I want to show that the higher the Sun's altitude, the higher the solar irradiation through the atmosphere, through simple experiments. However, I don't have the instrument that measures irradiation. Is there a good approach?
I believe you could use pyranometer data at various elevations to get a non-spectral measurement of the total irradiance received at the Earth's surface. From here: http://en.wikipedia.org/wiki/Pyranometer
A pyranometer is a type of actinometer used to measure broadband solar irradiance on a planar surface and is a sensor that is designed to measure the solar radiation flux density (W/m2) from a field of view of 180 degrees. Pyranometers are frequently used in meteorology, climatology, solar energy studies and building physics. They can be seen in many meteorological stations - typically installed horizontally and next to solar panels - typically mounted with the sensor surface in the plane of the panel.
If you want something more accurate, a pyrheliometer or radiometer is used. From http://www.esrl.noaa.gov/gmd/grad/starmeas.html
Direct solar irradiance is the solar radiation that passes directly though the atmosphere from the sun without being scattered or absorbed by the atmosphere. Typically it is measured on a surface that is kept normal to the direction of the center of the sun's disk. That surface is held normal to the sun's position by a solar tracker. Pyrheliometers are most commonly used to measure total direct solar irradiance. The most accurate measurements of this quantity are obtained from absolute cavity radiometers which serve as calibration standards and are being increasingly used operationally.