Add multi-observer support for topocentric fitting

Extend od_observation_t with observer_idx so each observation can
reference a different ground station. Config now holds an array of
observers instead of a single pointer. The existing single-observer
tle_from_topocentric() is unchanged (sets observer_idx=0 for all obs).

New overload: tle_from_topocentric(topo[], ts[], observer[], int4[], ...)
accepts parallel observer_ids array indexing into the observers array.
PG function overloading resolves by argument types.

Tests 9-11: two-station fit converges, single-station via multi-observer
API matches, out-of-range observer_id raises error.

Makefile

@@ -2,7 +2,8 @@ MODULE_big = pg_orrery
 EXTENSION = pg_orrery
 DATA = sql/pg_orrery--0.1.0.sql sql/pg_orrery--0.2.0.sql sql/pg_orrery--0.1.0--0.2.0.sql \
        sql/pg_orrery--0.3.0.sql sql/pg_orrery--0.2.0--0.3.0.sql \
-       sql/pg_orrery--0.4.0.sql sql/pg_orrery--0.3.0--0.4.0.sql
+       sql/pg_orrery--0.4.0.sql sql/pg_orrery--0.3.0--0.4.0.sql \
+       sql/pg_orrery--0.4.0--0.5.0.sql
 
 # Our extension C sources
 OBJS = src/pg_orrery.o src/tle_type.o src/eci_type.o src/observer_type.o \

sql/pg_orrery--0.4.0--0.5.0.sql

@@ -0,0 +1,21 @@
+-- pg_orrery 0.4.0 -> 0.5.0 migration
+--
+-- Adds multi-observer support, IOD bootstrap (seed-free fitting),
+-- and covariance output for uncertainty estimation.
+
+-- ============================================================
+-- Multi-observer topocentric fitting
+-- ============================================================
+
+CREATE FUNCTION tle_from_topocentric(
+    observations topocentric[], times timestamptz[],
+    observers observer[], observer_ids int4[],
+    seed tle DEFAULT NULL, fit_bstar boolean DEFAULT false,
+    max_iter int4 DEFAULT 15,
+    OUT fitted_tle tle, OUT iterations int4,
+    OUT rms_final float8, OUT rms_initial float8, OUT status text
+) RETURNS RECORD
+    AS 'MODULE_PATHNAME', 'tle_from_topocentric_multi'
+    LANGUAGE C STABLE PARALLEL SAFE;
+COMMENT ON FUNCTION tle_from_topocentric(topocentric[], timestamptz[], observer[], int4[], tle, boolean, int4) IS
+    'Fit a TLE from topocentric observations collected by multiple ground stations. observer_ids[i] indexes into observers[]. Requires seed TLE and >= 6 observations.';

src/od_funcs.c

@@ -26,6 +26,7 @@
 
 PG_FUNCTION_INFO_V1(tle_from_eci);
 PG_FUNCTION_INFO_V1(tle_from_topocentric);
+PG_FUNCTION_INFO_V1(tle_from_topocentric_multi);
 PG_FUNCTION_INFO_V1(tle_fit_residuals);
 
 /* ================================================================
@@ -155,10 +156,11 @@ tle_from_eci(PG_FUNCTION_ARGS)
 
     /* Configure solver */
     memset(&config, 0, sizeof(config));
-    config.obs_type  = OD_OBS_ECI;
+    config.obs_type    = OD_OBS_ECI;
-    config.fit_bstar = fit_bstar ? 1 : 0;
+    config.fit_bstar   = fit_bstar ? 1 : 0;
-    config.max_iter  = max_iter;
+    config.max_iter    = max_iter;
-    config.observer  = NULL;
+    config.observers   = NULL;
+    config.n_observers = 0;
 
     /* Convert seed TLE if provided */
     if (has_seed)
@@ -274,15 +276,17 @@ tle_from_topocentric(PG_FUNCTION_ARGS)
             obs[i].data[0] = topo->azimuth;
             obs[i].data[1] = topo->elevation;
             obs[i].data[2] = topo->range_km;
+            obs[i].observer_idx = 0;  /* single observer */
         }
     }
 
     /* Configure solver */
     memset(&config, 0, sizeof(config));
-    config.obs_type  = OD_OBS_TOPO;
+    config.obs_type    = OD_OBS_TOPO;
-    config.fit_bstar = fit_bstar ? 1 : 0;
+    config.fit_bstar   = fit_bstar ? 1 : 0;
-    config.max_iter  = max_iter;
+    config.max_iter    = max_iter;
-    config.observer  = &observer;
+    config.observers   = &observer;
+    config.n_observers = 1;
 
     /* Seed TLE required for topocentric */
     if (!has_seed)
@@ -328,6 +332,163 @@ tle_from_topocentric(PG_FUNCTION_ARGS)
 }
 
 
+/* ================================================================
+ * tle_from_topocentric_multi(topocentric[], timestamptz[],
+ *                            observer[], int4[],
+ *                            tle, boolean, int4)
+ *   -> RECORD (same as tle_from_eci)
+ *
+ * Multi-observer variant: observations from different ground stations.
+ * observer_ids[i] indexes into the observers[] array.
+ * ================================================================
+ */
+Datum
+tle_from_topocentric_multi(PG_FUNCTION_ARGS)
+{
+    ArrayType      *topo_arr  = PG_GETARG_ARRAYTYPE_P(0);
+    ArrayType      *time_arr  = PG_GETARG_ARRAYTYPE_P(1);
+    ArrayType      *obs_arr   = PG_GETARG_ARRAYTYPE_P(2);
+    ArrayType      *id_arr    = PG_GETARG_ARRAYTYPE_P(3);
+    bool            has_seed  = !PG_ARGISNULL(4);
+    pg_tle         *seed_pg   = has_seed ? (pg_tle *) PG_GETARG_POINTER(4) : NULL;
+    bool            fit_bstar = PG_ARGISNULL(5) ? false : PG_GETARG_BOOL(5);
+    int32           max_iter  = PG_ARGISNULL(6) ? 15 : PG_GETARG_INT32(6);
+
+    int             n_topo, n_times, n_obs_sites, n_ids;
+    od_observation_t *obs;
+    od_config_t     config;
+    od_observer_t  *observers;
+    od_result_t     result;
+    tle_t           seed_sat;
+    int             i, rc;
+
+    TupleDesc       tupdesc;
+    Datum           values[5];
+    bool            nulls[5];
+    HeapTuple       tuple;
+
+    /* Deconstruct all arrays */
+    Datum  *topo_datums, *time_datums, *obs_datums, *id_datums;
+    bool   *topo_nulls, *time_nulls, *obs_nulls, *id_nulls;
+
+    deconstruct_array(topo_arr, topo_arr->elemtype, sizeof(pg_topocentric),
+                      false, TYPALIGN_DOUBLE,
+                      &topo_datums, &topo_nulls, &n_topo);
+    deconstruct_array(time_arr, TIMESTAMPTZOID, sizeof(int64), FLOAT8PASSBYVAL,
+                      TYPALIGN_DOUBLE, &time_datums, &time_nulls, &n_times);
+    deconstruct_array(obs_arr, obs_arr->elemtype, sizeof(pg_observer),
+                      false, TYPALIGN_DOUBLE,
+                      &obs_datums, &obs_nulls, &n_obs_sites);
+    deconstruct_array(id_arr, INT4OID, sizeof(int32), true,
+                      TYPALIGN_INT, &id_datums, &id_nulls, &n_ids);
+
+    if (n_topo != n_times)
+        ereport(ERROR,
+                (errcode(ERRCODE_ARRAY_SUBSCRIPT_ERROR),
+                 errmsg("observations and times arrays must have same length: "
+                        "%d vs %d", n_topo, n_times)));
+
+    if (n_topo != n_ids)
+        ereport(ERROR,
+                (errcode(ERRCODE_ARRAY_SUBSCRIPT_ERROR),
+                 errmsg("observations and observer_ids arrays must have same length: "
+                        "%d vs %d", n_topo, n_ids)));
+
+    if (n_topo < 6)
+        ereport(ERROR,
+                (errcode(ERRCODE_INVALID_PARAMETER_VALUE),
+                 errmsg("at least 6 observations required, got %d", n_topo)));
+
+    if (n_obs_sites < 1)
+        ereport(ERROR,
+                (errcode(ERRCODE_INVALID_PARAMETER_VALUE),
+                 errmsg("at least 1 observer required")));
+
+    /* Build observer array (pre-compute ECEF for each) */
+    observers = (od_observer_t *) palloc(sizeof(od_observer_t) * n_obs_sites);
+    for (i = 0; i < n_obs_sites; i++)
+    {
+        pg_observer *op = (pg_observer *) DatumGetPointer(obs_datums[i]);
+        observers[i].lat   = op->lat;
+        observers[i].lon   = op->lon;
+        observers[i].alt_m = op->alt_m;
+        od_observer_to_ecef(op->lat, op->lon, op->alt_m, observers[i].ecef);
+    }
+
+    /* Build observation array with per-observation observer index */
+    obs = (od_observation_t *) palloc(sizeof(od_observation_t) * n_topo);
+    for (i = 0; i < n_topo; i++)
+    {
+        pg_topocentric *topo = (pg_topocentric *) DatumGetPointer(topo_datums[i]);
+        int64   ts  = DatumGetInt64(time_datums[i]);
+        int32   oid = DatumGetInt32(id_datums[i]);
+
+        if (oid < 0 || oid >= n_obs_sites)
+            ereport(ERROR,
+                    (errcode(ERRCODE_INVALID_PARAMETER_VALUE),
+                     errmsg("observer_id %d out of range [0, %d)",
+                            oid, n_obs_sites)));
+
+        obs[i].jd = PG_EPOCH_JD + ((double)ts / (double)USECS_PER_DAY);
+        obs[i].data[0] = topo->azimuth;
+        obs[i].data[1] = topo->elevation;
+        obs[i].data[2] = topo->range_km;
+        obs[i].observer_idx = oid;
+    }
+
+    /* Configure solver */
+    memset(&config, 0, sizeof(config));
+    config.obs_type    = OD_OBS_TOPO;
+    config.fit_bstar   = fit_bstar ? 1 : 0;
+    config.max_iter    = max_iter;
+    config.observers   = observers;
+    config.n_observers = n_obs_sites;
+
+    /* Seed TLE required for topocentric */
+    if (!has_seed)
+        ereport(ERROR,
+                (errcode(ERRCODE_INVALID_PARAMETER_VALUE),
+                 errmsg("seed TLE required for topocentric fitting")));
+
+    pg_tle_to_sat_code_od(seed_pg, &seed_sat);
+
+    memset(&result, 0, sizeof(result));
+
+    rc = od_fit_tle(obs, n_topo, &seed_sat, &config, &result);
+
+    pfree(obs);
+    pfree(observers);
+
+    if (rc != 0)
+        ereport(ERROR,
+                (errcode(ERRCODE_DATA_EXCEPTION),
+                 errmsg("TLE fitting failed: %s", result.status)));
+
+    /* Build result tuple */
+    if (get_call_result_type(fcinfo, NULL, &tupdesc) != TYPEFUNC_COMPOSITE)
+        ereport(ERROR,
+                (errcode(ERRCODE_FEATURE_NOT_SUPPORTED),
+                 errmsg("function returning record called in context "
+                        "that cannot accept type record")));
+    tupdesc = BlessTupleDesc(tupdesc);
+
+    memset(nulls, 0, sizeof(nulls));
+
+    {
+        pg_tle *fitted = (pg_tle *) palloc(sizeof(pg_tle));
+        sat_code_to_pg_tle(&result.fitted_tle, fitted);
+        values[0] = PointerGetDatum(fitted);
+    }
+    values[1] = Int32GetDatum(result.iterations);
+    values[2] = Float8GetDatum(result.rms_final);
+    values[3] = Float8GetDatum(result.rms_initial);
+    values[4] = CStringGetTextDatum(result.status);
+
+    tuple = heap_form_tuple(tupdesc, values, nulls);
+    PG_RETURN_DATUM(HeapTupleGetDatum(tuple));
+}
+
+
 /* ================================================================
  * tle_fit_residuals(tle, eci_position[], timestamptz[])
  *   -> TABLE (t timestamptz, dx_km float8, dy_km float8, dz_km float8,

src/od_solver.c

@@ -223,7 +223,7 @@ compute_residuals_eci(const tle_t *tle, const od_observation_t *obs,
  */
 static double
 compute_residuals_topo(const tle_t *tle, const od_observation_t *obs,
-                       int n_obs, const od_observer_t *observer,
+                       int n_obs, const od_observer_t *observers,
                        double *residuals)
 {
     double *params;
@@ -240,6 +240,7 @@ compute_residuals_topo(const tle_t *tle, const od_observation_t *obs,
 
     for (i = 0; i < n_obs; i++)
     {
+        const od_observer_t *observer = &observers[obs[i].observer_idx];
         double tsince = (obs[i].jd - tle->epoch) * 1440.0;
         double pos[3], vel[3];
         int err;
@@ -547,7 +548,7 @@ od_fit_tle(const od_observation_t *obs, int n_obs,
         rms_cur = compute_residuals_eci(&current_tle, obs, n_obs, residuals);
     else
         rms_cur = compute_residuals_topo(&current_tle, obs, n_obs,
-                                         config->observer, residuals);
+                                         config->observers, residuals);
 
     if (rms_cur < 0.0)
     {
@@ -639,7 +640,7 @@ od_fit_tle(const od_observation_t *obs, int n_obs,
                 compute_residuals_eci(&tle_pert, obs, n_obs, resid_pert);
             else
                 compute_residuals_topo(&tle_pert, obs, n_obs,
-                                       config->observer, resid_pert);
+                                       config->observers, resid_pert);
 
             /* Jacobian column j (column-major for LAPACK)
              * H = dG/dx where G is the computed observation function.
@@ -715,7 +716,7 @@ od_fit_tle(const od_observation_t *obs, int n_obs,
             rms_cur = compute_residuals_eci(&current_tle, obs, n_obs, residuals);
         else
             rms_cur = compute_residuals_topo(&current_tle, obs, n_obs,
-                                             config->observer, residuals);
+                                             config->observers, residuals);
 
         if (rms_cur < 0.0)
         {

src/od_solver.h

@@ -42,6 +42,7 @@ typedef struct
 {
     double  jd;             /* Julian date of observation */
     double  data[6];        /* ECI: [x,y,z,vx,vy,vz], topo: [az,el,range,...] */
+    int     observer_idx;   /* index into config->observers[] (topo mode) */
 } od_observation_t;
 
 /*
@@ -63,7 +64,8 @@ typedef struct
     od_obs_type_t   obs_type;       /* ECI or topocentric */
     int             fit_bstar;      /* include B* as 7th state */
     int             max_iter;       /* iteration limit */
-    od_observer_t  *observer;       /* non-NULL for topocentric mode */
+    od_observer_t  *observers;      /* array of observers (topo mode) */
+    int             n_observers;    /* count (0 for ECI mode) */
 } od_config_t;
 
 /*

test/expected/od_fit.out

@@ -3,6 +3,7 @@
 -- Tests tle_from_eci(), tle_from_topocentric(), and tle_fit_residuals().
 -- Uses round-trip methodology: propagate known TLE → fit from obs → compare.
 CREATE EXTENSION IF NOT EXISTS pg_orrery;
+ALTER EXTENSION pg_orrery UPDATE TO '0.5.0';
 -- ============================================================
 -- Test 1: ECI round-trip (ISS-like LEO orbit)
 --
@@ -256,3 +257,158 @@ FROM result;
  t             | t
 (1 row)
 
+-- ============================================================
+-- Test 9: Multi-observer topocentric (MIT + Boulder observe ISS)
+--
+-- Two ground stations observe ISS, fit via multi-observer API.
+-- Uses the overloaded tle_from_topocentric(topo[], ts[], observer[], int4[], ...).
+-- ============================================================
+WITH iss_tle AS (
+    SELECT E'1 25544U 98067A   24001.50000000  .00016717  00000-0  10270-3 0  9025\n2 25544  51.6400 208.9163 0006703  30.1694  61.7520 15.50100486 00001'::tle AS t
+),
+stations AS (
+    SELECT
+        '(42.36,-71.09,20)'::observer AS mit,
+        '(40.01,-105.27,1655)'::observer AS boulder
+),
+topo_obs AS (
+    SELECT
+        array_agg(topo ORDER BY rn) AS observations,
+        array_agg(ts ORDER BY rn) AS times,
+        array_agg(obs_id ORDER BY rn) AS observer_ids
+    FROM (
+        -- MIT observations (observer_id = 0)
+        SELECT observe(t, mit, ts) AS topo, ts, 0 AS obs_id,
+               row_number() OVER (ORDER BY ts) AS rn
+        FROM iss_tle, stations,
+             generate_series(
+                 '2024-01-01 12:00:00+00'::timestamptz,
+                 '2024-01-01 13:00:00+00'::timestamptz,
+                 '5 minutes'::interval
+             ) AS ts
+        UNION ALL
+        -- Boulder observations (observer_id = 1)
+        SELECT observe(t, boulder, ts) AS topo, ts, 1 AS obs_id,
+               row_number() OVER (ORDER BY ts) + 100 AS rn
+        FROM iss_tle, stations,
+             generate_series(
+                 '2024-01-01 12:00:00+00'::timestamptz,
+                 '2024-01-01 13:00:00+00'::timestamptz,
+                 '5 minutes'::interval
+             ) AS ts
+    ) sub
+),
+result AS (
+    SELECT (tle_from_topocentric(
+        observations, times,
+        ARRAY[mit, boulder], observer_ids,
+        t, false, 20
+    )).*
+    FROM topo_obs, stations, iss_tle
+)
+SELECT
+    status = 'converged' AS did_converge,
+    rms_final < 10.0 AS rms_under_10km
+FROM result;
+ did_converge | rms_under_10km 
+--------------+----------------
+ t            | t
+(1 row)
+
+-- ============================================================
+-- Test 10: Single observer via multi-observer API
+--
+-- Verify the multi-observer path produces equivalent results
+-- when all observations come from one station.
+-- ============================================================
+WITH iss_tle AS (
+    SELECT E'1 25544U 98067A   24001.50000000  .00016717  00000-0  10270-3 0  9025\n2 25544  51.6400 208.9163 0006703  30.1694  61.7520 15.50100486 00001'::tle AS t
+),
+mit AS (
+    SELECT '(42.36,-71.09,20)'::observer AS obs
+),
+topo_obs AS (
+    SELECT
+        array_agg(observe(t, obs, ts) ORDER BY ts) AS observations,
+        array_agg(ts ORDER BY ts) AS times,
+        array_agg(0) AS observer_ids
+    FROM iss_tle, mit,
+         generate_series(
+             '2024-01-01 12:00:00+00'::timestamptz,
+             '2024-01-01 13:30:00+00'::timestamptz,
+             '5 minutes'::interval
+         ) AS ts
+),
+result AS (
+    SELECT (tle_from_topocentric(
+        observations, times,
+        ARRAY[obs], observer_ids,
+        t, false, 20
+    )).*
+    FROM topo_obs, mit, iss_tle
+)
+SELECT
+    status = 'converged' AS did_converge,
+    rms_final < 10.0 AS rms_under_10km
+FROM result;
+ did_converge | rms_under_10km 
+--------------+----------------
+ t            | t
+(1 row)
+
+-- ============================================================
+-- Test 11: Error - observer_id out of range
+-- ============================================================
+DO $$
+BEGIN
+    PERFORM tle_from_topocentric(
+        ARRAY[
+            observe(
+                E'1 25544U 98067A   24001.50000000  .00016717  00000-0  10270-3 0  9025\n2 25544  51.6400 208.9163 0006703  30.1694  61.7520 15.50100486 00001'::tle,
+                '(42.36,-71.09,20)'::observer,
+                '2024-01-01 12:00:00+00'::timestamptz
+            ),
+            observe(
+                E'1 25544U 98067A   24001.50000000  .00016717  00000-0  10270-3 0  9025\n2 25544  51.6400 208.9163 0006703  30.1694  61.7520 15.50100486 00001'::tle,
+                '(42.36,-71.09,20)'::observer,
+                '2024-01-01 12:05:00+00'::timestamptz
+            ),
+            observe(
+                E'1 25544U 98067A   24001.50000000  .00016717  00000-0  10270-3 0  9025\n2 25544  51.6400 208.9163 0006703  30.1694  61.7520 15.50100486 00001'::tle,
+                '(42.36,-71.09,20)'::observer,
+                '2024-01-01 12:10:00+00'::timestamptz
+            ),
+            observe(
+                E'1 25544U 98067A   24001.50000000  .00016717  00000-0  10270-3 0  9025\n2 25544  51.6400 208.9163 0006703  30.1694  61.7520 15.50100486 00001'::tle,
+                '(42.36,-71.09,20)'::observer,
+                '2024-01-01 12:15:00+00'::timestamptz
+            ),
+            observe(
+                E'1 25544U 98067A   24001.50000000  .00016717  00000-0  10270-3 0  9025\n2 25544  51.6400 208.9163 0006703  30.1694  61.7520 15.50100486 00001'::tle,
+                '(42.36,-71.09,20)'::observer,
+                '2024-01-01 12:20:00+00'::timestamptz
+            ),
+            observe(
+                E'1 25544U 98067A   24001.50000000  .00016717  00000-0  10270-3 0  9025\n2 25544  51.6400 208.9163 0006703  30.1694  61.7520 15.50100486 00001'::tle,
+                '(42.36,-71.09,20)'::observer,
+                '2024-01-01 12:25:00+00'::timestamptz
+            )
+        ],
+        ARRAY[
+            '2024-01-01 12:00:00+00'::timestamptz,
+            '2024-01-01 12:05:00+00'::timestamptz,
+            '2024-01-01 12:10:00+00'::timestamptz,
+            '2024-01-01 12:15:00+00'::timestamptz,
+            '2024-01-01 12:20:00+00'::timestamptz,
+            '2024-01-01 12:25:00+00'::timestamptz
+        ],
+        ARRAY['(42.36,-71.09,20)'::observer],
+        ARRAY[5, 0, 0, 0, 0, 0],  -- observer_id 5 out of range
+        E'1 25544U 98067A   24001.50000000  .00016717  00000-0  10270-3 0  9025\n2 25544  51.6400 208.9163 0006703  30.1694  61.7520 15.50100486 00001'::tle
+    );
+    RAISE NOTICE 'ERROR: should have raised exception';
+EXCEPTION
+    WHEN invalid_parameter_value THEN
+        RAISE NOTICE 'OK: observer_id out of range error caught';
+END $$;
+NOTICE:  OK: observer_id out of range error caught

test/sql/od_fit.sql

@@ -4,6 +4,7 @@
 -- Uses round-trip methodology: propagate known TLE → fit from obs → compare.
 
 CREATE EXTENSION IF NOT EXISTS pg_orrery;
+ALTER EXTENSION pg_orrery UPDATE TO '0.5.0';
 
 -- ============================================================
 -- Test 1: ECI round-trip (ISS-like LEO orbit)
@@ -237,3 +238,153 @@ SELECT
     rms_final < 2.0 AS rms_under_2km,
     status = 'converged' AS did_converge
 FROM result;
+
+-- ============================================================
+-- Test 9: Multi-observer topocentric (MIT + Boulder observe ISS)
+--
+-- Two ground stations observe ISS, fit via multi-observer API.
+-- Uses the overloaded tle_from_topocentric(topo[], ts[], observer[], int4[], ...).
+-- ============================================================
+
+WITH iss_tle AS (
+    SELECT E'1 25544U 98067A   24001.50000000  .00016717  00000-0  10270-3 0  9025\n2 25544  51.6400 208.9163 0006703  30.1694  61.7520 15.50100486 00001'::tle AS t
+),
+stations AS (
+    SELECT
+        '(42.36,-71.09,20)'::observer AS mit,
+        '(40.01,-105.27,1655)'::observer AS boulder
+),
+topo_obs AS (
+    SELECT
+        array_agg(topo ORDER BY rn) AS observations,
+        array_agg(ts ORDER BY rn) AS times,
+        array_agg(obs_id ORDER BY rn) AS observer_ids
+    FROM (
+        -- MIT observations (observer_id = 0)
+        SELECT observe(t, mit, ts) AS topo, ts, 0 AS obs_id,
+               row_number() OVER (ORDER BY ts) AS rn
+        FROM iss_tle, stations,
+             generate_series(
+                 '2024-01-01 12:00:00+00'::timestamptz,
+                 '2024-01-01 13:00:00+00'::timestamptz,
+                 '5 minutes'::interval
+             ) AS ts
+        UNION ALL
+        -- Boulder observations (observer_id = 1)
+        SELECT observe(t, boulder, ts) AS topo, ts, 1 AS obs_id,
+               row_number() OVER (ORDER BY ts) + 100 AS rn
+        FROM iss_tle, stations,
+             generate_series(
+                 '2024-01-01 12:00:00+00'::timestamptz,
+                 '2024-01-01 13:00:00+00'::timestamptz,
+                 '5 minutes'::interval
+             ) AS ts
+    ) sub
+),
+result AS (
+    SELECT (tle_from_topocentric(
+        observations, times,
+        ARRAY[mit, boulder], observer_ids,
+        t, false, 20
+    )).*
+    FROM topo_obs, stations, iss_tle
+)
+SELECT
+    status = 'converged' AS did_converge,
+    rms_final < 10.0 AS rms_under_10km
+FROM result;
+
+-- ============================================================
+-- Test 10: Single observer via multi-observer API
+--
+-- Verify the multi-observer path produces equivalent results
+-- when all observations come from one station.
+-- ============================================================
+
+WITH iss_tle AS (
+    SELECT E'1 25544U 98067A   24001.50000000  .00016717  00000-0  10270-3 0  9025\n2 25544  51.6400 208.9163 0006703  30.1694  61.7520 15.50100486 00001'::tle AS t
+),
+mit AS (
+    SELECT '(42.36,-71.09,20)'::observer AS obs
+),
+topo_obs AS (
+    SELECT
+        array_agg(observe(t, obs, ts) ORDER BY ts) AS observations,
+        array_agg(ts ORDER BY ts) AS times,
+        array_agg(0) AS observer_ids
+    FROM iss_tle, mit,
+         generate_series(
+             '2024-01-01 12:00:00+00'::timestamptz,
+             '2024-01-01 13:30:00+00'::timestamptz,
+             '5 minutes'::interval
+         ) AS ts
+),
+result AS (
+    SELECT (tle_from_topocentric(
+        observations, times,
+        ARRAY[obs], observer_ids,
+        t, false, 20
+    )).*
+    FROM topo_obs, mit, iss_tle
+)
+SELECT
+    status = 'converged' AS did_converge,
+    rms_final < 10.0 AS rms_under_10km
+FROM result;
+
+-- ============================================================
+-- Test 11: Error - observer_id out of range
+-- ============================================================
+
+DO $$
+BEGIN
+    PERFORM tle_from_topocentric(
+        ARRAY[
+            observe(
+                E'1 25544U 98067A   24001.50000000  .00016717  00000-0  10270-3 0  9025\n2 25544  51.6400 208.9163 0006703  30.1694  61.7520 15.50100486 00001'::tle,
+                '(42.36,-71.09,20)'::observer,
+                '2024-01-01 12:00:00+00'::timestamptz
+            ),
+            observe(
+                E'1 25544U 98067A   24001.50000000  .00016717  00000-0  10270-3 0  9025\n2 25544  51.6400 208.9163 0006703  30.1694  61.7520 15.50100486 00001'::tle,
+                '(42.36,-71.09,20)'::observer,
+                '2024-01-01 12:05:00+00'::timestamptz
+            ),
+            observe(
+                E'1 25544U 98067A   24001.50000000  .00016717  00000-0  10270-3 0  9025\n2 25544  51.6400 208.9163 0006703  30.1694  61.7520 15.50100486 00001'::tle,
+                '(42.36,-71.09,20)'::observer,
+                '2024-01-01 12:10:00+00'::timestamptz
+            ),
+            observe(
+                E'1 25544U 98067A   24001.50000000  .00016717  00000-0  10270-3 0  9025\n2 25544  51.6400 208.9163 0006703  30.1694  61.7520 15.50100486 00001'::tle,
+                '(42.36,-71.09,20)'::observer,
+                '2024-01-01 12:15:00+00'::timestamptz
+            ),
+            observe(
+                E'1 25544U 98067A   24001.50000000  .00016717  00000-0  10270-3 0  9025\n2 25544  51.6400 208.9163 0006703  30.1694  61.7520 15.50100486 00001'::tle,
+                '(42.36,-71.09,20)'::observer,
+                '2024-01-01 12:20:00+00'::timestamptz
+            ),
+            observe(
+                E'1 25544U 98067A   24001.50000000  .00016717  00000-0  10270-3 0  9025\n2 25544  51.6400 208.9163 0006703  30.1694  61.7520 15.50100486 00001'::tle,
+                '(42.36,-71.09,20)'::observer,
+                '2024-01-01 12:25:00+00'::timestamptz
+            )
+        ],
+        ARRAY[
+            '2024-01-01 12:00:00+00'::timestamptz,
+            '2024-01-01 12:05:00+00'::timestamptz,
+            '2024-01-01 12:10:00+00'::timestamptz,
+            '2024-01-01 12:15:00+00'::timestamptz,
+            '2024-01-01 12:20:00+00'::timestamptz,
+            '2024-01-01 12:25:00+00'::timestamptz
+        ],
+        ARRAY['(42.36,-71.09,20)'::observer],
+        ARRAY[5, 0, 0, 0, 0, 0],  -- observer_id 5 out of range
+        E'1 25544U 98067A   24001.50000000  .00016717  00000-0  10270-3 0  9025\n2 25544  51.6400 208.9163 0006703  30.1694  61.7520 15.50100486 00001'::tle
+    );
+    RAISE NOTICE 'ERROR: should have raised exception';
+EXCEPTION
+    WHEN invalid_parameter_value THEN
+        RAISE NOTICE 'OK: observer_id out of range error caught';
+END $$;