The cooling effect of a volcano is primarily controlled by
1. The volume of SO2 that is released. Some volcanoes have a higher proportion of SO2 in their ejected material than others.
2. The height it is released at, particular the proportion that is released into the stratosphere where its life span is much longer.
3. The location. Tropical volcanoes cause more global cooling than high-latitude ones.
It typically takes a VEI 5 or higher to have significant climate impacts. El Chichon was a mid-range VEI 5 in 1982 that had observable global cooling effects. By contrast, Saint Helens in 1980 was a low-end VEI 5 and had very little if any global temperature impact. Pinatubo was a low-end VEI 6 in 1991 that had a peak global cooling of over .5C. Krakatoa in 1883 was a mid-range VEI 6 that had a global cooling of around 1C. Tambora in 1815 was a lower-end VEI 7 that had a global cooling of nearly 1C and caused the "Year Without a Summer" and global famine. High-end 7s and 8s have even more catastrophic potential because they use up all the available oxidants in the stratosphere which neutralize the SO2. Thus the SO2 has a much longer lifespan, allowing for much more cooling to occur and also for more positive feedbacks. By the time cooling ceases for VEI 5s and 6s, most of the SO2 has already dissipated. By lengthening the lifespan of the SO2, the climate is able to approach equilibrium much closer to the peak RF value. Toba 74,000 years ago was a VEI 8 and is believed to have nearly caused the extinction of humans. Such an eruption is theorized to have a global cooling effect of up to 5C.
However, there have been VEI 4s known to cause climactic effects. This is because Laki emitted 6X more SO2 than Pinatubo, even though the total volume of Tephra was less than a tenth of Pinatubo's and the height of the ash cloud reached only 15,000 feet instead of 100,000 feet for Pinatubo. The cooling effects were concentrated more regionally because the huge volume of SO2 was emitted into the troposphere instead of the stratosphere.
The global cooling effect of any volcano would likely outweigh any albedo change of the ice from soot. An eruption like Pinatubo or Krakatoa has the power to temporarily return us to early 20th century climate when ice was far thicker and more expansive. The type of volcano might be important too with less explosive but higher SO2 content volcanoes like Laki potentially having a different effect than explosive volcanoes like Pinatubo, Chichon, Krakatoa, Tambora or Toba. The location is potentially important too because tropical volcanoes block more sunlight in total, but also might have cooling effects and soot deposition rates that vary by region.
A VEI 5 occurs about once a decade. A VEI 6 occurs about once or twice a century. 7s occur once every 1000 years.
