Moment Tensors and other Phenomenological Descriptions of Seismic Sources--I. Continuous Displacements

If a seismic event involves no external bodies, its source can be described phenomenologically by a vector field or by any one of three kinds of symmetric second-order tensor fields. The vector field is the equivalent force. The tensor fields are the stress-free strain, the stress glut, and any other moment tensor density. A particular source uniquely determines all of its descriptions except the moment tensor densities, but the motion it produces determines only the equivalent force. The source can also be described by the polynomial moments of any of its four field descriptions, and for small sources exciting long waves, a good approximate description is obtained from a finite number of low-degree moments of the equivalent force or the stress glut. ‘The’ seismic moment tensor is the zeroth degree moment tensor of the stress glut or any moment tensor density and also the first degree moment tensor of the equivalent force. The n'th degree moments of any moment tensor density are uniquely determined by the motion if n = 0, 1 but not if n ⩾ 2. In the Earth, the stress drop is not a moment tensor density, and its volume integral is not ‘the’ seismic moment tensor. In a later paper these conclusions are shown to be unaffected by relaxing the assumption of a continuous displacement field.