Looking into the options of adding and retaining heat in the outdoor pool (and saving wallet) - heat inverter, roof solar water heater, gas heaters etc. the pool cover seems to rule them all (at least here in Australia).
There is a very nice (though pretty old - 1978) paper by CSIRO on this topic:
This paper very well proves the efficiency of solar covers (blankets) in retaining heat, that part is very clear. What I am not able to figure out from this paper, and could not find answer to anywhere else is:
From heat absorption and retention point of view, is it better to keep cover (blanket) on through the day or take it off?
Please if you are going to answer this question in other way than finding/recommending paper on this topic, consider there are many factors like day vs water temperature, sunny vs cloudy, humidity, wind, clear (opaque) solar blanket vs dark vs two layered (top layer clear, bottom layer), so you most likely will need to pick some most common scenario unless you need to cover all the factors and their combination of effects in general.
E.g. sunny day in Sydney, air temperature between 25° - 30°, water temperature 20°, standard humidity, two layered blanket with top layer transparent (light blue, bottom opaque silver) - for some reason this is the combination I see the most in e-Shops.
Also there seems to be a lot of bias about it including claims like bubbles work like magnifying glass so they transfer more heat from sunlight into the pool, or that double layered (combination of transparent layer over opaque bottom layer) absorbs more heat into the water than fully transparent layer which allows most sunlight go through into the water - which seems highly unlikely to me.