aircraft at the moment of exposure, which significantly reduces, or even eliminates entirely, the need for surveyed ground control. The use of two tightly-coupled technologies: (1) the global positioning systems (GPS) and (2) inertial measurement unit (IMU), now allow the exterior orientation parameters for each camera station to be obtained without the need for ground control, an approach known as direct georeferencing [34]. The GPS instrument uses a system of satellites to triangulate the position of the camera, while the inertial measurement unit uses a system of accelerometers and gyroscopes to determine the orientation of the camera. Furthermore, because the GPS acquires accurate, but relatively noisy, positional information, while the IMU provides trajectory and orientation information with relatively little noise but with systematic drift, the positional error can be dramatically reduced by merging these two complementary sources of positional information via a Kalman filter signal processing procedure. In the Kalman filter, the estimate of the position at time k + 1 is given by the so-called state estimate equation:

xk+1 = Axk + Buk + wk

yk = Cxk + zk

Kk = APk CT CPk CT + Sz −1

xˆ k+1 = (Axˆ k + Buk ) + Kk (yk+1 − Cxˆ k )

Pk+1 = APk AT + Sw − APk CT S−z 1CPk AT

In the above equation, A, B, and C are matrices that describe how the state changes and can be measured, k is the time index, x is the state of the system, u is the known input to the system, y is the measured output, w is the process noise, z is the measurement noise, Kk is the Kalman gain, Sw is the process noise covariance: Sw = E(wk wTk ), and Sz is the measurement noise covariance: Sz = E(zk zTk ), and P is the estimation error covariance. The first term in the fourth equation is basically A times the estimated position xˆ at time k, plus B times the known input u (IMU-based acceleration information) at time k. The second term is K (the so-called Kalman gain that minimizes the error covariance of the position at time k + 1) times the difference (residual) between the measured position yk+1 and the prediction of the