Ankh-Morpork is mostly built on Ankh-Morpork

Theoretically the city is built on black loam, but years of fires, flooding, and rebuilding mean that the city is mostly built on older versions of itself. If you want a couple of floors of basements, you just have to dig out the old buildings that lie underneath. Of course flooding is a good reason not to build downwards.

There's an important point to remember here. Most cities are built on fertile and riverside ground. You get a reasonable amount of soft ground under your feet that you can dig down into without excessive difficulty.

Draconian planning laws

Paris was built according to a plan. That plan forced all the buildings to look relatively uniform, have balconies that lined up with their neighbours and limited the height of buildings. If you want a larger building, but you're strictly limited in how far up you can go, perhaps down is your future.

Soft drifting sand

You can't really build large structures on it, but you can build on the harder ground underneath. It's not so much that you dig down to build as that periodically dunes roll through town and raise the effective ground level a few metres.

Related to this:

Wind

The howling debris laden winds that pass through mean that people want to expose as little of their building as possible to the risk of weather damage. A vast warehouse represents a massive exposure to the weather and hence risk of damage, sinking it into the ground protects the bulk of the structure from damage.

Generally sticking to low rise buildings will reduce the wind tunnel effect of larger buildings as the air at low levels is channeled down the narrow gaps through a city. If high winds are the norm in a region, keeping the buildings lower will make a much less hostile environment for pedestrians.

The city is built on its building material

London is built on a particular clay that is shown in the colours of London brick. You dig, you bake, you build, all using one hole. Paris equally is built on its own building material, the famous catacombs are originally the quarries for the stone from which the city is built.

However

No matter how you approach this, the sheer quantity of removed material far outweighs any usefulness on the scales you're asking for. If perhaps you were building small mud huts and using the excavated clay to build the walls then it's practical. However the good old square/cube law mean that this method fails to scale up to larger buildings and certainly not the large empty volume of a warehouse.

A customised solution

Match the Parisian style draconian planning laws to Parisian geology.

The city is bound to a two storey building height simply as a result of an arbitrary leader at an arbitrary point in history declaring this must be so. The warehouses are built over the quarries that the rest of the sprawling city is built out of. Most people don't build down, they build out. The city is huge and requires vast amounts of stone allowing you to get round the square cube problems from the size of your warehouses.

These already exist

This is more common than you think, most modern countries have cities with multilevel underground complexes. Tokyo is famous for how much of the city can be transversed by foot without ever coming to to the surface. Turkey in particular has had dozens of underground cities that are several thousand years old. Kaymakli City, Derinkuyu city, Özkonak City, Mazı City are just some of the examples, some of which go down many levels. removed stone was used to build additional above ground homes. Soft volcanic rock in the region makes excavating easy while still being strong enough for many levels of undercity. Dry hot conditions minimize the risk of flooding and makes the stable temperatures desirable, several cities are actually self-cooling using underground wells and evaporation. there is is a steady gradient from ancient to modern underground construction, you can easily find the particular "feel" you are looking for.

To summarizes the conditions you need.

1. The area needs to be dry and high enough that there is little risk of flooding.




2. The underlying material needs to be either a soft but solid rock or it needs to be several meters of loose earth overlying a shallow hard bedrock. the former makes building as easily as digging a hole, the latter makes deep excavation necessary for large buildings but makes a dug out solid foundation easier than pilings, (once you have dug it out why not use the space)




3. extremely hot or cold environments will also encourage building down, as the ground has a steady temprature and makes a good insulator.

As for reusing the material, stone was rarely waste, it can be reused for building well buildings as well as roads, sculptures, and depending on the rock many specialized uses. The type of rock will determine what is easier, but soft rock that makes digging easy also makes turning it into blocks easy, so it is almost as easy to build up as down is some places.

The below example in Cappadocia Turkey is almost exactly what you describe, it even makes your alliteration literal as part of the city is build in a canyon, with surface structures built from stone removed from the underground. The underground construction is rather rough but that is because it is mostly unused in modern times. We see in other modern underground construction that these tunnels can easily be further refines and modernized if they stay occupied.

If the setting is desert-like then there are practical reasons and real-world examples for building into the ground rather that on top.

The prerequisite is a hard-packed and dry loam as the predominant soil. People can dig caves into the ground with the most promitive tools, so the tradition of building houses this way can date back into prehistoric times. Over time the caves evolved to multi-room buildings with inner courtyards and storage space in a second storey.

The obvious advantage is the stable climate you have inside the cave-house. The thick walls insulate against the scorching heat of noon and the freezing cold of night. The entrances are all build facing the one direction where the cool breeze comes from to trap cool air in the courtyards. The building itself provides shade for the yard.

The dug-out can be used to construct buildings (or parts thereof) above ground by mixing some water in and forming the loam into shape. Over time the ground level will rise naturally because you cannot use all of this dugout. You can also use it for repairs after rainfalls.

I saw a village built in this way in the Sahara in Tunesia. The people there preferred their cave-houses over modern concrete buildings. I don't know how big one room can be made. You probably have to include pillars to stabilize the ceiling in factory halls.

There is a small town in Australia that is mostly underground due to harsh environment.

https://en.wikipedia.org/wiki/Coober_Pedy

Note the upfront cost is the same but running costs (ie air con) is much reduced. It is economically smart to build underground in that environment.

Also note multistorey buildings only make sense if land is valuable and scarce. So it makes sense to spend a lot extra on building. Note in a multistorey building each floor has less usable area than a single storey (stairs etc).

Any building activity requires either trees to die or massive holes in the ground called quarries. Your world seems to lack trees. So quarries are the only local option for building material. Instead of digging quarries away from urban centres, they turn each quarry into a building when finished.

Natural canyons, if the ground is cut by deep canyons, either erosional or tectonic in origin, then cutting buildings into the walls of those breaks makes more sense than quarrying and building above. This is particularly applicable when the rock is industrially useful, Rocksalt or other evaporites, Limestone, Dolomite, Gypsum, any of a number of Iron ores that form thick geological units (banded iron stone, hematite, magnetite for a start) or coal measures. There may also be a demand for building stone elsewhere that could fuel the cutting and transport of the local rock, this may be because the rock is superior to that found elsewhere (from either an engineering or aesthetic standpoint) or it may be because the local rock is socially significant (bringing good luck or of religious significance).

The natural canyons provide ease of access making the process more economically appealing and logistically manageable but may not provide an incentive of themselves, a hostile environmental setting does. If surface of the area the buildings are in is inhospitable due to climate, vegetation, or fauna, then building down rather than up is preferred. There are a number of modern cities that have active or historical tunnel networks created for this reason, Montreal's "Winter City" is quite famous.

The other option is that the underground building is a matter of social necessity rather than being in any way cost effective. Never underestimate the power of religion; if the locals have a strict belief that any building that rises higher than a sacred quall bush is an affront to the gods they simply can't build up. Thus to have the high density population centres needed for industrialisation they will have to dig down for living space.

You're in a desert? Your buildings function as solar stills. It's really very expensive to haul water over the vast desert sands, even by your implausible airships.
Therefore, and to contribute to sustainability and economic independence, each building is build in a cunning fashion so as to route outside air down to the lowest levels, where the air is simultaneously cooled and also relieved of the undue burden of its water vapours. This water, along with careful water management practices, helps to ensure the independence of this desert nation from outside forces who would try to influence it through the monopoly of vital resources.

The ground-cooled air also contributes to the comfort levels of the buildings, lowering energy expenditures on air conditioning (if your civilization has that technology).

Realistically, the amount fo digging and spoil, as well as issues of potential flooding make this very unlikely at best.

However, you could have the effect of an underground city by roofing over a narrow canyon or large ravine and building along the sides, leaving the open space like an atrium. Depending on the technological abilities of the nation, this technique could also be adapted for worked out quarries and open pit mines.

The interior would have this effect

The giant roofs would provide shelter from extreme heat, cold or raging winds (or whatever other environmental issues plague the population) and moderate the temperature. The interior space between the walls would function like an atrium in a building, and could be used as public space, for gardens and other outdoor activities. Due to shading issues, it is unlikely that you could practice intensive agriculture.

A more naturalistic space, but in reality the trees turned out to be too expensive to maintain

So a roofed built over a natural canyon, ravine or man made "hole" like a quarry or an open pit mine might fulfill the bill. It will provide shelter from extreme weather, and might even be a pleasant place to live, so long as provisions are made to prevent flooding, and a secure source of food is nearby to feed the population.

Underground structures have many advantages. They have better insulation against extreme solar heat (from a binary star system, for example) and extreme cold. They can withstand hurricanes, tornadoes, forest fires, blizzards, dust storms, pterodactyl-size predatory birds, and earthquakes. They are a natural defense against early 20th century weaponry, meaning artillery, catapults, cannons, aerial bombs, al Qaeda suicide zeppelins, and snipers. Defensive anti-aircraft guns and missiles will have clear unobstructed shots at enemy aircraft without buildings blocking their shots. Water does not need to be pumped up to a reservoir since all the water users will be at lower levels. Plus, once the population grows, it would be easy to enlarge the living spaces by digging down, rather than building more levels to the top of a highrise building.

Underground buildings are vulnerable to floods so assume the elevation of the cities are high enough to avoid such catastrophes. Waste water and sewage would have to be disposed of in some way, so assume the lowest levels of the structures can still use gravity to drain waste water to a waste treatment facility elsewhere. Once the structure drops below 200 feet the natural heat from the crust will become uncomfortable.

So what should be done with the excavated rocks? Iron ore and uranium can be processed for useful tools and energy. Coal can be used to power the steam engines. Granite, marble, and quartz can be used in kitchen and bathroom countertops. Less useful rocks can be dropped underwater to form artificial reefs, which can increase the population of fish for aquatic farming.

Space for Airships.

This is an airship heavy world, there could be rules against building tall buildings or there are social and economic advantages to not having tall buildings. For example if most buildings in an area are low and of similar height it becomes easier to approach, moor and load/unload Airships there. A high building would make that harder and lower the local property prices.

Emergency infrastructure designed around Airships.

Similar in execution as the above but for differenr reasons. In our world we can reach just about anything by car. Cant reach it by car? We'll build a road so you can. Now imagine a world where the primary transport choices are walking, horses&carriages (no cars especially with the lack of fuel), bicycles (if any), trains or Airships. City planning might simply not include large roadways for transport, cargo and evacuation purposes. With Airships being so important for getting around a city the heights of the buildings are limited. I heard that there were actual rules that you were not allowed to build higher than the mooring masts of Airships but I cant find references to it.

Protection against warfare

This is set around the industrial age, with a different history and Airships so prevalent the act of stragetic bombing can be both feared and practiced a lot. As alternative you could have sabotage and terror attacks similar to Vietnam or Partizans in WWII by commandeering an airship and having it ram tall buildings with a large payload. As a defense mechanism building into the ground could be less expensive than armoring tall buildings against damage and having a widespread network against such attacks.

Scientific misconceptions.

There have been tons of misconceptions in the past, such as that trains going more than 30km/h would mean its occupants would suffocate. The misconception could simply be that building too high would make the buildings catch too much wind and they would topple.

There are still plenty of reasons why building underground is going to be expensive and challenging, from getting enough fresh air into the buildings to their construction. But is it important enough to take into consideration? It might be like sound in space: it doesnt make sense in our world but damn doesnt it add to the feel of a well fleshed out world.

Think about lighting

One of the reasons for building up is to have windows to let in light. Old factories and schools often had more window than wall. Unlit space is only useful for storage.

One possibility is to have your people's eyes more sensitive than ours, relative to outdoor ambient light levels. Then maybe they build underground to get away from excessive light.

Otherwise, while lighting considerations can't explain why people build underground, you need to explain why they don't keep people above ground. To do that, you need cheap, effective lighting. In the era you're taking your tech from, we had skylights and deck prisms to distribute natural light, but they couldn't consistently keep a desk light enough to work on, especially on the second story down. We also had candles and whale oil for artificial light, but they weren't all that expensive and weren't cheap. Gas light may also have been available; I don't know how good it was. So, you need to introduce something special.

On Earth, building underground has been suggested for several reasons:

• to protect the houses from hurricanes


• to protect the houses from cold


• to preserve the surface on the roof of the houses as arable land or air filter or oxygen source


• because skyscrapers are ugly so we hide them in the ground

The most plausible reasons are solar radiation protection (sudden flares) and thermal management (less sun facing surface) as in the answers by hyperion4 and CatCat.

Basically unless you decide that the environment has a inconvenient sun you will have to make some other reason that the surface must be kept flat. The desert lets you stay below ground level because you do not have to worry about rain but you have a lot of other problems, you need to maintain airship wind speeds, you have to keep your structures clean of sand. You need to manage sewerage and lighting and a host of other things we take for granted with gravity pulling things down hill.

The only convincing reason is the immense cost and irritation of protecting structures from the regular and debilitating explosive tumble weeds.

However as these tumble weeds are explosive to help disperse seeds and penetrate natural barriers they are a nuisance on this continent. All the flat areas have been levelled to an almost perfect runway flatness by explosions on any bumps and filling with wind driven sand, clay and dust. Eradicating the plants is near impossible and wildlife and crops are only grown on the slopes and mountains that the tumble weed does not overrun. The flat land is no use for anything except solar collectors and underground structures that maintain a flat top when closed. Periscopes and wind sensors are used to check before opening doors. Once on the surface the area can be inspected and scanned by lasers to determine if there are any bushes moving closer.

Temporary work and larger construction in the correct non-tumbling season is safe enough but having any protrusion is an invitation to erosion by inevitable explosions.

The surface is flat enough to land aircraft but parking them there or landing in a wind is asking for trouble so this continent has elected to use airships and work underground.

Teenage rites of passage include catching a tumble weed bush while windy (must grab and hold the roots) and juggling one or more to gain rights of not chaperoned dates with consenting partners. Household chores include taking the garbage out to the local explosive detonation pit and vacuum cleaning the seals and hinges in the doors and surfaces of windows and solar collectors.

There is enough subsurface water to provide for small green houses and domestic use but no rain to cause flooding.