diff --git a/weatherbench2/metrics_test.py b/weatherbench2/metrics_test.py
index fda07f1..f4aa755 100644
--- a/weatherbench2/metrics_test.py
+++ b/weatherbench2/metrics_test.py
@@ -26,7 +26,12 @@
 
 
 def get_random_truth_and_forecast(
-    variables=('geopotential',), ensemble_size=None, seed=802701, **data_kwargs
+    variables=('geopotential',),
+    ensemble_size=None,
+    seed=802701,
+    lead_start='0 day',
+    lead_stop='10 day',
+    **data_kwargs,
 ):
   """Makes the tuple (truth, forecast) from kwargs."""
   data_kwargs_to_use = dict(
@@ -43,7 +48,10 @@ def get_random_truth_and_forecast(
   )
   forecast = utils.random_like(
       schema.mock_forecast_data(
-          ensemble_size=ensemble_size, **data_kwargs_to_use
+          ensemble_size=ensemble_size,
+          lead_start=lead_start,
+          lead_stop=lead_stop,
+          **data_kwargs_to_use,
       ),
       seed=seed + 1,
   )
@@ -1147,6 +1155,87 @@ def test_versus_large_ensemble_and_ensure_skipna_works(self):
         atol=4 * stderr,
     )
 
+  def test_integral_of_brier_score_is_crps(self):
+    # The integral over threshold of debiased brier score is unbiased CRPS.
+    truth, forecast = get_random_truth_and_forecast(
+        ensemble_size=2,
+        spatial_resolution_in_degrees=60,
+        time_start='2019-01-01',
+        time_stop='2019-12-31',
+        time_resolution='12 hours',
+        lead_start='0 day',
+        lead_stop='0 day',
+        levels=[500, 700, 850],
+    )
+
+    # Make forecasts (i) different mean/variance than truth, and (ii) depend on
+    # level.
+    forecast = (
+        forecast
+        + np.abs(forecast) ** 0.2
+        + xr.DataArray(
+            [-1, 0, 1], dims=['level'], coords={'level': forecast.level.data}
+        )
+    )
+
+    # climatology has the same stats as Normal(0, 1). So truth/forecast should
+    # be "perfect".
+    climatology_mean = xr.zeros_like(
+        truth.isel(time=0, drop=True).expand_dims(dayofyear=366)
+    )
+    climatology_std = xr.ones_like(
+        truth.isel(time=0, drop=True)
+        .expand_dims(
+            dayofyear=366,
+        )
+        .rename({'geopotential': 'geopotential_std'})
+    )
+    climatology = xr.merge([climatology_mean, climatology_std])
+    quantiles = np.linspace(0.005, 0.995, num=200)
+    threshold_objects = [
+        thresholds.GaussianQuantileThreshold(
+            climatology=climatology, quantile=q
+        )
+        for q in quantiles
+    ]
+    bs = metrics.DebiasedEnsembleBrierScore(threshold_objects).compute(
+        forecast, truth
+    )['geopotential']
+
+    # Now integrate BS, with respect to the threshold. To do that, we first
+    # build a DataArray of thresholds corresponding to the quantiles.
+    precip_thresholds = []
+    for q, thresh in zip(quantiles, threshold_objects):
+      t = thresh.compute(truth)['geopotential']
+      # To simplify integration, we ensured threshold depends only on level.
+      # This "assert_array_less" checks that we did this correctly.
+      np.testing.assert_array_less(
+          t.std(['time', 'longitude', 'latitude']), 1e-4
+      )
+      precip_thresholds.append(
+          t.isel(time=0, longitude=0, latitude=0, drop=True).expand_dims(
+              quantile=[q]
+          )
+      )
+    precip_thresholds = xr.concat(precip_thresholds, dim='quantile')
+
+    # Second, do the integral, one level at a time.
+    bs = bs.assign_coords(threshold=precip_thresholds)
+    integrals = []
+    for level in bs.level:
+      integrals.append(bs.sel(level=level).integrate('threshold'))
+    bs_integral = xr.concat(integrals, dim='level')
+
+    crps = metrics.CRPS().compute(forecast, truth)['geopotential']
+
+    # Tolerance is due to (i) finite samples, and (ii) integration error. The
+    # integation error is going to be tiny, due to using 200 points to
+    # interpolate a function that we know is bounded to ≈ [-5, 5].
+    stderr = 1 / np.sqrt(
+        np.prod([v for k, v in truth.sizes.items() if k != 'level'])
+    )
+    xr.testing.assert_allclose(bs_integral, crps, atol=4 * stderr)
+
 
 class EnsembleIgnoranceScoreTest(parameterized.TestCase):