a fault is observed in a road cut, but there is no obvious rock units to correlate on either side of the fault to determine relative movement. How might you determine or even infer whether the hanging wall has moved up or down relative to the footwall

To determine or infer whether the hanging wall has moved up or down relative to the footwall in the absence of obvious rock units for correlation, you can employ a few methods:

1. Study of Offset Features: Look for any identifiable features, such as layers, dikes, or veins that are offset across the fault. Measure the displacement of these features along the fault plane. If the offset feature horizontally offsets from left to right when you face the fault, the hanging wall has moved up relative to the footwall. Conversely, if the offset feature horizontally offsets from right to left, the hanging wall has moved down relative to the footwall.

2. Scarp Morphology: Analyze the morphology of the fault scarp. A scarp is a vertical or near-vertical exposure of the fault. If the scarp is higher on one side of the fault than the other, it suggests that the hanging wall has moved up relative to the footwall.

3. Fault Block Relationships: Examine the relationship between different fault blocks. Fault blocks are the segments of rock on each side of a fault. Determine the relative heights of the fault blocks. If the fault block on the hanging wall side is elevated compared to the footwall block, it implies that the hanging wall has moved up relative to the footwall.

4. Geologic Maps and Cross-Sections: Consult existing geologic maps and cross-sections of the area to find indications of the fault's movement history. These resources may provide evidence of fault displacement and the relative movement of the hanging wall and footwall, even if there are no immediate rock units for correlation at your specific location.

In cases where there is no clear evidence or multiple interpretations are possible, it is essential to gather more data, such as structural measurements, stratigraphic data, or conducting geophysical surveys like seismic reflection, in order to increase confidence in determining the relative movement of the hanging wall and footwall.