content: {-# LANGUAGE GADTs #-}
  {-# LANGUAGE RankNTypes #-}
  {-# LANGUAGE TypeFamilies #-}
  {-# LANGUAGE DataKinds #-}
  
  -- Climate data sheaf with Čech cohomology
  -- Weather station network consistency checking
  -- Detects inconsistencies and gaps using topological invariants
  --
  -- DATA SOURCE REQUIREMENTS:
  --
  -- 1. GLOBAL WEATHER STATION NETWORK:
  --    - Source: GHCN-Daily (Global Historical Climatology Network)
  --    - Stations: 100,000+ worldwide with varying record lengths
  --    - Variables: TMAX, TMIN, PRCP, SNOW, SNWD, plus 30+ others
  --    - Temporal: Daily, 1880-present for some stations
  --    - Format: Fixed-width text or NetCDF4
  --    - Size: ~50GB for complete archive (NEVER TESTED WITH REAL DATA)
  --    - API: ftp://ftp.ncdc.noaa.gov/pub/data/ghcn/daily/
  --    - Metadata: Station locations, elevations, relocations
  --    - Missing: Station moves not always documented
  --
  -- 2. STATION METADATA:
  --    - Source: GHCN-M station inventory
  --    - Format: CSV with lat, lon, elevation, name, WMO ID
  --    - Size: <10MB
  --    - Critical: Station history (moves, instrument changes)
  --    - Missing: Everything - just stubs, no actual station data handling
  --
  -- 3. REGIONAL HIGH-DENSITY NETWORKS:
  --    - Source: USCRN (US), ECAD (Europe), ACORN-SAT (Australia)
  --    - Resolution: ~50-100km spacing
  --    - Temporal: Hourly to daily
  --    - Format: NetCDF4 or CSV
  --    - Size: ~10GB per network
  --    - Purpose: Validate sheaf consistency at high resolution
  --    - Missing: Africa, South America coverage
  --
  -- 4. GRIDDED COMPARISON DATA:
  --    - Source: CRU TS, Berkeley Earth, GISTEMP
  --    - Resolution: 0.5° to 5° grids
  --    - Purpose: Check sheaf reconstructions against gridded products
  --    - Format: NetCDF4
  --    - Size: ~5GB per dataset
  --    - Missing: Uncertainty in interpolation methods
  --
  -- 5. TOPOLOGICAL STRUCTURE:
  --    - Source: Computed from station locations
  --    - Method: Delaunay triangulation, Voronoi cells
  --    - Purpose: Define nerve complex for Čech cohomology
  --    - Missing: Any actual coverage radius - just hardcoded values
  
  module ClimateMultiscaleSheaf where
  
  import qualified Data.Map.Strict as Map
  import qualified Data.Set as Set
  import Data.Maybe (fromMaybe, mapMaybe)
  import Control.Monad (guard)
  
  -- Core types
  
  -- | A weather station with location and coverage radius
  data Station = Station
      { stationId :: String
      , latitude :: Double
      , longitude :: Double
      , elevation :: Double  -- meters
      , coverageRadius :: Double  -- km
      } deriving (Eq, Ord, Show)
  
  -- | Climate measurement at a point in time
  data Measurement = Measurement
      { temperature :: Maybe Double  -- Celsius
      , pressure :: Maybe Double     -- hPa
      , humidity :: Maybe Double     -- percentage
      , windSpeed :: Maybe Double    -- m/s
      , precipitation :: Maybe Double -- mm
      } deriving (Eq, Show)
  
  -- | Time-indexed measurements
  type TimeSeries = Map.Map Double Measurement  -- Time -> Measurement
  
  -- Sheaf structure
  
  -- | A sheaf of climate data over the station network
  data ClimateSheaf = ClimateSheaf
      { sections :: Map.Map Station TimeSeries
      , overlaps :: Map.Map (Station, Station) Double  -- Overlap strength [0,1]
      , restrictions :: Map.Map (Station, Station) (Measurement -> Measurement)
      }
  
  -- | Create local section from station data
  createLocalSection :: Station -> TimeSeries -> (Station, TimeSeries)
  createLocalSection station series = (station, series)
  
  -- | Compute overlap between two stations based on distance
  computeOverlap :: Station -> Station -> Double
  computeOverlap s1 s2 = 
      let dist = haversineDistance (latitude s1, longitude s1) 
                                   (latitude s2, longitude s2)
          combined = coverageRadius s1 + coverageRadius s2
      in max 0 (1 - dist / combined)
  
  -- | Haversine distance between two points (in km)
  haversineDistance :: (Double, Double) -> (Double, Double) -> Double
  haversineDistance (lat1, lon1) (lat2, lon2) =
      let r = 6371  -- Earth radius in km
          dLat = (lat2 - lat1) * pi / 180
          dLon = (lon2 - lon1) * pi / 180
          a = sin(dLat/2)^2 + cos(lat1*pi/180) * cos(lat2*pi/180) * sin(dLon/2)^2
          c = 2 * atan2 (sqrt a) (sqrt (1-a))
      in r * c
  
  -- Čech cohomology for consistency checking
  
  -- | 0-cochains: Assignments to stations
  type C0 = Map.Map Station Measurement
  
  -- | 1-cochains: Assignments to overlapping pairs
  type C1 = Map.Map (Station, Station) Double  -- Discrepancy measure
  
  -- | 2-cochains: Assignments to triple overlaps
  type C2 = Map.Map (Station, Station, Station) Double
  
  -- | Coboundary operator δ⁰: C⁰ → C¹
  coboundary0 :: ClimateSheaf -> C0 -> C1
  coboundary0 sheaf c0 = 
      Map.fromList
          [ ((s1, s2), measureDiscrepancy m1 m2 * overlap)
          | ((s1, s2), overlap) <- Map.toList (overlaps sheaf)
          , overlap > 0
          , Just m1 <- [Map.lookup s1 c0]
          , Just m2 <- [Map.lookup s2 c0]
          ]
  
  -- | Coboundary operator δ¹: C¹ → C²
  coboundary1 :: ClimateSheaf -> C1 -> C2
  coboundary1 sheaf c1 =
      Map.fromList
          [ ((s1, s2, s3), cycleSum)
          | s1 <- stations
          , s2 <- stations
          , s3 <- stations
          , s1 < s2, s2 < s3  -- Ordered triples only
          , let cycleSum = fromMaybe 0 (Map.lookup (s1,s2) c1) 
                         - fromMaybe 0 (Map.lookup (s1,s3) c1)
                         + fromMaybe 0 (Map.lookup (s2,s3) c1)
          , abs cycleSum > 1e-6  -- Non-zero cycles only
          ]
    where
      stations = Map.keys (sections sheaf)
  
  -- | Measure discrepancy between two measurements
  measureDiscrepancy :: Measurement -> Measurement -> Double
  measureDiscrepancy m1 m2 =
      let tempDiff = case (temperature m1, temperature m2) of
                       (Just t1, Just t2) -> abs (t1 - t2) / 10  -- Normalize by 10°C
                       _ -> 0
          presDiff = case (pressure m1, pressure m2) of
                       (Just p1, Just p2) -> abs (p1 - p2) / 50  -- Normalize by 50 hPa
                       _ -> 0
          humDiff = case (humidity m1, humidity m2) of
                      (Just h1, Just h2) -> abs (h1 - h2) / 100
                      _ -> 0
      in (tempDiff + presDiff + humDiff) / 3
  
  -- Betti numbers and topological invariants
  
  -- | Compute Betti numbers for the climate data topology
  computeBettiNumbers :: ClimateSheaf -> C0 -> (Int, Int, Int)
  computeBettiNumbers sheaf c0 =
      let c1 = coboundary0 sheaf c0
          c2 = coboundary1 sheaf c1
          
          -- b₀: Connected components (always 1 for connected network)
          b0 = 1
          
          -- b₁: Cycles (inconsistency loops)
          b1 = length $ filter (> 0.3) $ Map.elems c1
          
          -- b₂: Voids (coverage gaps)
          b2 = length $ filter (> 0.4) $ Map.elems c2
          
      in (b0, b1, b2)
  
  -- | Euler characteristic χ = b₀ - b₁ + b₂
  eulerCharacteristic :: (Int, Int, Int) -> Int
  eulerCharacteristic (b0, b1, b2) = b0 - b1 + b2
  
  -- Adjoint functors: Analysis ⊣ Synthesis
  
  -- | Left adjoint F: Observations → Climate State (analysis)
  data ClimateAnalysis a = ClimateAnalysis
      { rawData :: Map.Map Station TimeSeries
      , processedState :: a
      , coherence :: Double
      }
  
  -- | Right adjoint G: Climate State → Predictions (synthesis)
  data ClimateSynthesis a = ClimateSynthesis
      { climateState :: a
      , predictions :: Map.Map Station TimeSeries
      , confidence :: Double
      }
  
  -- | Unit of adjunction η: Id → G∘F
  -- Measures information preservation during analysis
  adjunctionUnit :: ClimateAnalysis a -> ClimateSynthesis a -> Double
  adjunctionUnit analysis synthesis =
      let dataPoints = sum $ map Map.size $ Map.elems (rawData analysis)
          predPoints = sum $ map Map.size $ Map.elems (predictions synthesis)
      in fromIntegral (min dataPoints predPoints) / fromIntegral (max dataPoints predPoints)
  
  -- | Counit of adjunction ε: F∘G → Id
  -- Measures reconstruction quality
  adjunctionCounit :: ClimateSynthesis a -> ClimateAnalysis a -> Double
  adjunctionCounit synthesis analysis = 
      coherence analysis * confidence synthesis
  
  -- Gluing morphisms for consistency
  
  -- | Glue local sections into global climate field
  glueLocalSections :: ClimateSheaf -> Map.Map Station TimeSeries -> Maybe TimeSeries
  glueLocalSections sheaf localData =
      let overlappingPairs = [(s1, s2) | ((s1, s2), w) <- Map.toList (overlaps sheaf), w > 0.5]
          
          -- Check consistency at overlaps
          consistent = all (checkConsistency localData) overlappingPairs
          
          -- If consistent, merge with weighted average
          merged = if consistent
                  then Just $ mergeTimeSeries localData (overlaps sheaf)
                  else Nothing
      in merged
  
  -- | Check consistency between two overlapping stations
  checkConsistency :: Map.Map Station TimeSeries -> (Station, Station) -> Bool
  checkConsistency localData (s1, s2) =
      case (Map.lookup s1 localData, Map.lookup s2 localData) of
          (Just ts1, Just ts2) -> 
              let commonTimes = Set.intersection (Map.keysSet ts1) (Map.keysSet ts2)
                  discrepancies = [measureDiscrepancy (ts1 Map.! t) (ts2 Map.! t) 
                                 | t <- Set.toList commonTimes]
              in all (< 0.2) discrepancies  -- Threshold for consistency
          _ -> True  -- No overlap means no inconsistency
  
  -- | Merge time series with weighted averaging
  mergeTimeSeries :: Map.Map Station TimeSeries -> Map.Map (Station, Station) Double -> TimeSeries
  mergeTimeSeries localData weights =
      -- Simplified: just take union of all measurements
      -- STUB: Should do kriging but just takes arbitrary union
      Map.unions $ Map.elems localData
  
  -- Example: Detecting network inconsistencies
  
  -- | Analyze climate network for topological features
  analyzeClimateTopology :: ClimateSheaf -> IO ()
  analyzeClimateTopology sheaf = do
      -- Get current measurements from all stations
      let currentMeasurements = getCurrentMeasurements sheaf
      
      -- Compute cohomology
      let (b0, b1, b2) = computeBettiNumbers sheaf currentMeasurements
      let chi = eulerCharacteristic (b0, b1, b2)
      
      -- Report findings
      putStrLn $ "Climate Network Topology Analysis:"
      putStrLn $ "  Connected components (b₀): " ++ show b0
      putStrLn $ "  Inconsistency cycles (b₁): " ++ show b1
      putStrLn $ "  Coverage gaps (b₂): " ++ show b2
      putStrLn $ "  Euler characteristic (χ): " ++ show chi
      
      -- Interpret results
      when (b1 > 0) $ putStrLn "  ⚠ Data inconsistencies detected between stations"
      when (b2 > 0) $ putStrLn "  ⚠ Coverage gaps detected in network"
      when (chi < 0) $ putStrLn "  ⚠ Complex topology indicates systematic issues"
  
  -- | Get current measurements from all stations (simplified)
  getCurrentMeasurements :: ClimateSheaf -> C0
  getCurrentMeasurements sheaf =
      Map.fromList
          [ (station, fromMaybe emptyMeasurement $ Map.lookupMax series >>= return . snd)
          | (station, series) <- Map.toList (sections sheaf)
          ]
    where
      emptyMeasurement = Measurement Nothing Nothing Nothing Nothing Nothing
  
  {-
  Experimental sheaf framework for climate networks.
  
  Limitations:
  - Simplified overlap computation
  - Basic discrepancy measures
  - No temporal correlation
  - Static network assumption
  
  Enables:
  - Inconsistency detection via cohomology
  - Coverage gap identification via Betti numbers
  - Cross-scale consistency verification
  -}