**Explanation:**

**Benchmark** in surveying refers to a point of reference with a known elevation used to determine the elevation of other points or features in the survey area. It is an essential component in establishing a vertical control network for any surveying project.

The correct answer for the establishment of a benchmark is **c) spirit leveling**. Spirit leveling, also known as differential leveling or simply leveling, is a method used to determine the relative heights or elevations of different points on the earth's surface. It involves the use of a spirit level instrument, such as a surveyor's level or an automatic level, and a graduated leveling staff.

Here is a detailed explanation of why spirit leveling is used to establish a benchmark:

**1. Accuracy and Precision:**
- Spirit leveling is considered one of the most accurate and precise methods for determining elevations.
- The use of a spirit level instrument allows for precise measurements of height differences between different points, ensuring reliable and consistent results.

**2. Relative Measurement:**
- Spirit leveling is a relative measurement technique, where the height differences between points are determined.
- A benchmark is established by leveling a known reference point to a high degree of accuracy.
- The elevation of the benchmark is then used as a reference to determine the elevations of other points in the survey area.

**3. Levelling Instrument:**
- The spirit leveling technique requires the use of a leveling instrument, which consists of a telescope, a spirit level, and a leveling staff.
- The telescope is used to observe a leveling staff held vertically at different points.
- The spirit level ensures that the telescope remains horizontal during the measurements.

**4. Leveling Staff:**
- A leveling staff is a graduated rod that is held vertically at different points to determine the height differences.
- The staff is marked with graduations in metric or imperial units, allowing for accurate readings.

**5. Procedure:**
- To establish a benchmark, the leveling instrument is set up at the benchmark location.
- The telescope is focused on the leveling staff held upright at the benchmark.
- The staff reading is recorded, which represents the elevation of the benchmark.
- The instrument is then moved to other points in the survey area, and the height differences are measured relative to the benchmark.

**6. Importance:**
- Establishing a benchmark is crucial for ensuring accurate and consistent elevation measurements throughout the survey.
- It allows for the creation of a vertical control network, which is essential for mapping, construction, and other civil engineering projects.
- The benchmark serves as a reference point for future surveys and can be used to establish additional benchmarks in the survey area.

In conclusion, spirit leveling is used to establish a benchmark due to its accuracy, precision, and relative measurement technique. It provides reliable elevation measurements that serve as a reference for other points in the survey area.

C